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+Incremental Dead State Detection in
+Logarithmic Time
+Caleb Stanford1 and Margus Veanes2
+1 University of California, San Diego and University of California, Davis
+cdstanford@ucdavis.edu
+2 Microsoft Research, Redmond margus@microsoft.com
+Abstract. Identifying live and dead states in an abstract transition sys-
+tem is a recurring problem in formal verification. However, state-of-the-
+art graph algorithms for maintaining reachability information incremen-
+tally (that is, as states are visited and before the entire state space is
+explored) assume that new edges can be added from any state at any
+time, whereas in many applications, outgoing edges are added from each
+state as it is explored. To formalize the latter situation, we propose guided
+incremental digraphs (GIDs), incremental graphs which support labeling
+closed states (states which will not receive further outgoing edges). Our
+main result is that dead state detection in GIDs is solvable in O(log m)
+time per edge update for m edges, improving upon O(√m) per edge due
+to Bender, Fineman, Gilbert, and Tarjan (BFGT) for general incremen-
+tal directed graphs.
+We introduce two algorithms for GIDs: one establishing the logarithmic
+time bound, and a second algorithm to explore a lazy heuristics-based ap-
+proach. To demonstrate applicability, we show how GIDs can be used to
+lazily decide regular expression constraints in SMT applications. To en-
+able an apples-to-apples experimental comparison, we implemented both
+algorithms, two naive baselines, and the state-of-the-art BFGT baseline
+using a common directed graph interface in Rust. Our evaluation shows
+110-530x speedups over BFGT for the largest input graphs over a range
+of graph classes, random graphs, and graphs arising from regular expres-
+sion benchmarks.
+Keywords: Dead State Detection · Graph Algorithms · Online Algo-
+rithms · SMT.
+1
+Introduction
+Classifying states in a transition system as live or dead is a recurring problem
+in formal verification. For example, given an expression, can it be simplified
+to the identity? Given an input to a nondeterministic program, can it reach a
+terminal state, or can it reach an infinitely looping state? Given a state in an
+automaton, can it reach an accepting state? Domain-specific variations on this
+problem have led to many general and specialized algorithms used by automated
+arXiv:2301.05308v1  [cs.DS]  12 Jan 2023
+
+2
+Caleb Stanford and Margus Veanes
+verification tools. State classification is relevant in the context of satisfiability
+modulo theories (SMT) [17,41,3,55,15,36], where theory-specific partial decision
+procedures often work by exploring the state space to find a reachable path that
+corresponds to a satisfying string or, more generally, a sequence of constructors.
+In all of these cases, the core problem is live and dead state detection in a
+directed graph.
+Motivating application. For example, recent approaches for the SMT theory of
+regular expressions [34,50] rely on regular expression derivatives to explore the
+states of the finite state machine corresponding to the regex incrementally, rather
+than expanding all states initially which is often prohibitively expensive. This
+requires solving the incremental live and dead state detection problem in the
+finite state machine (a directed graph). This is particularly important for regexes
+with intersection and complement (extended regexes [12,22,19]), which have been
+shown to arise natively in applications of SMT string solvers to security [2,50]).
+Concretely, consider the regex (�*α�100)C ∩ (�α), where � matches any character,
+∩ is regex intersection, C is regex complement, and α matches any digit (0-9). A
+traditional solver would expand the left and right operands as state machines,
+but the left operand (�*α�100)C is astronomically large as a DFA, causing the
+solver to hang. The derivative-based technique instead constructs the derivative
+regex: (�*α�100)C ∩(�100)C ∩α. At this stage we have a graph of two states and one
+edge, where the states are regexes and the edge is the derivative relation. After
+one more derivative operation, the regex becomes one that is clearly satisfiable
+as it accepts the empty string.
+In order to be efficient, a derivative-based solver needs to identify satisfiable
+(live) and unsatisfiable (dead) regexes incrementally (as the graph is built), be-
+cause it does not generally construct the entire space before terminating (see the
+graph update inference rule Upd, p. 626 [50]). Indeed, satisfiability (nonempti-
+ness) for extended regexes is non-elementary, and is still PSPACE-complete for
+more restrictive fragments, strongly incentivizing the incremental approach. Be-
+yond regexes, we believe that GIDs are general enough to be applicable in a
+range of future applications.
+Prior work. Traditionally, while live state detection can be done incrementally,
+dead state detection is often done exhaustively (i.e., after the entire state space
+is explored). For example, bounded and finite-state model checkers based on
+translations to automata [32,48,13], as well as classical dead-state elimination
+algorithms [28,7,10], generally work on a fixed state space after it has been fully
+enumerated. However, exhaustive exploration is prohibitive for large (e.g., ex-
+ponential or infinite) state spaces which arise in an SMT verification context.
+Moreover, exhaustive exploration may simply be unnecessary if partial infor-
+mation can be deduced about the states seen so far which already leads to a
+satisfiable or unsatisfiable result along a given solver path. We also have good
+evidence that incremental feedback can improve SMT solver performance: a
+representative success story is the e-graph data structure [54,16] for congruence
+closure [18,42], which maintains an equivalence relation among expressions in-
+
+Incremental Dead State Detection in Logarithmic Time
+3
+crementally; because it applies to general expressions, it is theory-independent
+and re-usable. Incremental state space exploration could lead to similar benefits
+if applied to SMT procedures which still rely on exhaustive search.
+However, in order to perform incremental dead state detection, we currently
+lack algorithms which match offline performance. As we discuss in Section 2,
+the best-known existing solutions would require maintaining strong connected
+components (SCCs) incrementally. For SCC maintenance and the related sim-
+pler problem of cycle detection, O(m3/2) amortized algorithms are known for m
+edge additions [23,4], with some recently announced improvements [5,8]. Note
+that this is in sharp contrast to O(m) for the offline variants of these problems,
+which can be solved by breadth-first or depth-first search. More generally, re-
+search suggests that there is a computational barrier to what can be determined
+incrementally in the worst case [20,21,1].
+This paper. To improve on prior results, our key observation is that in many
+applications, edges are not added adversarially, but from one state at a time as
+the states are explored. As a result, we know when a state will have no further
+outgoing edges; we may use this information to (i) provide information about
+dead states incrementally, rather than after the whole state space is explored;
+and (ii) obtain more efficient algorithms than currently exist for general graph
+reachability.
+We introduce guided incremental digraphs (GIDs), a variation on incremental
+graphs. Like an incremental directed graph, a guided incremental digraph may be
+updated by adding new edges between states, or a state may be labeled as closed,
+meaning it will receive no further outgoing edges. Some states are designated as
+terminal, and we say that a state is live if it can reach a terminal state and dead
+if it will never reach a terminal state in any extension – i.e. if all reachable states
+from it are closed. To our knowledge, the problem of detecting dead states in
+such a system has not been studied by existing work in graph algorithms. Our
+problem can be solved through solving SCC maintenance, but not necessarily
+the other way around (see Proposition 1). We provide two new algorithms for
+dead-state detection in GIDs.
+First, we show that the dead-state detection problem for GIDs can be solved
+in time O(m · log m) for m edge additions, within a logarithmic factor of the
+O(m) cost for offline search. The worst-case performance of our algorithm thus
+strictly improves on the O(m3/2) upper bound for SCC maintenance in general
+incremental graphs. Our algorithm utilizes several data structures and existing
+results in online algorithms: in particular, Union-Find [52] and Henzinger and
+King’s Euler Tour Trees [26]. The main idea of our algorithm is that, rather than
+explicitly computing the set of SCCs, for closed states we maintain a single path
+to a non-closed (open) state. This turns out to reduce the problem to quickly
+determining whether two states are currently assigned a path to the same open
+state. On the other hand, Euler Tour Trees can solve undirected reachability for
+graphs that are forests in logarithmic time; the challenge then lies in figuring
+out how to reduce directed connectivity in the graph of paths to undirected
+connectivity in an Euler Tour Trees forest. At the same time, we must maintain
+
+4
+Caleb Stanford and Margus Veanes
+this structure under Union-Find state merges, in order to deal with cycles that
+are found in the graph.
+While as theorists we would like to believe that asymptotic complexity is
+enough, the truth is that the use of complex data structures (1) can be pro-
+hibitively expensive in practice due to constant-time overheads, and (2) can
+make algorithms substantially more difficult to implement, leading practition-
+ers to prefer simpler approaches. To address these needs, in addition to the
+logarithmic-time algorithm, we provide a second lazy algorithm which avoids
+the user of Euler Tour Trees, and only uses union-find. This algorithm is based
+on an optimization of adding shortcut jump edges for long paths in the graph to
+quickly determine reachability. This approach aims to perform well in practice
+on typical graphs, and is evaluated in our evaluation along with the logarithmic
+time algorithm, though we do not prove its asymptotic complexity.
+Finally, we implement and empirically evaluate both of our algorithms for
+GIDs against several baselines in 5.5k lines of code in Rust [37,33]. Our evaluation
+focuses on the performance of the GID data structure itself, rather than its end-
+to-end performance in applications. To ensure an apples-to-apples comparison
+with existing approaches, we put particular focus on providing a directed graph
+data structure backend shared by all algorithms, so that the cost of graph search
+as well as state and edge merges is identical across algorithms. We implement
+two naive baselines, as well as an implementation of the state-of-the-art solution
+based on maintaining SCCs, BFGT [4] in our framework. To our knowledge,
+the latter is the first implementation of BFGT for SCC maintenance. On a
+collection of generated benchmark GIDs and GIDs directly pulled from the regex
+application, we demonstrate a substantial improvement over BFGT for both of
+our algorithms. For example, for larger GIDs (those with over 100K updates),
+we observe a 110-530x speedup over BFGT.
+Our primary contributions are:
+– Guided incremental digraphs (GIDs), a formalization of incremental live and
+dead state detection which supports labeling closed states. (Section 2)
+– Two algorithms for the state classification problem in GIDs: first, an algo-
+rithm that works in amortized O(log m) time per update, improving upon
+the state-of-the-art amortized O(√m) per update for incremental graphs;
+and second, a simpler algorithm based on lazy heuristics. (Section 3)
+– An open-source implementation of GIDs in Rust3, including an implementa-
+tion the BFGT baseline and supporting data structures for a fair comparison,
+and an evaluation which demonstrates that our algorithm outperforms the
+best known incremental SCC algorithm by two orders of magnitude for a
+range of GID benchmarks. (Section 4)
+Following these contributions, we expand on the application of GIDs to regex
+solving in SMT (Section 5), survey related work (Section 6), and conclude (Sec-
+tion 7).
+3 https://github.com/cdstanford/gid
+
+Incremental Dead State Detection in Logarithmic Time
+5
+1
+2
+3
+Fig. 1.
+Guided incremental digraph consisting of the sequence of updates E(1, 2),
+E(1, 3), T(2). Terminal states are denoted with double circles. After the update T(2),
+states 1 and 2 are known to be live. However, for this graph state 3 is not dead, as a
+future edge may cause it to be live.
+2
+Guided Incremental Digraphs
+2.1
+Problem Statement
+An incremental digraph is a sequence of edge updates E(u, v), where the algo-
+rithmic challenge in this context is to produce some output after each edge is
+received (e.g., whether or not a cycle exists). If the graph also contains updates
+T(u) labeling a state as terminal, then we say that a state is live if it can reach
+a terminal state in the current graph. In a guided incremental digraph, we also
+include updates C(u) labeling a state as closed, meaning that will not receive
+any further outgoing edges.
+Definition 1. Define a guided incremental digraph (GID) to be a sequence of
+updates, where each update is one of the following:
+(i) a new directed edge E(u, v);
+(ii) a label T(u) which indicates that u is terminal; or
+(iii) a label C(u) which indicates that u is closed, i.e. no further edges will be
+added going out from u (or labels to u).
+The guided incremental digraph is valid if the closed labels are correct: there
+are no instances of E(u, v) or T(u) after an update C(u). The denotation of G is
+the directed graph (V, E) where V is the set of all states u which have occurred
+in any update in the sequence, and E is the set of all (u, v) such that E(u, v)
+occurs in G.
+An extension of a valid GID G is a valid GID G′ such that G is a prefix of
+G′. In a valid GID G, we say that a state u is live if there is a path from u to
+a terminal state in the denotation of G; and a state u is dead if it is not live in
+any extension of G. Notice that in a GID without any C(u) updates, no states
+are dead as an edge may be added in an extension which makes them live.
+We provide an example of a valid GID in Figures 1 and 2 resulting from
+the following sequence of updates: E(1, 2), E(1, 3), T(2), E(4, 3), E(4, 5), C(4),
+C(5). The states are denoted based on whether they are marked as terminal T(u)
+(double circle) or closed C(u); states that are not closed are denoted with dashed
+circles. Notice that after C(4), state 4 is marked closed but can still reach 3 and
+5, so it is not dead. Additionally, notice that once 1 and 2 are live (after T(2)),
+further edges from them do not matter.
+
+6
+Caleb Stanford and Margus Veanes
+1
+2
+3
+4
+5
+Fig. 2.
+Guided incremental digraph consisting of the sequence of updates E(1, 2),
+E(1, 3), T(2), E(4, 3), E(4, 5), C(4), C(5). Closed states (from which future edges may not
+be added) are denoted with solid circles. After the update C(5) (but not earlier), state
+5 is dead.
+Definition 2. Given as input a valid GID, the guided incremental state clas-
+sification problem is to output, in an online fashion after each update, the set
+of new live and new dead states. That is, output Live(u) or Dead(u) on the
+smallest prefix of updates such that u is live or dead on that prefix, respectively.
+2.2
+Existing Approaches
+In many applications, one might choose to classify dead states offline, after the
+entire state space is enumerated. This leads to a linear-time algorithm via either
+DFS or BFS, but it does not solve the state classification problem (Definition 2)
+because it is not incremental. Naive application of this idea leads to an O(m)
+time algorithm per update for m updates (O(m2) total), as we may have to
+search the entire graph after each update.
+For acyclic graphs, there exists an amortized O(1)-time per update algorithm
+for the problem (Definition 2): maintain the graph as a list of forward and
+backward edges at each state. When a state v is marked terminal, do a DFS
+along backward edges to determine all states u that can reach v not already
+marked as live, and mark them live. When a state v is marked closed, visit
+all forward-edges from v; if all are dead, mark v as dead and recurse along all
+backwards edges from v. As each edge is visited only when marking a state live
+or dead, it is only visited a constant number of times overall (though we may
+use more than O(1) time on some particular update pass). Additionally, the live
+state detection part of this procedure still works for graphs containing cycles.
+The challenge, therefore, lies primarily in detecting dead states in graphs
+which may contain cycles. For this, the breakthrough state-of-the-art approach
+from [4] enables maintaining the graph as a condensed graph which is acyclic,
+where the vertices in the condensed graph represent strongly connected compo-
+nents (SCCs) of states. The mapping from states to SCCs is maintained using
+a Union-Find [52] data structure. Maintaining this requires O(√m) time per
+update. To ensure that vertices in the condensed graph correspond to SCCs in
+the original, we have to also make sure that closed and non-closed states are not
+merged into the same SCC; the easiest solution to this is to withhold all edges
+from each state u in the graph until u are closed, which ensures that u must
+be its own SCC. Once we have the condensed graph with these modifications,
+
+Incremental Dead State Detection in Logarithmic Time
+7
+the same algorithm as in the previous paragraph works to identify live and dead
+states. Since each edge is only visited when a state is marked closed or live, each
+edge is visited only once throughout the algorithm, we use only amortized O(1)
+additional time to calculate live and dead states. While this SCC maintenance
+algorithm ignores the fact that edges do not occur from closed states C(u), this
+still proves the following result:
+Proposition 1. Guided incremental state classification reduces to SCC main-
+tenance. That is, suppose we have an algorithm over incremental graphs that
+maintains the set of SCCs in O(f(m, n)) total time given n states and m edge
+additions, where “maintains” means that (i) we can check whether two states
+are in the same SCC in O(1) time, and (ii) we can iterate over all the states
+in an SCC, or iterate over the forward-edges or backward-edges from an SCC
+(to or from other SCCs, respectively) in O(1) time per edge. Then there exists
+an algorithm to solve guided incremental state classification in O(f(m, n)) total
+time.
+Despite this reduction one way, there is no obvious reduction the other way –
+from cycle detection or SCCs to Definition 2. This is because, while the existence
+of a cycle of non-live states implies bi-reachability between all states in the cycle,
+it does not necessarily imply that all of the bi-reachable states are dead. In fact,
+consider a GID consisting only of E(u, v) and C(u) updates. In order to identify
+dead states, rather than enumerating all cycles or SCCs, it is enough to maintain
+a single path from each non-dead state to a non-closed state. This observation
+forms the starting point for our approach in the following sections, improving
+on the upper bound given by Proposition 1.
+3
+Algorithms
+This section presents Algorithm 2, which solves the state classification problem in
+logarithmic time (Theorem 2); and Algorithm 3, an alternative lazy approach.
+Both algorithms are optimized versions of Algorithm 1, a first-cut algorithm
+which establishes the structure of our approach. We begin by establishing some
+basic terminology shared by all of the algorithms (see Figure 3).
+States in a GID can be usefully classified as exactly one of four statuses: live,
+dead, unknown, or open, where an unknown state is one that is closed but not
+yet live or dead, and an open state is one that is not closed and not live. Note
+that a state may be live and neither open nor closed; this terminology keeps the
+classification disjoint. Pragmatically, for live states it does not matter if they
+are classified as open or closed, since edges from those states no longer have any
+effect. However, all dead and unknown states are closed, and no states are both
+open and closed.
+Given this classification, the intuition is that for each unknown state u, we
+only need one path from u to an open state to prove that it is not dead; we want
+to maintain one such path for all unknown states. To maintain all of these paths
+simultaneously, we maintain an acyclic directed forest structure on unknown and
+
+8
+Caleb Stanford and Margus Veanes
+Live
+Some reachable state from u is terminal.
+Dead
+All reachable states from u (including u itself) are closed and
+not terminal.
+Unknown
+u is closed, but not live or dead.
+Open
+u is not live and not closed.
+Terminal
+A state u labeled by T(u).
+Closed
+A state u labeled by C(u).
+Canonical
+A state x that is the uniquely chosen representative of its
+union-find equivalence class, i.e. UF.find(x) = x.
+u, v, w
+States in the original graph.
+x, y, z
+Canonical states in the condensed graph.
+Successor succ(x)
+For an unknown, canonical state x, a uniquely chosen state v
+such that (x, v) is an edge, and following the path of successors
+leads to an open state.
+Fig. 3. Top: Basic classification of GID states into four disjoint categories. Bottom:
+Additional terminology used in this paper.
+open states where the roots are open states, and all non-root states have a single
+edge to another state, called its successor. Edges other than successor edges can
+be temporarily ignored, except for when marking live states; these are kept as
+reserve edges. Specifically, we add every edge (u, v) as a backward-edge from v
+(to allow propagating live states), but for edges not in the forest we keep (u, v)
+in a reserve list from u. We store all edges, including backward-edges, in the
+original order (u, v). The reserve list edge becomes relevant only when either (i)
+u is marked as closed, or (ii) u’s successor is marked as dead.
+In order to deal with cycles, we need to maintain the forest of unknown states
+not on the original graph, but on a union-find condensed graph, similar to [52].
+When we find a cycle of unknown states, we merge all states in the cycle by
+calling the union method in the union-find. We refer to a state as canonical if
+it is the canonical representative of its equivalence class in the union find; the
+condensed graph is a forest on canonical states. We use x, y, z to denote canonical
+states (states in the condensed graph), and u, v, w to denote the original states
+(not known to be canonical).
+Following [52], we maintain edges as linked lists rather than sets, and using
+the original states instead of canonical states; this is important as it allows
+combining edge lists in O(1) time when merging states.
+3.1
+First-Cut Algorithm
+A first-cut algorithm based on these ideas is shown in Algorithm 1. The union-
+find data structure UF provides UF.union(v1, v2), UF.find(v), and UF.iter(v):
+UF.union merges v1 and v2 to refer to the same canonical state, UF.find returns
+the canonical state for v, and UF.iter iterates over states equivalent to v. The
+running time of each of these operations is amortized α(n) for n updates, where
+α(n) ∈ o(log n) is the inverse Ackermann function.
+
+Incremental Dead State Detection in Logarithmic Time
+9
+Algorithm 1 First-cut algorithm.
+V: a type for states (integers) (variables u, v, . . .)
+E: the type of edges, equal to (V, V)
+UF: a union-find data structure over V
+X: the set of canonical states in UF (variables x, y, z, . . .)
+status: a map from X to Live, Dead, Unknown, or Open
+succ: a map from X to V
+res and bck: maps from X to linked lists of E
+procedure OnEdge(E(u, v))
+x ← UF.find(u); y ← UF.find(v)
+if status(y) = Live then
+OnTerminal(T(x))
+▷ mark x and its ancestors live
+else if status(x) ̸= Live then
+▷ status(x) must be Open
+append (u, v) to res(x)
+append (u, v) to bck(y)
+procedure OnTerminal(T(v))
+y ← UF.find(v)
+for all x in DFS backwards (along bck) from y not already Live do
+status(x) ← Live
+output Live(x′) for all x′ in UF.iter(x)
+procedure OnClosed(C(v))
+y ← UF.find(v)
+if status(y) ̸= Open then return
+▷ y is already live or closed
+while res(y) is nonempty do
+pop (v, w) from res(y); z ← UF.find(w)
+if status(z) = Dead then continue
+else if CheckCycle(y, z) then
+for all z′ in cycle from z to y do z ← Merge(z, z′)
+else
+status(y) ← Unknown; succ(y) ← z;
+return
+status(y) ← Dead; output Dead(y′) for all y′ in UF.iter(y)
+ToRecurse ← ∅
+for all (u, v) in bck(y) do
+x ← UF.find(u)
+if status(x) = Unknown and UF.find(succ(x)) = y then
+status(x) ← Open
+▷ temporary – marked closed on recursive call
+add x to ToRecurse
+for all x in ToRecurse do OnClosed(C(x))
+procedure CheckCycle(y, z) returning bool
+while status(z) = Unknown do z ← UF.find(succ(z))
+▷ get root state from z
+return y = z
+procedure Merge(x, y) returning V
+z ← UF.union(x, y)
+bck(z) ← bck(x) + bck(y)
+▷ O(1) linked list append
+res(z) ← res(x) + res(y)
+▷ O(1) linked list append
+return z
+
+10
+Caleb Stanford and Margus Veanes
+We will only merge states if they are bi-reachable from each other, and both
+unknown. This implies that all states in the equivalence class of a canonical state
+x have the same status. We also maintain the set of canonical states in UF as a
+set X.
+The edge maps res and bck are stored as maps from X to linked lists of edges.
+Each edge (u, v) is always stored using its original states (i.e., edge labels are not
+updated when states are merged); but we can easily obtain the corresponding
+edge on canonical states via (UF.find(u), UF.find(v)). On an input edge (u, v),
+the edge is immediately added to bck as a backward-edge from v (this is used
+for detecting live states), but forward-edges are only added selectively to succ
+when they are needed to prove unknown status, and are otherwise stored in the
+reserve list res.
+Invariants. In total, we maintain the following invariants. For live and dead
+states, we no longer care about forward or backward edges from them, so there
+are no invariants about live and dead states other than that status(x) is Live or
+Dead for these states, respectively. The successor edges and no cycles invariants
+are about the forest structure. The last invariant, edge representation, is about
+making sure that all edges in the input GID are represented somehow in the
+current graph.
+– Merge equivalence: For all states u and v, if UF.find(u) = UF.find(v), then
+u and v are bi-reachable and both closed. (This implies that u and v are
+both live, both dead, or both unknown.)
+– Status correctness: For all u, status(UF.find(u)) is equal to the status of
+u.
+– Successor edges: If x is unknown, then succ(x) is defined and is an unknown
+or open state. If x is open, then succ(x) is not defined.
+– No cycles: There are no cycles among the set of edges (x, UF.find(succ(x))),
+over all unknown and open canonical states x.
+– Edge representation: Lastly, for all edges (u, v) in the input GID, at least one
+of the following holds: (i) (u, v) is stored in res, i.e. (u, v) ∈ res(UF.find(v));
+(ii) (u, v) is stored in succ, i.e. v = succ(UF.find(u)); (iii) u and v are
+equivalent in the condensed graph, i.e. UF.find(u) = UF.find(v); (iv) u is
+live; or (v) v is dead.
+Proposition 2. Algorithm 1 is correct.
+Proof. We need to argue that all of the invariants described above are preserved.
+This implies correctness of the algorithm by the status correctness invariant,
+since it implies that live and dead states are labeled (and output) correctly.
+Upon receiving E(u, v) or T(u), this may cause some dead, unknown, or open
+states to change status to live, but does not change the status of any other
+states. As bck stores all backwards-edges (not just succ edges), live states are
+marked correctly. This preserves the forest invariants because if an unknown or
+open state is marked live, so are all its predecessors. This also preserves the edge
+representation invariant, either by adding new edges to case (i) and case (iv)
+
+Incremental Dead State Detection in Logarithmic Time
+11
+(for E updates) or moving edges from cases (i)-(iii) to case (iv) (for both E and
+T updates).
+The procedure C(u) is recursive; during the procedure and on recursive calls,
+some states are temporarily marked Open, meaning they are roots in the forest
+structure. During these recursive calls, we need a slightly generalized invariant:
+each forest root corresponds to a pending future call to OnClosed(C(u)) (i.e.,
+an element of ToRecurse for some call on the stack) and is a state that needs to
+either find a successor or be marked dead. This means that the status labels for
+unknown- and open-labeled states are not necessarily correct during recursive
+calls; we are only concerned with preserving the forest structure, so that they will
+be correct after all calls complete. Note in particular that after merging cycles
+of unknown states via Merge(x, y), the new root is marked Open to preserve
+the generalized invariant on recursive calls.
+Upon receiving C(u), without loss of generality we may assume status(u) =
+Open (the opposite only occurs if the input GID has duplicate C(u) updates, or
+if C(u) occurs for a live state). At the top of the while loop, there are three cases:
+– If a reserve edge is available, and its target is dead, then we discard it. This
+preserves edge representation because that edge still satisfies (iv).
+– If a reserve edge is available and its target is not dead, then we see if adding
+that edge creates a cycle by calling CheckCycle. If no, then the two trees are
+distinct, so we add an edge between them. This preserves edge representation
+by moving that edge from case (i) to case (ii). If yes, then we collapse the
+cycle in the forest to a single state by repeated calls to Merge(x, y). This
+preserves edge representation by moving the edges in the cycle from case (ii)
+to case (iii). The forest structure of the succ is also preserved because if a
+graph is a forest, then it remains a forest after merging adjacent states.
+– Finally, if there are no reserve edges left, then because of edge representation,
+and because there is no successor from u, all edges from u must lead to dead
+states (case (v)) and therefore u is dead. This case splits the tree rooted at
+u into one tree for each of its predecessors, preserving the forest structure.
+Each of the succ edges from predecessors are deleted; this preserves edge
+representation by moving those edges from case (ii) to case (v).
+In any case, the generalized invariant for recursive calls is preserved, and all dead
+states are labeled correctly (given the invariant), so we are done.
+⊓⊔
+Complexity. The core inefficiency in Algorithm 1 lies in CheckCycle. The proce-
+dure repeatedly sets z ← succ(z) to find the tree root, which in general could be
+linear time in the number of edges. For example, suppose we have a line graph
+where we add edges in a backwards fashion and close them: E(2, 1), C(2), E(3,
+2), C(3), . . . , E(n, n-1), C(n). In such a graph, this inefficiency results in O(m2)
+work. We therefore want to improve upon this step in Algorithms 2 and 3.
+All other parts of the algorithm run in amortized α(m) time per update for
+m updates (using vectors to represent the maps fwd, bck, and succ for O(1)
+lookups). The OnTerminal calls and loop iterations only run once per edge
+in the graph when the target of that edge is marked live or terminal. Likewise,
+
+12
+Caleb Stanford and Margus Veanes
+the procedure OnClosed is called conservatively: the cost of each call can be
+assigned either to the target of an edge being marked dead, or to an edge being
+merged as part of a cycle. Both marking dead and merging can only happen
+once for a given edge, so this is O(1) per edge.
+3.2
+Logarithmic Algorithm
+At its core, CheckCycle requires solving an undirected reachability problem on
+a graph that is restricted to a forest. However, the forest is changed not just by
+edge additions, but edge additions and deletions. While undirected reachability
+and reachability in directed graphs are both difficult to solve incrementally,
+reachability in dynamic forests can be solved in O(log m) time per operation.
+Our algorithm uses an Euler Tour Forest data structure EF of Henzinger and
+King [26], and is shown in Algorithm 2.
+However, this idea does not work straightforwardly – once again because of
+the presence of cycles in the original graph. We cannot simply store the forest as
+a condensed graph with edges on condensed states. As we saw in Algorithm 1,
+it was important to store successor edges as edges into V, rather than edges into
+X – this is the only way that we can merge states in O(1), without actually
+inspecting the edge lists. If we needed to update the forest edges to be in X, this
+could require O(m) work to merge two O(m)-sized edge lists as each edge might
+need to be relabeled in the EF graph.
+To solve this challenge, we instead store the EF data structure on the original
+states, rather than the condensed graph; but we ensure that each condensed state
+is represented by a tree of original states in the original graph. When adding
+edges between condensed graph states, we need to make sure to remember the
+original state labels (u, v), so that we can later remove it using the original
+labels (this happens when its target becomes dead). When an edge would create
+a cycle, we instead simply ignore it in the EF graph: because a line of connected
+trees forms a tree.
+In summary, algorithm borrows most data and procedures from Algorithm 1,
+with a few important changes: (1) we maintain the EF data structure EF, a forest
+on V (2) the successor edges are stored as their original edge labels (u, v), rather
+than just as a target state; (3) the procedure OnClosed is updated in several
+locations to maintain the graph EF.
+Invariants. We continue all invariants from Algorithm 1, with the small mod-
+ification that the successor edges and no cycles invariants use the new succ
+representation: that is, they are constraints on the edges (x, UF.find(v)), where
+succ(x) = (u, v). In addition, we have two constraints on edges in EF, de-
+pending on whether those states are equivalent in the union-find structure. We
+call edges between inequivalent states inter-edges and those between equivalent
+states intra-edges.
+– EF inter-edges: For all states u and v not in the same canonical state, (u, v)
+is in the EF graph if and only if (u, v) = succ(UF.find(u)) or (v, u) =
+succ(UF.find(v)).
+
+Incremental Dead State Detection in Logarithmic Time
+13
+Algorithm 2 Logarithmic time algorithm.
+All data from Algorithm 1
+succ: a map from X to E (instead of to V)
+EF: Euler Tour Trees data structure providing: EF.add, EF.remove, and EF.connected
+procedure OnEdge(E(u, v)) as in Algorithm 1
+procedure Merge(x, y) returning V as in Algorithm 1
+procedure OnTerminal(T(v))
+y ← UF.find(v)
+for all x in DFS backwards (along bck) from y not already Live do
+if status(x) = Unknown then
+▷ The following line is not strictly necessary, but simplifies the analysis
+(u, v) ← succ(x); delete succ(x); EF.remove(u, v)
+status(x) ← Live; output Live(x′) for all x′ in UF.iter(x)
+procedure OnClosed(C(v))
+y ← UF.find(v)
+if status(y) ̸= Open then return
+while res(y) is nonempty do
+pop (v, w) from res(y); z ← UF.find(w)
+if status(z) = Dead then continue
+else if CheckCycle(y, z) then
+for all z′ in cycle from z to y do z ← Merge(z, z′)
+else
+status(x) ← Unknown; succ(x) ← (v, w)
+EF.add(v, w)
+▷ undirected edge; use original labels (not (x, y))
+return
+status(y) ← Dead; output Dead(y′) for all y′ in UF.iter(y)
+ToRecurse ← ∅
+for all (u, v) in bck(y) do
+x ← UF.find(u)
+if status(x) = Unknown then
+(u′, v′) ← succ(x)
+if UF.find(v′) = y then
+EF.remove(u′, v′)
+▷ undirected edge; use original labels
+status(x) ← Open; delete succ(x); add x to ToRecurse
+for all x in ToRecurse do OnClosed(C(x))
+procedure CheckCycle(y, z) returning bool
+return EF.connected(y, z)
+
+14
+Caleb Stanford and Margus Veanes
+– EF intra-edges: For all unknown canonical states x, the set of edges (u, v)
+in the EF between states belonging to x forms a tree.
+Theorem 1. Algorithm 2 is correct.
+Proof. Observe that the EF inter-edges constraint implies that EF only contains
+edges between unknown and open states, together with isolated trees. In the
+modified OnTerminal procedure, when marking states as live we remove inter-
+edges, so we preserve this invariant.
+Next we argue that given the invariants about EF, for an open state y the
+CheckCycle procedure returns true if and only if (y, z) would create a directed
+cycle. If there is a cycle of canonical states, then because canonical states are
+connected trees in EF, the cycle can be lifted to a cycle on original states, so y and
+z must already be connected in this cycle without the edge (y, z). Conversely, if
+y and z are connected in EF, then there is a path from y to z, and this can be
+projected to a path on canonical states. However, because y is open, it is a root
+in the successor forest, so any path from y along successor edges travels only
+on backwards edges; hence z is an ancestor of y in the directed graph, and thus
+(y, z) creates a directed cycle.
+This leaves the OnClosed procedure. Other than the EF lines, the structure
+is the same as in Algorithm 1, so the previous invariants are still preserved,
+and it remains to check the EF invariants. When we delete the successor edge
+and temporarily mark status(x) = Open for recursive calls, we also remove it
+from EF, preserving the inter-edge invariant. Similarly, when we add a successor
+edge to x, we add it to EF, preserving the inter-edge invariant. So it remains to
+consider when the set of canonical states changes, which is when merging states
+in a cycle. Here, a line of canonical states is merged into a single state, and a
+line of connected trees is still a tree, so the intra-edge invariant still holds for
+the new canonical state, and we are done.
+⊓⊔
+Theorem 2. Algorithm 2 uses amortized logarithmic time per edge update.
+Proof. By the analysis of Algorithm 1, each line of the algorithm runs amortized
+O(1) time other than those in CheckCycle. For the CheckCycle, procedure
+it now avoids the loop and so also runs amortized O(1) times. Each line is either
+constant-time, α(m) = o(log n) time for the UF calls, or O(log n) time for the EF
+calls, so in total the algorithm runs in amortized O(log n) time per update.
+⊓⊔
+3.3
+Lazy Algorithm
+While the asymptotic complexity of log n could be the end of the story, in prac-
+tice, the cost of the EF calls could be a significant overhead. The technical details
+of Euler-Tour Trees include building an AVL-tree cycle for each tree, where the
+cycle contains each state of the graph and each edge in the graph twice. It turns
+out that adding one edge to EF results in up to eight modifications to the AVL
+tree: it needs to be split at the source, split at the target, then the edge needs to
+
+Incremental Dead State Detection in Logarithmic Time
+15
+Algorithm 3 Lazy algorithm.
+All data from Algorithm 1
+jumps: a map from X to lists of V
+procedure OnEdge(E(u, v)) as in Algorithm 1
+procedure OnTerminal(T(v)) as in Algorithm 1
+procedure OnClosed(C(v)) as in Algorithm 1
+procedure CheckCycle(y, z) returning bool
+return y = GetRoot(z)
+procedure GetRoot(z) returning V
+if status(z) = Open then return z
+if jumps(z) is empty then
+push succ(z) to jumps(z)
+▷ set 0th jump
+repeat
+pop w from jumps(z); z′ = UF.find(w)
+▷ remove dead jumps
+until status(z′) ̸= Dead
+push z′ to jumps(z)
+result ← GetRoot(z′)
+n ← length(jumps(z)); n′ ← length(jumps(z′))
+if n ≤ n′ then
+push jumps(z′)[n − 1] to jumps(z)
+▷ set nth jump
+return result
+procedure Merge(x, y) returning V
+z ← UF.union(x, y)
+bck(z) ← bck(x) + bck(y); res(z) ← res(x) + res(y)
+jumps(z) ← empty
+return z
+be added in both directions (u, v) and (v, u) to the cycle, and then these trees
+need to be glued together. Every one of these operations comes with a rebalanc-
+ing operation which could do Ω(log n) tree rotations and pointer dereferences to
+visit the nodes in the AVL tree.
+As a result of these (constant-time) overheads, in this section, we investigate
+a simpler, lazy algorithm which avoids Euler Tour Trees, and works directly by
+modifying Algorithm 1. To optimize Algorithm 1, one idea in the right direction
+is to store for each state a direct pointer to the root which results from repeatedly
+calling succ. But there are two issues with this. First, maintaining this may be
+difficult (when the root changes, potentially updating a linear number of root
+pointers). Second, the root may sometimes be marked dead, in which case we
+have to re-compute all the successor pointers.
+Instead, we introduce a jump list from each state: intuitively, it will contain
+states after calling successor once, twice, four times, eight times, and so on, and
+will be updated at most once for every visit to the state. When a jump becomes
+obsolete (the target dead), we just pop off the largest jump, so we do not lose all
+of our work in building the list. When states are merged, we do lose our work,
+discarding the jump lists, and this also means the exact power-of-two distances
+are no longer preserved. We maintain the following additional information: for
+
+16
+Caleb Stanford and Margus Veanes
+each unknown canonical state x, a nonempty list of jumps [v0, v1, v2, . . . , vk],
+such that v0 is reachable from x, v1 is reachable from v0, v2 is reachable from
+v1, and so on, and v1 = succ(x). The algorithm is shown in Algorithm 3.
+The key procedure is GetRoot(z), which is called when adding a reserve
+edge (y, z) to the graph. It finds the root in the successor forest when repeatedly
+calling successor from z. It shortcuts by using the jump list, and also updates
+the set of jumps to be approximately at powers of two: to do this, the nth jump
+from a state z is set to the (n − 1)st jump from the (n − 1)th jump from z.
+Invariants. In addition to the invariants from Algorithm 1: for each unknown
+canonical state x, jumps(x) is a list of states v0, v1, v2, . . . , vk such that:
+– First jump: if the jump list is nonempty, then v0 = succ(v).
+– Reachability: vi+1 is reachable from vi for all i.
+– Powers of two: on the path of canonical states from v0 to vi, the total number
+of states (including all the states in each equivalence class) is at least 2i.
+The powers of two invariant is not necessary for correctness, but is the key
+to the efficiency of this algorithm: it ensures that the nth jump travels at least
+2n states at once in the successor forest. It follows from this that if the jump list
+is fully saturated for every state, querying GetRoot(z) will take logarithmic
+time. But because the jump lists are updated lazily (and have dead states that
+need to be discarded), this does not establish an asymptotic complexity for the
+algorithm.
+Theorem 3. Algorithm 3 is correct.
+Proof. We use the invariant on the jump list mentioned: that v1 is the successor,
+v2 is reachable from v1, v3 is reachable from v2, and so on, using only forward
+edges added to the graph (not reserve edges). I.e., v1, v2, . . . is some sublist of
+the states along the path from an unknown state to its root, potentially followed
+by some dead states. We need to argue that the subprocedure GetRoot (i)
+receives the same verdict as repeatedly calling succ to find a cycle in the first-cut
+algorithm and (ii) preserves the jump list invariant. Popping dead states from the
+jump list clearly preserves the invariant, as does adding on a state along the path
+to the root, which is done when k′ ≥ k. Merging states preserves the jump list
+invariant trivially because we throw the jump list away, and marking states live
+preserves the jump list invariant trivially since the jump list is only maintained
+and used for unknown states. Finally, marking states as closed initializes the
+jump list correctly assuming that the successor is calculated correctly.
+⊓⊔
+4
+Experimental Evaluation
+The primary goal of our evaluation has been to experimentally validate the
+performance of GIDs as a data structure in isolation, rather than their use in a
+particular application. Our evaluation seeks to answer the following questions:
+Q1 How does our approach (Algorithms 2 and 3) compare to the state-of-the-art
+approach based on maintaining SCCs?
+
+Incremental Dead State Detection in Logarithmic Time
+17
+Q2 How does the performance of the studied algorithms vary when the class of
+input graphs changes (e.g., sparse vs. dense graphs, structured vs. random
+graphs)?
+Q3 Finally, how do the studied algorithms perform on GIDs taken from the
+example application to regexes described in Section 5?
+To answer Q1, we put substantial implementation effort into a common
+framework on which a fair comparison could be made between different ap-
+proaches. To this end, we implemented GIDs as a data structure in Rust which
+includes a graph data structure on top of which all algorithms are built. In par-
+ticular, this equalizes performance across algorithms for the following baseline
+operations: state and edge addition and retrieval, DFS and BFS search, edge
+iteration, and state merging. We chose Rust for our implementation for its per-
+formance, and because there does not appear to be an existing publicly available
+implementation of BFGT in any other language. The number of lines of code
+used to implement these various structures is summarized in Figure 4. We im-
+plement Algorithms 2 and 3 and compare them with the following baselines:
+BFGT This algorithm is a careful implementation of the state-of-the-art ap-
+proach based on SCC maintenance, using worst-case amortized O(√m) time
+per update [4].
+Simple In addition to BFGT, we provide a simpler SCC-based baseline, using
+a DFS instead of using the BFGT algorithm. It updates the SCCs after each
+state is marked closed by searching for all cycles from that state using a
+forward-DFS. This takes Θ(m2) in the worst case, but can be very efficient
+when edges are added in a “forward-facing” topologically sorted manner, so
+represents an important point of comparison.
+Naive Finally, the naive algorithm is a greedy baseline that represents a worst-
+case upper bound: it re-computes the entire set of dead states using a linear-
+time DFS after each update. It is Θ(m2) for m updates.
+To answer Q2, first, we compiled a range of basic graph classes which are
+designed to expose edge case behavior in the algorithms, as well as randomly
+generated graphs. We focus on graphs with no live states, as live states are
+treated similarly by all algorithms. Most of the generated graphs come in 2×2 =
+4 variants: (i) the states are either read in a forwards- or backwards- order; and
+(ii) they are either dead graphs, where there are no open states at the end and
+so everything gets marked dead; or unknown graphs, where there is a single
+open state at the end, so most states are unknown. In the unknown case, it is
+sufficient to have one open state at the end, as many open states can be reduced
+to the case of a single open state where all edges point to that one. We include
+GIDs from line graphs and cycle graphs (up to 100K states in multiples of 3),
+as well as complete graphs, complete acyclic graphs (with only forward-edges),
+and bipartite graphs (up to 1K states). These are important cases, for example,
+because the reverse-order version of the line and cycle graphs is an edge case for
+Simple: it often times out on these, but performs well on the forward-version of
+these GIDs.
+
+18
+Caleb Stanford and Margus Veanes
+Implementation
+Component
+LoC
+Common Framework
+1197
+Naive Algorithm
+78
+Simple Algorithm
+98
+BFGT Algorithm
+265
+Algorithm 2 (Ours)
+253
+Algorithm 3 (Ours)
+283
+Euler Tour Trees
+1510
+Experimental Scripts
+556
+Separated Unit Tests
+798
+Util
+217
+Other
+69
+Total
+5324
+Category
+Benchmark
+Source
+Qty
+Basic
+Line
+24
+Cycle
+24
+Complete
+18
+Bipartite
+14
+Total
+80
+Random
+Sparse
+260
+Dense
+130
+Total
+390
+Regex
+RegExLib [9,49]
+2061
+37
+Handwritten [50]
+70
+26
+Additional
+11
+Total
+74
+Fig. 4. Left: Lines of code for each algorithm and other implementation components.
+Right: Benchmark GIDs used in our evaluation. Where present, the source column
+indicates the quantity prior to filtering out trivially small graphs.
+Time (ms)
+Benchmarks Solved
+0
+30
+60
+90
+1
+10
+100
+1000
+10000
+Naive
+Simple
+BFGT
+Alg 2
+Alg 3
+Time (ms)
+Benchmarks Solved
+0
+100
+200
+300
+400
+1
+10
+100
+1000
+10000
+Naive
+Simple
+BFGT
+Alg 2
+Alg 3
+Time (ms)
+Benchmarks Solved
+0
+20
+40
+60
+80
+1
+10
+100
+1000
+10000
+Naive
+Simple
+BFGT
+Alg 2
+Alg 3
+Benchmark Size
+Time (ms)
+1
+10
+100
+1000
+10000
+100
+1000
+10000
+100000
+1000000
+Naive
+Simple
+BFGT
+Alg 2
+Alg 3
+Benchmark Size
+Average Time (ms)
+1
+10
+100
+1000
+10000
+100
+1000
+10000
+100000
+1000000
+Naive
+Simple
+BFGT
+Alg 2
+Alg 3
+Fig. 5. Evaluation results. Left: Cumulative plot showing the number of benchmarks
+solved in time t or less for basic GID classes (top), randomly generated GIDs (middle),
+and regex-derived GIDs (bottom). Top right: Scatter plot showing the size of each
+benchmark vs time to solve. Bottom right: Average time to solve benchmarks of size
+closest to s, where values of s are chosen in increments of 1/3 on a log scale.
+
+Incremental Dead State Detection in Logarithmic Time
+19
+Second, to exhibit more dynamic behavior, we generated random graphs:
+sparse graphs with a fixed out-degree from each state, which can be either 1, 2, 3,
+or 10 (up to 100K states); and dense graphs with a fixed probability of each edge,
+which can be .01, .02, or .03 (up to 10K states). As with the basic graphs, states
+are read in some order and marked closed; at most one state is left open at the
+end. Each random graph is generated using 10 different random seeds.
+To answer Q3, we needed to isolate the performance of the GID data struc-
+ture itself, rather than the performance of the other components of the regex
+SMT solver. Prior work on regex solving integrated with Z3 [50] implements
+an earlier version of GIDs, which is essentially the Simple algorithm in C++.
+However, preliminary tests suggest that on some examples, GIDs are already
+quite efficient and contribute only a fraction of the performance (1-10%), as
+a substantial amount of time is spent manipulating and rewriting expressions
+and computing derivatives. While we would expect GIDs to be the bottleneck on
+larger examples where the asymptotic complexity of Simple becomes prohibitive,
+this fact makes it difficult to isolate the performance of the GID data structure
+itself, which is the focus of this paper.
+To isolate the GID performance, we therefore instrumented the modified Z3
+solver code to export the (incremental) sequence of graph updates that would
+be performed during a run of Z3 on existing regex benchmarks. For each regex
+benchmark, this instrumented code produces a faithful representation of the se-
+quence of graph updates that would occur in a run of the SMT solver on this
+particular benchmark. It is also important to use ERE benchmarks, rather than
+plain regular expressions as these are the ones for which dead state detection is
+relevant (see Section 5). For each regex benchmark, we thus get a GID bench-
+mark for the present paper. In our regex-derived GID examples, we include the
+RegExLib benchmarks from [9,49] and the handcrafted Boolean benchmarks re-
+ported in [50]. We add to these 11 additional examples designed to stress test the
+GID side of a regex solver. The collection of regex SMT benchmarks is available
+on GitHub4.
+From both the Q2 and Q3 benchmarks, we filter out those that are too easy:
+any benchmark which takes under 10 milliseconds for all of the algorithms to
+solve (including Naive). We enforce a timeout of 60 seconds. The evaluation was
+run on a 2020 MacBook Air running MacOS Monterey, containing an Apple M1
+processor and 8GB of memory.
+Correctness. To ensure that all of our implementations our correct, we also
+invested time into unit testing and checked output correctness on all of our
+collected benchmarks, including several cases which exposed bugs in previous
+versions of one or more algorithms. In total, all algorithms are vetted against 25
+unit tests from handwritten edge cases that exposed prior bugs, 373 unit tests
+from benchmarks, and 30 module-level unit tests for specific functions.
+Results. Figure 5 shows the results. Algorithm 3 shows significant improve-
+ments over the state-of-the-art, solving more benchmarks in a smaller amount
+4 https://github.com/cdstanford/regex-smt-benchmarks
+
+20
+Caleb Stanford and Margus Veanes
+of time across basic GIDs, random GIDs, and regex GIDs. Algorithm 2 also
+shows state-of-the-art performance, similar to BFGT on basic and regex GIDs
+and significantly better on random GIDs. On the bottom right, since looking at
+average time is not meaningful for benchmarks of widely varying size, we stratify
+the size of benchmarks into buckets, and plot time-to-solve as a function of size.
+note that as both x-axis and y-axis are on a log scale, a total running time of
+O(mp) for m updates corresponds to a line with slope p. The plot shows that
+Algorithm 3 exhibits up to two orders of magnitude speedup over BFGT for
+larger GIDs – we see speedups of 110x to 530x for GIDs in the top five size
+buckets (GIDs of size nearest to 100K, 200K, 500K, 1M, and 2M).
+New implementations of existing work. Our implementation contributes, to our
+knowledge, the first implementation of BFGT for SCC maintenance. In addition,
+it is one of the first implementations of Euler Tour Trees (EF) for undirected
+reachability in forests, including the AVL tree backing for EF which maintains
+a disjoint collection of lists of state and edge identifiers, and the first implemen-
+tation in Rust.
+5
+Application to Extended Regular Expressions
+In this section, we describe one application of GIDs in the context of deciding
+regex constraints in SMT. In particular, we explain how precisely the GID state
+classification problem arises in the context of derivative-based solvers [50,34].
+We first define extended regexes (regexes extended with intersection & and
+complement ~) modulo a (symbolic) alphabet A that intuitively provides char-
+acter classes as predicates that are closed under Boolean operations. We explain
+the main idea behind (symbolic) derivatives [50] that provides the foundation
+for incrementally creating a GID. Then we show, through an example, how a
+solver can incrementally expand derivatives to reduce the satisfiability problem
+to the GID state classification problem (Definition 2).
+EREs or regexes are defined as follows where ϕ is a predicate in A.
+RE ::= ϕ | ε | RE1 · RE2 | RE* | RE1 | RE2 | RE1 & RE2 | ~RE
+Let Rk represent the concatenation of R k times. A regex R is nullable if it
+matches the empty string. Nullability is defined inductively over the structure
+of regexes with ε and R* being nullable by definition. Observe in particular that
+~R is nullable iff R is not nullable, and R1 & R2 is nullable iff both R1 and R2
+are nullable. Terminal states are precisely the nullable regexes.
+The false predicate ⊥ of A serves as the regex that matches nothing and is
+a trivially dead state. Thus ~⊥ is equivalent to �* that matches everything and
+is a trivially live state, where � is the true predicate of A.
+A symbolic derivative δ(R) of a regex R is a binary tree or term t whose
+leaves are regexes and internal nodes are labeled by predicates from A – the two
+immediate subtrees t1 and t2 of a node with label ϕ, denoted by (ϕ ? t1 : t2),
+correspond to ϕ being true in t1 and false in t2. A branch with the accumulated
+
+Incremental Dead State Detection in Logarithmic Time
+21
+branch condition ψ from the root of t to one of its leaves R′ then defines a
+transition R
+ψ−→R′ – provided ψ is satisfiable in A – and the edge (R, R′) is added
+to the GID where the actual predicate ψ is no longer needed because the edge
+itself represents its feasibility. The number of leaves of δ(R) is the out-degree
+deg+(R) of R that is independent of the actual size of the concrete alphabet.
+Thus R becomes closed when deg+(R) many edges have been added from R.
+The formal definition of a symbolic derivative is as follows where the op-
+erations ⃝| , ⃝& , ⃝~ and ⃝· are here for the purposes of this paper and w.l.o.g.
+normalizing variants of | , & , ~ and ·, by distributing the operations into if-then-
+elses and build a single binary tree as a nested if-then-else as a result (see [50]
+for details):
+δ(ε) = δ(⊥) = ⊥
+δ(ϕ) = (ϕ ? ε : ⊥)
+δ(R∗) = δ(R) ⃝· R∗
+δ(R | R′) = δ(R) ⃝| δ(R′)
+δ(R & R′) = δ(R) ⃝& δ(R′)
+δ(~R) = ⃝~ δ(R)
+δ(R · R′) = if Nullable(R) then (δ(R) ⃝· R′) ⃝| δ(R′) else δ(R) ⃝· R′
+If c is a term of type character then δc(R) is obtained from δ(R) by instantiating
+all internal nodes (conditions of the if-then-elses) ϕ by tests c ∈ ϕ. This means
+that in the context of SMT, δc(R) is a term of type regex, while δ(R) represents
+the lambda-term λc.δc(R) that is constructed independently of c.
+Concretely, to apply Definition 1 to regexes, states will be regexes, and edges
+will represent transitions to their derivatives. A live state here is thus a regex
+that reaches a nullable regex via 0 or more edges. This implies that there exists
+a concrete string matching it. Conversely dead states are always empty, i.e. they
+match no strings, but can reach other dead states, creating strongly connected
+components of closed states none of which are live.
+5.1
+Reduction from Incremental Regex Emptiness to GIDs
+For a solver based on derivatives of EREs, suppose we want to determine the
+satisfiability of a single regex constraint s ∈ R, where s is a string variable and
+R is a concrete regex. E.g., let L = ~(�*α�100) and R = L & (�α) where α is some
+character predicate in A s.t. both α and ¬α are satisfiable, e.g., take α to mean
+“is digit” to be concrete.5 The solver manipulates regex membership constraints
+on strings by unfolding them [50]. The constraint s ∈ R, that essentially tests
+nonemptiness of R with s as a witness, becomes
+(s = ϵ ∧ Nullable(R)) ∨ (s ̸= ϵ ∧ s1.. ∈ δs0(R))
+where, s ̸= ϵ since R is not nullable, si.. is the suffix of s from index i, and
+δ(R) = δ(L) ⃝& δ(�α) = (α ? L & ~(�100) : L) ⃝& α = (α ? L & ~(�100) & α : L & α)
+Let R1 = L & ~(�100) & α and R2 = L & α. So R has two outgoing transitions
+R
+α−→R1 and R
+¬α
+−−→R2 that contribute the edges (R, R1) and (R, R2) into the
+GID. Note that these edges depend only on R and not on s0.
+5 Using standard notations: \d denotes all digits, and [^\d] denotes all non-digits.
+
+22
+Caleb Stanford and Margus Veanes
+We now continue the search incrementally by checking the two branches of
+the if-then-else constraint, where R1 and R2 are again both not nullable (so
+s1.. ̸= ϵ):
+s0 ∈ α ∧ s2.. ∈ δs1(R1)
+∨
+s0 ∈ ¬α ∧ s2.. ∈ δs1(R2)
+δ(R1) = (α ? L & ~(�100) & ~(�99) : L & ~(�99)) ⃝& (α ? ε : ⊥) = (α ? ε : ⊥)
+δ(R2) = (α ? L & ~(�100) : L) ⃝& (α ? ε : ⊥) = (α ? ε : ⊥)
+It follows that R1
+α−→ε and R2
+α−→ε, so the edges (R1, ε) and (R2, ε) are added to
+the GID where ϵ is a trivial terminal state. In fact, after R1 the search already
+terminates because we then have the path (R, R1)(R1, ϵ) that implies that R is
+live. The associated constraints s0 ∈ α and s1 ∈ α and the final constraint that
+s2.. = ϵ can be used to extract a concrete witness, e.g., s = "42". From the point
+of view of deciding nonemptiness the role of the witness s is immaterial, and
+can be omitted, as it poses no additional constraints on its own if s is a variable
+(a.k.a., an uninterpreted constant in SMT).
+Soundness of the algorithm follows from that if R is nonempty, i.e., s ∈ R
+is satisfiable, then by expanding the search, we eventually arrive at a nullable
+(terminal) regex, as in the example run above. To achieve completeness – and
+to eliminate dead states as early as possible – we incrementally construct a
+GID corresponding to the set of regexes seen so far (as above). After all the
+feasible transitions from R to its derivatives in δ(R) are added to the GID as
+edges (w.l.o.g. in one batch), the state R becomes closed. Crucially, due to the
+symbolic form of δ(R), no derivative is missing. Therefore R is known to be
+empty precisely as soon as R is detected as a dead state in the GID. We get the
+following theorem that uses finiteness of the closure of symbolic derivatives [50,
+Theorem 7.1]:
+Theorem 4. For any regex R, (1) If R is nonempty, then the decision procedure
+eventually marks R live. If n is the shortest distance from R to a terminal state,
+then a concrete witness of length n can be generated. (2) If R is empty, then the
+decision procedure marks R dead after a finite number of steps and terminates.
+Observe that any number of simultaneous regex constraints for a string s can be
+combined into single regex constraint by using the Boolean operations of ERE.
+More crucially, the algorithm is independent of the size of the universe of A,
+that may be very large, like the Unicode character set, or even infinite.
+6
+Related Work
+Online graph algorithms: Online graph algorithms are typically divided into
+problems over incremental graphs (where edges are added), decremental graphs
+(where edges are deleted), and dynamic graphs (where edges are both added
+and deleted), with core data structures discussed in [38]. For dynamic directed
+graphs, important problems include transitive closure, cycle detection, topological
+ordering, and strongly connected component (SCC) maintenance. Maintaining
+
+Incremental Dead State Detection in Logarithmic Time
+23
+a topological order of dynamic DAGs is studied in [35]. An online algorithm
+for topological sorting that is experimentally shown to be preferable for sparse
+graphs is discussed in [46], and in a related article [45] it is also discussed how
+to extend such an algorithm to detect strongly connected components. The key
+applications of such algorithms have traditionally been in pointer analysis [44].
+Topological order of incremental DAGs is studied in [24], presenting two different
+algorithms, one for sparse graphs and one for dense graphs – the algorithms
+are also extended to work with SCCs. The sparse algorithm was subsequently
+simplified in [4] and is the basis of our implementation named BFGT in the
+Evaluation section. A unified approach of several algorithms based on [4] is
+presented in
+[14] that uses a notion of weak topological order and a labeling
+technique that estimates transitive closure size. Further extensions of [4] are
+studied in [6,8] based on randomization. Transitive closure of decremental DAGs
+is studied in [47] that improve upon some algorithms presented earlier in [25].
+Although we have not looked at decremental graphs in this paper, the problem
+may become relevant in the context of backtracking in SMT string theory solver
+during proof search in the presence of non-ground regexes, where properties such
+as nullability may be theory-dependent and thus affect liveness of states.
+Data structures for SMT: UnionFind [52] is a frequently used data structure
+that we also rely on in our algorithm. E-graphs [54,16,18,42] are used to ensure
+functional extensionality, where two expressions are equivalent if their subex-
+pressions are equivalent. In both cases the maintained relation is an equivalence
+relation. In contrast, maintaining live and dead states involves tracking reacha-
+bility rather than equivalence. While reachability is a basic problem in many core
+SMT algorithms, to the best of our knowledge, the formulation of the problem
+we consider here is new.
+Dead state elimination: Dead state elimination for automata, also known as
+trimming [7], is a specialized version of useless symbol elimination from context-
+free grammars [28, pp 88-89]. It plays an important role in minimization of
+automata [31,40,30,29]. The classical minimization algorithms have been studied
+extensively, e.g., in [10,7]. Brzozowski [11] observed that a DFA can be minimized
+by determinizing the reversal of the determinization of the reversal of the DFA.
+Regular expression matching and analysis: Decision problems for regexes are
+studied in the context of Satisfiability Modulo Theories (SMT), where the first-
+order objects are strings and formulas consist of string constraints such as x = yz
+or x ∈ R, where R is a regex. Over such theories, the problem of nonemptiness
+of a regex reduces to live state detection in the automaton (or set of derivatives)
+for R. String and regex constraints have been the focus of both SMT and the
+Constraint Programming (CP) solving communities, with a wide range of tools
+focusing on different problem classes and applications [27]. Regexes and are now a
+standard part of the SMTLIB2 format [53]. In SMT, string solvers are integrated
+in the CDCL(T) architecture [43]. In the CP community, the MiniZinc format
+integrates membership constraints over regular languages presented as either
+
+24
+Caleb Stanford and Margus Veanes
+DFAs or NFAs [39]. The solvers presented in [34] and [50] are based on computing
+Antimirov derivatives – a state graph of regexes is used in [50] to terminate search
+from dead state regular expressions that arise in ERE constraints, which helps
+the solver terminate search from goals that would otherwise not terminate.
+7
+Conclusion
+GIDs are a form of incremental abstract transition system in which states are
+closed when they will receive no further outgoing edges; Algorithm 2 and Al-
+gorithm 3 solve the incremental live and dead detection problem in GIDs. The
+former runs in logarithmic worst-case time, and both algorithms achieve orders
+of magnitude speedup over the state-of-the-art based on maintaining strong con-
+nected components. Our implementation is open-source and publicly available6.
+Acknowledgments
+We would like to thank the anonymous reviewers of CAV 2021 and TACAS 2022
+for feedback leading to substantial improvements in both the paper and results.
+We also extend a special thanks to Nikolaj Bjørner, for his collaboration and
+involvement with the Z3 implementation, and Yu Chen, for helpful discussions
+in which he proposed the idea for the first-cut algorithm.
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+A
+Appendix
+A.1
+Extension to the Framework: Non-Reachable Updates
+When GIDs are viewed as an abstract data type, other application-specific
+heuristics can be incorporated. To illustrate this point, we consider another piece
+of application-specific information: assertions which state that one vertex is not
+(and will never be) reachable from another. We discuss how to incorporate such
+updates to further improve the optimized algorithm in practice to exploit such
+updates.
+First, we augment the definition of guided incremental digraph to allow up-
+dates of the form N(u, v), which labels that v is not reachable from u. We then
+say that the graph is valid if, in addition to the previous conditions, for every
+update N(u, v), v is not reachable from u via a sequence of edges in the final
+digraph. Figure 6 extends the example presented in Figures 1 and 2 with a not-
+reachable update. Notice that N(3, 5) does not affect the set of live and dead
+states, but may be exploited for more efficient search.
+To incorporate N(u, v) updates in our algorithm, when moving a reserve edge
+into the graph, N(u, v) can be used to shortcut the whole procedure: if we add
+an edge (v, u) in particular, then we know that this doesn’t create a cycle, and
+we don’t need to repeatedly call succz at all. We store N(u, v) updates in a set
+instead of a list for O(1) querying; if we need to merge them, we just discard the
+set if it gets too large. Therefore, in addition what was maintained previously,
+the extended algorithm tracks, for each unknown canonical state x, a set of
+non-reachable edges (x, y), where each corresponds to some update N(u, v) with
+x = find(u) and y = find(v) (but not necessarily vice versa); and a constant κ
+which bounds the amount of work when merging non-reachable sets (additional
+
+28
+Caleb Stanford and Margus Veanes
+1
+2
+3
+4
+5
+Fig. 6. Extension to our framework: guided incremental digraph from Figures 1 and 2,
+with an added update N(3, 5) which states that 5 is not reachable from 3. In this graph,
+it would be invalid to then add E(3, 4) to the graph. This does not change the set of
+live or dead states, but may be exploited for more efficient search: when checking if
+state 3 is live or dead, we may ignore states 4 and 5.
+elements will be discarded). We modify the code for is-root(y, x) with a single
+check in the beginning of the procedure: if x is in the not-reachable set from y,
+return false. Otherwise, we continue as normal. Because this algorithm uses the
+N(u, v) information directly and conservatively, assuming the input GID is valid,
+it remains correct.
+A.2
+Reachability Problems on EREs
+For extended regexes X and Y , we say that Y is reachable from X if Y ∈ ∂⋆(X).
+We say that X and Y are strongly connected, denoted X ⟳ Y , when both
+Y ∈ ∂⋆(X) and X ∈ ∂⋆(Y ), i.e., when both Y is reachable from X and X is
+reachable from Y through derivation as defined above.
+We also consider the following subclasses of extended regexes:
+– Regexes (REs) are EREs that do not contain complement or intersection,
+that is, classical regexes.
+– Semi-extended regexes (SEREs) are EREs that do not contain complement.
+– Intersections of regexes (IREs) are SEREs that are equal to an intersection
+of REs.
+– Boolean combinations of regexes (BCREs) are EREs that are equal to a
+Boolean combination (combination of intersection and complement) of REs.
+Semantically, all of these capture the set of regular languages. The syntactic
+relationships between these different subclasses is summarized as follows, where
+all inclusions are strict:
+RE ⊂ IRE ⊂ BCRE
+∩
+∩
+SERE ⊂ ERE
+We consider the following three decision problems. They are closely related
+and in order of generality: the first is a special case of both the second and
+thirds, and the second reduces to two queries of the third. For each problem,
+we also consider problems on all the subclasses of EREs that we defined: REs,
+SEREs, IREs, and BCREs. We summarize the complexity results in the table in
+Figure 7.
+
+Incremental Dead State Detection in Logarithmic Time
+29
+Subclass
+Edge in Cycle
+Strong Connectedness
+Reachability
+RE
+Linear
+Linear
+Linear
+IRE
+PSPACE-C
+PSPACE-C
+PSPACE-C
+BCRE
+PSPACE-C
+PSPACE-C
+PSPACE-C
+SERE
+PSPACE-C
+PSPACE-C
+PSPACE-C
+ERE
+non-elementary
+non-elementary
+non-elementary
+Fig. 7. Complexity results for reachability of various subclasses of extended regexes.
+Edge in Cycle: Given X and Y such that X ∈ ∂(Y ), is Y ∈ ∂⋆(X)?
+Strong Connectedness: Given X and Y , is X ⟳ Y ?
+Reachability: Given X and Y , is Y ∈ ∂⋆(X)?
+Theorem 5. For RE, the edge-in-cycle, strong-connectedness, and reachability
+problems can be solved in linear time.
+Proof. It suffices to show that reachability can be solved in linear time, since as
+stated above, it is the most general of the three. We give a decision procedure for
+deciding Y ∈ ∂⋆(X) by recursing on the structure of X. Prior to the procedure,
+we ensure (i) that all regexes are normalized, (ii) that equal regexes are repre-
+sented by the same pointer, and (iii) that concatenations are represented as lists
+and precompute the length of each list, so that for any subexpression X0 of X,
+whether Y = Y1·X0 can be checked in constant time. This precomputation takes
+O(|X| + |Y |) time. The recursion then works as follows, on input (X, Y ). (1) If
+X = ϕ, ε, or ⊥, we can check directly if Y is one of the finitely many derivatives.
+(2) If X = X1 | X2, we observe that Y ∈ ∂⋆(X) if and only if Y ∈ ∂⋆(X1) or
+Y ∈ ∂⋆(X2). So we recurse on (X1, Y ) and (X2, Y ). (3) If X = X∗
+1, we observe
+that Y ∈ ∂⋆(X) if and only if Y = X′
+1(X∗
+1) for X′
+1 ∈ ∂⋆(X1). So we first check
+if Y has the form Y1X∗
+1, then recurse on (X1, Y1). (4) Finally, if X = X1 · X2,
+we observe that Y ∈ ∂⋆(X) if and only if either Y = X′
+1 · X2 for X′
+1 ∈ ∂⋆(X1),
+or Y = X′
+2 for X′
+2 ∈ ∂⋆(X2). (Note that this relies on the fact that X1 is an
+RE, not an ERE, so we know it is nonempty and in particular ε ∈ ∂⋆(X1).) So
+we first check if Y has the form Y1 · X2, if so recursing on both (X1, Y1) and
+(X2, Y ), otherwise recursing on just (X2, Y ). Since the procedure never recurses
+twice on X or twice on the same subexpression of X, it takes O(|X|) time.
+⊓⊔
+Theorem 6. For the classes IRE, BCRE, and SERE, the edge-in-cycle, strong-
+connectedness, and reachability problems are PSPACE-complete.
+Proof. It suffices to show: (1) the edge-in-cycle problem for IRE is PSPACE-
+hard; (2) the reachability problem for BCRE is in PSPACE; and (3) the reach-
+ability problem for SERE is in PSPACE. (1) We reduce from intersection-
+nonemptiness of REs, which is known to be PSPACE-hard. Given a list of
+REs R1, R2, . . . , Rk, then let b be a fresh character that does not appear in
+
+30
+Caleb Stanford and Margus Veanes
+R1, . . . , Rk, and construct the IRE X = (bR1)∗ & (bR2)∗ & · · · & (bRk)∗. X has
+only one derivative X′, which is an intersection where the ith term is Ri(bRi)∗.
+Then the edge-in-cycle property for (X, X′) holds if and only if the intersection
+of Ri is nonempty, because ∂b is empty for all subexpressions of Ri except ε,
+so the only way to get back to X is by stepping all Ri to ε simultaneously. (2)
+We first convert the BCRE X to an alternating automaton (AFA): first convert-
+ing the RE subexpressions to NFAs, then using the standard constructions for
+Boolean operations on AFAs. (3) For an SERE X, for any X′ ∈ ∂⋆(X) we show
+that the size of X′ is bounded linearly in the size of the X. This can be seen
+by induction on the structure of X; for example, elements of ∂⋆(X1 & X2) are a
+pair of an element of ∂⋆(X1) and an element of ∂⋆(X2). It follows that we can
+provide the following NPSPACE algorithm for reachability of (X, Y ): at each
+step, pick a derivative nondeterministically of X and recurse.
+⊓⊔
+Theorem 7. For ERE, the edge-in-cycle, strong-connectedness, and reachability
+problems are non-elementary.
+Proof. It suffices to show that the edge-in-cycle problem is non-elementary. We
+reduce from the nonemptiness problem for EREs, which is known to be non-
+elementary [51]. Given an ERE R, we let b be a fresh character not appearing
+in R, and we let X = (bR)∗. Then X has one derivative, X′ = R(bR)∗. The
+edge-in-cycle property for (X, X′) holds and only if R is nonempty, because R
+is nonempty exactly when ε ∈ ∂⋆(R).
+⊓⊔
+
diff --git a/-NE4T4oBgHgl3EQf3w1M/content/tmp_files/load_file.txt b/-NE4T4oBgHgl3EQf3w1M/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..44a54828f8311584e8a988f952a44efdcbc8f837
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@@ -0,0 +1,1233 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf,len=1232
+page_content='Incremental Dead State Detection in  Logarithmic Time  Caleb Stanford1 and Margus Veanes2  1 University of California, San Diego and University of California, Davis  cdstanford@ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='edu  2 Microsoft Research, Redmond margus@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='com  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Identifying live and dead states in an abstract transition sys-  tem is a recurring problem in formal verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' However, state-of-the-  art graph algorithms for maintaining reachability information incremen-  tally (that is, as states are visited and before the entire state space is  explored) assume that new edges can be added from any state at any  time, whereas in many applications, outgoing edges are added from each  state as it is explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To formalize the latter situation, we propose guided  incremental digraphs (GIDs), incremental graphs which support labeling  closed states (states which will not receive further outgoing edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Our  main result is that dead state detection in GIDs is solvable in O(log m)  time per edge update for m edges, improving upon O(√m) per edge due  to Bender, Fineman, Gilbert, and Tarjan (BFGT) for general incremen-  tal directed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  We introduce two algorithms for GIDs: one establishing the logarithmic  time bound, and a second algorithm to explore a lazy heuristics-based ap-  proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To demonstrate applicability, we show how GIDs can be used to  lazily decide regular expression constraints in SMT applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To en-  able an apples-to-apples experimental comparison, we implemented both  algorithms, two naive baselines, and the state-of-the-art BFGT baseline  using a common directed graph interface in Rust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Our evaluation shows  110-530x speedups over BFGT for the largest input graphs over a range  of graph classes, random graphs, and graphs arising from regular expres-  sion benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Keywords: Dead State Detection · Graph Algorithms · Online Algo-  rithms · SMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  1  Introduction  Classifying states in a transition system as live or dead is a recurring problem  in formal verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For example, given an expression, can it be simplified  to the identity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Given an input to a nondeterministic program, can it reach a  terminal state, or can it reach an infinitely looping state?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Given a state in an  automaton, can it reach an accepting state?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Domain-specific variations on this  problem have led to many general and specialized algorithms used by automated  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='05308v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='DS]  12 Jan 2023  2  Caleb Stanford and Margus Veanes  verification tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' State classification is relevant in the context of satisfiability  modulo theories (SMT) [17,41,3,55,15,36], where theory-specific partial decision  procedures often work by exploring the state space to find a reachable path that  corresponds to a satisfying string or, more generally, a sequence of constructors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  In all of these cases, the core problem is live and dead state detection in a  directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Motivating application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For example, recent approaches for the SMT theory of  regular expressions [34,50] rely on regular expression derivatives to explore the  states of the finite state machine corresponding to the regex incrementally, rather  than expanding all states initially which is often prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This  requires solving the incremental live and dead state detection problem in the  finite state machine (a directed graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This is particularly important for regexes  with intersection and complement (extended regexes [12,22,19]), which have been  shown to arise natively in applications of SMT string solvers to security [2,50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Concretely, consider the regex (�*α�100)C ∩ (�α), where � matches any character,  ∩ is regex intersection, C is regex complement, and α matches any digit (0-9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' A  traditional solver would expand the left and right operands as state machines,  but the left operand (�*α�100)C is astronomically large as a DFA, causing the  solver to hang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The derivative-based technique instead constructs the derivative  regex: (�*α�100)C ∩(�100)C ∩α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' At this stage we have a graph of two states and one  edge, where the states are regexes and the edge is the derivative relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' After  one more derivative operation, the regex becomes one that is clearly satisfiable  as it accepts the empty string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  In order to be efficient, a derivative-based solver needs to identify satisfiable  (live) and unsatisfiable (dead) regexes incrementally (as the graph is built), be-  cause it does not generally construct the entire space before terminating (see the  graph update inference rule Upd, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' 626 [50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Indeed, satisfiability (nonempti-  ness) for extended regexes is non-elementary, and is still PSPACE-complete for  more restrictive fragments, strongly incentivizing the incremental approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Be-  yond regexes, we believe that GIDs are general enough to be applicable in a  range of future applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Traditionally, while live state detection can be done incrementally,  dead state detection is often done exhaustively (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', after the entire state space  is explored).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For example, bounded and finite-state model checkers based on  translations to automata [32,48,13], as well as classical dead-state elimination  algorithms [28,7,10], generally work on a fixed state space after it has been fully  enumerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' However, exhaustive exploration is prohibitive for large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', ex-  ponential or infinite) state spaces which arise in an SMT verification context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Moreover, exhaustive exploration may simply be unnecessary if partial infor-  mation can be deduced about the states seen so far which already leads to a  satisfiable or unsatisfiable result along a given solver path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We also have good  evidence that incremental feedback can improve SMT solver performance: a  representative success story is the e-graph data structure [54,16] for congruence  closure [18,42], which maintains an equivalence relation among expressions in-  Incremental Dead State Detection in Logarithmic Time  3  crementally;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' because it applies to general expressions, it is theory-independent  and re-usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Incremental state space exploration could lead to similar benefits  if applied to SMT procedures which still rely on exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  However, in order to perform incremental dead state detection, we currently  lack algorithms which match offline performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' As we discuss in Section 2,  the best-known existing solutions would require maintaining strong connected  components (SCCs) incrementally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For SCC maintenance and the related sim-  pler problem of cycle detection, O(m3/2) amortized algorithms are known for m  edge additions [23,4], with some recently announced improvements [5,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Note  that this is in sharp contrast to O(m) for the offline variants of these problems,  which can be solved by breadth-first or depth-first search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' More generally, re-  search suggests that there is a computational barrier to what can be determined  incrementally in the worst case [20,21,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  This paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To improve on prior results, our key observation is that in many  applications, edges are not added adversarially, but from one state at a time as  the states are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' As a result, we know when a state will have no further  outgoing edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' we may use this information to (i) provide information about  dead states incrementally, rather than after the whole state space is explored;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  and (ii) obtain more efficient algorithms than currently exist for general graph  reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  We introduce guided incremental digraphs (GIDs), a variation on incremental  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Like an incremental directed graph, a guided incremental digraph may be  updated by adding new edges between states, or a state may be labeled as closed,  meaning it will receive no further outgoing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Some states are designated as  terminal, and we say that a state is live if it can reach a terminal state and dead  if it will never reach a terminal state in any extension – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' if all reachable states  from it are closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To our knowledge, the problem of detecting dead states in  such a system has not been studied by existing work in graph algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Our  problem can be solved through solving SCC maintenance, but not necessarily  the other way around (see Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We provide two new algorithms for  dead-state detection in GIDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  First, we show that the dead-state detection problem for GIDs can be solved  in time O(m · log m) for m edge additions, within a logarithmic factor of the  O(m) cost for offline search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The worst-case performance of our algorithm thus  strictly improves on the O(m3/2) upper bound for SCC maintenance in general  incremental graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Our algorithm utilizes several data structures and existing  results in online algorithms: in particular, Union-Find [52] and Henzinger and  King’s Euler Tour Trees [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The main idea of our algorithm is that, rather than  explicitly computing the set of SCCs, for closed states we maintain a single path  to a non-closed (open) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This turns out to reduce the problem to quickly  determining whether two states are currently assigned a path to the same open  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' On the other hand, Euler Tour Trees can solve undirected reachability for  graphs that are forests in logarithmic time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' the challenge then lies in figuring  out how to reduce directed connectivity in the graph of paths to undirected  connectivity in an Euler Tour Trees forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' At the same time, we must maintain  4  Caleb Stanford and Margus Veanes  this structure under Union-Find state merges, in order to deal with cycles that  are found in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  While as theorists we would like to believe that asymptotic complexity is  enough, the truth is that the use of complex data structures (1) can be pro-  hibitively expensive in practice due to constant-time overheads, and (2) can  make algorithms substantially more difficult to implement, leading practition-  ers to prefer simpler approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To address these needs, in addition to the  logarithmic-time algorithm, we provide a second lazy algorithm which avoids  the user of Euler Tour Trees, and only uses union-find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This algorithm is based  on an optimization of adding shortcut jump edges for long paths in the graph to  quickly determine reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This approach aims to perform well in practice  on typical graphs, and is evaluated in our evaluation along with the logarithmic  time algorithm, though we do not prove its asymptotic complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Finally, we implement and empirically evaluate both of our algorithms for  GIDs against several baselines in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='5k lines of code in Rust [37,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Our evaluation  focuses on the performance of the GID data structure itself, rather than its end-  to-end performance in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To ensure an apples-to-apples comparison  with existing approaches, we put particular focus on providing a directed graph  data structure backend shared by all algorithms, so that the cost of graph search  as well as state and edge merges is identical across algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We implement  two naive baselines, as well as an implementation of the state-of-the-art solution  based on maintaining SCCs, BFGT [4] in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To our knowledge,  the latter is the first implementation of BFGT for SCC maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' On a  collection of generated benchmark GIDs and GIDs directly pulled from the regex  application, we demonstrate a substantial improvement over BFGT for both of  our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For example, for larger GIDs (those with over 100K updates),  we observe a 110-530x speedup over BFGT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Our primary contributions are:  – Guided incremental digraphs (GIDs), a formalization of incremental live and  dead state detection which supports labeling closed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (Section 2)  – Two algorithms for the state classification problem in GIDs: first, an algo-  rithm that works in amortized O(log m) time per update, improving upon  the state-of-the-art amortized O(√m) per update for incremental graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  and second, a simpler algorithm based on lazy heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (Section 3)  – An open-source implementation of GIDs in Rust3, including an implementa-  tion the BFGT baseline and supporting data structures for a fair comparison,  and an evaluation which demonstrates that our algorithm outperforms the  best known incremental SCC algorithm by two orders of magnitude for a  range of GID benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (Section 4)  Following these contributions, we expand on the application of GIDs to regex  solving in SMT (Section 5), survey related work (Section 6), and conclude (Sec-  tion 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='com/cdstanford/gid  Incremental Dead State Detection in Logarithmic Time  5  1  2  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Guided incremental digraph consisting of the sequence of updates E(1, 2),  E(1, 3), T(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Terminal states are denoted with double circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' After the update T(2),  states 1 and 2 are known to be live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' However, for this graph state 3 is not dead, as a  future edge may cause it to be live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  2  Guided Incremental Digraphs  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='1  Problem Statement  An incremental digraph is a sequence of edge updates E(u, v), where the algo-  rithmic challenge in this context is to produce some output after each edge is  received (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', whether or not a cycle exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' If the graph also contains updates  T(u) labeling a state as terminal, then we say that a state is live if it can reach  a terminal state in the current graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In a guided incremental digraph, we also  include updates C(u) labeling a state as closed, meaning that will not receive  any further outgoing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Define a guided incremental digraph (GID) to be a sequence of  updates, where each update is one of the following:  (i) a new directed edge E(u, v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  (ii) a label T(u) which indicates that u is terminal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' or  (iii) a label C(u) which indicates that u is closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' no further edges will be  added going out from u (or labels to u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  The guided incremental digraph is valid if the closed labels are correct: there  are no instances of E(u, v) or T(u) after an update C(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The denotation of G is  the directed graph (V, E) where V is the set of all states u which have occurred  in any update in the sequence, and E is the set of all (u, v) such that E(u, v)  occurs in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  An extension of a valid GID G is a valid GID G′ such that G is a prefix of  G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In a valid GID G, we say that a state u is live if there is a path from u to  a terminal state in the denotation of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' and a state u is dead if it is not live in  any extension of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Notice that in a GID without any C(u) updates, no states  are dead as an edge may be added in an extension which makes them live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  We provide an example of a valid GID in Figures 1 and 2 resulting from  the following sequence of updates: E(1, 2), E(1, 3), T(2), E(4, 3), E(4, 5), C(4),  C(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The states are denoted based on whether they are marked as terminal T(u)  (double circle) or closed C(u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' states that are not closed are denoted with dashed  circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Notice that after C(4), state 4 is marked closed but can still reach 3 and  5, so it is not dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Additionally, notice that once 1 and 2 are live (after T(2)),  further edges from them do not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  6  Caleb Stanford and Margus Veanes  1  2  3  4  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Guided incremental digraph consisting of the sequence of updates E(1, 2),  E(1, 3), T(2), E(4, 3), E(4, 5), C(4), C(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Closed states (from which future edges may not  be added) are denoted with solid circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' After the update C(5) (but not earlier), state  5 is dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Given as input a valid GID, the guided incremental state clas-  sification problem is to output, in an online fashion after each update, the set  of new live and new dead states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' That is, output Live(u) or Dead(u) on the  smallest prefix of updates such that u is live or dead on that prefix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='2  Existing Approaches  In many applications, one might choose to classify dead states offline, after the  entire state space is enumerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This leads to a linear-time algorithm via either  DFS or BFS, but it does not solve the state classification problem (Definition 2)  because it is not incremental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Naive application of this idea leads to an O(m)  time algorithm per update for m updates (O(m2) total), as we may have to  search the entire graph after each update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  For acyclic graphs, there exists an amortized O(1)-time per update algorithm  for the problem (Definition 2): maintain the graph as a list of forward and  backward edges at each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' When a state v is marked terminal, do a DFS  along backward edges to determine all states u that can reach v not already  marked as live, and mark them live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' When a state v is marked closed, visit  all forward-edges from v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' if all are dead, mark v as dead and recurse along all  backwards edges from v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' As each edge is visited only when marking a state live  or dead, it is only visited a constant number of times overall (though we may  use more than O(1) time on some particular update pass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Additionally, the live  state detection part of this procedure still works for graphs containing cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  The challenge, therefore, lies primarily in detecting dead states in graphs  which may contain cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For this, the breakthrough state-of-the-art approach  from [4] enables maintaining the graph as a condensed graph which is acyclic,  where the vertices in the condensed graph represent strongly connected compo-  nents (SCCs) of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The mapping from states to SCCs is maintained using  a Union-Find [52] data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Maintaining this requires O(√m) time per  update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To ensure that vertices in the condensed graph correspond to SCCs in  the original, we have to also make sure that closed and non-closed states are not  merged into the same SCC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' the easiest solution to this is to withhold all edges  from each state u in the graph until u are closed, which ensures that u must  be its own SCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Once we have the condensed graph with these modifications,  Incremental Dead State Detection in Logarithmic Time  7  the same algorithm as in the previous paragraph works to identify live and dead  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Since each edge is only visited when a state is marked closed or live, each  edge is visited only once throughout the algorithm, we use only amortized O(1)  additional time to calculate live and dead states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' While this SCC maintenance  algorithm ignores the fact that edges do not occur from closed states C(u), this  still proves the following result:  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Guided incremental state classification reduces to SCC main-  tenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' That is, suppose we have an algorithm over incremental graphs that  maintains the set of SCCs in O(f(m, n)) total time given n states and m edge  additions, where “maintains” means that (i) we can check whether two states  are in the same SCC in O(1) time, and (ii) we can iterate over all the states  in an SCC, or iterate over the forward-edges or backward-edges from an SCC  (to or from other SCCs, respectively) in O(1) time per edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Then there exists  an algorithm to solve guided incremental state classification in O(f(m, n)) total  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Despite this reduction one way, there is no obvious reduction the other way –  from cycle detection or SCCs to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This is because, while the existence  of a cycle of non-live states implies bi-reachability between all states in the cycle,  it does not necessarily imply that all of the bi-reachable states are dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In fact,  consider a GID consisting only of E(u, v) and C(u) updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In order to identify  dead states, rather than enumerating all cycles or SCCs, it is enough to maintain  a single path from each non-dead state to a non-closed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This observation  forms the starting point for our approach in the following sections, improving  on the upper bound given by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  3  Algorithms  This section presents Algorithm 2, which solves the state classification problem in  logarithmic time (Theorem 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' and Algorithm 3, an alternative lazy approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Both algorithms are optimized versions of Algorithm 1, a first-cut algorithm  which establishes the structure of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We begin by establishing some  basic terminology shared by all of the algorithms (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  States in a GID can be usefully classified as exactly one of four statuses: live,  dead, unknown, or open, where an unknown state is one that is closed but not  yet live or dead, and an open state is one that is not closed and not live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Note  that a state may be live and neither open nor closed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' this terminology keeps the  classification disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Pragmatically, for live states it does not matter if they  are classified as open or closed, since edges from those states no longer have any  effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' However, all dead and unknown states are closed, and no states are both  open and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Given this classification, the intuition is that for each unknown state u, we  only need one path from u to an open state to prove that it is not dead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' we want  to maintain one such path for all unknown states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To maintain all of these paths  simultaneously, we maintain an acyclic directed forest structure on unknown and  8  Caleb Stanford and Margus Veanes  Live  Some reachable state from u is terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Dead  All reachable states from u (including u itself) are closed and  not terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Unknown  u is closed, but not live or dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Open  u is not live and not closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Terminal  A state u labeled by T(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Closed  A state u labeled by C(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Canonical  A state x that is the uniquely chosen representative of its  union-find equivalence class, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  u, v, w  States in the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  x, y, z  Canonical states in the condensed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Successor succ(x)  For an unknown, canonical state x, a uniquely chosen state v  such that (x, v) is an edge, and following the path of successors  leads to an open state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Top: Basic classification of GID states into four disjoint categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Bottom:  Additional terminology used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  open states where the roots are open states, and all non-root states have a single  edge to another state, called its successor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Edges other than successor edges can  be temporarily ignored, except for when marking live states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' these are kept as  reserve edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Specifically, we add every edge (u, v) as a backward-edge from v  (to allow propagating live states), but for edges not in the forest we keep (u, v)  in a reserve list from u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We store all edges, including backward-edges, in the  original order (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The reserve list edge becomes relevant only when either (i)  u is marked as closed, or (ii) u’s successor is marked as dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  In order to deal with cycles, we need to maintain the forest of unknown states  not on the original graph, but on a union-find condensed graph, similar to [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  When we find a cycle of unknown states, we merge all states in the cycle by  calling the union method in the union-find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We refer to a state as canonical if  it is the canonical representative of its equivalence class in the union find;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' the  condensed graph is a forest on canonical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We use x, y, z to denote canonical  states (states in the condensed graph), and u, v, w to denote the original states  (not known to be canonical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Following [52], we maintain edges as linked lists rather than sets, and using  the original states instead of canonical states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' this is important as it allows  combining edge lists in O(1) time when merging states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='1  First-Cut Algorithm  A first-cut algorithm based on these ideas is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The union-  find data structure UF provides UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='union(v1, v2), UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v), and UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='iter(v):  UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='union merges v1 and v2 to refer to the same canonical state, UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find returns  the canonical state for v, and UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='iter iterates over states equivalent to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The  running time of each of these operations is amortized α(n) for n updates, where  α(n) ∈ o(log n) is the inverse Ackermann function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Incremental Dead State Detection in Logarithmic Time  9  Algorithm 1 First-cut algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  V: a type for states (integers) (variables u, v, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=')  E: the type of edges, equal to (V, V)  UF: a union-find data structure over V  X: the set of canonical states in UF (variables x, y, z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=')  status: a map from X to Live, Dead, Unknown, or Open  succ: a map from X to V  res and bck: maps from X to linked lists of E  procedure OnEdge(E(u, v))  x ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' y ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v)  if status(y) = Live then  OnTerminal(T(x))  ▷ mark x and its ancestors live  else if status(x) ̸= Live then  ▷ status(x) must be Open  append (u, v) to res(x)  append (u, v) to bck(y)  procedure OnTerminal(T(v))  y ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v)  for all x in DFS backwards (along bck) from y not already Live do  status(x) ← Live  output Live(x′) for all x′ in UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='iter(x)  procedure OnClosed(C(v))  y ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v)  if status(y) ̸= Open then return  ▷ y is already live or closed  while res(y) is nonempty do  pop (v, w) from res(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' z ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(w)  if status(z) = Dead then continue  else if CheckCycle(y, z) then  for all z′ in cycle from z to y do z ← Merge(z, z′)  else  status(y) ← Unknown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' succ(y) ← z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  return  status(y) ← Dead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' output Dead(y′) for all y′ in UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='iter(y)  ToRecurse ← ∅  for all (u, v) in bck(y) do  x ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(u)  if status(x) = Unknown and UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(succ(x)) = y then  status(x) ← Open  ▷ temporary – marked closed on recursive call  add x to ToRecurse  for all x in ToRecurse do OnClosed(C(x))  procedure CheckCycle(y, z) returning bool  while status(z) = Unknown do z ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(succ(z))  ▷ get root state from z  return y = z  procedure Merge(x, y) returning V  z ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='union(x, y)  bck(z) ← bck(x) + bck(y)  ▷ O(1) linked list append  res(z) ← res(x) + res(y)  ▷ O(1) linked list append  return z  10  Caleb Stanford and Margus Veanes  We will only merge states if they are bi-reachable from each other, and both  unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This implies that all states in the equivalence class of a canonical state  x have the same status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We also maintain the set of canonical states in UF as a  set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  The edge maps res and bck are stored as maps from X to linked lists of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Each edge (u, v) is always stored using its original states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', edge labels are not  updated when states are merged);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' but we can easily obtain the corresponding  edge on canonical states via (UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(u), UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' On an input edge (u, v),  the edge is immediately added to bck as a backward-edge from v (this is used  for detecting live states), but forward-edges are only added selectively to succ  when they are needed to prove unknown status, and are otherwise stored in the  reserve list res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In total, we maintain the following invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For live and dead  states, we no longer care about forward or backward edges from them, so there  are no invariants about live and dead states other than that status(x) is Live or  Dead for these states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The successor edges and no cycles invariants  are about the forest structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The last invariant, edge representation, is about  making sure that all edges in the input GID are represented somehow in the  current graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  – Merge equivalence: For all states u and v, if UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(u) = UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v), then  u and v are bi-reachable and both closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (This implies that u and v are  both live, both dead, or both unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=')  – Status correctness: For all u, status(UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(u)) is equal to the status of  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  – Successor edges: If x is unknown, then succ(x) is defined and is an unknown  or open state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' If x is open, then succ(x) is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  – No cycles: There are no cycles among the set of edges (x, UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(succ(x))),  over all unknown and open canonical states x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  – Edge representation: Lastly, for all edges (u, v) in the input GID, at least one  of the following holds: (i) (u, v) is stored in res, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (u, v) ∈ res(UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  (ii) (u, v) is stored in succ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' v = succ(UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(u));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (iii) u and v are  equivalent in the condensed graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(u) = UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (iv) u is  live;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' or (v) v is dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Algorithm 1 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We need to argue that all of the invariants described above are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  This implies correctness of the algorithm by the status correctness invariant,  since it implies that live and dead states are labeled (and output) correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Upon receiving E(u, v) or T(u), this may cause some dead, unknown, or open  states to change status to live, but does not change the status of any other  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' As bck stores all backwards-edges (not just succ edges), live states are  marked correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This preserves the forest invariants because if an unknown or  open state is marked live, so are all its predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This also preserves the edge  representation invariant, either by adding new edges to case (i) and case (iv)  Incremental Dead State Detection in Logarithmic Time  11  (for E updates) or moving edges from cases (i)-(iii) to case (iv) (for both E and  T updates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  The procedure C(u) is recursive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' during the procedure and on recursive calls,  some states are temporarily marked Open, meaning they are roots in the forest  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' During these recursive calls, we need a slightly generalized invariant:  each forest root corresponds to a pending future call to OnClosed(C(u)) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=',  an element of ToRecurse for some call on the stack) and is a state that needs to  either find a successor or be marked dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This means that the status labels for  unknown- and open-labeled states are not necessarily correct during recursive  calls;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' we are only concerned with preserving the forest structure, so that they will  be correct after all calls complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Note in particular that after merging cycles  of unknown states via Merge(x, y), the new root is marked Open to preserve  the generalized invariant on recursive calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Upon receiving C(u), without loss of generality we may assume status(u) =  Open (the opposite only occurs if the input GID has duplicate C(u) updates, or  if C(u) occurs for a live state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' At the top of the while loop, there are three cases:  – If a reserve edge is available, and its target is dead, then we discard it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This  preserves edge representation because that edge still satisfies (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  – If a reserve edge is available and its target is not dead, then we see if adding  that edge creates a cycle by calling CheckCycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' If no, then the two trees are  distinct, so we add an edge between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This preserves edge representation  by moving that edge from case (i) to case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' If yes, then we collapse the  cycle in the forest to a single state by repeated calls to Merge(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This  preserves edge representation by moving the edges in the cycle from case (ii)  to case (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The forest structure of the succ is also preserved because if a  graph is a forest, then it remains a forest after merging adjacent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  – Finally, if there are no reserve edges left, then because of edge representation,  and because there is no successor from u, all edges from u must lead to dead  states (case (v)) and therefore u is dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This case splits the tree rooted at  u into one tree for each of its predecessors, preserving the forest structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Each of the succ edges from predecessors are deleted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' this preserves edge  representation by moving those edges from case (ii) to case (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  In any case, the generalized invariant for recursive calls is preserved, and all dead  states are labeled correctly (given the invariant), so we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  ⊓⊔  Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The core inefficiency in Algorithm 1 lies in CheckCycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The proce-  dure repeatedly sets z ← succ(z) to find the tree root, which in general could be  linear time in the number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For example, suppose we have a line graph  where we add edges in a backwards fashion and close them: E(2, 1), C(2), E(3,  2), C(3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' , E(n, n-1), C(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In such a graph, this inefficiency results in O(m2)  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We therefore want to improve upon this step in Algorithms 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  All other parts of the algorithm run in amortized α(m) time per update for  m updates (using vectors to represent the maps fwd, bck, and succ for O(1)  lookups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The OnTerminal calls and loop iterations only run once per edge  in the graph when the target of that edge is marked live or terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Likewise,  12  Caleb Stanford and Margus Veanes  the procedure OnClosed is called conservatively: the cost of each call can be  assigned either to the target of an edge being marked dead, or to an edge being  merged as part of a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Both marking dead and merging can only happen  once for a given edge, so this is O(1) per edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='2  Logarithmic Algorithm  At its core, CheckCycle requires solving an undirected reachability problem on  a graph that is restricted to a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' However, the forest is changed not just by  edge additions, but edge additions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' While undirected reachability  and reachability in directed graphs are both difficult to solve incrementally,  reachability in dynamic forests can be solved in O(log m) time per operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Our algorithm uses an Euler Tour Forest data structure EF of Henzinger and  King [26], and is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  However, this idea does not work straightforwardly – once again because of  the presence of cycles in the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We cannot simply store the forest as  a condensed graph with edges on condensed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' As we saw in Algorithm 1,  it was important to store successor edges as edges into V, rather than edges into  X – this is the only way that we can merge states in O(1), without actually  inspecting the edge lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' If we needed to update the forest edges to be in X, this  could require O(m) work to merge two O(m)-sized edge lists as each edge might  need to be relabeled in the EF graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  To solve this challenge, we instead store the EF data structure on the original  states, rather than the condensed graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' but we ensure that each condensed state  is represented by a tree of original states in the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' When adding  edges between condensed graph states, we need to make sure to remember the  original state labels (u, v), so that we can later remove it using the original  labels (this happens when its target becomes dead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' When an edge would create  a cycle, we instead simply ignore it in the EF graph: because a line of connected  trees forms a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  In summary, algorithm borrows most data and procedures from Algorithm 1,  with a few important changes: (1) we maintain the EF data structure EF, a forest  on V (2) the successor edges are stored as their original edge labels (u, v), rather  than just as a target state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (3) the procedure OnClosed is updated in several  locations to maintain the graph EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We continue all invariants from Algorithm 1, with the small mod-  ification that the successor edges and no cycles invariants use the new succ  representation: that is, they are constraints on the edges (x, UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v)), where  succ(x) = (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In addition, we have two constraints on edges in EF, de-  pending on whether those states are equivalent in the union-find structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We  call edges between inequivalent states inter-edges and those between equivalent  states intra-edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  – EF inter-edges: For all states u and v not in the same canonical state, (u, v)  is in the EF graph if and only if (u, v) = succ(UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(u)) or (v, u) =  succ(UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Incremental Dead State Detection in Logarithmic Time  13  Algorithm 2 Logarithmic time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  All data from Algorithm 1  succ: a map from X to E (instead of to V)  EF: Euler Tour Trees data structure providing: EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='add, EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='remove, and EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='connected  procedure OnEdge(E(u, v)) as in Algorithm 1  procedure Merge(x, y) returning V as in Algorithm 1  procedure OnTerminal(T(v))  y ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v)  for all x in DFS backwards (along bck) from y not already Live do  if status(x) = Unknown then  ▷ The following line is not strictly necessary, but simplifies the analysis  (u, v) ← succ(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' delete succ(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='remove(u, v)  status(x) ← Live;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' output Live(x′) for all x′ in UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='iter(x)  procedure OnClosed(C(v))  y ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v)  if status(y) ̸= Open then return  while res(y) is nonempty do  pop (v, w) from res(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' z ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(w)  if status(z) = Dead then continue  else if CheckCycle(y, z) then  for all z′ in cycle from z to y do z ← Merge(z, z′)  else  status(x) ← Unknown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' succ(x) ← (v, w)  EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='add(v, w)  ▷ undirected edge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' use original labels (not (x, y))  return  status(y) ← Dead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' output Dead(y′) for all y′ in UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='iter(y)  ToRecurse ← ∅  for all (u, v) in bck(y) do  x ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(u)  if status(x) = Unknown then  (u′, v′) ← succ(x)  if UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(v′) = y then  EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='remove(u′, v′)  ▷ undirected edge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' use original labels  status(x) ← Open;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' delete succ(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' add x to ToRecurse  for all x in ToRecurse do OnClosed(C(x))  procedure CheckCycle(y, z) returning bool  return EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='connected(y, z)  14  Caleb Stanford and Margus Veanes  – EF intra-edges: For all unknown canonical states x, the set of edges (u, v)  in the EF between states belonging to x forms a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Algorithm 2 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Observe that the EF inter-edges constraint implies that EF only contains  edges between unknown and open states, together with isolated trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In the  modified OnTerminal procedure, when marking states as live we remove inter-  edges, so we preserve this invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Next we argue that given the invariants about EF, for an open state y the  CheckCycle procedure returns true if and only if (y, z) would create a directed  cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' If there is a cycle of canonical states, then because canonical states are  connected trees in EF, the cycle can be lifted to a cycle on original states, so y and  z must already be connected in this cycle without the edge (y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Conversely, if  y and z are connected in EF, then there is a path from y to z, and this can be  projected to a path on canonical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' However, because y is open, it is a root  in the successor forest, so any path from y along successor edges travels only  on backwards edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' hence z is an ancestor of y in the directed graph, and thus  (y, z) creates a directed cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  This leaves the OnClosed procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Other than the EF lines, the structure  is the same as in Algorithm 1, so the previous invariants are still preserved,  and it remains to check the EF invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' When we delete the successor edge  and temporarily mark status(x) = Open for recursive calls, we also remove it  from EF, preserving the inter-edge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Similarly, when we add a successor  edge to x, we add it to EF, preserving the inter-edge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' So it remains to  consider when the set of canonical states changes, which is when merging states  in a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Here, a line of canonical states is merged into a single state, and a  line of connected trees is still a tree, so the intra-edge invariant still holds for  the new canonical state, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  ⊓⊔  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Algorithm 2 uses amortized logarithmic time per edge update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' By the analysis of Algorithm 1, each line of the algorithm runs amortized  O(1) time other than those in CheckCycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For the CheckCycle, procedure  it now avoids the loop and so also runs amortized O(1) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Each line is either  constant-time, α(m) = o(log n) time for the UF calls, or O(log n) time for the EF  calls, so in total the algorithm runs in amortized O(log n) time per update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  ⊓⊔  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='3  Lazy Algorithm  While the asymptotic complexity of log n could be the end of the story, in prac-  tice, the cost of the EF calls could be a significant overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The technical details  of Euler-Tour Trees include building an AVL-tree cycle for each tree, where the  cycle contains each state of the graph and each edge in the graph twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' It turns  out that adding one edge to EF results in up to eight modifications to the AVL  tree: it needs to be split at the source, split at the target, then the edge needs to  Incremental Dead State Detection in Logarithmic Time  15  Algorithm 3 Lazy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  All data from Algorithm 1  jumps: a map from X to lists of V  procedure OnEdge(E(u, v)) as in Algorithm 1  procedure OnTerminal(T(v)) as in Algorithm 1  procedure OnClosed(C(v)) as in Algorithm 1  procedure CheckCycle(y, z) returning bool  return y = GetRoot(z)  procedure GetRoot(z) returning V  if status(z) = Open then return z  if jumps(z) is empty then  push succ(z) to jumps(z)  ▷ set 0th jump  repeat  pop w from jumps(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' z′ = UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='find(w)  ▷ remove dead jumps  until status(z′) ̸= Dead  push z′ to jumps(z)  result ← GetRoot(z′)  n ← length(jumps(z));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' n′ ← length(jumps(z′))  if n ≤ n′ then  push jumps(z′)[n − 1] to jumps(z)  ▷ set nth jump  return result  procedure Merge(x, y) returning V  z ← UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='union(x, y)  bck(z) ← bck(x) + bck(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' res(z) ← res(x) + res(y)  jumps(z) ← empty  return z  be added in both directions (u, v) and (v, u) to the cycle, and then these trees  need to be glued together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Every one of these operations comes with a rebalanc-  ing operation which could do Ω(log n) tree rotations and pointer dereferences to  visit the nodes in the AVL tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  As a result of these (constant-time) overheads, in this section, we investigate  a simpler, lazy algorithm which avoids Euler Tour Trees, and works directly by  modifying Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To optimize Algorithm 1, one idea in the right direction  is to store for each state a direct pointer to the root which results from repeatedly  calling succ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' But there are two issues with this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' First, maintaining this may be  difficult (when the root changes, potentially updating a linear number of root  pointers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Second, the root may sometimes be marked dead, in which case we  have to re-compute all the successor pointers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Instead, we introduce a jump list from each state: intuitively, it will contain  states after calling successor once, twice, four times, eight times, and so on, and  will be updated at most once for every visit to the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' When a jump becomes  obsolete (the target dead), we just pop off the largest jump, so we do not lose all  of our work in building the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' When states are merged, we do lose our work,  discarding the jump lists, and this also means the exact power-of-two distances  are no longer preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We maintain the following additional information: for  16  Caleb Stanford and Margus Veanes  each unknown canonical state x, a nonempty list of jumps [v0, v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' , vk],  such that v0 is reachable from x, v1 is reachable from v0, v2 is reachable from  v1, and so on, and v1 = succ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The algorithm is shown in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  The key procedure is GetRoot(z), which is called when adding a reserve  edge (y, z) to the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' It finds the root in the successor forest when repeatedly  calling successor from z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' It shortcuts by using the jump list, and also updates  the set of jumps to be approximately at powers of two: to do this, the nth jump  from a state z is set to the (n − 1)st jump from the (n − 1)th jump from z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In addition to the invariants from Algorithm 1: for each unknown  canonical state x, jumps(x) is a list of states v0, v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' , vk such that:  – First jump: if the jump list is nonempty, then v0 = succ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  – Reachability: vi+1 is reachable from vi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  – Powers of two: on the path of canonical states from v0 to vi, the total number  of states (including all the states in each equivalence class) is at least 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  The powers of two invariant is not necessary for correctness, but is the key  to the efficiency of this algorithm: it ensures that the nth jump travels at least  2n states at once in the successor forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' It follows from this that if the jump list  is fully saturated for every state, querying GetRoot(z) will take logarithmic  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' But because the jump lists are updated lazily (and have dead states that  need to be discarded), this does not establish an asymptotic complexity for the  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Algorithm 3 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We use the invariant on the jump list mentioned: that v1 is the successor,  v2 is reachable from v1, v3 is reachable from v2, and so on, using only forward  edges added to the graph (not reserve edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' is some sublist of  the states along the path from an unknown state to its root, potentially followed  by some dead states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We need to argue that the subprocedure GetRoot (i)  receives the same verdict as repeatedly calling succ to find a cycle in the first-cut  algorithm and (ii) preserves the jump list invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Popping dead states from the  jump list clearly preserves the invariant, as does adding on a state along the path  to the root, which is done when k′ ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Merging states preserves the jump list  invariant trivially because we throw the jump list away, and marking states live  preserves the jump list invariant trivially since the jump list is only maintained  and used for unknown states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Finally, marking states as closed initializes the  jump list correctly assuming that the successor is calculated correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  ⊓⊔  4  Experimental Evaluation  The primary goal of our evaluation has been to experimentally validate the  performance of GIDs as a data structure in isolation, rather than their use in a  particular application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Our evaluation seeks to answer the following questions:  Q1 How does our approach (Algorithms 2 and 3) compare to the state-of-the-art  approach based on maintaining SCCs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Incremental Dead State Detection in Logarithmic Time  17  Q2 How does the performance of the studied algorithms vary when the class of  input graphs changes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', sparse vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' dense graphs, structured vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' random  graphs)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Q3 Finally, how do the studied algorithms perform on GIDs taken from the  example application to regexes described in Section 5?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  To answer Q1, we put substantial implementation effort into a common  framework on which a fair comparison could be made between different ap-  proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To this end, we implemented GIDs as a data structure in Rust which  includes a graph data structure on top of which all algorithms are built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In par-  ticular, this equalizes performance across algorithms for the following baseline  operations: state and edge addition and retrieval, DFS and BFS search, edge  iteration, and state merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We chose Rust for our implementation for its per-  formance, and because there does not appear to be an existing publicly available  implementation of BFGT in any other language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The number of lines of code  used to implement these various structures is summarized in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We im-  plement Algorithms 2 and 3 and compare them with the following baselines:  BFGT This algorithm is a careful implementation of the state-of-the-art ap-  proach based on SCC maintenance, using worst-case amortized O(√m) time  per update [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Simple In addition to BFGT, we provide a simpler SCC-based baseline, using  a DFS instead of using the BFGT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' It updates the SCCs after each  state is marked closed by searching for all cycles from that state using a  forward-DFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This takes Θ(m2) in the worst case, but can be very efficient  when edges are added in a “forward-facing” topologically sorted manner, so  represents an important point of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Naive Finally, the naive algorithm is a greedy baseline that represents a worst-  case upper bound: it re-computes the entire set of dead states using a linear-  time DFS after each update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' It is Θ(m2) for m updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  To answer Q2, first, we compiled a range of basic graph classes which are  designed to expose edge case behavior in the algorithms, as well as randomly  generated graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We focus on graphs with no live states, as live states are  treated similarly by all algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Most of the generated graphs come in 2×2 =  4 variants: (i) the states are either read in a forwards- or backwards- order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' and  (ii) they are either dead graphs, where there are no open states at the end and  so everything gets marked dead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' or unknown graphs, where there is a single  open state at the end, so most states are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In the unknown case, it is  sufficient to have one open state at the end, as many open states can be reduced  to the case of a single open state where all edges point to that one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We include  GIDs from line graphs and cycle graphs (up to 100K states in multiples of 3),  as well as complete graphs, complete acyclic graphs (with only forward-edges),  and bipartite graphs (up to 1K states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' These are important cases, for example,  because the reverse-order version of the line and cycle graphs is an edge case for  Simple: it often times out on these, but performs well on the forward-version of  these GIDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  18  Caleb Stanford and Margus Veanes  Implementation  Component  LoC  Common Framework  1197  Naive Algorithm  78  Simple Algorithm  98  BFGT Algorithm  265  Algorithm 2 (Ours)  253  Algorithm 3 (Ours)  283  Euler Tour Trees  1510  Experimental Scripts  556  Separated Unit Tests  798  Util  217  Other  69  Total  5324  Category  Benchmark  Source  Qty  Basic  Line  24  Cycle  24  Complete  18  Bipartite  14  Total  80  Random  Sparse  260  Dense  130  Total  390  Regex  RegExLib [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='49]  2061  37  Handwritten [50]  70  26  Additional  11  Total  74  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Left: Lines of code for each algorithm and other implementation components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Right: Benchmark GIDs used in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Where present, the source column  indicates the quantity prior to filtering out trivially small graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='Time (ms)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='Benchmarks Solved  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content='Alg 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='Alg 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='Time (ms)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content='Alg 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content='Time (ms)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content='Simple  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='BFGT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='Alg 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='Alg 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Left: Cumulative plot showing the number of benchmarks  solved in time t or less for basic GID classes (top), randomly generated GIDs (middle),  and regex-derived GIDs (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Top right: Scatter plot showing the size of each  benchmark vs time to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Bottom right: Average time to solve benchmarks of size  closest to s, where values of s are chosen in increments of 1/3 on a log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Incremental Dead State Detection in Logarithmic Time  19  Second, to exhibit more dynamic behavior, we generated random graphs:  sparse graphs with a fixed out-degree from each state, which can be either 1, 2, 3,  or 10 (up to 100K states);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' and dense graphs with a fixed probability of each edge,  which can be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='01, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='02, or .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='03 (up to 10K states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' As with the basic graphs, states  are read in some order and marked closed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' at most one state is left open at the  end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Each random graph is generated using 10 different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  To answer Q3, we needed to isolate the performance of the GID data struc-  ture itself, rather than the performance of the other components of the regex  SMT solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Prior work on regex solving integrated with Z3 [50] implements  an earlier version of GIDs, which is essentially the Simple algorithm in C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  However, preliminary tests suggest that on some examples, GIDs are already  quite efficient and contribute only a fraction of the performance (1-10%), as  a substantial amount of time is spent manipulating and rewriting expressions  and computing derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' While we would expect GIDs to be the bottleneck on  larger examples where the asymptotic complexity of Simple becomes prohibitive,  this fact makes it difficult to isolate the performance of the GID data structure  itself, which is the focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  To isolate the GID performance, we therefore instrumented the modified Z3  solver code to export the (incremental) sequence of graph updates that would  be performed during a run of Z3 on existing regex benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For each regex  benchmark, this instrumented code produces a faithful representation of the se-  quence of graph updates that would occur in a run of the SMT solver on this  particular benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' It is also important to use ERE benchmarks, rather than  plain regular expressions as these are the ones for which dead state detection is  relevant (see Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For each regex benchmark, we thus get a GID bench-  mark for the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In our regex-derived GID examples, we include the  RegExLib benchmarks from [9,49] and the handcrafted Boolean benchmarks re-  ported in [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We add to these 11 additional examples designed to stress test the  GID side of a regex solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The collection of regex SMT benchmarks is available  on GitHub4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  From both the Q2 and Q3 benchmarks, we filter out those that are too easy:  any benchmark which takes under 10 milliseconds for all of the algorithms to  solve (including Naive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We enforce a timeout of 60 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The evaluation was  run on a 2020 MacBook Air running MacOS Monterey, containing an Apple M1  processor and 8GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To ensure that all of our implementations our correct, we also  invested time into unit testing and checked output correctness on all of our  collected benchmarks, including several cases which exposed bugs in previous  versions of one or more algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In total, all algorithms are vetted against 25  unit tests from handwritten edge cases that exposed prior bugs, 373 unit tests  from benchmarks, and 30 module-level unit tests for specific functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Figure 5 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Algorithm 3 shows significant improve-  ments over the state-of-the-art, solving more benchmarks in a smaller amount  4 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='com/cdstanford/regex-smt-benchmarks  20  Caleb Stanford and Margus Veanes  of time across basic GIDs, random GIDs, and regex GIDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Algorithm 2 also  shows state-of-the-art performance, similar to BFGT on basic and regex GIDs  and significantly better on random GIDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' On the bottom right, since looking at  average time is not meaningful for benchmarks of widely varying size, we stratify  the size of benchmarks into buckets, and plot time-to-solve as a function of size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  note that as both x-axis and y-axis are on a log scale, a total running time of  O(mp) for m updates corresponds to a line with slope p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The plot shows that  Algorithm 3 exhibits up to two orders of magnitude speedup over BFGT for  larger GIDs – we see speedups of 110x to 530x for GIDs in the top five size  buckets (GIDs of size nearest to 100K, 200K, 500K, 1M, and 2M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  New implementations of existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Our implementation contributes, to our  knowledge, the first implementation of BFGT for SCC maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In addition,  it is one of the first implementations of Euler Tour Trees (EF) for undirected  reachability in forests, including the AVL tree backing for EF which maintains  a disjoint collection of lists of state and edge identifiers, and the first implemen-  tation in Rust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  5  Application to Extended Regular Expressions  In this section, we describe one application of GIDs in the context of deciding  regex constraints in SMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In particular, we explain how precisely the GID state  classification problem arises in the context of derivative-based solvers [50,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  We first define extended regexes (regexes extended with intersection & and  complement ~) modulo a (symbolic) alphabet A that intuitively provides char-  acter classes as predicates that are closed under Boolean operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We explain  the main idea behind (symbolic) derivatives [50] that provides the foundation  for incrementally creating a GID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Then we show, through an example, how a  solver can incrementally expand derivatives to reduce the satisfiability problem  to the GID state classification problem (Definition 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  EREs or regexes are defined as follows where ϕ is a predicate in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  RE ::= ϕ | ε | RE1 · RE2 | RE* | RE1 | RE2 | RE1 & RE2 | ~RE  Let Rk represent the concatenation of R k times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' A regex R is nullable if it  matches the empty string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Nullability is defined inductively over the structure  of regexes with ε and R* being nullable by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Observe in particular that  ~R is nullable iff R is not nullable, and R1 & R2 is nullable iff both R1 and R2  are nullable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Terminal states are precisely the nullable regexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  The false predicate ⊥ of A serves as the regex that matches nothing and is  a trivially dead state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Thus ~⊥ is equivalent to �* that matches everything and  is a trivially live state, where � is the true predicate of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  A symbolic derivative δ(R) of a regex R is a binary tree or term t whose  leaves are regexes and internal nodes are labeled by predicates from A – the two  immediate subtrees t1 and t2 of a node with label ϕ, denoted by (ϕ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' t1 : t2),  correspond to ϕ being true in t1 and false in t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' A branch with the accumulated  Incremental Dead State Detection in Logarithmic Time  21  branch condition ψ from the root of t to one of its leaves R′ then defines a  transition R  ψ−→R′ – provided ψ is satisfiable in A – and the edge (R, R′) is added  to the GID where the actual predicate ψ is no longer needed because the edge  itself represents its feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The number of leaves of δ(R) is the out-degree  deg+(R) of R that is independent of the actual size of the concrete alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Thus R becomes closed when deg+(R) many edges have been added from R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  The formal definition of a symbolic derivative is as follows where the op-  erations ⃝| , ⃝& , ⃝~ and ⃝· are here for the purposes of this paper and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  normalizing variants of | , & , ~ and ·, by distributing the operations into if-then-  elses and build a single binary tree as a nested if-then-else as a result (see [50]  for details):  δ(ε) = δ(⊥) = ⊥  δ(ϕ) = (ϕ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' ε : ⊥)  δ(R∗) = δ(R) ⃝· R∗  δ(R | R′) = δ(R) ⃝| δ(R′)  δ(R & R′) = δ(R) ⃝& δ(R′)  δ(~R) = ⃝~ δ(R)  δ(R · R′) = if Nullable(R) then (δ(R) ⃝· R′) ⃝| δ(R′) else δ(R) ⃝· R′  If c is a term of type character then δc(R) is obtained from δ(R) by instantiating  all internal nodes (conditions of the if-then-elses) ϕ by tests c ∈ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This means  that in the context of SMT, δc(R) is a term of type regex, while δ(R) represents  the lambda-term λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='δc(R) that is constructed independently of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Concretely, to apply Definition 1 to regexes, states will be regexes, and edges  will represent transitions to their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' A live state here is thus a regex  that reaches a nullable regex via 0 or more edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This implies that there exists  a concrete string matching it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Conversely dead states are always empty, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' they  match no strings, but can reach other dead states, creating strongly connected  components of closed states none of which are live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='1  Reduction from Incremental Regex Emptiness to GIDs  For a solver based on derivatives of EREs, suppose we want to determine the  satisfiability of a single regex constraint s ∈ R, where s is a string variable and  R is a concrete regex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', let L = ~(�*α�100) and R = L & (�α) where α is some  character predicate in A s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' both α and ¬α are satisfiable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', take α to mean  “is digit” to be concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='5 The solver manipulates regex membership constraints  on strings by unfolding them [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The constraint s ∈ R, that essentially tests  nonemptiness of R with s as a witness, becomes  (s = ϵ ∧ Nullable(R)) ∨ (s ̸= ϵ ∧ s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='. ∈ δs0(R))  where, s ̸= ϵ since R is not nullable, si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='. is the suffix of s from index i, and  δ(R) = δ(L) ⃝& δ(�α) = (α ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' L & ~(�100) : L) ⃝& α = (α ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' L & ~(�100) & α : L & α)  Let R1 = L & ~(�100) & α and R2 = L & α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' So R has two outgoing transitions  R  α−→R1 and R  ¬α  −−→R2 that contribute the edges (R, R1) and (R, R2) into the  GID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Note that these edges depend only on R and not on s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  5 Using standard notations: \\d denotes all digits, and [^\\d] denotes all non-digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  22  Caleb Stanford and Margus Veanes  We now continue the search incrementally by checking the two branches of  the if-then-else constraint, where R1 and R2 are again both not nullable (so  s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='. ̸= ϵ):  s0 ∈ α ∧ s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='. ∈ δs1(R1)  ∨  s0 ∈ ¬α ∧ s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='. ∈ δs1(R2)  δ(R1) = (α ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' L & ~(�100) & ~(�99) : L & ~(�99)) ⃝& (α ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' ε : ⊥) = (α ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' ε : ⊥)  δ(R2) = (α ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' L & ~(�100) : L) ⃝& (α ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' ε : ⊥) = (α ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' ε : ⊥)  It follows that R1  α−→ε and R2  α−→ε, so the edges (R1, ε) and (R2, ε) are added to  the GID where ϵ is a trivial terminal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In fact, after R1 the search already  terminates because we then have the path (R, R1)(R1, ϵ) that implies that R is  live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The associated constraints s0 ∈ α and s1 ∈ α and the final constraint that  s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='. = ϵ can be used to extract a concrete witness, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', s = "42".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' From the point  of view of deciding nonemptiness the role of the witness s is immaterial, and  can be omitted, as it poses no additional constraints on its own if s is a variable  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', an uninterpreted constant in SMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Soundness of the algorithm follows from that if R is nonempty, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', s ∈ R  is satisfiable, then by expanding the search, we eventually arrive at a nullable  (terminal) regex, as in the example run above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To achieve completeness – and  to eliminate dead states as early as possible – we incrementally construct a  GID corresponding to the set of regexes seen so far (as above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' After all the  feasible transitions from R to its derivatives in δ(R) are added to the GID as  edges (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' in one batch), the state R becomes closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Crucially, due to the  symbolic form of δ(R), no derivative is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Therefore R is known to be  empty precisely as soon as R is detected as a dead state in the GID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We get the  following theorem that uses finiteness of the closure of symbolic derivatives [50,  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='1]:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For any regex R, (1) If R is nonempty, then the decision procedure  eventually marks R live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' If n is the shortest distance from R to a terminal state,  then a concrete witness of length n can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (2) If R is empty, then the  decision procedure marks R dead after a finite number of steps and terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Observe that any number of simultaneous regex constraints for a string s can be  combined into single regex constraint by using the Boolean operations of ERE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  More crucially, the algorithm is independent of the size of the universe of A,  that may be very large, like the Unicode character set, or even infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  6  Related Work  Online graph algorithms: Online graph algorithms are typically divided into  problems over incremental graphs (where edges are added), decremental graphs  (where edges are deleted), and dynamic graphs (where edges are both added  and deleted), with core data structures discussed in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For dynamic directed  graphs, important problems include transitive closure, cycle detection, topological  ordering, and strongly connected component (SCC) maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Maintaining  Incremental Dead State Detection in Logarithmic Time  23  a topological order of dynamic DAGs is studied in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' An online algorithm  for topological sorting that is experimentally shown to be preferable for sparse  graphs is discussed in [46], and in a related article [45] it is also discussed how  to extend such an algorithm to detect strongly connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The key  applications of such algorithms have traditionally been in pointer analysis [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Topological order of incremental DAGs is studied in [24], presenting two different  algorithms, one for sparse graphs and one for dense graphs – the algorithms  are also extended to work with SCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The sparse algorithm was subsequently  simplified in [4] and is the basis of our implementation named BFGT in the  Evaluation section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' A unified approach of several algorithms based on [4] is  presented in  [14] that uses a notion of weak topological order and a labeling  technique that estimates transitive closure size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Further extensions of [4] are  studied in [6,8] based on randomization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Transitive closure of decremental DAGs  is studied in [47] that improve upon some algorithms presented earlier in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Although we have not looked at decremental graphs in this paper, the problem  may become relevant in the context of backtracking in SMT string theory solver  during proof search in the presence of non-ground regexes, where properties such  as nullability may be theory-dependent and thus affect liveness of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Data structures for SMT: UnionFind [52] is a frequently used data structure  that we also rely on in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' E-graphs [54,16,18,42] are used to ensure  functional extensionality, where two expressions are equivalent if their subex-  pressions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In both cases the maintained relation is an equivalence  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In contrast, maintaining live and dead states involves tracking reacha-  bility rather than equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' While reachability is a basic problem in many core  SMT algorithms, to the best of our knowledge, the formulation of the problem  we consider here is new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Dead state elimination: Dead state elimination for automata, also known as  trimming [7], is a specialized version of useless symbol elimination from context-  free grammars [28, pp 88-89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' It plays an important role in minimization of  automata [31,40,30,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The classical minimization algorithms have been studied  extensively, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', in [10,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Brzozowski [11] observed that a DFA can be minimized  by determinizing the reversal of the determinization of the reversal of the DFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Regular expression matching and analysis: Decision problems for regexes are  studied in the context of Satisfiability Modulo Theories (SMT), where the first-  order objects are strings and formulas consist of string constraints such as x = yz  or x ∈ R, where R is a regex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Over such theories, the problem of nonemptiness  of a regex reduces to live state detection in the automaton (or set of derivatives)  for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' String and regex constraints have been the focus of both SMT and the  Constraint Programming (CP) solving communities, with a wide range of tools  focusing on different problem classes and applications [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Regexes and are now a  standard part of the SMTLIB2 format [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In SMT, string solvers are integrated  in the CDCL(T) architecture [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In the CP community, the MiniZinc format  integrates membership constraints over regular languages presented as either  24  Caleb Stanford and Margus Veanes  DFAs or NFAs [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The solvers presented in [34] and [50] are based on computing  Antimirov derivatives – a state graph of regexes is used in [50] to terminate search  from dead state regular expressions that arise in ERE constraints, which helps  the solver terminate search from goals that would otherwise not terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  7  Conclusion  GIDs are a form of incremental abstract transition system in which states are  closed when they will receive no further outgoing edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Algorithm 2 and Al-  gorithm 3 solve the incremental live and dead detection problem in GIDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The  former runs in logarithmic worst-case time, and both algorithms achieve orders  of magnitude speedup over the state-of-the-art based on maintaining strong con-  nected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Our implementation is open-source and publicly available6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Acknowledgments  We would like to thank the anonymous reviewers of CAV 2021 and TACAS 2022  for feedback leading to substantial improvements in both the paper and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  We also extend a special thanks to Nikolaj Bjørner, for his collaboration and  involvement with the Z3 implementation, and Yu Chen, for helpful discussions  in which he proposed the idea for the first-cut algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=', Luckow, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=',  Rungta, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', Tkachuk, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', Varming, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=': Semantic-based automated reason-  ing  for  AWS  access  policies  using  SMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  In:  Bjørner,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=',  Gurfinkel,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=')  2018  Formal  Methods  in  Computer  Aided  Design,  FMCAD  2018,  Austin,  TX,  USA,  October  30 November  2,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content='  IEEE  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=' Barrett, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=' Springer (2011)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=' : A new approach to incre-  mental cycle detection and related problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' ACM Transactions on Algorithms  12(2), 14:1–14:22 (Dec 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=': Incremental topological sort and cycle detection in ex-  pected total time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Sym-  posium on Discrete Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=' SIAM (2018)  6 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=' Bernstein, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=': Incremental topological sort and cycle detection in  o(m∗sqrt(n)) expected total time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In: Proceedings of the 29th Annual ACM-SIAM  Symposium on Discrete Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=' SODA’18, Society for Industrial  and Applied Mathematics (2018)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=', Vardi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' : LTL satisfiability checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In: International SPIN  Workshop on Model Checking of Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=' Springer (2007)  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' SMT:  (2012),  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='com/en-us/research/wp-  content/uploads/2016/02/nbjorner-microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='smtbenchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='zip  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Stanford, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', Veanes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', Bjørner, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=': Symbolic Boolean derivatives for efficiently  solving extended regular expression constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In: Proceedings of the 42nd ACM  SIGPLAN International Conference on Programming Language Design and Imple-  mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=' Stockmeyer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+page_content=' : Efficiency of a good but not linear set union algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' JACM 22,  215–225 (1975)  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Tinelli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', Barrett, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', Fontaine, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=': (2020), http://smtlib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='uiowa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='edu/theories-  UnicodeStrings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='shtml  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Willsey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', Nandi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', Flatt, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', Tatlock, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', Panchekha, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=': egg:  fast and extensible equality saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Proceedings of the ACM on Programming  Languages 5(POPL), 1–29 (2021)  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Z3: (2020), http://research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='com/projects/z3  A  Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='1  Extension to the Framework: Non-Reachable Updates  When GIDs are viewed as an abstract data type, other application-specific  heuristics can be incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' To illustrate this point, we consider another piece  of application-specific information: assertions which state that one vertex is not  (and will never be) reachable from another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We discuss how to incorporate such  updates to further improve the optimized algorithm in practice to exploit such  updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  First, we augment the definition of guided incremental digraph to allow up-  dates of the form N(u, v), which labels that v is not reachable from u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We then  say that the graph is valid if, in addition to the previous conditions, for every  update N(u, v), v is not reachable from u via a sequence of edges in the final  digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Figure 6 extends the example presented in Figures 1 and 2 with a not-  reachable update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Notice that N(3, 5) does not affect the set of live and dead  states, but may be exploited for more efficient search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  To incorporate N(u, v) updates in our algorithm, when moving a reserve edge  into the graph, N(u, v) can be used to shortcut the whole procedure: if we add  an edge (v, u) in particular, then we know that this doesn’t create a cycle, and  we don’t need to repeatedly call succz at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We store N(u, v) updates in a set  instead of a list for O(1) querying;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' if we need to merge them, we just discard the  set if it gets too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Therefore, in addition what was maintained previously,  the extended algorithm tracks, for each unknown canonical state x, a set of  non-reachable edges (x, y), where each corresponds to some update N(u, v) with  x = find(u) and y = find(v) (but not necessarily vice versa);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' and a constant κ  which bounds the amount of work when merging non-reachable sets (additional  28  Caleb Stanford and Margus Veanes  1  2  3  4  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Extension to our framework: guided incremental digraph from Figures 1 and 2,  with an added update N(3, 5) which states that 5 is not reachable from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' In this graph,  it would be invalid to then add E(3, 4) to the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This does not change the set of  live or dead states, but may be exploited for more efficient search: when checking if  state 3 is live or dead, we may ignore states 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  elements will be discarded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We modify the code for is-root(y, x) with a single  check in the beginning of the procedure: if x is in the not-reachable set from y,  return false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Otherwise, we continue as normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Because this algorithm uses the  N(u, v) information directly and conservatively, assuming the input GID is valid,  it remains correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='2  Reachability Problems on EREs  For extended regexes X and Y , we say that Y is reachable from X if Y ∈ ∂⋆(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  We say that X and Y are strongly connected, denoted X ⟳ Y , when both  Y ∈ ∂⋆(X) and X ∈ ∂⋆(Y ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=', when both Y is reachable from X and X is  reachable from Y through derivation as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  We also consider the following subclasses of extended regexes:  – Regexes (REs) are EREs that do not contain complement or intersection,  that is, classical regexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  – Semi-extended regexes (SEREs) are EREs that do not contain complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  – Intersections of regexes (IREs) are SEREs that are equal to an intersection  of REs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  – Boolean combinations of regexes (BCREs) are EREs that are equal to a  Boolean combination (combination of intersection and complement) of REs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Semantically, all of these capture the set of regular languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The syntactic  relationships between these different subclasses is summarized as follows, where  all inclusions are strict:  RE ⊂ IRE ⊂ BCRE  ∩  ∩  SERE ⊂ ERE  We consider the following three decision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' They are closely related  and in order of generality: the first is a special case of both the second and  thirds, and the second reduces to two queries of the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For each problem,  we also consider problems on all the subclasses of EREs that we defined: REs,  SEREs, IREs, and BCREs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We summarize the complexity results in the table in  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Incremental Dead State Detection in Logarithmic Time  29  Subclass  Edge in Cycle  Strong Connectedness  Reachability  RE  Linear  Linear  Linear  IRE  PSPACE-C  PSPACE-C  PSPACE-C  BCRE  PSPACE-C  PSPACE-C  PSPACE-C  SERE  PSPACE-C  PSPACE-C  PSPACE-C  ERE  non-elementary  non-elementary  non-elementary  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Complexity results for reachability of various subclasses of extended regexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Edge in Cycle: Given X and Y such that X ∈ ∂(Y ), is Y ∈ ∂⋆(X)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Strong Connectedness: Given X and Y , is X ⟳ Y ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Reachability: Given X and Y , is Y ∈ ∂⋆(X)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For RE, the edge-in-cycle, strong-connectedness, and reachability  problems can be solved in linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' It suffices to show that reachability can be solved in linear time, since as  stated above, it is the most general of the three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We give a decision procedure for  deciding Y ∈ ∂⋆(X) by recursing on the structure of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Prior to the procedure,  we ensure (i) that all regexes are normalized, (ii) that equal regexes are repre-  sented by the same pointer, and (iii) that concatenations are represented as lists  and precompute the length of each list, so that for any subexpression X0 of X,  whether Y = Y1·X0 can be checked in constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This precomputation takes  O(|X| + |Y |) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The recursion then works as follows, on input (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (1) If  X = ϕ, ε, or ⊥, we can check directly if Y is one of the finitely many derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  (2) If X = X1 | X2, we observe that Y ∈ ∂⋆(X) if and only if Y ∈ ∂⋆(X1) or  Y ∈ ∂⋆(X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' So we recurse on (X1, Y ) and (X2, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (3) If X = X∗  1, we observe  that Y ∈ ∂⋆(X) if and only if Y = X′  1(X∗  1) for X′  1 ∈ ∂⋆(X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' So we first check  if Y has the form Y1X∗  1, then recurse on (X1, Y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (4) Finally, if X = X1 · X2,  we observe that Y ∈ ∂⋆(X) if and only if either Y = X′  1 · X2 for X′  1 ∈ ∂⋆(X1),  or Y = X′  2 for X′  2 ∈ ∂⋆(X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (Note that this relies on the fact that X1 is an  RE, not an ERE, so we know it is nonempty and in particular ε ∈ ∂⋆(X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=') So  we first check if Y has the form Y1 · X2, if so recursing on both (X1, Y1) and  (X2, Y ), otherwise recursing on just (X2, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Since the procedure never recurses  twice on X or twice on the same subexpression of X, it takes O(|X|) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  ⊓⊔  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For the classes IRE, BCRE, and SERE, the edge-in-cycle, strong-  connectedness, and reachability problems are PSPACE-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' It suffices to show: (1) the edge-in-cycle problem for IRE is PSPACE-  hard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (2) the reachability problem for BCRE is in PSPACE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' and (3) the reach-  ability problem for SERE is in PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (1) We reduce from intersection-  nonemptiness of REs, which is known to be PSPACE-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Given a list of  REs R1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' , Rk, then let b be a fresh character that does not appear in  30  Caleb Stanford and Margus Veanes  R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' , Rk, and construct the IRE X = (bR1)∗ & (bR2)∗ & · · · & (bRk)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' X has  only one derivative X′, which is an intersection where the ith term is Ri(bRi)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Then the edge-in-cycle property for (X, X′) holds if and only if the intersection  of Ri is nonempty, because ∂b is empty for all subexpressions of Ri except ε,  so the only way to get back to X is by stepping all Ri to ε simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (2)  We first convert the BCRE X to an alternating automaton (AFA): first convert-  ing the RE subexpressions to NFAs, then using the standard constructions for  Boolean operations on AFAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' (3) For an SERE X, for any X′ ∈ ∂⋆(X) we show  that the size of X′ is bounded linearly in the size of the X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' This can be seen  by induction on the structure of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' for example, elements of ∂⋆(X1 & X2) are a  pair of an element of ∂⋆(X1) and an element of ∂⋆(X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' It follows that we can  provide the following NPSPACE algorithm for reachability of (X, Y ): at each  step, pick a derivative nondeterministically of X and recurse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  ⊓⊔  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' For ERE, the edge-in-cycle, strong-connectedness, and reachability  problems are non-elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' It suffices to show that the edge-in-cycle problem is non-elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' We  reduce from the nonemptiness problem for EREs, which is known to be non-  elementary [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Given an ERE R, we let b be a fresh character not appearing  in R, and we let X = (bR)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' Then X has one derivative, X′ = R(bR)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content=' The  edge-in-cycle property for (X, X′) holds and only if R is nonempty, because R  is nonempty exactly when ε ∈ ∂⋆(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
+page_content='  ⊓⊔' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQf3w1M/content/2301.05308v1.pdf'}
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+1
+Deep Injective Prior for Inverse Scattering
+AmirEhsan Khorashadizadeh , Sepehr Eskandari , Vahid Khorashadi-Zadeh
+and Ivan Dokmani´c
+Abstract—In electromagnetic inverse scattering, we aim
+to reconstruct object permittivity from scattered waves. Deep
+learning is a promising alternative to traditional iterative solvers,
+but it has been used mostly in a supervised framework to regress
+the permittivity patterns from scattered fields or back-projections.
+While such methods are fast at test-time and achieve good results
+for specific data distributions, they are sensitive to the distribution
+drift of the scattered fields, common in practice. If the distribu-
+tion of the scattered fields changes due to changes in frequency,
+the number of transmitters and receivers, or any other real-world
+factor, an end-to-end neural network must be re-trained or fine-
+tuned on a new dataset. In this paper, we propose a new data-
+driven framework for inverse scattering based on deep generative
+models. We model the target permittivities by a low-dimensional
+manifold which acts as a regularizer and learned from data.
+Unlike supervised methods which require both scattered fields
+and target signals, we only need the target permittivities for
+training; it can then be used with any experimental setup. We
+show that the proposed framework significantly outperforms the
+traditional iterative methods especially for strong scatterers while
+having comparable reconstruction quality to state-of-the-art deep
+learning methods like U-Net.
+I. INTRODUCTION
+E
+LECTROMAGNETIC inverse scattering is the problem of
+determining the electromagnetic properties of unknown
+objects from how they scatter incident fields. As it is non-
+destructive, it has a variety of applications in different areas,
+such as early detection of breast cancer [1], mineral prospect-
+ing [2], detecting defects and cracks inside objects [3], imaging
+through the wall [4] and remote sensing [5].
+We consider the reconstruction of the finite number of
+parameters of the object from the scattered fields. While inverse
+scattering is well-posed and Lipschitz stable in theory, when
+full-aperture continuous measurements are available [6], it is
+Manuscript received December 29, 2022.
+AmirEhsan Khorashadizadeh and Ivan Dokmani´c were supported by the
+European Research Council Starting Grant 852821–SWING.
+AmirEhsan Khorashadizadeh is with the Department of Mathematics and
+Computer Science of the University of Basel, 4001 Basel, Switzerland (e-mail:
+amir.kh@unibas.ch).
+Sepehr Eskandari is with Microwave Systems, Sensors, and Imaging Lab
+(MiXIL), University of Southern California, Los Angeles, CA 90089 USA
+(e-mail: sepehres@usc.edu).
+Vahid Khorashadi-Zadeh is with School of Electric and Computer
+Engineering, University of Tehran, Tehran 14399-57131, Iran (e-mail:
+v.khorashadi@ut.ac.ir).
+Ivan Dokmani´c is with the Department of Mathematics and Computer
+Science of the University of Basel, 4001 Basel, Switzerland, and also
+with the Department of Electrical, Computer Engineering, the Univer-
+sity of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail:
+ivan.dokmanic@unibas.ch).
+Our
+implementation
+is
+available
+at
+https://github.com/swing-research/
+scattering injective prior.
+a severely ill-posed inverse problem for a finite number of
+measurements. This means that a small perturbation in the
+scattered fields may lead to a large error in the reconstructed
+permittivity pattern [7]. Moreover, the forward operator from
+the permittivity to the scattered fields is nonlinear, which
+further complicates the inversion. The nonlinearity is due to the
+multiple scattering; the problem becomes more nonlinear as the
+permittivity contrast increases [7]. All these together make the
+inverse scattering challenging, especially for strong scatterers
+(objects with large permittivity) and noisy measurements, which
+require an effective regularization to restrict the search space
+and enable accurate reconstruction.
+A number of optimization-based methods are proposed to
+address nonlinearity and ill-posedness of the inverse scattering
+including Born iterative [8], distorted Born iterative method
+(DBIM) [9], contrast source inversion (CSI) [10], and subspace-
+based optimization (SOM) [11]. Although these methods have
+been shown to be effective for objects with small permittivity,
+they do not give an accurate reconstruction for large permittivity.
+All the above methods iteratively optimize a regularized
+objective, with a hand-crafted regularization term.
+Recently, thanks to the significant increase in computing
+power and the availability of sufficient data, data-driven
+approaches to inverse scattering have started receiving attention.
+Most deep learning models for inverse scattering use a super-
+vised learning approach, which takes the scattered fields or a
+simple back-projection as input and trains a deep neural network
+to produce the target permittivity pattern. The authors of [12]–
+[14] used scattered fields as input. While this approach provides
+good reconstructions even for objects with strong scatterers [14],
+it is sensitive to the experiment configuration such as the
+frequency or the number of incident waves and receivers; if
+the distribution of the scattered fields in test-time slightly
+changes, the quality of reconstructions remarkably degrades;
+the model requires new training data which is too costly.
+One strategy to tackle this issue is to use back-projections
+as input instead of the raw scattered fields, thus enabling
+the use of convolutional neural networks [15]–[17]. While
+this approach is shown to be effective for objects with small
+and moderate permittivity, the quality of the back-projections
+significantly drops in large permittivity (Figure 3), which leads
+to a drop in the reconstruction quality [14]. On the other hand,
+supervised learning methods are also potentially vulnerable
+to adversarial attacks [18], which is problematic in medical
+applications [19]. Moreover, incorporating the well-known
+physics of the scattering problem (forward operator), which
+can strikingly improve the accuracy of the reconstructions, is
+not easy in supervised learning models [20].
+arXiv:2301.03092v1  [cs.LG]  8 Jan 2023
+
+2
+To tackle these issues, we propose a deep learning
+approach to inverse scattering using injective generative models.
+The proposed method benefits from an unsupervised learning
+framework—the training phase uses only the target permittivity
+patterns and the physics of scattering is fully incorporated into
+the solution. Deep generative models are a class of unsupervised
+learning methods that model the probability distribution of
+data by using deep neural networks. Latent-variable generative
+models such as generative adversarial networks (GAN) [21],
+[22], variational autoencoders [23], normalizing flows [24]–
+[26] and diffusion models [27] train a deep neural network to
+transform the samples of a simple (Gaussian) distribution to
+those of target data distribution. We expect a trained generator
+produce plausible samples similar to the training data when
+taking samples of a Gaussian distribution as input.
+Bora et al. [28] used GANs to constrain the solution
+domain to the range of the trained generator for solving
+compressed sensing inverse problems. The strength of this
+idea is that it requires only the target signals for training and
+does not know anything about the inverse problem is going to
+be solved. As soon as the generator is trained, it can be used
+to solve any inverse problem with a known forward operator.
+This property makes the model robust to the distribution shift
+of the measurements and adversarial attack.
+Several methods have tried to improve the quality of
+reconstructions; Kelkar et al. [29] used the popular style-
+GAN generator [30] while Hussein et al. [31] jointly optimize
+the generator weights with latent codes to further reduce
+the reconstructions error. However, GANs are known to be
+unstable in training, and the optimization process over latent
+space sticks in the local minimum which requires many
+restarts [28]. Recently, normalizing flows as a generator instead
+of GANs have been shown to have a stable optimization
+landscape [32], [33]. Nevertheless, normalizing flows have
+their own drawbacks; having a latent code with the same
+dimension as data space makes them too expensive in training
+and does not provide an appropriate constraint in the generator
+output, leading to poor reconstruction for ill-posed inverse
+problems. More recently, injective normalizing flows [34], [35]
+are proposed to alleviate these issues. Injective flows are a
+class of deep generative models that are well-suited for solving
+ill-posed inverse problems. These models unlike GANs give us
+access to the approximate likelihood of the generated samples,
+which provides a likelihood-based regularization. Moreover,
+injective flows unlike regular normalizing flows benefit from
+a low-dimensional latent space which provides an additional
+effective regularizer for ill-posed inverse problems. Injective
+flows have been shown in [35] to effectively solve linear inverse
+problems.
+In this paper, we use injective flows as the generator
+to solve full-wave inverse scattering. we will show that the
+proposed method effectively solves inverse scattering even for
+objects with large permittivity for which traditional methods
+fail. Moreover, we use a data-driven initial guess for starting
+the iterative optimization, which significantly outperforms the
+simple back-projection initialization used in the traditional
+Fig. 1: The setup for the inverse scattering problem, red
+arrows show the incident plane waves; the green circles are
+the receivers.
+methods. Finally, we show that the proposed framework yields
+reconstructions of comparable or better quality to highly
+successful supervised methods like the U-Net [36].
+II. FORWARD AND INVERSE SCATTERING
+We begin our discussion with equations governing the
+2D forward and inverse scattering problem. We consider
+two-dimensional transverse magnetic scattering, where the
+longitudinal direction is along the z direction (TMz). As shown
+in Figure 1, non-magnetic scatterers with permittivity ϵr in
+a vacuum background with permittivity ϵ0 and permeability
+µ0 are located in investigation domain Dinv which is a D×D
+square, and are illuminated by Ni plane waves with equispaced
+directions. We have Nr receivers placed uniformly on a circle S
+with radius R which measure the scattered fields. The forward
+scattering problem can be derived from the time-harmonic form
+of Maxwell’s equations and stated as [37],
+∇ × (∇ × Et(r)) − k2
+0ϵr(r)Et(r) = iωµ0J(r)
+(1)
+where Et is the total electric field, k0 = ω√µ0ϵ0 is the
+wavenumber of the homogeneous background and J is the
+contrast current density and can be calculated using equivalence
+theorem [38] as J(r) = χ(r)Et(r) where χ(r) = ϵr(r)−1 and
+is called the contrast. The time-dependence factor exp(iωt)
+with angular working frequency ω is assumed and will be
+suppressed throughout this paper. By using Dyadic Green’s
+function [39], Equation (1) can be formalized by two coupled
+integral equations. The first one, called Lippmann–Schwinger
+equation, relates total electric fields Et in an unknown domain
+to contrast current density J,
+Et(r) = Ei(r) + k2
+0
+�
+Dinv
+g(r, r′)J(r′)dr′,
+(2)
+
+3
+where r ∈ Dinv, and
+g(r, r′) = 1
+4iH(2)
+0 (k0|r − r′|),
+and H(2)
+0
+is the Hankel function of the second kind, denotes
+2D free space Green’s function. The second equation, referred
+to as the data equation, maps the contrast current density J
+to the scattered electric fields Es at the receivers locations,
+Es(r) = k2
+0
+�
+Dinv
+g(r, r′)J(r′)dr′, r ∈ S.
+(3)
+We discretize the investigation domain Dinv with N × N units
+and rewrite Equations (2) and (3) as
+Et = Ei + GdχEt
+(4)
+Es = GsχEt + δ
+(5)
+where Gd ∈ RN 2×N 2 and Gs ∈ RNr×N 2 have analytical
+closed form [7]. Moreover, Et, Ei and Es respectively
+correspond to total, incident and scattered electric fields
+and χ is a diagonal matrix with the diagonal elements
+χ(n, n) = ϵr(rn) − 1. We also consider additive noise δ to the
+measurements.
+We merge Equations (4) and (5) to make a single
+expression for the forward equation [7],
+Es = Gsχ(I − Gdχ)−1Ei + δ,
+(6)
+which is a nonlinear mapping χ �→ Es. It is convenient to
+define a forward operator A mapping χ to Es,
+y = A(x) + δ,
+(7)
+where A(·) is the nonlinear forward scattering operator
+A(χ) = Gsχ(I − Gdχ)−1Ei,
+(8)
+y = Es and x = χ. The task of inverse scattering is to
+reconstruct the contrast signal χ from the scattered fields Es
+where we assume Gd, Gs, incident electric waves Ei and
+consequently the forward operator A(·) are known. In the next
+section, we briefly review deep generative models as the prior
+model for inverse problems.
+III. DEEP GENERATIVE MODELS
+One major division in machine learning is generative and
+discriminative models. In discriminative models, one aims
+to learn a direct mapping from the measurements to the
+target signals. Discriminative approaches have been extensively
+used in solving inverse problems, specifically U-Net [36] has
+shown great success in many applications such as computed
+tomography (CT) [40], magnetic resonance imaging (MRI) [41],
+optoacoustic tomography [42] and electromagnetic inverse
+scattering [16]. The key idea of this success might be ascribed to
+having a multiscale architecture [43]. Recently, the variational
+version of U-Net has shown excellent performance for posterior
+sampling and uncertainty quantification of inverse scattering
+problem [44].
+On the other hand, the task of generative models is to learn
+the probability distribution of data. Latent-variable generative
+models including generative adversarial networks (GANs) [21],
+[22], variational autoencoders [23] and normalizing flows [24]–
+[26] are a subcategory of deep generative models (DGMs)
+which train a deep neural network, called the generator, to
+transform the samples of a simple and known distribution
+(often Gaussian) to data distribution samples. DGMs have
+many applications such as image generation [22], image-to-
+image translation [45], density estimation [46] and variational
+inference [47], [48]. Recently, DGMs have been used as a prior
+for solving inverse problems [28], [35], [49]–[52]; consider a
+DGM trained on a training set of target signals (the solutions
+of a given inverse problem), one can search in the latent space
+of the trained generator for the latent code yielding a solution
+aligns with the given measurements. The pre-trained generator
+plays the role of an effective regularizer for generating plausible
+target signals. As discussed earlier, the key advantage of this
+approach is that the measurements are not used in the training
+phase. As a result, once the DGM is trained, it can be used
+for solving any inverse problems.
+However, the choice of DGM is of paramount importance
+to provide stable training, high-quality sample generation,
+and an effective regularizer for solving ill-posed inverse
+problems. While modern GANs with the various innovations
+in architectures and training protocols exhibit high-quality
+samples, they suffer from training instability [53], [54] and have
+shown poor reconstructions when used as a prior for solving ill-
+posed inverse problems [35], [55]. Normalizing flows alleviates
+many drawbacks of GAN; they are stable in training and can
+produce high-quality samples. Normalizing flows comprise a
+set of bijective transformations which have tractable inverse
+and log det Jacobian computations. They give access to the
+likelihood of the generated samples and can be trained based
+on maximum likelihood. Normalizing flows as a prior were
+shown to be more effective than GANs for solving inverse
+problems [32], [33]. However unlike GANs, normalizing flows
+are bijective mappings, having the same dimension in the
+latent space and data space, consequently, the network output
+is not constraint and leading to poor regularization for solving
+ill-posed inverse problems [14].
+More recently, the authors of [35] showed that injective
+normalizing flows are so effective for solving ill-posed inverse
+problems. Injective normalizing flows are an extension of
+normalizing flows which have a low-dimensional latent space.
+Injective flows provide a low-dimensional representation of the
+data (like GANs) which perform as a strong regularization for
+solving inverse problems while giving access to the likelihood
+of the generated samples (like normalizing flows) which can
+be seen as the second regularization. In the next section, we
+briefly review injective flows called Trumpets.
+IV. INJECTIVE NORMALIZING FLOWS
+Injective normalizing flows [35] map a low-dimensional
+latent space to the high-dimensional data space using a set of
+
+4
+MOG (µz)
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+Fig. 2: Injective normalizing flows [35] comprise two submodules, a low-dimensional bijective flow hη and an injective network
+with expansive layers gγ. The MOG initialization is the mean of the Gaussian in the latent space zinit = µz (red circle).
+invertible neural networks. Injective normalizing flows have
+a fast inverse on the range and give access to the likelihood
+of the generated samples. As shown in Figure 2, fθ(z) =
+gγ(hη(z)) with weights θ = (γ, η) comprises two subnetworks:
+an injective part (with expansive layers) gγ that maps a low-
+dimensional space Rd to high-dimensional data space RD
+where d ≪ D, and a bijective mapping hη that keeps the
+dimension in low dimensional space Rd. The training of the
+injective normalizing flows consists of two phases, at first, the
+range of the injective generator must be adjusted to lie on the
+training data by optimizing over the weights of the injective
+subnetwork gγ, when we ensure the training data are close to
+the range of the generator, in the second phase the likelihood
+of the pre-image of the training data should be maximized
+over the weights of the bijective subnetwork hη. Further details
+are explained in [35]. When the network is trained, we can
+generate random samples similar to the training data
+xgen = f(zgen)
+(9)
+where zgen ∼ N(µz, σ2
+zI). Further information about the
+universality of density and manifold approximation of injective
+normalizing flows are studied in [56].
+Due to having a low-dimensional latent space, injec-
+tive flows provide a low-dimensional manifold in the high-
+dimensional data space. When they are trained, the manifold
+would contain plausible samples which can be used as an
+effective regularizer for solving ill-posed inverse problems.
+The injective part provides a projection operator gγ(g†
+γ(x))
+which maps the data samples x to the intermediate space by
+z′ = g†
+γ(x) and projects them back to the data space by g†
+γ(z′).
+The authors of [35] used the projection operator to project a
+sample on the manifold to suppress the noise and artifacts,
+as the learned manifold contains high-quality samples. In the
+next section, we will present our method of solving inverse
+scattering problems using injective normalizing flows.
+V. INJECTIVE FLOWS FOR INVERSE SCATTERING
+Inverse scattering is a severely ill-posed inverse problem,
+which means a small perturbation in the scattered fields leads
+to an exponentially large error in the contrast [7]. This makes
+the inversion unstable even for a small amount of noise. As
+discussed in section II, inverse scattering is a nonlinear inverse
+problem whose nonlinearity heavily relies on the maximum
+contrast value. For objects with large contrasts, the problem
+becomes highly nonlinear, which makes the inversion extremely
+difficult. In such a scenario, the significance of having a strong
+regularizer to restrict the search domain is crucially important.
+We consider the contrast signal χ = x ∈ X and
+the scattered fields Es = y ∈ Y, described by forward
+operator (7), as random vectors. While our framework admits
+other distributions beyond Gaussian for the additive noise δ in
+(7), in this paper we assume that δ is a random vector with
+Gaussian distribution δ ∼ N(0, σ2I) yielding
+Y |X ∼ N(A(X), σ2I).
+(10)
+One effective approach for solving ill-posed inverse problems is
+computing MAP estimate, where we look for the solution x that
+has the highest posterior likelihood for a given measurement
+y,
+xMAP = arg maxx log(pX|Y (x|y)),
+(11)
+
+85
+where pX|Y (x|y) is the posterior distribution for the given
+measurement y. Using Bayes theorem yields,
+xMAP
+=
+arg minx − log(pX|Y (x|y))
+=
+arg minx − log(pY |X(y|x)pX(x)
+pY (y)
+)
+=
+arg minx − log(pY |X(y|x)) − log(pX(x)).
+(12)
+The first term can be obtained from (10),
+xMAP = arg minx
+1
+2∥y − A(x)∥2
+2 − λ log(pX(x)),
+(13)
+where the first term is the data-consistency loss while
+log(pX(x)) is the prior distribution of the contrast and plays
+the role of the regularization. We also have λ which is a
+hyperparameter to adjust the amount of regularization term
+(although it comes from the noise standard deviation which
+we assume is unknown). In principle, the prior distribution
+pX(x) is not available and must be estimated. For example,
+traditionally pX(x) is approximated with a Gaussian distribu-
+tion with zero mean leading to the Tikhonov regularization.
+However, a Gaussian distribution is often too far from the true
+prior distribution and leads to poor reconstructions.
+In this paper, we study a new regularization family for
+inverse scattering based on deep generative models. We consider
+a training set of contrast signals {x(i)}N
+i=1 is available, and we
+train a deep generative model x = f(z) to produce high-quality
+contrast samples. we expect the trained generator f to produce
+high-quality contrast samples when it takes random samples
+from the Gaussian distribution in the latent space z ∈ Z.
+As discussed in section III, this property of deep generative
+models makes them an effective regularizer for solving inverse
+problems [28].
+We use injective normalizing flows as the generator of the
+contrast signal since they are well-suited for solving ill-posed
+inverse problems [35]. We perform an optimization in the latent
+space to find the latent code that aligns with scattered fields y,
+zMAP = arg max
+z
+1
+2∥y −A(f(z))∥2
+2 +λ log(pX(f(z))), (14)
+Where an approximation to pX is also provided by the injective
+flows and acts as the second regularizer. The reconstructed
+contrast signal can be obtained as xMAP = f(zMAP). We
+call this method latent space optimization (LSO). It is worth
+mentioning that (14) has been previously proposed by [32],
+[33] for solving compressed sensing inverse problems using
+regular normalizing flows.
+Unlike the supervised learning methods for inverse scatter-
+ing [13], [15]–[17] which use a paired training set of contrast
+and scattered fields {(x(i), y(i))}N
+i=1, our framework benefits
+from an unsupervised learning paradigm where scattered fields
+are not used in the training of injective flows. This means, if the
+distribution of the scattered fields changes (due to the change
+in experimental configuration), the generative network is not
+required to be retrained unlike the supervised methods; as soon
+as the injective generator is trained over the contrast signals, we
+can optimize (14) for each new scattered fields to reconstruct
+Ground truth
+BP (𝜖! = 1.2) 
+BP (𝜖! = 2) 
+BP (𝜖! = 4) 
+BA (𝜖! = 1.2) 
+BA (𝜖! = 2) 
+BA (𝜖! = 4) 
+Fig. 3: Performance analysis of back-propagation (BP) and
+Born approximation (BA) methods for different ϵr; while BA
+and BP reconstructions are visually meaningful for small ϵr,
+their performance sharply drop for objects with large ϵr.
+Random ellipses 
+MNIST
+Fig. 4: Illustration of the MOG initializer in the data space
+f(µz) for random ellipses and MNIST datasets
+the corresponding contrast. Apart from the advantages of an
+unsupervised learning paradigm, the proposed method fully
+exploits the underlying physics of the scattering problem by
+optimizing over the complex-valued scattered fields in (14).
+Kothari et al. [57] have shown incorporating wave physics
+in the neural network architecture can significantly improve
+the quality of reconstructions, especially for out-of-distribution
+data.
+We solve the optimization problem (14) by Adam op-
+timizer [58] and compute the gradients with respect to z
+by automatic differentiation provided by TensorFlow [59] in
+Python. We also use an alternative method for Equation (14)
+proposed by [35] which performs the optimization in the data
+space,
+xMAP = arg min
+x
+1
+2∥y −A(g(g†(x)))∥2
+2 −λ log(pX(x)) (15)
+where g(g†(x)) is the projection operator explained in sec-
+tion IV. We call this method data space optimization (DSO).
+This method is shown in [35] to be effective for solving
+linear inverse problems. While in each of the LSO iterations
+the reconstructed point x = f(z) is always on the learned
+manifold, this is not the case for the DSO method, where the
+reconstructed image might be off the manifold. Moreover, the
+DSO method requires more computations than LSO as we need
+to take derivatives over the reverse direction of the injective
+subnetwork g†(x).
+The choice of initial guess is crucially important in inverse
+
+0.07
+0.41
+0.08
+0.01
+0.02
+-0.33
+0.08
+0.14
+0.12
+0.00
+-0.04
+-0.156
+Ground truths
+Projections on the manifold
+Generated samples
+Fig. 5: Performance evaluation of the trained injective normalizing flows for random ellipses dataset; ground truth contrasts,
+their projections on the learned manifold and some randomly generated samples.
+scattering solvers. While a poor initialization misleads the
+optimization process, a good initial step helps the algorithm to
+converge more accurately and faster. The authors of [9] used
+Born approximation as the initialization for the distorted Born
+iterative method (DBIM). A back-propagation (BP) solution
+was also used in [10], [60] as an initial guess of the contrast
+source inversion (CSI) method. Figure 3 shows the ground
+truth, back-propagation (BP) and Born approximations (BA)
+for an object with different maximum ϵr values. While BP
+and BA can present fuzzy reconstructions for objects with
+small permittivity, their performance sharply drops for large ϵr
+(especially numerically) which makes them a poor initialization
+for strong scatterers.
+In order to circumvent this issue, we use a data-driven
+initialization suggested in [32]; mean of the Gaussian distri-
+bution (MOG) in the latent space as shown in Figure 2. The
+MOG initializer zinit = µz provides a fixed initialization with
+respect to the measurements (scattered fields); thereby being
+independent of the maximum contrast value and the problem
+configuration. This property helps (14) and (15) to converge
+better even for a large ϵr. In section VI, we will show that
+MOG initialization significantly improves the quality of the
+reconstructions compared to BP. To our best knowledge, this
+is the first time that we propose a data-driven initializer for
+inverse scattering.
+VI. COMPUTATIONAL EXPERIMENTS
+In the following, we will evaluate the performance of DSO
+and LSO for inverse scattering. We consider the MOG and BP
+initializations for DSO while using just MOG initialization for
+LSO. We compare the performance of our proposed methods
+with a traditional iterative method DBIM [9]. While our method
+is based on an unsupervised-learning paradigm; scattered fields
+are not used during training, we also compare its performance
+with a successful supervised learning method, U-Net [36] which
+has shown great success for inverse scattering [16], to show the
+effectivity of our method. U-Net takes the back-propagation
+(BP) as input and returns the permittivity pattern of the object
+in the output.
+We use MNIST [61] with 60000 training samples in the
+resolution N = 32. We also use a custom dataset of 60000
+training samples with resolution N = 64 of overlapping ellipses
+used in [14] to have a more challenging task for accurate
+reconstructions.
+We use Ni = 12 incident plane waves and Nr = 12
+receivers, uniformly distributed on a circle with radius R =
+20 cm around the object with maximum permittivity ϵr and
+dimension D = 20 cm. The working frequency is 3 GHz and
+we added 30 dB noise to the scattered fields.
+We used the injective normalizing flows with the same
+architecture described in [35]. We trained the injective subnet-
+work gγ for 150 epochs to ensure the training samples (contrast
+signals) are close to the range of the generator. Figure 5 shows
+some test samples and their projections on the learned manifold.
+Then we trained the bijective subnetwork hη for 150 epochs
+to maximize the likelihood of the pre-image of the training
+samples in the intermediate space. Figure 5 illustrates some
+randomly generated samples which confirms that the model
+can produce plausible samples to be used as an effective prior
+for solving inverse scattering.
+We optimize (14) and (15) by using Adam optimizer with a
+learning rate of 0.05 for 300 iterations. We use λ = 0.01 for BP
+and λ = 0 for MOG initialization. It is worth mentioning that
+when we use MOG initializer, we start from high-likelihood
+regions (mean of the Gaussian) which can be viewed as a
+hidden regularizer, we thus use λ = 0 in this case. Figure 4
+shows the MOG initialization for MNIST and random ellipses
+datasets.
+Figures 6 and 7 show the performance of different methods
+for inverse scattering for ϵr = 4 over five test samples from
+random ellipses and MNIST datasets. While the traditional
+iterative method (DBIM) meets with complete failure in this
+challenging task (ϵr = 4 and 30 dB noise), DSO and LSO
+have strikingly more accurate reconstructions, which clearly
+
+7
+BP
+DBIM
+U-Net
+DSO (BP)
+DSO (MOG)
+LSO
+Ground truth
+Fig. 6: Performance comparison of different methods over random ellipses dataset in resolution 64 × 64
+BP
+DBIM
+U-Net
+DSO (BP)
+DSO (MOG)
+LSO
+Ground truth
+Fig. 7: Performance comparison of different methods over MNIST dataset in resolution 32 × 32
+
+0.5
+2.5
+3.7
+4.0
+4.0
+3.5
+3.6
+-3.5
+1.0
+1.0
+1.0
+1.0
+0.5
+5.3
+4.0
+4.0
+4.0
+4.0
+3.5
+1.0
+1.0
+1.0
+1.0
+1.0
+0.5
+4.5
+4.0
+4.0
+3.9
+ 3.9
+-3.2
+1.0
+1.0
+1.0
+1.0
+1.0
+0.5
+2.9
+4.0
+4.0
+4.0
+4.0
+ 4.0
+-1.8
+1.0
+1.0
+1.0
+1.0
+1.0
+0.5
+1.5
+3.9
+4.0
+4.0
+4.0
+ 4.0
+0.3
+-4.5
+1.0
+1.0
+1.0
+1.0
+1.00.5
+4.0
+9.4
+4.2
+3.9
+4.0
+0.4
+0.7
+0.9
+5.7
+3.9
+4.1
+4.1
+3.9
+4.0
+5
+01
+3.3
+1.0
+0.9
+0.9
+1.0
+1.0
+0.6
+5.6
+4.0
+11.8
+6.6
+4.0
+ 4.0
+-3.2
+1.0
+3.2
+-1.6
+0.9
+1.0
+0.5
+5.8
+4.0
+11.9
+7.8
+4.1
+ 4.0
+-3.8
+1.0
+.1.4
+-0.8
+0.9
+1.0
+4.4
+4.0
+12.1
+4.2
+3.9
+ 4.0
+0.7
+9
+1
+0.3
+2.8
+1.0
+-7.1
+0.8
+0.9
+1.08
+TABLE I: Performance of different methods for solving inverse
+scattering (ϵr = 4) averaged over 25 test samples
+PSNR
+SSIM
+MNIST
+Ellipses
+MNIST
+Ellipses
+BP
+7.75
+7.00
+0.01
+0.01
+DBIM [9]
+5.77
+4.67
+0.01
+0.01
+U-Net [36]
+24.26
+21.94
+0.90
+0.82
+DSO (BP)
+8.73
+7.89
+0.16
+0.16
+DSO (MOG)
+21.50
+14.56
+0.56
+0.44
+LSO (MOG)
+25.22
+20.50
+0.89
+0.85
+Fig. 8: The performance of different methods for different ϵr
+over MNIST dataset; SSIM is computed over the normalized
+signals between −1 and 1.
+shows the significance of a data-driven regularization. Moreover,
+the MOG initialization, as we expected, exhibits remarkably
+better reconstructions compared to BP for this high permittivity
+(ϵr = 4). Furthermore, both figures show that LSO outperforms
+DSO; running optimization in the latent space results in better
+reconstructions as discussed in section V. Finally, while LSO
+does not use scattered fields in the training phase, it could
+present the reconstructions with comparable quality (or even
+better) to the highly successful supervised method U-Net.
+Table I shows the numerical results in PSNR and SSIM
+averaged over 25 test samples.
+As discussed in section V, the maximum ϵr of the
+object plays a significant role in the performance of inverse
+scattering solvers; objects with large ϵr are more difficult to
+be reconstructed. Figure 8 demonstrates the performance of
+different methods per ϵr over the MNIST dataset. This figure
+discovers that LSO with MOG initialization is effective even
+for objects with large, ϵr which clearly signifies the power of
+a data-driven initialization and performing the optimization in
+the latent space.
+VII. LIMITATIONS AND CONCLUSIONS
+We proposed a learning-based framework for inverse
+scattering using an injective prior. The proposed method fully
+exploits the physical domain of the scattering problem while
+benefiting from a data-driven initialization which makes a
+powerful solver even for objects with a large contrast. The
+invertible generator admits performing optimization in both
+latent and data space and uses a data-driven or back-projection
+as the initializer. We showed that performing optimization in
+the latent space and using the mean of the Gaussian as the
+initial guess significantly outperforms the traditional iterative
+methods and even gives reconstructions comparable to the
+successful supervised learning method, U-Net.
+Limitations: The proposed framework has several limitations
+and entails discussion. It requires running an iterative method
+in test-time, which is slow and cannot be used for real-time
+applications. To speed up the convergence, one may consider a
+more accurate initial guess by exploiting the physical domain
+in the data-driven initializer; a combination of traditional back-
+projection (like BP) and the data-driven initializers (like MOG).
+Recently, Hussein et all. [31] optimized the generator weights
+with a small rate after finding the optimal latent code in
+Equation (14) to further improve the reconstructions. This
+idea might be taken in our framework to go beyond the quality
+of the generator, but we leave it for future works.
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diff --git a/-dE1T4oBgHgl3EQfUgPr/content/tmp_files/load_file.txt b/-dE1T4oBgHgl3EQfUgPr/content/tmp_files/load_file.txt
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@@ -0,0 +1,854 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf,len=853
+page_content='1  Deep Injective Prior for Inverse Scattering  AmirEhsan Khorashadizadeh , Sepehr Eskandari , Vahid Khorashadi-Zadeh  and Ivan Dokmani´c  Abstract—In electromagnetic inverse scattering, we aim  to reconstruct object permittivity from scattered waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Deep  learning is a promising alternative to traditional iterative solvers,  but it has been used mostly in a supervised framework to regress  the permittivity patterns from scattered fields or back-projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  While such methods are fast at test-time and achieve good results  for specific data distributions, they are sensitive to the distribution  drift of the scattered fields, common in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' If the distribu-  tion of the scattered fields changes due to changes in frequency,  the number of transmitters and receivers, or any other real-world  factor, an end-to-end neural network must be re-trained or fine-  tuned on a new dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' In this paper, we propose a new data-  driven framework for inverse scattering based on deep generative  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We model the target permittivities by a low-dimensional  manifold which acts as a regularizer and learned from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Unlike supervised methods which require both scattered fields  and target signals, we only need the target permittivities for  training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' it can then be used with any experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We  show that the proposed framework significantly outperforms the  traditional iterative methods especially for strong scatterers while  having comparable reconstruction quality to state-of-the-art deep  learning methods like U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' INTRODUCTION  E  LECTROMAGNETIC inverse scattering is the problem of  determining the electromagnetic properties of unknown  objects from how they scatter incident fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' As it is non-  destructive, it has a variety of applications in different areas,  such as early detection of breast cancer [1], mineral prospect-  ing [2], detecting defects and cracks inside objects [3], imaging  through the wall [4] and remote sensing [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  We consider the reconstruction of the finite number of  parameters of the object from the scattered fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' While inverse  scattering is well-posed and Lipschitz stable in theory, when  full-aperture continuous measurements are available [6], it is  Manuscript received December 29, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  AmirEhsan Khorashadizadeh and Ivan Dokmani´c were supported by the  European Research Council Starting Grant 852821–SWING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  AmirEhsan Khorashadizadeh is with the Department of Mathematics and  Computer Science of the University of Basel, 4001 Basel, Switzerland (e-mail:  amir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='kh@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='ch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Sepehr Eskandari is with Microwave Systems, Sensors, and Imaging Lab  (MiXIL), University of Southern California, Los Angeles, CA 90089 USA  (e-mail: sepehres@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Vahid Khorashadi-Zadeh is with School of Electric and Computer  Engineering, University of Tehran, Tehran 14399-57131, Iran (e-mail:  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='khorashadi@ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='ir).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Ivan Dokmani´c is with the Department of Mathematics and Computer  Science of the University of Basel, 4001 Basel, Switzerland, and also  with the Department of Electrical, Computer Engineering, the Univer-  sity of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail:  ivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='dokmanic@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='ch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Our  implementation  is  available  at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='com/swing-research/  scattering injective prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  a severely ill-posed inverse problem for a finite number of  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' This means that a small perturbation in the  scattered fields may lead to a large error in the reconstructed  permittivity pattern [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Moreover, the forward operator from  the permittivity to the scattered fields is nonlinear, which  further complicates the inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The nonlinearity is due to the  multiple scattering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' the problem becomes more nonlinear as the  permittivity contrast increases [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' All these together make the  inverse scattering challenging, especially for strong scatterers  (objects with large permittivity) and noisy measurements, which  require an effective regularization to restrict the search space  and enable accurate reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  A number of optimization-based methods are proposed to  address nonlinearity and ill-posedness of the inverse scattering  including Born iterative [8], distorted Born iterative method  (DBIM) [9], contrast source inversion (CSI) [10], and subspace-  based optimization (SOM) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Although these methods have  been shown to be effective for objects with small permittivity,  they do not give an accurate reconstruction for large permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  All the above methods iteratively optimize a regularized  objective, with a hand-crafted regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Recently, thanks to the significant increase in computing  power and the availability of sufficient data, data-driven  approaches to inverse scattering have started receiving attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Most deep learning models for inverse scattering use a super-  vised learning approach, which takes the scattered fields or a  simple back-projection as input and trains a deep neural network  to produce the target permittivity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The authors of [12]–  [14] used scattered fields as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' While this approach provides  good reconstructions even for objects with strong scatterers [14],  it is sensitive to the experiment configuration such as the  frequency or the number of incident waves and receivers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' if  the distribution of the scattered fields in test-time slightly  changes, the quality of reconstructions remarkably degrades;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  the model requires new training data which is too costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  One strategy to tackle this issue is to use back-projections  as input instead of the raw scattered fields, thus enabling  the use of convolutional neural networks [15]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' While  this approach is shown to be effective for objects with small  and moderate permittivity, the quality of the back-projections  significantly drops in large permittivity (Figure 3), which leads  to a drop in the reconstruction quality [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' On the other hand,  supervised learning methods are also potentially vulnerable  to adversarial attacks [18], which is problematic in medical  applications [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Moreover, incorporating the well-known  physics of the scattering problem (forward operator), which  can strikingly improve the accuracy of the reconstructions, is  not easy in supervised learning models [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='03092v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='LG]  8 Jan 2023  2  To tackle these issues, we propose a deep learning  approach to inverse scattering using injective generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  The proposed method benefits from an unsupervised learning  framework—the training phase uses only the target permittivity  patterns and the physics of scattering is fully incorporated into  the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Deep generative models are a class of unsupervised  learning methods that model the probability distribution of  data by using deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Latent-variable generative  models such as generative adversarial networks (GAN) [21],  [22], variational autoencoders [23], normalizing flows [24]–  [26] and diffusion models [27] train a deep neural network to  transform the samples of a simple (Gaussian) distribution to  those of target data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We expect a trained generator  produce plausible samples similar to the training data when  taking samples of a Gaussian distribution as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Bora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' [28] used GANs to constrain the solution  domain to the range of the trained generator for solving  compressed sensing inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The strength of this  idea is that it requires only the target signals for training and  does not know anything about the inverse problem is going to  be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' As soon as the generator is trained, it can be used  to solve any inverse problem with a known forward operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  This property makes the model robust to the distribution shift  of the measurements and adversarial attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Several methods have tried to improve the quality of  reconstructions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Kelkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' [29] used the popular style-  GAN generator [30] while Hussein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' [31] jointly optimize  the generator weights with latent codes to further reduce  the reconstructions error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' However, GANs are known to be  unstable in training, and the optimization process over latent  space sticks in the local minimum which requires many  restarts [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Recently, normalizing flows as a generator instead  of GANs have been shown to have a stable optimization  landscape [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Nevertheless, normalizing flows have  their own drawbacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' having a latent code with the same  dimension as data space makes them too expensive in training  and does not provide an appropriate constraint in the generator  output, leading to poor reconstruction for ill-posed inverse  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' More recently, injective normalizing flows [34], [35]  are proposed to alleviate these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Injective flows are a  class of deep generative models that are well-suited for solving  ill-posed inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' These models unlike GANs give us  access to the approximate likelihood of the generated samples,  which provides a likelihood-based regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Moreover,  injective flows unlike regular normalizing flows benefit from  a low-dimensional latent space which provides an additional  effective regularizer for ill-posed inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Injective  flows have been shown in [35] to effectively solve linear inverse  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  In this paper, we use injective flows as the generator  to solve full-wave inverse scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' we will show that the  proposed method effectively solves inverse scattering even for  objects with large permittivity for which traditional methods  fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Moreover, we use a data-driven initial guess for starting  the iterative optimization, which significantly outperforms the  simple back-projection initialization used in the traditional  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' 1: The setup for the inverse scattering problem, red  arrows show the incident plane waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' the green circles are  the receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Finally, we show that the proposed framework yields  reconstructions of comparable or better quality to highly  successful supervised methods like the U-Net [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' FORWARD AND INVERSE SCATTERING  We begin our discussion with equations governing the  2D forward and inverse scattering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We consider  two-dimensional transverse magnetic scattering, where the  longitudinal direction is along the z direction (TMz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' As shown  in Figure 1, non-magnetic scatterers with permittivity ϵr in  a vacuum background with permittivity ϵ0 and permeability  µ0 are located in investigation domain Dinv which is a D×D  square, and are illuminated by Ni plane waves with equispaced  directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We have Nr receivers placed uniformly on a circle S  with radius R which measure the scattered fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The forward  scattering problem can be derived from the time-harmonic form  of Maxwell’s equations and stated as [37],  ∇ × (∇ × Et(r)) − k2  0ϵr(r)Et(r) = iωµ0J(r)  (1)  where Et is the total electric field, k0 = ω√µ0ϵ0 is the  wavenumber of the homogeneous background and J is the  contrast current density and can be calculated using equivalence  theorem [38] as J(r) = χ(r)Et(r) where χ(r) = ϵr(r)−1 and  is called the contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The time-dependence factor exp(iωt)  with angular working frequency ω is assumed and will be  suppressed throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' By using Dyadic Green’s  function [39], Equation (1) can be formalized by two coupled  integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The first one, called Lippmann–Schwinger  equation, relates total electric fields Et in an unknown domain  to contrast current density J,  Et(r) = Ei(r) + k2  0  �  Dinv  g(r, r′)J(r′)dr′,  (2)  3  where r ∈ Dinv, and  g(r, r′) = 1  4iH(2)  0 (k0|r − r′|),  and H(2)  0  is the Hankel function of the second kind, denotes  2D free space Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The second equation, referred  to as the data equation, maps the contrast current density J  to the scattered electric fields Es at the receivers locations,  Es(r) = k2  0  �  Dinv  g(r, r′)J(r′)dr′, r ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  (3)  We discretize the investigation domain Dinv with N × N units  and rewrite Equations (2) and (3) as  Et = Ei + GdχEt  (4)  Es = GsχEt + δ  (5)  where Gd ∈ RN 2×N 2 and Gs ∈ RNr×N 2 have analytical  closed form [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Moreover, Et, Ei and Es respectively  correspond to total, incident and scattered electric fields  and χ is a diagonal matrix with the diagonal elements  χ(n, n) = ϵr(rn) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We also consider additive noise δ to the  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  We merge Equations (4) and (5) to make a single  expression for the forward equation [7],  Es = Gsχ(I − Gdχ)−1Ei + δ,  (6)  which is a nonlinear mapping χ �→ Es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' It is convenient to  define a forward operator A mapping χ to Es,  y = A(x) + δ,  (7)  where A(·) is the nonlinear forward scattering operator  A(χ) = Gsχ(I − Gdχ)−1Ei,  (8)  y = Es and x = χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The task of inverse scattering is to  reconstruct the contrast signal χ from the scattered fields Es  where we assume Gd, Gs, incident electric waves Ei and  consequently the forward operator A(·) are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' In the next  section, we briefly review deep generative models as the prior  model for inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' DEEP GENERATIVE MODELS  One major division in machine learning is generative and  discriminative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' In discriminative models, one aims  to learn a direct mapping from the measurements to the  target signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Discriminative approaches have been extensively  used in solving inverse problems, specifically U-Net [36] has  shown great success in many applications such as computed  tomography (CT) [40], magnetic resonance imaging (MRI) [41],  optoacoustic tomography [42] and electromagnetic inverse  scattering [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The key idea of this success might be ascribed to  having a multiscale architecture [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Recently, the variational  version of U-Net has shown excellent performance for posterior  sampling and uncertainty quantification of inverse scattering  problem [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  On the other hand, the task of generative models is to learn  the probability distribution of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Latent-variable generative  models including generative adversarial networks (GANs) [21],  [22], variational autoencoders [23] and normalizing flows [24]–  [26] are a subcategory of deep generative models (DGMs)  which train a deep neural network, called the generator, to  transform the samples of a simple and known distribution  (often Gaussian) to data distribution samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' DGMs have  many applications such as image generation [22], image-to-  image translation [45], density estimation [46] and variational  inference [47], [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Recently, DGMs have been used as a prior  for solving inverse problems [28], [35], [49]–[52];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' consider a  DGM trained on a training set of target signals (the solutions  of a given inverse problem), one can search in the latent space  of the trained generator for the latent code yielding a solution  aligns with the given measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The pre-trained generator  plays the role of an effective regularizer for generating plausible  target signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' As discussed earlier, the key advantage of this  approach is that the measurements are not used in the training  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' As a result, once the DGM is trained, it can be used  for solving any inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  However, the choice of DGM is of paramount importance  to provide stable training, high-quality sample generation,  and an effective regularizer for solving ill-posed inverse  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' While modern GANs with the various innovations  in architectures and training protocols exhibit high-quality  samples, they suffer from training instability [53], [54] and have  shown poor reconstructions when used as a prior for solving ill-  posed inverse problems [35], [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Normalizing flows alleviates  many drawbacks of GAN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' they are stable in training and can  produce high-quality samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Normalizing flows comprise a  set of bijective transformations which have tractable inverse  and log det Jacobian computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' They give access to the  likelihood of the generated samples and can be trained based  on maximum likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Normalizing flows as a prior were  shown to be more effective than GANs for solving inverse  problems [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' However unlike GANs, normalizing flows  are bijective mappings, having the same dimension in the  latent space and data space, consequently, the network output  is not constraint and leading to poor regularization for solving  ill-posed inverse problems [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  More recently, the authors of [35] showed that injective  normalizing flows are so effective for solving ill-posed inverse  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Injective normalizing flows are an extension of  normalizing flows which have a low-dimensional latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Injective flows provide a low-dimensional representation of the  data (like GANs) which perform as a strong regularization for  solving inverse problems while giving access to the likelihood  of the generated samples (like normalizing flows) which can  be seen as the second regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' In the next section, we  briefly review injective flows called Trumpets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' INJECTIVE NORMALIZING FLOWS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='Injective normalizing flows [35] map a low-dimensional  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='latent space to the high-dimensional data space using a set of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='MOG (µz)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='<latexit sha1_base64="C4QjJmxveAJDFx/9MTdnNEoO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='sB8=">ACAnicbVA9SwNBEN3zM8avqJXYLAYhNuFOIloGLbQRI5gPSMKxt9kS3bvjt05MR6HjX/FxkIRW3+Fnf/GT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='XKFJj4YeLw3w8w8LxRcg21/W3PzC4tLy5mV7Ora+sZmbmu7poNIUValgQhUwyOaCe6zKnAQrBEqRqQnWN0bnI/8+h1T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='mgf+LQxD1pak5/MupwSM5OZ2W8DuIb6vsCFpCUj92EiHCZuLm8X7THwLHFSkcpKm7uq9UJaCSZD1QrZuOHUI7Jgo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='4FSzJtiLNQkIHpMeahvpEMt2Oxy8k+MAoHdwNlCkf8Fj9PRETqfVQeqZTEujraW8k/uc1I+ietmPuhxEwn04WdSOBIc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='CjPHCHK0ZBDA0hVHFzK6Z9ogFk1rWhOBMvzxLakdFp1Q8vinly2dpHBm0h/ZRATnoBJXRJaqgKqLoET2jV/RmPVkv1r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='v1MWmds9KZHfQH1ucPKAKXSg=</latexit>  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' 2: Injective normalizing flows [35] comprise two submodules, a low-dimensional bijective flow hη and an injective network  with expansive layers gγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The MOG initialization is the mean of the Gaussian in the latent space zinit = µz (red circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  invertible neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Injective normalizing flows have  a fast inverse on the range and give access to the likelihood  of the generated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' As shown in Figure 2, fθ(z) =  gγ(hη(z)) with weights θ = (γ, η) comprises two subnetworks:  an injective part (with expansive layers) gγ that maps a low-  dimensional space Rd to high-dimensional data space RD  where d ≪ D, and a bijective mapping hη that keeps the  dimension in low dimensional space Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The training of the  injective normalizing flows consists of two phases, at first, the  range of the injective generator must be adjusted to lie on the  training data by optimizing over the weights of the injective  subnetwork gγ, when we ensure the training data are close to  the range of the generator, in the second phase the likelihood  of the pre-image of the training data should be maximized  over the weights of the bijective subnetwork hη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Further details  are explained in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' When the network is trained, we can  generate random samples similar to the training data  xgen = f(zgen)  (9)  where zgen ∼ N(µz, σ2  zI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Further information about the  universality of density and manifold approximation of injective  normalizing flows are studied in [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Due to having a low-dimensional latent space, injec-  tive flows provide a low-dimensional manifold in the high-  dimensional data space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' When they are trained, the manifold  would contain plausible samples which can be used as an  effective regularizer for solving ill-posed inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  The injective part provides a projection operator gγ(g†  γ(x))  which maps the data samples x to the intermediate space by  z′ = g†  γ(x) and projects them back to the data space by g†  γ(z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  The authors of [35] used the projection operator to project a  sample on the manifold to suppress the noise and artifacts,  as the learned manifold contains high-quality samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' In the  next section, we will present our method of solving inverse  scattering problems using injective normalizing flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' INJECTIVE FLOWS FOR INVERSE SCATTERING  Inverse scattering is a severely ill-posed inverse problem,  which means a small perturbation in the scattered fields leads  to an exponentially large error in the contrast [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' This makes  the inversion unstable even for a small amount of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' As  discussed in section II, inverse scattering is a nonlinear inverse  problem whose nonlinearity heavily relies on the maximum  contrast value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' For objects with large contrasts, the problem  becomes highly nonlinear, which makes the inversion extremely  difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' In such a scenario, the significance of having a strong  regularizer to restrict the search domain is crucially important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  We consider the contrast signal χ = x ∈ X and  the scattered fields Es = y ∈ Y, described by forward  operator (7), as random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' While our framework admits  other distributions beyond Gaussian for the additive noise δ in  (7), in this paper we assume that δ is a random vector with  Gaussian distribution δ ∼ N(0, σ2I) yielding  Y |X ∼ N(A(X), σ2I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  (10)  One effective approach for solving ill-posed inverse problems is  computing MAP estimate, where we look for the solution x that  has the highest posterior likelihood for a given measurement  y,  xMAP = arg maxx log(pX|Y (x|y)),  (11)  85  where pX|Y (x|y) is the posterior distribution for the given  measurement y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Using Bayes theorem yields,  xMAP  =  arg minx − log(pX|Y (x|y))  =  arg minx − log(pY |X(y|x)pX(x)  pY (y)  )  =  arg minx − log(pY |X(y|x)) − log(pX(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  (12)  The first term can be obtained from (10),  xMAP = arg minx  1  2∥y − A(x)∥2  2 − λ log(pX(x)),  (13)  where the first term is the data-consistency loss while  log(pX(x)) is the prior distribution of the contrast and plays  the role of the regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We also have λ which is a  hyperparameter to adjust the amount of regularization term  (although it comes from the noise standard deviation which  we assume is unknown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' In principle, the prior distribution  pX(x) is not available and must be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' For example,  traditionally pX(x) is approximated with a Gaussian distribu-  tion with zero mean leading to the Tikhonov regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  However, a Gaussian distribution is often too far from the true  prior distribution and leads to poor reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  In this paper, we study a new regularization family for  inverse scattering based on deep generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We consider  a training set of contrast signals {x(i)}N  i=1 is available, and we  train a deep generative model x = f(z) to produce high-quality  contrast samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' we expect the trained generator f to produce  high-quality contrast samples when it takes random samples  from the Gaussian distribution in the latent space z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  As discussed in section III, this property of deep generative  models makes them an effective regularizer for solving inverse  problems [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  We use injective normalizing flows as the generator of the  contrast signal since they are well-suited for solving ill-posed  inverse problems [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We perform an optimization in the latent  space to find the latent code that aligns with scattered fields y,  zMAP = arg max  z  1  2∥y −A(f(z))∥2  2 +λ log(pX(f(z))), (14)  Where an approximation to pX is also provided by the injective  flows and acts as the second regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The reconstructed  contrast signal can be obtained as xMAP = f(zMAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We  call this method latent space optimization (LSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' It is worth  mentioning that (14) has been previously proposed by [32],  [33] for solving compressed sensing inverse problems using  regular normalizing flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Unlike the supervised learning methods for inverse scatter-  ing [13], [15]–[17] which use a paired training set of contrast  and scattered fields {(x(i), y(i))}N  i=1, our framework benefits  from an unsupervised learning paradigm where scattered fields  are not used in the training of injective flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' This means, if the  distribution of the scattered fields changes (due to the change  in experimental configuration), the generative network is not  required to be retrained unlike the supervised methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' as soon  as the injective generator is trained over the contrast signals, we  can optimize (14) for each new scattered fields to reconstruct  Ground truth  BP (𝜖!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='2)  BP (𝜖!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' = 2)  BP (𝜖!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' = 4)  BA (𝜖!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='2)  BA (𝜖!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' = 2)  BA (𝜖!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' = 4)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' 3: Performance analysis of back-propagation (BP) and  Born approximation (BA) methods for different ϵr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' while BA  and BP reconstructions are visually meaningful for small ϵr,  their performance sharply drop for objects with large ϵr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Random ellipses  MNIST  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' 4: Illustration of the MOG initializer in the data space  f(µz) for random ellipses and MNIST datasets  the corresponding contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Apart from the advantages of an  unsupervised learning paradigm, the proposed method fully  exploits the underlying physics of the scattering problem by  optimizing over the complex-valued scattered fields in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Kothari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' [57] have shown incorporating wave physics  in the neural network architecture can significantly improve  the quality of reconstructions, especially for out-of-distribution  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  We solve the optimization problem (14) by Adam op-  timizer [58] and compute the gradients with respect to z  by automatic differentiation provided by TensorFlow [59] in  Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We also use an alternative method for Equation (14)  proposed by [35] which performs the optimization in the data  space,  xMAP = arg min  x  1  2∥y −A(g(g†(x)))∥2  2 −λ log(pX(x)) (15)  where g(g†(x)) is the projection operator explained in sec-  tion IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We call this method data space optimization (DSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  This method is shown in [35] to be effective for solving  linear inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' While in each of the LSO iterations  the reconstructed point x = f(z) is always on the learned  manifold, this is not the case for the DSO method, where the  reconstructed image might be off the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Moreover, the  DSO method requires more computations than LSO as we need  to take derivatives over the reverse direction of the injective  subnetwork g†(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  The choice of initial guess is crucially important in inverse  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='156  Ground truths  Projections on the manifold  Generated samples  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' 5: Performance evaluation of the trained injective normalizing flows for random ellipses dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' ground truth contrasts,  their projections on the learned manifold and some randomly generated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  scattering solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' While a poor initialization misleads the  optimization process, a good initial step helps the algorithm to  converge more accurately and faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The authors of [9] used  Born approximation as the initialization for the distorted Born  iterative method (DBIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' A back-propagation (BP) solution  was also used in [10], [60] as an initial guess of the contrast  source inversion (CSI) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Figure 3 shows the ground  truth, back-propagation (BP) and Born approximations (BA)  for an object with different maximum ϵr values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' While BP  and BA can present fuzzy reconstructions for objects with  small permittivity, their performance sharply drops for large ϵr  (especially numerically) which makes them a poor initialization  for strong scatterers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  In order to circumvent this issue, we use a data-driven  initialization suggested in [32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' mean of the Gaussian distri-  bution (MOG) in the latent space as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The  MOG initializer zinit = µz provides a fixed initialization with  respect to the measurements (scattered fields);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' thereby being  independent of the maximum contrast value and the problem  configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' This property helps (14) and (15) to converge  better even for a large ϵr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' In section VI, we will show that  MOG initialization significantly improves the quality of the  reconstructions compared to BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' To our best knowledge, this  is the first time that we propose a data-driven initializer for  inverse scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' COMPUTATIONAL EXPERIMENTS  In the following, we will evaluate the performance of DSO  and LSO for inverse scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We consider the MOG and BP  initializations for DSO while using just MOG initialization for  LSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We compare the performance of our proposed methods  with a traditional iterative method DBIM [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' While our method  is based on an unsupervised-learning paradigm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' scattered fields  are not used during training, we also compare its performance  with a successful supervised learning method, U-Net [36] which  has shown great success for inverse scattering [16], to show the  effectivity of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' U-Net takes the back-propagation  (BP) as input and returns the permittivity pattern of the object  in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  We use MNIST [61] with 60000 training samples in the  resolution N = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We also use a custom dataset of 60000  training samples with resolution N = 64 of overlapping ellipses  used in [14] to have a more challenging task for accurate  reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  We use Ni = 12 incident plane waves and Nr = 12  receivers, uniformly distributed on a circle with radius R =  20 cm around the object with maximum permittivity ϵr and  dimension D = 20 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The working frequency is 3 GHz and  we added 30 dB noise to the scattered fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  We used the injective normalizing flows with the same  architecture described in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We trained the injective subnet-  work gγ for 150 epochs to ensure the training samples (contrast  signals) are close to the range of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Figure 5 shows  some test samples and their projections on the learned manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Then we trained the bijective subnetwork hη for 150 epochs  to maximize the likelihood of the pre-image of the training  samples in the intermediate space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Figure 5 illustrates some  randomly generated samples which confirms that the model  can produce plausible samples to be used as an effective prior  for solving inverse scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  We optimize (14) and (15) by using Adam optimizer with a  learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='05 for 300 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We use λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='01 for BP  and λ = 0 for MOG initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' It is worth mentioning that  when we use MOG initializer, we start from high-likelihood  regions (mean of the Gaussian) which can be viewed as a  hidden regularizer, we thus use λ = 0 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Figure 4  shows the MOG initialization for MNIST and random ellipses  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Figures 6 and 7 show the performance of different methods  for inverse scattering for ϵr = 4 over five test samples from  random ellipses and MNIST datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' While the traditional  iterative method (DBIM) meets with complete failure in this  challenging task (ϵr = 4 and 30 dB noise), DSO and LSO  have strikingly more accurate reconstructions, which clearly  7  BP  DBIM  U-Net  DSO (BP)  DSO (MOG)  LSO  Ground truth  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' 6: Performance comparison of different methods over random ellipses dataset in resolution 64 × 64  BP  DBIM  U-Net  DSO (BP)  DSO (MOG)  LSO  Ground truth  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
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+page_content='08  TABLE I: Performance of different methods for solving inverse  scattering (ϵr = 4) averaged over 25 test samples  PSNR  SSIM  MNIST  Ellipses  MNIST  Ellipses  BP  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='75  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
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+page_content='01  DBIM [9]  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='77  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='67  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='01  U-Net [36]  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='26  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='82  DSO (BP)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='73  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='16  DSO (MOG)  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='50  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='44  LSO (MOG)  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='22  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='85  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' 8: The performance of different methods for different ϵr  over MNIST dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' SSIM is computed over the normalized  signals between −1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  shows the significance of a data-driven regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Moreover,  the MOG initialization, as we expected, exhibits remarkably  better reconstructions compared to BP for this high permittivity  (ϵr = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Furthermore, both figures show that LSO outperforms  DSO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' running optimization in the latent space results in better  reconstructions as discussed in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Finally, while LSO  does not use scattered fields in the training phase, it could  present the reconstructions with comparable quality (or even  better) to the highly successful supervised method U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Table I shows the numerical results in PSNR and SSIM  averaged over 25 test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  As discussed in section V, the maximum ϵr of the  object plays a significant role in the performance of inverse  scattering solvers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' objects with large ϵr are more difficult to  be reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' Figure 8 demonstrates the performance of  different methods per ϵr over the MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' This figure  discovers that LSO with MOG initialization is effective even  for objects with large, ϵr which clearly signifies the power of  a data-driven initialization and performing the optimization in  the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' LIMITATIONS AND CONCLUSIONS  We proposed a learning-based framework for inverse  scattering using an injective prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The proposed method fully  exploits the physical domain of the scattering problem while  benefiting from a data-driven initialization which makes a  powerful solver even for objects with a large contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' The  invertible generator admits performing optimization in both  latent and data space and uses a data-driven or back-projection  as the initializer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' We showed that performing optimization in  the latent space and using the mean of the Gaussian as the  initial guess significantly outperforms the traditional iterative  methods and even gives reconstructions comparable to the  successful supervised learning method, U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Limitations: The proposed framework has several limitations  and entails discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' It requires running an iterative method  in test-time, which is slow and cannot be used for real-time  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' To speed up the convergence, one may consider a  more accurate initial guess by exploiting the physical domain  in the data-driven initializer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' a combination of traditional back-  projection (like BP) and the data-driven initializers (like MOG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content='  Recently, Hussein et all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' [31] optimized the generator weights  with a small rate after finding the optimal latent code in  Equation (14) to further improve the reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
+page_content=' This  idea might be taken in our framework to go beyond the quality  of the generator, but we leave it for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE1T4oBgHgl3EQfUgPr/content/2301.03092v1.pdf'}
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+arXiv:2301.00263v1  [math.CA]  31 Dec 2022
+ON BOCHNER’S ALMOST-PERIODICITY CRITERION
+PHILIPPE CIEUTAT
+Abstract. We give an extension of Bochner’s criterion for the almost periodic func-
+tions. By using our main result, we extend two results of A. Haraux. The first is a
+generalization of Bochner’s criterion which is useful for periodic dynamical systems. The
+second is a characterization of periodic functions in term of Bochner’s criterion.
+2020 Mathematic Subject Classification: 35B10, 35B40, 42A75, 47H20.
+Keywords: Bochner almost periodicity, periodic function, almost periodic function,
+asymptotically almost periodic function, nonlinear semigroup, periodic dynamical sys-
+tem.
+1. Introduction
+The almost periodic functions in the sense of Bohr have been characterized by Bochner
+by means of a compactness criterion in the space of the bounded and continuous functions
+[2, 3]. The Bochner’s criterion plays an essential role in the theory and in applications.
+We give a new almost-periodicity criterion for functions with values in a given complete
+metric space which is useful to study the almost periodicity of solutions of dynamical
+systems governed by a family of operators with a positive parameter. This criterion is an
+extension of Bochner’s criterion. Then Haraux gave a generalization of Bochner’s criterion
+[9, Theorem 1], called a simple almost-periodicity criterion which is useful for periodic
+dynamical systems. From our result, we deduce an extension of this criterion. We also
+obtain an extension of an other result of Haraux which characterizes the periodic functions
+in terms of the Bochner’s criterion [8]. In the same spirit, we treat the asymptotically
+almost periodic case.
+We give a description of this article, the precise definitions will be given in Section 2.
+Throughout this section (X, d) is a complete metric space. An almost periodic function
+u : R → X in the sense of Bohr is characterized by the Bochner’s criterion which is the
+following: u is bounded and continuous, and from any real sequence of real numbers (τn)n,
+there exists a subsequence (τφ(n))n such that the sequence of functions (u(t + τφ(n)))n is
+uniformly convergent on R. In Section 3, we give two extensions of Bochner’s criterion.
+First u : R → X is an almost periodic if and if only if in the Bochner’s criterion, we impose
+that the terms of the sequence of real numbers (τn)n are all positive. Second u : R → X
+is an almost periodic if and if only if in the Bochner’s criterion, the convergence of the
+subsequence of functions (u(t + τφ(n)))n is uniform only on [0, +∞). These improvements
+are useful to study the almost periodicity of solutions of an evolution equation governed by
+a family of operators with a positive parameter, in particular for a C0-semigroup of linear
+operators or more generally, for an autonomous dynamical system (nonlinear semigroup).
+Universit´e Paris-Saclay, UVSQ, CNRS, Laboratoire de math´ematiques de Versailles, 78000, Versailles,
+France. E-mail address: philippe.cieutat@uvsq.fr
+1
+
+2
+P. Cieutat
+From our extension of Bochner’s criterion, we give new proofs which are direct and simpler
+on known results on the almost periodicity of solutions of autonomous dynamic systems.
+Haraux gave a generalization of Bochner’s criterion the called a simple almost-periodicity
+criterion [9, Theorem 1]. This criterion makes it possible to choose in the Bochner’s cri-
+terion, the sequence of real numbers (τn)n in a set of the type ωZ which is very useful
+for periodic dynamical systems. From our extension of Bochner’s criterion, in Section
+4, we deduce an improvement of this result. An asymptotically almost periodic function
+u : R+ → X is a perturbation of almost periodic. A such function is characterized by a
+property of the type of the Bochner’s criterion. In the same spirit, we extend this char-
+acterization of asymptotically almost periodic functions. Then we apply these results to
+study the almost periodicity of solutions of periodic dynamical systems.
+Bochner’s criterion can also be expressed in terms of the relative compactness of the
+set {u(· + τ); τ ∈ R} in a suitable set of continuous functions. A periodic function is a
+special case of almost periodic function. A direct consequence of [8, Proposition 2] given
+by Haraux characterizes a periodic function in terms of the Bochner’s criterion. This
+characterization is the following: u : R → X is continuous is periodic if and if only if the
+set {u(· + τ); τ ∈ R} is compact. In Section 5, By using our improvement of Bochner’s
+criterion, we give an extension of the Haraux’s characterization of periodic functions. We
+will also give a result on asymptotically periodic functions of the type of Haraux result
+described above. Then we apply these results to study the periodicity of solutions of
+autonomous dynamical systems.
+2. Notation
+Let us now give some notations, definitions and properties which will be used.
+Throughout this section (X, d) is a complete metric space.
+R, Z and N stand re-
+spectively for the real numbers, the integers and the natural integers.
+We denote by
+R+ := {t ∈ R; t ≥ 0}.
+Let E be a topological space.
+We denote by C(E, X) the
+space of all continuous functions from E into X.
+When J = R or J = R+, we de-
+note by BC(J, X) the space of all bounded and continuous functions from J into X
+equipped with the sup-distance, denoted by d∞(u, v) := sup
+t∈R
+d(u(t), v(t)) when J = R and
+d∞,+(u, v) := sup
+t≥0
+d(u(t), v(t))) when J = R+ for u, v ∈ BC(J, X). The metric spaces
+(BC(R, X), d∞) and (BC(R+, X), d∞,+)) are complete.
+We now give some definitions and properties on almost periodic, asymptotically almost
+periodic functions with values in a given complete metric space.
+A subset D of R (respectively of R+) is said to be relatively dense if there exists ℓ > 0
+such that D ∩ [α, α + ℓ] ̸= ∅ for all α ∈ R (respectively α ≥ 0). A continuous function
+u : R → X is said to be almost periodic (in the sense of Bohr) if for each ε > 0,
+the set of ε-almost periods: P(u, ε) =
+�
+τ ∈ R ; sup
+t∈R
+d(u(t + τ), u(t)) ≤ ε
+�
+is relatively
+dense in R. An almost periodic function u has its range u(R) relatively compact, that
+is its closure denoted by cl (u(R)) is a compact set of (X, d). We denote the space of
+all such functions by AP(R, X). It is a closed metric subspace of (BC(R, X), d∞). An
+almost periodic function u is uniformly recurrent, that is there exists a sequence of real
+
+On Bochner’s almost-periodicity criterion
+3
+numbers (τn)n such that
+lim
+n→+∞ sup
+t∈R
+d(u(t + τn), u(t)) = 0 and
+lim
+n→+∞ τn = +∞. To see
+that consider the Bohr’s definition of u ∈ AP(R, X), then the set of
+1
+n-almost periods
+satisfies P(u, 1
+n)∩[n, +∞) ̸= ∅, for each integer n > 0. A useful characterization of almost
+periodic functions was given by Bochner. The Bochner’s criterion which may be found in
+[12, Bochner’s theorem, p. 4] in the context of metric spaces. Before to cite this criterion,
+we need to introduce the translation mapping of a function of BC(R, X). For τ ∈ R and
+u ∈ BC(R, X), we define the translation mapping Tτu ∈ BC(R, X) by Tτu(t) = u(t + τ)
+for t ∈ R.
+Theorem 2.1 (Bochner’s criterion). For u ∈ BC(R, X), the following statements are
+equivalent.
+i) u ∈ AP(R, X).
+ii) The set {Tτu; τ ∈ R} is relatively compact in (BC(R, X), d∞).
+Haraux gave a generalization of Bochner’ criterion the called a simple almost-periodicity
+criterion [9, Theorem 1] which is useful for periodic dynamical systems.
+Theorem 2.2 (Haraux’s criterion). Let D be a relatively dense subset of R. The following
+statements are equivalent for u ∈ BC(R, X).
+i) u ∈ AP(R, X).
+ii) The set {Tτu; τ ∈ D} is relatively compact in (BC(R, X), d∞).
+Periodic functions, which are a special case of almost periodic functions, are also char-
+acterized in terms of Bochner’s criterion. This criterion is a direct consequence of a result
+of Haraux.
+Theorem 2.3. [8, Consequence of Proposition 2] The following statements are equivalent
+for u ∈ BC(R, X).
+i) u is periodic.
+ii) The set {Tτu; τ ∈ R} is a compact set of (BC(R, X), d∞).
+For some preliminary results on almost periodic functions with values in a given com-
+plete metric space, we refer to the book of Levitan-Zhikov [12] and in the special case of
+Banach spaces to the book of Amerio-Prouse [1].
+The notion of asymptotic almost periodicity was first introduced by Fr´echet [6] in 1941
+in the case where X = C. A continuous function u : R+ → X is said to be asymp-
+totically almost periodic if there exists v ∈ AP(R, X) such that lim
+t→∞ d(u(t), v(t)) = 0.
+An asymptotic almost periodic function u has its range u(R+) relatively compact. We
+denote the space of all such functions by AAP(R+, X). It is a closed metric subspace
+of (BC(R+, X), d∞,+). An asymptotic almost periodicity function u : R+ → X is char-
+acterized by u ∈ APP(R+, X) if and only if u ∈ C(R+, X) and for each ε > 0, there
+exists M ≥ 0 such that the
+�
+τ ≥ 0 ; sup
+t≥M
+d(u(t + τ), u(t)) ≤ ε
+�
+is relatively dense in R+
+[15, Theorems 1.3]. In the context of metric spaces, Ruess and Summers give a charac-
+terization of asymptotically almost periodic functions in the spirit of Bochner’s criterion.
+
+4
+P. Cieutat
+To prove this characterization, Ruess and Summers use results from the paper [16] by
+the same authors. For τ ≥ 0 and u ∈ BC(R+, X), we define the translation mapping
+T +
+τ u ∈ BC(R+, X) by T +
+τ u(t) = u(t + τ) for t ≥ 0.
+Theorem 2.4. [15, a part of Theorems 1.2 & 1.3] Let (X, d) be a complete metric space.
+For u ∈ BC(R+, X), the following statements are equivalent.
+i) u ∈ AAP(R+, X).
+ii) The set {T +
+τ u; τ ≥ 0} is relatively compact in (BC(R+, X), d∞,+).
+For some preliminary results on asymptotically almost periodic functions, we refer to
+the book of Yoshizawa [17] in the case where X is a finite dimensional space, to the book
+of Zaidman [18] where X is a Babach space and to Ruess and Summers [14, 15, 16] in the
+general case: X is a complete metric space.
+3. An improvement of Bochner’s criterion
+An almost periodic function is characterized by the Bochner’s criterion, recalled in Sec-
+tion 2. Our main result is an extension of Bochner’s criterion. Then we deduce new proofs
+which are direct and simpler on known results on the solutions of autonomous dynamic
+systems. Before to state our extension of Bochner’s criterion, we need to introduce the
+restriction operator R : BC(R, X) → BC(R+, X) defined by R(u)(t) := u(t) for t ≥ 0
+and u ∈ BC(R, X).
+Theorem 3.1. Let (X, d) be a complete metric space. For u ∈ BC(R, X) the following
+statements are equivalent.
+i) u ∈ AP(R, X).
+ii) The set {Tτu; τ ≥ 0} is relatively compact in (BC(R, X), d∞).
+iii) The set {R(Tτu); τ ∈ R} is relatively compact in (BC(R+, X), d∞,+).
+In our results, the compactness and the relative compactness of a set often intervene.
+To prove them, we will often use the following result whose proof is obvious. Recall that
+a set A of a metric space (E, d) is relatively compact if its closure denoted by cl (A) is a
+compact set of (E, d).
+Lemma 3.2. Let E be a set, (G1, d1) and (G2, d2) be two metric spaces. Let u : E → G1
+and v : E → G2 be two functions. Assume there exists M > 0 such that
+∀x1, x2 ∈ E,
+d1(u(x1), u(x2)) ≤ Md2(v(x1), v(x2)).
+Then the following statements hold.
+i) If the metric space (G1, d1) is complete and v(E) is relatively compact in (G2, d2),
+then u(E) is relatively compact in (G1, d1).
+ii) If v(E) is a compact set of (G2, d2), then u(E) is a compact set of (G1, d1).
+Proof of Theorem 3.1. i) =⇒ iii). It is obvious by using the Bochner’s criterion and the
+continuity of the restriction operator R.
+iii) =⇒ ii). The set u(R) = {R(Tτu)(0); τ ∈ R} is relatively compact in X as the
+range of {R(Tτu); τ ∈ R} by the continuous evaluation map at 0 from BC(R+, X) into
+X. By assumption, H := cl ({R(Ttu) ; t ∈ R}) is a compact set of (BC(R+, X), d∞,+).
+
+On Bochner’s almost-periodicity criterion
+5
+For all τ ≥ 0, we define φτ : H → X by φτ(h) = h(τ). The functions φτ are 1-Lipschitz
+continuous and for each t ∈ R, the set {φτ(R(Ttu)) = u(τ + t) ; τ ≥ 0} is included in the
+relatively compact set u(R). By density of {R(Ttu) ; t ∈ R} in H and the continuity of
+φτ, it follows that {φτ(h) ; τ ≥ 0} is relatively compact in X for each h ∈ H. According
+to Arzel`a-Ascoli’s theorem [11, Theorem 3.1, p. 57], the set {φτ ; τ ≥ 0} is relatively
+compact in C(H, X) equipped with the sup-norm denoted by dC.
+From the density
+of {R(Ttu) ; t ∈ R} in H and the continuity of φτ, we deduce that for τ1 and τ2 ≥ 0,
+sup
+h∈H
+d(φτ1(h), φτ2(h)) = sup
+t∈R
+d (φτ1(R(Ttu)), φτ2(R(Ttu))) = sup
+t∈R
+d (u(τ1 + t), u(τ2 + t)) =
+sup
+t∈R
+d (Tτ1u(t), Tτ2u(t)), then dC(φτ1, φτ2) = d∞ (Tτ1u, Tτ2u). From Lemma 3.2, it follows
+that {Tτu; τ ≥ 0} is relatively compact in the complete metric space (BC(R, X), d∞)
+since {φτ ; τ ≥ 0} is also one in (C(H, X), dC).
+ii) =⇒ i). For τ1, τ2 ≥ 0, d∞(Tτ1u, Tτ2u) := sup
+t∈R
+d(u(τ1 + t), u(τ2 + t)). Replacing t
+by t − τ1 − τ2 in the upper bound, we get d∞(Tτ1u, Tτ2u) = d∞(T−τ1u, T−τ2u). Then the
+set {Tτu; τ ≤ 0} = {T−τu; τ ≥ 0} is relatively compact in BC(R, X) since {Tτu; τ ≥ 0}
+is also one. Therefore the set {Tτu; τ ∈ R} is relatively compact in BC(R, X) as the
+union of two relatively compact sets in BC(R, X).
+According to Bochner’s criterion,
+u ∈ AP(R, X).
+□
+The connection between the almost periodicity of a solution of a dynamical system and
+its stability is well known (see the monograph by Nemytskii & Stepanov [13, Ch. 5]. This
+weakened form of Bochner’s criterion: Theorem 3.1 makes it possible to obtain direct and
+simpler proofs on these questions. Let us start by recalling some definitions on dynamical
+systems.
+A dynamical system or nonlinear semigroup on a complete metric space (X, d) is a one
+parameter family (S(t))t≥0 of maps from X into itself such that i) S(t) ∈ C(X, X) for all
+t ≥ 0, ii) S(0)x = x for all x ∈ X, iii) S(t + s) = S(t) ◦ S(s) for all s, t ≥ 0 and iv) the
+mapping S(·)x ∈ C([0, +∞), X) for all x ∈ X.
+For each x ∈ X, the positive trajectory of x is the map S(·)x : R+ → X. A function
+u : R → X is called a complete trajectory if we have u(t + τ) = S(τ)u(t), for all t ∈ R
+and τ ≥ 0.
+We will need a notion of Lagrange-type stability to ensure that a solution with a
+relatively compact range is almost periodic. Recall that (S(t))t≥0 is equicontinuous on a
+compact set K of X, if forall ε > 0, there exists δ > 0, such that
+∀x1, x2 ∈ K, d(x1, x2) ≤ δ =⇒ sup
+t≥0
+d(x1, x2) ≤ ε.
+Using Theorem 3.1, we give a new proof which is direct and simpler of the following
+result which can be found in [10, Theorem 4.3.2, p. 51] or partly in [12, Markov’s theorem,
+p. 10].
+Corollary 3.3. Let (S(t))t≥0 be a dynamical system on a complete metric space (X, d)
+and u be a complete trajectory such that u(R) is relatively compact. Then u is almost
+periodic if and only if (S(t))t≥0 is equicontinuous on cl (u(R)) the closure of u(R).
+Proof. Let us denote the compact set K := cl (u(R)). It follows by density of u(R) in
+K and the continuity of S(t), that {S(t)x; t ≥ 0} ⊂ K for each x ∈ K.
+According
+
+6
+P. Cieutat
+to Arzel`a-Ascoli’s theorem, (S(t))t≥0 is equicontinuous on K if and only if (S(t))t≥0 is
+relatively compact in C(K, X). From Theorem 3.1, we have u ∈ AP(R, X) if and only if
+{Tτu; τ ≥ 0} is relatively compact in BC(R, X). Then it remains to prove that (S(t))t≥0
+is relatively compact in C(K, X) equipped with the sup-norm if and only if {Tτu; τ ≥ 0}
+is relatively compact in (BC(R, X), d∞). This results from the following equalities, for τ1
+and τ2 ≥ 0, sup
+t∈R
+d (Tτ1u(t), Tτ2u(t)) = sup
+t∈R
+d (S(τ1)u(t), S(τ2)u(t)) = sup
+x∈K
+d (S(τ1)x, S(τ2)x)
+and Lemma 3.2.
+□
+Remark 3.4. a) The condition of equicontinuity required by Corollary 3.3 is satisfied by
+a bounded dynamical system : d (S(t)x1, S(t)x2) ≤ Md (x1, x2) for some M ≥ 1 and
+in particular for a C0 semigroup of contractions. In this case, the almost periodicity of
+a complete trajectory u having a relatively compact range results from Corollary 3.3.
+We can also obtain this result with the implication iii) =⇒ i) of Theorem 3.1 and the
+inequality sup
+t≥0
+d(R(Tτ1u)(t), R(Tτ2u)(t)) = sup
+t≥0
+d (S(t)u(τ1), S(t)u(τ2) ≤ Md(u(τ1), u(τ2))
+for τ1, τ2 ∈ R.
+b) For a bounded C0-semigroup (S(t))t≥0, the main result of Zaidman [19] asserts
+that a positive trajectory u with relatively compact range satisfies a condition called the
+generalized normality property in Bochner’s sense, without concluding that u is almost
+periodic. This condition is nothing but hypothesis iii) of Theorem 3.1, so u is almost
+periodic.
+Using Theorem 2.4, we give a new proof which is direct and simpler of the following
+result which can be found in [15, Theorem 2.2, p. 149].
+Corollary 3.5. Let (S(t))t≥0 be a dynamical system on a complete metric space (X, d) and
+u be a positive trajectory such that u(R+) is relatively compact. Then u is asymptotically
+almost periodic if and only if (S(t))t≥0 is equicontinuous on cl (u(R+)).
+Proof. The proof is analogous to that of Corollary 3.3, using Corollary 2.4 instead of
+Theorem 3.1 and by replacing R by R+ and AP(R, X) by AAP(R+, X).
+□
+4. An improvement of Haraux’s criterion
+Haraux gave a generalization of Bochner’scriterion [9, Theorem 1], the called a simple
+almost-periodicity criterion which is useful for periodic dynamical systems.
+From our
+main result, Theorem 3.1, we deduce an extension of the Haraux’s criterion, recalled in
+Section 2. In the same spirit, we extend the well-known characterization of asymptotically
+almost periodic functions. To end this section, we give an exemple of application on a
+periodic dynamical system.
+We give an extension of the Haraux’s criterion (see Theorem 2.2). Recall that we denote
+by R the restriction operator R : BC(R, X) → BC(R+, X) defined by R(u)(t) := u(t) for
+t ≥ 0 and u ∈ BC(R, X).
+Corollary 4.1. Let (X, d) be a complete metric space. For u ∈ BC(R, X) the following
+statements are equivalent.
+i) u ∈ AP(R, X).
+ii) The set {Tτu; τ ∈ D} is relatively compact in (BC(R, X), d∞) where D be a relatively
+dense subset of R+.
+
+On Bochner’s almost-periodicity criterion
+7
+iii) The set {R(Tτu); τ ∈ D} is relatively compact in (BC(R+, X), d∞,+) where D be a
+relatively dense subset of R.
+Remark 4.2. Our main result, Theorem 3.1 is obviously a particular case of Corollary 4.1.
+But to present our results, it was easier to start with Theorem 3.1. To prove Corollary
+4.1, we use Haraux’s criterion and Theorem 3.1.
+Proof of Corollary 4.1. i) =⇒ iii). It is a consequence of Theorem 3.1.
+iii) =⇒ ii). To establish this implication, using Theorem 3.1, it suffices to show that
+assertion iii) implies that {R(Tτu); τ ∈ R} is relatively compact in BC(R+, X). The proof
+of this last implication is a slight adaptation of those of those of the Haraux’s criterion
+given in [9, Theorem 1]. A similar proof will be detailed in the following result as there
+will be technical issues. To demonstrate that {R(Tτu); τ ∈ R} is relatively compact, it
+suffices in the proof of ii) =⇒ i) of Corollary 4.3 to take ℓ = 0 and replace {T +
+τ u; τ ∈ D}
+by {R(Tτu); τ ∈ D}.
+ii) =⇒ i). For τ1, τ2 ≥ 0, d∞(Tτ1u, Tτ2u) = sup
+t∈R
+d(u(τ1 + t), u(τ2 + t)). Replacing t by
+t − τ1 − τ2 in the upper bound, we get d∞(Tτ1u, Tτ2u) = d∞(T−τ1u, T−τ2u). Then the set
+{Tτu; τ ∈ −D} = {T−τu; τ ∈ D} is relatively compact in BC(R, X) since {Tτu; τ ∈ D}
+is also one. Therefore the set {Tτu; τ ∈ D ∪ (−D)} is relatively compact in BC(R, X).
+Moreover D ∪(−D) is a relatively dense subset of R. According to Haraux’s criterion, we
+have u ∈ AP(R, X).
+□
+We extend Theorem 2.4, the well-known characterization of asymptotically almost pe-
+riodic functions. For τ ∈ R+ and u ∈ BC(R+, X), we define the translation mapping
+T +
+τ u ∈ BC(R+, X) by T +
+τ u(t) = u(t + τ) for t ≥ 0.
+Corollary 4.3. Let (X, d) be a complete metric space and let D be a relatively dense
+subset of R+. For u ∈ BC(R+, X) the following statements are equivalent.
+i) u ∈ AAP(R+, X).
+ii) The set {T +
+τ u; τ ∈ D} is relatively compact in (BC(R+, X), d∞,+).
+Remark 4.4. To establish implication ii) =⇒ i), by using Theorem 2.4, it suffices to prove
+that assertion ii) implies that {T +
+τ u; τ ≥ 0} is relatively compact in (BC(R+, X), d∞,+).
+The proof of this last implication is an adaptation of those of the Haraux’s criterion. But
+contrary to the proof of implication iii) =⇒ ii) in Corollary 4.1, there are technical issues.
+These technical difficulties come from the fact that when D is a relatively dense subset
+in R+, the sets D and [t − ℓ, t] can be disjoint for some 0 ≤ t ≤ ℓ. For this reason we give
+the complete proof of this implication.
+Proof of Corollary 4.3. i) =⇒ ii). It is a consequence of Theorem 2.4.
+ii) =⇒ i). We will prove that assumption ii) implies {T +
+τ u; τ ≥ 0} is relatively compact
+in BC(R+, X), then we conclude by using Theorem 2.4. The subset D being relatively
+dense in R+, there exists ℓ > 0 such that D ∩ [α, α + ℓ] ̸= ∅ for all α ≥ 0.
+• We prove that u is uniformly continuous on [ℓ, +∞). Let us fix ε > 0. By assump-
+tion the set {T +
+τ u; τ ∈ D} is in particular relatively compact in C([0, 2ℓ], X), then it is
+uniformly equicontinuous on [0, 2ℓ], that is there exists δ > 0 such that
+s1, s2 ∈ [0, 2l],
+|s1 − s2| ≤ δ =⇒ sup
+τ∈D
+d(u(s1 + τ), u(s2 + τ)) ≤ ε.
+(4.1)
+
+8
+P. Cieutat
+Let t1, t2 be two real numbers such that t1, t2 ≥ ℓ and |t1 − t2| ≤ δ. We can assume
+without loss of generality that ℓ ≤ t1 ≤ t2 ≤ t1 + ℓ. We have D ∩ [t1 − ℓ, t1] ̸= ∅ since
+t1 − ℓ ≥ 0, then there exists τ ∈ D such that 0 ≤ t1 − τ ≤ t2 − τ ≤ 2l. Taking account
+(4.1), we deduce that d(u(t1), u(t2)) = d(u((t1 − τ) + τ), u((t2 − τ) + τ)) ≤ ε. Hence u is
+uniformly continuous on [ℓ, +∞).
+• We prove that {T +
+τ u; τ ≥ ℓ} is relatively compact in BC(R+, X) . Let (tn)n be a
+sequence of real numbers such that tn ≥ ℓ. We have D ∩ [tn − ℓ, tn] ̸= ∅ for each n ∈ N,
+since tn − ℓ ≥ 0, then there exist τn ∈ D and σn ∈ [0, l] such that tn = τn + σn. By
+compactness of the sequences (σn)n in [0, ℓ] and (T +
+τnu)n in BC(R+, X), it follows that
+lim
+n→+∞ σn = σ and
+lim
+n→+∞ sup
+t≥0
+d(u(t + τn), v(t)) = 0 (up to a subsequence).
+From the
+following inequality
+sup
+t≥0
+d(u(tn + t), v(σ + t)) ≤ sup
+t≥0
+d(u(τn + σn + t), u(τn + σ + t) + sup
+t≥0
+d(u(τn + t), v(t))
+and the uniform continuity of u, we deduce that lim
+n→+∞ sup
+t≥0
+d(u(tn +t), v(σ +t)) = 0. Then
+{T +
+τ u; τ ≥ ℓ} is relatively compact in BC(R+, X).
+• We prove that u ∈ AAP(R+, X). The function u is uniformly continuous on R+,
+since u is continuous on [0, ℓ] and uniformly continuous on [ℓ, +∞). Then the map ˆu :
+R+ → BC(R+, X) defined by ˆu(τ) = T +
+τ u for τ ≥ 0 is continuous, consequently the
+set {T +
+τ u; 0 ≤ τ ≤ ℓ} is relatively compact in BC(R+, X). The set {T +
+τ u; τ ≥ 0} is
+relatively compact in BC(R+, X) as the union of two relatively compact sets. According
+to Theorem 2.4, u ∈ AAP(R, X).
+□
+Using corollaries 4.1 and 4.3, we give a proof which is direct and simpler of the following
+result which can be found in [7, 9, 10]. Before we recall some definitions on process.
+A process on a complete metric space (X, d) according to Dafermos [4] is a two pa-
+rameter family U(t, τ) of maps from X into itself defined for (t, τ) ∈ R × R+ and such
+that i) U(t, 0)x = x for all (t, x) ∈ R × X, ii) U(t, σ + τ) = U(t + σ, τ) ◦ U(t, σ) for all
+(t, σ, τ) ∈ R×R+×R+ and iii) the mapping U(t, ·)x ∈ C([0, +∞), X) for all (t, x) ∈ R×X.
+For each x ∈ X, the positive trajectory starting of x is the map U(0, ·)x : R+ → X. A
+function u : R → X is called a complete trajectory if we have u(t + τ) = U(t, τ)u(t) for
+all (t, τ) ∈ R × R+.
+A process U is said ω-periodic (ω > 0) if U(t + ω, τ) = U(t, τ) for all (t, τ) ∈ R × R+.
+A process U is said bounded if we have for some M ≥ 1 for all (τ, x1, x2) ∈ R+ ×X ×X
+d (U(0, τ)x1, U(0, τ)x2) ≤ Md (x1, x2).
+Corollary 4.5. [7, 9], [10, Th´eor`eme 6.4.6, p. 84] Let U be a ω-periodic process on a
+complete metric space (X, d). If U is bounded, then the following statements hold.
+i) If u is a complete trajectory of U such that u(−ωN) is relatively compact, then u is
+almost periodic.
+ii) If u is a positive trajectory of U such that u(ωN) is relatively compact, then u is
+asymptotically almost periodic.
+Proof. i) The process U is ω-periodic, then we have u(nω) = U(−mω, (n+m)ω)u(−mω) =
+U(0, (n + m)ω)u(−mω) for all n, m ∈ N. From the boundedness assumption on U, we
+
+On Bochner’s almost-periodicity criterion
+9
+deduce that d (u(nω), u(mω)) ≤ Md (u(−mω), u(−nω)), then u(ωN) is relatively compact
+since u(−ωN) is also one, therefore u(ωZ) is relatively compact. From assumptions on
+the process U, it follows that for all n, m ∈ Z,
+sup
+τ≥0
+d (u(τ + nω), u(τ + mω)) ≤ Md (u(nω), u(mω)) .
+(4.2)
+From Lemma 3.2, {R(Tnωu); n ∈ Z} is relatively compact in (BC(R+, X), d∞,+) since
+u(ωZ) is also one in (X, d). We conclude with Corollary 4.1 by setting D = ωZ.
+ii) For all n, m ∈ N, (4.2) holds on the positive trajectory u, then from Lemma 3.2,
+{T +
+nωu; n ∈ N} is relatively compact in (BC(R+, X), d∞,+) since u(ωN) is also one in
+(X, d). We conclude with Corollary 4.3 by setting D = ωN.
+□
+5. Bochner’s criterion in the periodic case
+Periodic functions are a special case of almost periodic functions. Haraux gave a charac-
+terization of periodic functions in terms of Bochner’s criterion which is recalled in Section
+2.
+This criterion is a direct consequence of [8, Proposition 2].
+Haraux established a
+general result [8, Th´eor`eme 1] implying as a special case a characterization of periodic
+functions and the fact that any compact trajectory of a one-parameter continuous group
+is automatically periodic.
+In this section, we give an extension of this characterization of periodic functions in the
+spirit of the main result of this article. We also treat the asymptotically periodic case.
+Then we apply these results to study the periodicity of solutions of dynamical systems.
+Recall that we denote by R the restriction operator R : BC(R, X) → BC(R+, X)
+defined by R(u)(t) := u(t) for t ≥ 0 and u ∈ BC(R, X).
+Corollary 5.1. Let (X, d) be a complete metric space. For u ∈ BC(R, X) the following
+statements are equivalent.
+i) The function u is ω-periodic (ω > 0).
+ii) The set {Tτu; τ ≥ 0} is a compact set of (BC(R, X), d∞).
+iii) The set {R(Tτu); τ ∈ R} is a compact set of (BC(R+, X), d∞,+).
+Proof. i) =⇒ ii). From assumption, it follows that the function τ → Tτu from R into
+BC(R, X) is continuous and ω-periodic. Then the {Tτu; τ ≥ 0} = {Tτu; 0 ≤ τ ≤ ω} is a
+compact set of (BC(R, X), d∞) as the range of a compact set by a continuous map.
+ii) =⇒ i).
+For τ1, τ2 ∈ R, d∞(Tτ1u, Tτ2u) := sup
+t∈R
+d(u(τ1 + t), u(τ2 + t)), we get
+d∞(Tτ1u, Tτ2u) = d∞(T−τ1u, T−τ2u).
+Then the set {Tτu; τ ≤ 0} = {T−τu; τ ≥ 0} is
+compact in BC(R, X) since {Tτu; τ ≥ 0} is also one. Therefore the set {Tτu; τ ∈ R} is a
+compact set of BC(R, X) as the union of two compact sets in BC(R, X). According to
+Theorem 2.3, u is periodic.
+i) =⇒ iii). It is obvious by using Theorem 2.3 and the continuity of the restriction
+operator R.
+iii) =⇒ i). By using Theorem 2.3, we have to prove that K := {Tτu; τ ∈ R} is a
+compact set of (BC(R, X), d∞). As consequence of Theorem 3.1 and Bohner’s criterion,
+the set K is relatively compact in (BC(R, X), d∞) and the function u is almost periodic.
+
+10
+P. Cieutat
+It remains to prove that K is closed in (BC(R, X), d∞). Let (τn)n be a sequence of real
+numbers such that
+lim
+n→+∞ d∞(Tτnu, v) = 0. Let us prove that v = Tτu for some τ ∈ R. By
+continuity of the operator R, we have lim
+n→+∞ d∞,+(R(Tτnu), R(v)) = 0. By assumption, the
+set {R(Tτu); τ ∈ R} is in particular closed in (BC(R+, X), d∞,+), then R(v) = R(Tτu)
+for some τ ∈ R, that is
+∀t ≥ 0,
+v(t) = Tτu(t).
+(5.1)
+We have to prove that (5.1) holds on the whole real line. The function Tτu is almost
+periodic as translation of an almost periodic function and v is also one as uniform limit
+on R of almost periodic functions. Let us denote by φ : R → R the function defined by
+φ(t) := d(Tτu(t), v(t)). The function φ is almost periodic [12, Property 4, p. 3 & 7, p.6].
+An almost periodic function is uniformly recurrent, then there exists a sequence of real
+numbers such that
+lim
+n→+∞ τn = +∞ and
+lim
+n→+∞ φ(t+ τn) = φ(t) for all t ∈ R. From (5.1), it
+follows φ(t) = 0 for all t ≥ 0, so we deduce that φ(t) =
+lim
+n→+∞ φ(t + τn) = 0 for all t ∈ R.
+Then v(t) = Tτu(t) for all t ∈ R. This ends the proof.
+□
+According Theorem 2.4, if the set {T +
+τ u; τ ≥ 0} is relatively compact in BC(R+, X),
+then the function u is asymptotically almost periodic. We now give an answer to the
+question what can be said about the function u when {T +
+τ u; τ ≥ 0} is a compact set of
+BC(R+, X). For u ∈ BC(R+, X), we say that u is ω-periodic (ω > 0) on [t0, +∞) for
+some t0 ≥ 0 if u(t + ω) = u(t) for all t ≥ t0.
+Corollary 5.2. Let (X, d) be a complete metric space. For u ∈ BC(R+, X) the following
+statements are equivalent.
+i) There exists t0 ≥ 0 such that u is ω-periodic on [t0, +∞).
+ii) The set {T +
+τ u; τ ≥ 0} is a compact set of (BC(R+, X), d∞,+).
+Remark 5.3. Let u be a function which satisfies condition i) of Corollary 5.2.
+i) Let us denote by v ∈ C(R, X) the ω-periodic function satisfying u(t) = v(t) for
+t ≥ t0. A such function v exists and is unique, v is defined by v(t) = u(t − [ t−t0
+ω ]ω) where
+[ t−t0
+ω ] denotes the integer part of t−t0
+ω .
+ii) The function u is a special case of asymptotic almost periodic function where the
+almost periodic function v is periodic and d(u(t), v(t)) = 0 for t ≥ t0.
+Proof of Corollary 5.2. i) =⇒ ii). Let us denote by v the function defined in Remark 5.3.
+By Corollary 5.1 and the periodicity of v, we have {R(Tτv); τ ≥ t0} = {R(Tτv); τ ∈ R} is
+a compact set of (BC(R+, X), d∞,+).
+First, we have T +
+τ u = R(Tτv) for τ ≥ t0, then {T +
+τ u; τ ≥ t0} is a compact set.
+Second, the function u is uniformly continuous on R+, then the function from R+ to
+BC(R+, X) defined by τ → T +
+τ u is continuous.
+Then the set {T +
+τ u; 0 ≤ τ ≤ t0} is
+compact.
+Therefore the set {T +
+τ u; τ ≥ 0} is a compact set of BC(R+, X) as the union of two
+compact set.
+ii) =⇒ i). As consequence of Theorem 2.4, the function u is asymptotically almost
+periodic, that is lim
+t→∞ d(u(t), v(t)) = 0 for some v ∈ AP(R, X).
+An almost periodic
+
+On Bochner’s almost-periodicity criterion
+11
+function is uniformly recurrent, then there exists a sequence of real numbers (tn)n such
+that
+lim
+n→+∞ tn = +∞ and
+lim
+n→+∞ v(t + tn) = v(t) for all t ∈ R. We deduce that
+∀t ∈ R,
+lim
+n→+∞ u(t + tn) = v(t).
+(5.2)
+First we prove that v is periodic. For t ∈ R, τ1, τ2 ≥ 0, we have for n enough large
+d(u(t + tn + τ1), u(t + tn + τ2)) ≤ sup
+s≥0
+d(u(s + τ1), u(s + τ2)). From (5.2), it follows that
+sup
+t∈R
+d(v(t+τ1), v(t+τ2)) ≤ sup
+s≥0
+d(u(s+τ1), u(s+τ2)) for each τ1 and τ2 ≥ 0. According to
+Lemma 3.2, {Tτv ; τ ≥ 0} is a compact set of (BC(R, X), d∞) since {T +
+τ u ; τ ≥ 0} is also
+one in (BC(R+, X), d∞,+). As consequence of Corollary 5.1, the function v is periodic.
+Second we prove that: ∃t0 ≥ 0 such that ∀t ≥ 0, v(t) = u(t + t0). By compactness of
+{T +
+τ u; τ ≥ 0}, there exists a subsequence (T +
+tφ(n)u)n such that lim
+n→+∞ d∞,+(T +
+tφ(n)u, T +
+t0 u) = 0
+for some t0 ≥ 0. From (5.2) we deduce that R(v) = T +
+t0 u, that is v(t) = u(t + t0) for all
+t ≥ 0.
+Then u(t) = v(t − t0) for each t ≥ t0 where the function v(· − t0) is periodic on R.
+□
+Now we give an example of application on dynamical systems of Corollary of 5.1 and
+Corollary of 5.2. For the definition of a dynamical system, see above Corollary 3.3 in
+Section 3.
+Corollary 5.4. Let (S(t))t≥0 be a dynamical system on a complete metric space (X, d).
+i) If u is a positive trajectory, then u is periodic on [t0, +∞) for some t0 ≥ 0 if and
+only if u(R+) is a compact set and (S(t))t≥0 is equicontinuous on u(R+).
+ii) If u is a complete trajectory, then u is periodic if and only if u(R) is a compact set
+and (S(t))t≥0 is equicontinuous on u(R).
+iii) There exists a complete trajectory which is periodic if and only if there exists a
+positive trajectory u such that u(R+) is a compact set and (S(t))t≥0 is equicontinuous on
+u(R+).
+Remark 5.5. Thus under the assumption of equicontinuity, a complete trajectory of a
+dynamical system with a compact range is necessarily periodic, although there are almost
+periodic functions with a compact range, which are not periodic. An example of such
+function is given by Haraux in [8].
+Proof of Corollary 5.4. i) Remark that if u is a positive trajectory which is periodic on
+[t0, +∞) for some t0 ≥ 0, then first u(R+) is compact and second u ∈ AAP(R+, X)
+(see Remark 5.3). As consequence of Corollary 3.5, the set (S(t))t≥0 is equicontinuous on
+u(R+). Reciprocally assume the positive trajectory u is such that (S(t))t≥0 is equicontinu-
+ous on the compact set u(R+). It remains to prove that the positive trajectory u is periodic
+on [t0, +∞) for some t0 ≥ 0. For each x ∈ u(R+), the map S(·)x is continuous and satisfies
+S(t)x ∈ u(R+) for each t ≥ 0. Then the map S(·)x is bounded and continuous , so the map
+Φ : u(R+) → BC(R+, X) with Φ(x) = S(·)x is well-defined. The continuity of Φ results
+of the equicontinuity of (S(t))t≥0 on u(R+). Then the set Φ(u(R+)) = {Φ(u(τ)) ; τ ≥ 0}
+is a compact of BC(R+, X). Moreover Φ(u(τ))(t) = S(t)u(τ) = u(t + τ) for t and τ ≥ 0,
+
+12
+P. Cieutat
+then Φ(u(τ)) = T +
+τ u, so {T +
+τ u ; τ ≥ 0} is a compact set of BC(R+, X). According to
+Corollary 5.2, the function u is periodic on [t0, +∞) for some t0 ≥ 0.
+ii) The proof of ii) is similar to that of i) by using Corollary 3.3 instead of 3.5, Corollary
+5.1 instead of Corollary 5.2 and by replacing the map Φ : u(R+) → BC(R+, X) with
+Φ(x) = S(·)x by the map Φ : u(R) → BC(R+, X). This permits to prove that the set
+{Φ(u(τ)) = R(Tτu); τ ∈ R} is a compact set of (BC(R+, X), d∞,+).
+iii) If v is a complete trajectory which is periodic, then v(R) is compact and according
+to ii), (S(t))t≥0 is equicontinuous on v(R). So the restriction u of v on R+ is a posi-
+tive trajectory such that u(R+) = v(R+) = v(R) since v is periodic, then (S(t))t≥0 is
+equicontinuous on the compact set u(R+). Reciprocally, assume that u is a positive tra-
+jectory such that (S(t))t≥0 is equicontinuous on the compact set u(R+). According to
+i), u is ω-periodic on [t0, +∞) for some t0 ≥ 0. Let us denote by v the function defined
+in Remark 5.3. For t ≥ s, there exists n0 ∈ N such that s + n0ω ≥ t0. The function
+v is ω-periodic and u is a positive trajectory satisfying u(τ) = v(τ) for τ ≥ t0, then
+v(t) = v(t + n0ω) = u(t + n0ω) = T(t − s)u(s + n0ω) = T(t − s)v(s + n0ω) = T(t − s)v(s)
+for t ∈ R and n enough large. Then v is a periodic complete trajectory.
+□
+Remark 5.6. Under i) of Corollary 5.4, one can have t0 > 0, that is the positive trajectory
+u is not the restriction of a periodic complete trajectory.
+For example, consider the
+bounded dynamical system (S(t))t≥0 on L1(0, 1) defined by
+(S(t)x)(s) =
+
+
+
+x(s − t)
+if
+t < s < 1
+0
+if
+0 < s < t
+for x ∈ L1(0, 1) and 0 < t < 1. For t ≥ 1, we set S(t) = 0. Then all positive trajectories
+have a compact range and the alone complete trajectory is the null function. Thus all
+positive trajectories are not the restriction of a periodic complete trajectory except the
+null function.
+Not all dynamical systems have this pathology, some systems are such that if two
+positive trajectories have the same value at the same time, then they are equal. If we
+consider such systems, we get more refined results from Corollary 5.4.
+A dynamical system (S(t))t≥0 has the backward uniqueness property if any two positive
+trajectories having the same value at t = t0 ≥ 0 coincide for any other t ≥ 0. This
+property is equivalent to S(t) ∈ C(X, X) is injective for each t ≥ 0. We say that a
+positive trajectory u is extendable to a periodic complete trajectory, if there exists a periodic
+complete trajectory such that its restriction on R+ is u.
+Corollary 5.7. Let (S(t))t≥0 be a dynamical system on a complete metric space (X, d).
+Assume that (S(t))t≥0 has the backward uniqueness property.
+i) If u is a positive trajectory, then u is periodic on R+ if and only if u(R+) is a
+compact set and (S(t))t≥0 is equicontinuous on u(R+). In this case the positive trajectory
+u is extendable to a periodic complete trajectory v.
+ii) If v is a complete trajectory, then v is periodic if and only if v(R+) is a compact set
+and (S(t))t≥0 is equicontinuous on v(R+).
+
+On Bochner’s almost-periodicity criterion
+13
+Proof. i) The direct implication results of i) of Corollary 5.4. For the reciprocal implica-
+tion we use i) of Corollary 5.4. Then the positive trajectory u is ω-periodic on [t0, +∞) for
+some t0 ≥ 0. Let us denote by v ∈ C(R, X) the ω-periodic function satisfying u(t) = v(t)
+for t ≥ t0 (see Remark 5.3). The restriction of v on R+ and u are two positive trajectories
+having the same value at t = t0 (t0 ≥ 0). From the backward uniqueness property, we
+have u(t) = v(t) for t ≥ 0, then u is periodic on R+. By build, v is periodic and as in the
+proof of iii) of Corollary 5.4, we deduce that v is a complete trajectory.
+ii) The direct implication results of ii) Corollary 5.4, since v(R+) = v(R). For the
+reciprocal implication, we consider v a complete trajectory such that v(R+) is a compact
+set and (S(t))t≥0 is equicontinuous on v(R+). Then the restriction u of the complete
+trajectory v on R+ is a positive trajectory such that u(R+) is compact and (S(t))t≥0
+is equicontinuous on u(R+). According to i), u is ω-periodic on R+. Let us denote by
+w ∈ C(R, X) the ω-periodic function satisfying u(t) = w(t) for t ≥ 0. As in proof of iii)
+Corollary 5.4, we deduce that w is a complete trajectory. Fix T > 0. The two maps ˜v
+and ˜w : R+ → X defined by ˜v = v(·, −T) and ˜w = w(·, −T) are two positive trajectories
+having the same value at t = T. From the backward uniqueness property, we have ˜v = ˜w,
+that is v(t) = w(t) for t ≥ −T. Since T is arbitrary, then v(t) = w(t) for each t ∈ R
+where w is a periodic complete trajectory.
+This proves that v is a periodic complete
+trajectory.
+□
+References
+[1] L. Amerio, G. Prouse, Almost periodic functions and functional equations, Van Nostrand Reinhold
+Comp., New York, 1971.
+[2] S. Bochner, A new approach to almost periodicity, Proc. Natl. Acad. Sci. USA 48 (1962) 2039-2043.
+[3] S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, Proc. Natl.
+Acad. Sci. USA 52 (1964) 907-910.
+[4] C. M. Dafermos, Almost periodic processes and almost periodic solutions of evolution equations. Dy-
+namical systems (Proc. Internat. Sympos., Univ. Florida, Gainesville, Fla., 1976), pp. 43-57. Academic
+Press, New York, 1977.
+[5] A. M. Fink, Almost periodic differential equations, Lecture Notes in Math., vol. 377, Springer-Verlag,
+Berlin-New York, 1974.
+[6] M. Fr´echet, Les fonctions asymptotiquement presque-p´eriodiques continues, C.R. Math. Acad. Sci.
+Paris 213 (1941), pp. 520-522 (in French).
+[7] A. Haraux, Asymptotic behavior of trajectories for some nonautonomous, almost periodic processes,
+J. Differential Equations 49 (1983), 473-483.
+[8] A. Haraux, Sur les trajectoires compactes de syst`emes dynamiques autonomes. [On the compact tra-
+jectories of autonomous dynamical systems], Portugal. Math. 44 (1987), 253-259 (in French).
+[9] A. Haraux, A simple almost-periodicity criterion and applications, J. Differential Equations 66 (1987),
+51-61.
+[10] A. Haraux, Syst`emes dynamiques dissipatifs et applications. [Dissipative dynamical systems and ap-
+plications], Recherches en Math´ematiques Appliqu´ees [Research in Applied Mathematics], 17, Masson,
+Paris, 1991 (in French).
+[11] S. Lang Real and functional analysis, Third edition, Springer-Verlag, New York, 1993.
+[12] B. M. Levitan, V. V. Zhikov, Almost periodic functions and differential equations,Translated from
+the Russian by L. W. Longdon. Cambridge University Press, Cambridge-New York, 1982.
+[13] V. Nemytskii and V. Stepanov, Qualitative theory of differential equations, Princeton University
+Press, Princeton, New Jersey, 1960.
+[14] W. M. Ruess, W. H. Summers, Asymptotic almost periodicity and motions of semigroups of operators,
+Proceedings of the symposium on operator theory (Athens, 1985). Linear Algebra Appl. 84 (1986),
+335-351.
+
+14
+P. Cieutat
+[15] W. M. Ruess, W. H. Summers, Minimal sets of almost periodic motions Math. Ann. 276 (1986),
+145-158.
+[16] W. M. Ruess, W. H. Summers, Compactness in spaces of vector valued continuous functions and
+asymptotic almost periodicity, Math. Nachr. 135 (1988), 7-33.
+[17] T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions,
+Springer, New-york, 1975.
+[18] S. Zaidman, Almost-periodic functions in abstract spaces, Research Notes in Mathematics, 126,
+Boston, 1985.
+[19] S. Zaidman, On relatively compact trajectories of semigroups of class C0, Applicable Anal. 21 (1986),
+9-12.
+
diff --git a/0dAyT4oBgHgl3EQfbfdc/content/tmp_files/load_file.txt b/0dAyT4oBgHgl3EQfbfdc/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf,len=693
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='00263v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='CA]  31 Dec 2022  ON BOCHNER’S ALMOST-PERIODICITY CRITERION  PHILIPPE CIEUTAT  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We give an extension of Bochner’s criterion for the almost periodic func-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' By using our main result, we extend two results of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Haraux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The first is a  generalization of Bochner’s criterion which is useful for periodic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The  second is a characterization of periodic functions in term of Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  2020 Mathematic Subject Classification: 35B10, 35B40, 42A75, 47H20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Keywords: Bochner almost periodicity, periodic function, almost periodic function,  asymptotically almost periodic function, nonlinear semigroup, periodic dynamical sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Introduction  The almost periodic functions in the sense of Bohr have been characterized by Bochner  by means of a compactness criterion in the space of the bounded and continuous functions  [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The Bochner’s criterion plays an essential role in the theory and in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  We give a new almost-periodicity criterion for functions with values in a given complete  metric space which is useful to study the almost periodicity of solutions of dynamical  systems governed by a family of operators with a positive parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' This criterion is an  extension of Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then Haraux gave a generalization of Bochner’s criterion  [9, Theorem 1], called a simple almost-periodicity criterion which is useful for periodic  dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' From our result, we deduce an extension of this criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We also  obtain an extension of an other result of Haraux which characterizes the periodic functions  in terms of the Bochner’s criterion [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' In the same spirit, we treat the asymptotically  almost periodic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  We give a description of this article, the precise definitions will be given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Throughout this section (X, d) is a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' An almost periodic function  u : R → X in the sense of Bohr is characterized by the Bochner’s criterion which is the  following: u is bounded and continuous, and from any real sequence of real numbers (τn)n,  there exists a subsequence (τφ(n))n such that the sequence of functions (u(t + τφ(n)))n is  uniformly convergent on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' In Section 3, we give two extensions of Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  First u : R → X is an almost periodic if and if only if in the Bochner’s criterion, we impose  that the terms of the sequence of real numbers (τn)n are all positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Second u : R → X  is an almost periodic if and if only if in the Bochner’s criterion, the convergence of the  subsequence of functions (u(t + τφ(n)))n is uniform only on [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' These improvements  are useful to study the almost periodicity of solutions of an evolution equation governed by  a family of operators with a positive parameter, in particular for a C0-semigroup of linear  operators or more generally, for an autonomous dynamical system (nonlinear semigroup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Universit´e Paris-Saclay, UVSQ, CNRS, Laboratoire de math´ematiques de Versailles, 78000, Versailles,  France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' E-mail address: philippe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='cieutat@uvsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='fr  1  2  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Cieutat  From our extension of Bochner’s criterion, we give new proofs which are direct and simpler  on known results on the almost periodicity of solutions of autonomous dynamic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Haraux gave a generalization of Bochner’s criterion the called a simple almost-periodicity  criterion [9, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' This criterion makes it possible to choose in the Bochner’s cri-  terion, the sequence of real numbers (τn)n in a set of the type ωZ which is very useful  for periodic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' From our extension of Bochner’s criterion, in Section  4, we deduce an improvement of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' An asymptotically almost periodic function  u : R+ → X is a perturbation of almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' A such function is characterized by a  property of the type of the Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' In the same spirit, we extend this char-  acterization of asymptotically almost periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then we apply these results to  study the almost periodicity of solutions of periodic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Bochner’s criterion can also be expressed in terms of the relative compactness of the  set {u(· + τ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} in a suitable set of continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' A periodic function is a  special case of almost periodic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' A direct consequence of [8, Proposition 2] given  by Haraux characterizes a periodic function in terms of the Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' This  characterization is the following: u : R → X is continuous is periodic if and if only if the  set {u(· + τ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' In Section 5, By using our improvement of Bochner’s  criterion, we give an extension of the Haraux’s characterization of periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We  will also give a result on asymptotically periodic functions of the type of Haraux result  described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then we apply these results to study the periodicity of solutions of  autonomous dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Notation  Let us now give some notations, definitions and properties which will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Throughout this section (X, d) is a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  R, Z and N stand re-  spectively for the real numbers, the integers and the natural integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  We denote by  R+ := {t ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' t ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Let E be a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  We denote by C(E, X) the  space of all continuous functions from E into X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  When J = R or J = R+, we de-  note by BC(J, X) the space of all bounded and continuous functions from J into X  equipped with the sup-distance, denoted by d∞(u, v) := sup  t∈R  d(u(t), v(t)) when J = R and  d∞,+(u, v) := sup  t≥0  d(u(t), v(t))) when J = R+ for u, v ∈ BC(J, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The metric spaces  (BC(R, X), d∞) and (BC(R+, X), d∞,+)) are complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  We now give some definitions and properties on almost periodic, asymptotically almost  periodic functions with values in a given complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  A subset D of R (respectively of R+) is said to be relatively dense if there exists ℓ > 0  such that D ∩ [α, α + ℓ] ̸= ∅ for all α ∈ R (respectively α ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' A continuous function  u : R → X is said to be almost periodic (in the sense of Bohr) if for each ε > 0,  the set of ε-almost periods: P(u, ε) =  �  τ ∈ R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' sup  t∈R  d(u(t + τ), u(t)) ≤ ε  �  is relatively  dense in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' An almost periodic function u has its range u(R) relatively compact, that  is its closure denoted by cl (u(R)) is a compact set of (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We denote the space of  all such functions by AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' It is a closed metric subspace of (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' An  almost periodic function u is uniformly recurrent, that is there exists a sequence of real  On Bochner’s almost-periodicity criterion  3  numbers (τn)n such that  lim  n→+∞ sup  t∈R  d(u(t + τn), u(t)) = 0 and  lim  n→+∞ τn = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' To see  that consider the Bohr’s definition of u ∈ AP(R, X), then the set of  1  n-almost periods  satisfies P(u, 1  n)∩[n, +∞) ̸= ∅, for each integer n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' A useful characterization of almost  periodic functions was given by Bochner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The Bochner’s criterion which may be found in  [12, Bochner’s theorem, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' 4] in the context of metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Before to cite this criterion,  we need to introduce the translation mapping of a function of BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For τ ∈ R and  u ∈ BC(R, X), we define the translation mapping Tτu ∈ BC(R, X) by Tτu(t) = u(t + τ)  for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1 (Bochner’s criterion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For u ∈ BC(R, X), the following statements are  equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) u ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) The set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is relatively compact in (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Haraux gave a generalization of Bochner’ criterion the called a simple almost-periodicity  criterion [9, Theorem 1] which is useful for periodic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2 (Haraux’s criterion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let D be a relatively dense subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The following  statements are equivalent for u ∈ BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) u ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) The set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ D} is relatively compact in (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Periodic functions, which are a special case of almost periodic functions, are also char-  acterized in terms of Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' This criterion is a direct consequence of a result  of Haraux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' [8, Consequence of Proposition 2] The following statements are equivalent  for u ∈ BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) u is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) The set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is a compact set of (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  For some preliminary results on almost periodic functions with values in a given com-  plete metric space, we refer to the book of Levitan-Zhikov [12] and in the special case of  Banach spaces to the book of Amerio-Prouse [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  The notion of asymptotic almost periodicity was first introduced by Fr´echet [6] in 1941  in the case where X = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' A continuous function u : R+ → X is said to be asymp-  totically almost periodic if there exists v ∈ AP(R, X) such that lim  t→∞ d(u(t), v(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  An asymptotic almost periodic function u has its range u(R+) relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We  denote the space of all such functions by AAP(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' It is a closed metric subspace  of (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' An asymptotic almost periodicity function u : R+ → X is char-  acterized by u ∈ APP(R+, X) if and only if u ∈ C(R+, X) and for each ε > 0, there  exists M ≥ 0 such that the  �  τ ≥ 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' sup  t≥M  d(u(t + τ), u(t)) ≤ ε  �  is relatively dense in R+  [15, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' In the context of metric spaces, Ruess and Summers give a charac-  terization of asymptotically almost periodic functions in the spirit of Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  4  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Cieutat  To prove this characterization, Ruess and Summers use results from the paper [16] by  the same authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For τ ≥ 0 and u ∈ BC(R+, X), we define the translation mapping  T +  τ u ∈ BC(R+, X) by T +  τ u(t) = u(t + τ) for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' [15, a part of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2 & 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3] Let (X, d) be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  For u ∈ BC(R+, X), the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) u ∈ AAP(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) The set {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is relatively compact in (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  For some preliminary results on asymptotically almost periodic functions, we refer to  the book of Yoshizawa [17] in the case where X is a finite dimensional space, to the book  of Zaidman [18] where X is a Babach space and to Ruess and Summers [14, 15, 16] in the  general case: X is a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' An improvement of Bochner’s criterion  An almost periodic function is characterized by the Bochner’s criterion, recalled in Sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Our main result is an extension of Bochner’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then we deduce new proofs  which are direct and simpler on known results on the solutions of autonomous dynamic  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Before to state our extension of Bochner’s criterion, we need to introduce the  restriction operator R : BC(R, X) → BC(R+, X) defined by R(u)(t) := u(t) for t ≥ 0  and u ∈ BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let (X, d) be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For u ∈ BC(R, X) the following  statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) u ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) The set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is relatively compact in (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  iii) The set {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is relatively compact in (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  In our results, the compactness and the relative compactness of a set often intervene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  To prove them, we will often use the following result whose proof is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Recall that  a set A of a metric space (E, d) is relatively compact if its closure denoted by cl (A) is a  compact set of (E, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let E be a set, (G1, d1) and (G2, d2) be two metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let u : E → G1  and v : E → G2 be two functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Assume there exists M > 0 such that  ∀x1, x2 ∈ E,  d1(u(x1), u(x2)) ≤ Md2(v(x1), v(x2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) If the metric space (G1, d1) is complete and v(E) is relatively compact in (G2, d2),  then u(E) is relatively compact in (G1, d1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) If v(E) is a compact set of (G2, d2), then u(E) is a compact set of (G1, d1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' i) =⇒ iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' It is obvious by using the Bochner’s criterion and the  continuity of the restriction operator R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  iii) =⇒ ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The set u(R) = {R(Tτu)(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is relatively compact in X as the  range of {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} by the continuous evaluation map at 0 from BC(R+, X) into  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' By assumption, H := cl ({R(Ttu) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' t ∈ R}) is a compact set of (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  On Bochner’s almost-periodicity criterion  5  For all τ ≥ 0, we define φτ : H → X by φτ(h) = h(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The functions φτ are 1-Lipschitz  continuous and for each t ∈ R, the set {φτ(R(Ttu)) = u(τ + t) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is included in the  relatively compact set u(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' By density of {R(Ttu) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' t ∈ R} in H and the continuity of  φτ, it follows that {φτ(h) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is relatively compact in X for each h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' According  to Arzel`a-Ascoli’s theorem [11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' 57], the set {φτ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is relatively  compact in C(H, X) equipped with the sup-norm denoted by dC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  From the density  of {R(Ttu) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' t ∈ R} in H and the continuity of φτ, we deduce that for τ1 and τ2 ≥ 0,  sup  h∈H  d(φτ1(h), φτ2(h)) = sup  t∈R  d (φτ1(R(Ttu)), φτ2(R(Ttu))) = sup  t∈R  d (u(τ1 + t), u(τ2 + t)) =  sup  t∈R  d (Tτ1u(t), Tτ2u(t)), then dC(φτ1, φτ2) = d∞ (Tτ1u, Tτ2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2, it follows  that {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is relatively compact in the complete metric space (BC(R, X), d∞)  since {φτ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is also one in (C(H, X), dC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) =⇒ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For τ1, τ2 ≥ 0, d∞(Tτ1u, Tτ2u) := sup  t∈R  d(u(τ1 + t), u(τ2 + t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Replacing t  by t − τ1 − τ2 in the upper bound, we get d∞(Tτ1u, Tτ2u) = d∞(T−τ1u, T−τ2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then the  set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≤ 0} = {T−τu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is relatively compact in BC(R, X) since {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0}  is also one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Therefore the set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is relatively compact in BC(R, X) as the  union of two relatively compact sets in BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  According to Bochner’s criterion,  u ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  □  The connection between the almost periodicity of a solution of a dynamical system and  its stability is well known (see the monograph by Nemytskii & Stepanov [13, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' This  weakened form of Bochner’s criterion: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1 makes it possible to obtain direct and  simpler proofs on these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let us start by recalling some definitions on dynamical  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  A dynamical system or nonlinear semigroup on a complete metric space (X, d) is a one  parameter family (S(t))t≥0 of maps from X into itself such that i) S(t) ∈ C(X, X) for all  t ≥ 0, ii) S(0)x = x for all x ∈ X, iii) S(t + s) = S(t) ◦ S(s) for all s, t ≥ 0 and iv) the  mapping S(·)x ∈ C([0, +∞), X) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  For each x ∈ X, the positive trajectory of x is the map S(·)x : R+ → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' A function  u : R → X is called a complete trajectory if we have u(t + τ) = S(τ)u(t), for all t ∈ R  and τ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  We will need a notion of Lagrange-type stability to ensure that a solution with a  relatively compact range is almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Recall that (S(t))t≥0 is equicontinuous on a  compact set K of X, if forall ε > 0, there exists δ > 0, such that  ∀x1, x2 ∈ K, d(x1, x2) ≤ δ =⇒ sup  t≥0  d(x1, x2) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1, we give a new proof which is direct and simpler of the following  result which can be found in [10, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' 51] or partly in [12, Markov’s theorem,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let (S(t))t≥0 be a dynamical system on a complete metric space (X, d)  and u be a complete trajectory such that u(R) is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then u is almost  periodic if and only if (S(t))t≥0 is equicontinuous on cl (u(R)) the closure of u(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let us denote the compact set K := cl (u(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' It follows by density of u(R) in  K and the continuity of S(t), that {S(t)x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' t ≥ 0} ⊂ K for each x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  According  6  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Cieutat  to Arzel`a-Ascoli’s theorem, (S(t))t≥0 is equicontinuous on K if and only if (S(t))t≥0 is  relatively compact in C(K, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1, we have u ∈ AP(R, X) if and only if  {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is relatively compact in BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then it remains to prove that (S(t))t≥0  is relatively compact in C(K, X) equipped with the sup-norm if and only if {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0}  is relatively compact in (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' This results from the following equalities, for τ1  and τ2 ≥ 0, sup  t∈R  d (Tτ1u(t), Tτ2u(t)) = sup  t∈R  d (S(τ1)u(t), S(τ2)u(t)) = sup  x∈K  d (S(τ1)x, S(τ2)x)  and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' a) The condition of equicontinuity required by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3 is satisfied by  a bounded dynamical system : d (S(t)x1, S(t)x2) ≤ Md (x1, x2) for some M ≥ 1 and  in particular for a C0 semigroup of contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' In this case, the almost periodicity of  a complete trajectory u having a relatively compact range results from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  We can also obtain this result with the implication iii) =⇒ i) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1 and the  inequality sup  t≥0  d(R(Tτ1u)(t), R(Tτ2u)(t)) = sup  t≥0  d (S(t)u(τ1), S(t)u(τ2) ≤ Md(u(τ1), u(τ2))  for τ1, τ2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  b) For a bounded C0-semigroup (S(t))t≥0, the main result of Zaidman [19] asserts  that a positive trajectory u with relatively compact range satisfies a condition called the  generalized normality property in Bochner’s sense, without concluding that u is almost  periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' This condition is nothing but hypothesis iii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1, so u is almost  periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4, we give a new proof which is direct and simpler of the following  result which can be found in [15, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' 149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let (S(t))t≥0 be a dynamical system on a complete metric space (X, d) and  u be a positive trajectory such that u(R+) is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then u is asymptotically  almost periodic if and only if (S(t))t≥0 is equicontinuous on cl (u(R+)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The proof is analogous to that of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3, using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4 instead of  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1 and by replacing R by R+ and AP(R, X) by AAP(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' An improvement of Haraux’s criterion  Haraux gave a generalization of Bochner’scriterion [9, Theorem 1], the called a simple  almost-periodicity criterion which is useful for periodic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  From our  main result, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1, we deduce an extension of the Haraux’s criterion, recalled in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' In the same spirit, we extend the well-known characterization of asymptotically  almost periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' To end this section, we give an exemple of application on a  periodic dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  We give an extension of the Haraux’s criterion (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Recall that we denote  by R the restriction operator R : BC(R, X) → BC(R+, X) defined by R(u)(t) := u(t) for  t ≥ 0 and u ∈ BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let (X, d) be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For u ∈ BC(R, X) the following  statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) u ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) The set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ D} is relatively compact in (BC(R, X), d∞) where D be a relatively  dense subset of R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  On Bochner’s almost-periodicity criterion  7  iii) The set {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ D} is relatively compact in (BC(R+, X), d∞,+) where D be a  relatively dense subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Our main result, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1 is obviously a particular case of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  But to present our results, it was easier to start with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' To prove Corollary  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1, we use Haraux’s criterion and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' i) =⇒ iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' It is a consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  iii) =⇒ ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' To establish this implication, using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1, it suffices to show that  assertion iii) implies that {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is relatively compact in BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The proof  of this last implication is a slight adaptation of those of those of the Haraux’s criterion  given in [9, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' A similar proof will be detailed in the following result as there  will be technical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' To demonstrate that {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is relatively compact, it  suffices in the proof of ii) =⇒ i) of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3 to take ℓ = 0 and replace {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ D}  by {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) =⇒ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For τ1, τ2 ≥ 0, d∞(Tτ1u, Tτ2u) = sup  t∈R  d(u(τ1 + t), u(τ2 + t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Replacing t by  t − τ1 − τ2 in the upper bound, we get d∞(Tτ1u, Tτ2u) = d∞(T−τ1u, T−τ2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then the set  {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ −D} = {T−τu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ D} is relatively compact in BC(R, X) since {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ D}  is also one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Therefore the set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ D ∪ (−D)} is relatively compact in BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Moreover D ∪(−D) is a relatively dense subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' According to Haraux’s criterion, we  have u ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  □  We extend Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4, the well-known characterization of asymptotically almost pe-  riodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For τ ∈ R+ and u ∈ BC(R+, X), we define the translation mapping  T +  τ u ∈ BC(R+, X) by T +  τ u(t) = u(t + τ) for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let (X, d) be a complete metric space and let D be a relatively dense  subset of R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For u ∈ BC(R+, X) the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) u ∈ AAP(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) The set {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ D} is relatively compact in (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' To establish implication ii) =⇒ i), by using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4, it suffices to prove  that assertion ii) implies that {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is relatively compact in (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  The proof of this last implication is an adaptation of those of the Haraux’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' But  contrary to the proof of implication iii) =⇒ ii) in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1, there are technical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  These technical difficulties come from the fact that when D is a relatively dense subset  in R+, the sets D and [t − ℓ, t] can be disjoint for some 0 ≤ t ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For this reason we give  the complete proof of this implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' i) =⇒ ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' It is a consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) =⇒ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We will prove that assumption ii) implies {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is relatively compact  in BC(R+, X), then we conclude by using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The subset D being relatively  dense in R+, there exists ℓ > 0 such that D ∩ [α, α + ℓ] ̸= ∅ for all α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  We prove that u is uniformly continuous on [ℓ, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let us fix ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' By assump-  tion the set {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ D} is in particular relatively compact in C([0, 2ℓ], X), then it is  uniformly equicontinuous on [0, 2ℓ], that is there exists δ > 0 such that  s1, s2 ∈ [0, 2l],  |s1 − s2| ≤ δ =⇒ sup  τ∈D  d(u(s1 + τ), u(s2 + τ)) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1)  8  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Cieutat  Let t1, t2 be two real numbers such that t1, t2 ≥ ℓ and |t1 − t2| ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We can assume  without loss of generality that ℓ ≤ t1 ≤ t2 ≤ t1 + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We have D ∩ [t1 − ℓ, t1] ̸= ∅ since  t1 − ℓ ≥ 0, then there exists τ ∈ D such that 0 ≤ t1 − τ ≤ t2 − τ ≤ 2l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Taking account  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1), we deduce that d(u(t1), u(t2)) = d(u((t1 − τ) + τ), u((t2 − τ) + τ)) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Hence u is  uniformly continuous on [ℓ, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  We prove that {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ ℓ} is relatively compact in BC(R+, X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let (tn)n be a  sequence of real numbers such that tn ≥ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We have D ∩ [tn − ℓ, tn] ̸= ∅ for each n ∈ N,  since tn − ℓ ≥ 0, then there exist τn ∈ D and σn ∈ [0, l] such that tn = τn + σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' By  compactness of the sequences (σn)n in [0, ℓ] and (T +  τnu)n in BC(R+, X), it follows that  lim  n→+∞ σn = σ and  lim  n→+∞ sup  t≥0  d(u(t + τn), v(t)) = 0 (up to a subsequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  From the  following inequality  sup  t≥0  d(u(tn + t), v(σ + t)) ≤ sup  t≥0  d(u(τn + σn + t), u(τn + σ + t) + sup  t≥0  d(u(τn + t), v(t))  and the uniform continuity of u, we deduce that lim  n→+∞ sup  t≥0  d(u(tn +t), v(σ +t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then  {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ ℓ} is relatively compact in BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  We prove that u ∈ AAP(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The function u is uniformly continuous on R+,  since u is continuous on [0, ℓ] and uniformly continuous on [ℓ, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then the map ˆu :  R+ → BC(R+, X) defined by ˆu(τ) = T +  τ u for τ ≥ 0 is continuous, consequently the  set {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' 0 ≤ τ ≤ ℓ} is relatively compact in BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The set {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is  relatively compact in BC(R+, X) as the union of two relatively compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' According  to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4, u ∈ AAP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  □  Using corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3, we give a proof which is direct and simpler of the following  result which can be found in [7, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Before we recall some definitions on process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  A process on a complete metric space (X, d) according to Dafermos [4] is a two pa-  rameter family U(t, τ) of maps from X into itself defined for (t, τ) ∈ R × R+ and such  that i) U(t, 0)x = x for all (t, x) ∈ R × X, ii) U(t, σ + τ) = U(t + σ, τ) ◦ U(t, σ) for all  (t, σ, τ) ∈ R×R+×R+ and iii) the mapping U(t, ·)x ∈ C([0, +∞), X) for all (t, x) ∈ R×X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  For each x ∈ X, the positive trajectory starting of x is the map U(0, ·)x : R+ → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' A  function u : R → X is called a complete trajectory if we have u(t + τ) = U(t, τ)u(t) for  all (t, τ) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  A process U is said ω-periodic (ω > 0) if U(t + ω, τ) = U(t, τ) for all (t, τ) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  A process U is said bounded if we have for some M ≥ 1 for all (τ, x1, x2) ∈ R+ ×X ×X  d (U(0, τ)x1, U(0, τ)x2) ≤ Md (x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' [7, 9], [10, Th´eor`eme 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' 84] Let U be a ω-periodic process on a  complete metric space (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' If U is bounded, then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) If u is a complete trajectory of U such that u(−ωN) is relatively compact, then u is  almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) If u is a positive trajectory of U such that u(ωN) is relatively compact, then u is  asymptotically almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' i) The process U is ω-periodic, then we have u(nω) = U(−mω, (n+m)ω)u(−mω) =  U(0, (n + m)ω)u(−mω) for all n, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' From the boundedness assumption on U, we  On Bochner’s almost-periodicity criterion  9  deduce that d (u(nω), u(mω)) ≤ Md (u(−mω), u(−nω)), then u(ωN) is relatively compact  since u(−ωN) is also one, therefore u(ωZ) is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' From assumptions on  the process U, it follows that for all n, m ∈ Z,  sup  τ≥0  d (u(τ + nω), u(τ + mω)) ≤ Md (u(nω), u(mω)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2)  From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2, {R(Tnωu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' n ∈ Z} is relatively compact in (BC(R+, X), d∞,+) since  u(ωZ) is also one in (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We conclude with Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1 by setting D = ωZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) For all n, m ∈ N, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2) holds on the positive trajectory u, then from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2,  {T +  nωu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' n ∈ N} is relatively compact in (BC(R+, X), d∞,+) since u(ωN) is also one in  (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We conclude with Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3 by setting D = ωN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Bochner’s criterion in the periodic case  Periodic functions are a special case of almost periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Haraux gave a charac-  terization of periodic functions in terms of Bochner’s criterion which is recalled in Section  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  This criterion is a direct consequence of [8, Proposition 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Haraux established a  general result [8, Th´eor`eme 1] implying as a special case a characterization of periodic  functions and the fact that any compact trajectory of a one-parameter continuous group  is automatically periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  In this section, we give an extension of this characterization of periodic functions in the  spirit of the main result of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We also treat the asymptotically periodic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Then we apply these results to study the periodicity of solutions of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Recall that we denote by R the restriction operator R : BC(R, X) → BC(R+, X)  defined by R(u)(t) := u(t) for t ≥ 0 and u ∈ BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let (X, d) be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For u ∈ BC(R, X) the following  statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) The function u is ω-periodic (ω > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) The set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is a compact set of (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  iii) The set {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is a compact set of (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' i) =⇒ ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' From assumption, it follows that the function τ → Tτu from R into  BC(R, X) is continuous and ω-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then the {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} = {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' 0 ≤ τ ≤ ω} is a  compact set of (BC(R, X), d∞) as the range of a compact set by a continuous map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) =⇒ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  For τ1, τ2 ∈ R, d∞(Tτ1u, Tτ2u) := sup  t∈R  d(u(τ1 + t), u(τ2 + t)), we get  d∞(Tτ1u, Tτ2u) = d∞(T−τ1u, T−τ2u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Then the set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≤ 0} = {T−τu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is  compact in BC(R, X) since {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is also one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Therefore the set {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is a  compact set of BC(R, X) as the union of two compact sets in BC(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' According to  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3, u is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) =⇒ iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' It is obvious by using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3 and the continuity of the restriction  operator R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  iii) =⇒ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' By using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3, we have to prove that K := {Tτu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is a  compact set of (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' As consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1 and Bohner’s criterion,  the set K is relatively compact in (BC(R, X), d∞) and the function u is almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  10  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Cieutat  It remains to prove that K is closed in (BC(R, X), d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let (τn)n be a sequence of real  numbers such that  lim  n→+∞ d∞(Tτnu, v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let us prove that v = Tτu for some τ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' By  continuity of the operator R, we have lim  n→+∞ d∞,+(R(Tτnu), R(v)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' By assumption, the  set {R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is in particular closed in (BC(R+, X), d∞,+), then R(v) = R(Tτu)  for some τ ∈ R, that is  ∀t ≥ 0,  v(t) = Tτu(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1)  We have to prove that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1) holds on the whole real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The function Tτu is almost  periodic as translation of an almost periodic function and v is also one as uniform limit  on R of almost periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let us denote by φ : R → R the function defined by  φ(t) := d(Tτu(t), v(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The function φ is almost periodic [12, Property 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' 3 & 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  An almost periodic function is uniformly recurrent, then there exists a sequence of real  numbers such that  lim  n→+∞ τn = +∞ and  lim  n→+∞ φ(t+ τn) = φ(t) for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1), it  follows φ(t) = 0 for all t ≥ 0, so we deduce that φ(t) =  lim  n→+∞ φ(t + τn) = 0 for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Then v(t) = Tτu(t) for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  □  According Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4, if the set {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is relatively compact in BC(R+, X),  then the function u is asymptotically almost periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We now give an answer to the  question what can be said about the function u when {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is a compact set of  BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For u ∈ BC(R+, X), we say that u is ω-periodic (ω > 0) on [t0, +∞) for  some t0 ≥ 0 if u(t + ω) = u(t) for all t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let (X, d) be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For u ∈ BC(R+, X) the following  statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) There exists t0 ≥ 0 such that u is ω-periodic on [t0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) The set {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is a compact set of (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let u be a function which satisfies condition i) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) Let us denote by v ∈ C(R, X) the ω-periodic function satisfying u(t) = v(t) for  t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' A such function v exists and is unique, v is defined by v(t) = u(t − [ t−t0  ω ]ω) where  [ t−t0  ω ] denotes the integer part of t−t0  ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) The function u is a special case of asymptotic almost periodic function where the  almost periodic function v is periodic and d(u(t), v(t)) = 0 for t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' i) =⇒ ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let us denote by v the function defined in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1 and the periodicity of v, we have {R(Tτv);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ t0} = {R(Tτv);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is  a compact set of (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  First, we have T +  τ u = R(Tτv) for τ ≥ t0, then {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ t0} is a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Second, the function u is uniformly continuous on R+, then the function from R+ to  BC(R+, X) defined by τ → T +  τ u is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Then the set {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' 0 ≤ τ ≤ t0} is  compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Therefore the set {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is a compact set of BC(R+, X) as the union of two  compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) =⇒ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' As consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4, the function u is asymptotically almost  periodic, that is lim  t→∞ d(u(t), v(t)) = 0 for some v ∈ AP(R, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  An almost periodic  On Bochner’s almost-periodicity criterion  11  function is uniformly recurrent, then there exists a sequence of real numbers (tn)n such  that  lim  n→+∞ tn = +∞ and  lim  n→+∞ v(t + tn) = v(t) for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We deduce that  ∀t ∈ R,  lim  n→+∞ u(t + tn) = v(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2)  First we prove that v is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For t ∈ R, τ1, τ2 ≥ 0, we have for n enough large  d(u(t + tn + τ1), u(t + tn + τ2)) ≤ sup  s≥0  d(u(s + τ1), u(s + τ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2), it follows that  sup  t∈R  d(v(t+τ1), v(t+τ2)) ≤ sup  s≥0  d(u(s+τ1), u(s+τ2)) for each τ1 and τ2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' According to  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2, {Tτv ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is a compact set of (BC(R, X), d∞) since {T +  τ u ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is also  one in (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' As consequence of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1, the function v is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Second we prove that: ∃t0 ≥ 0 such that ∀t ≥ 0, v(t) = u(t + t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' By compactness of  {T +  τ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0}, there exists a subsequence (T +  tφ(n)u)n such that lim  n→+∞ d∞,+(T +  tφ(n)u, T +  t0 u) = 0  for some t0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2) we deduce that R(v) = T +  t0 u, that is v(t) = u(t + t0) for all  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Then u(t) = v(t − t0) for each t ≥ t0 where the function v(· − t0) is periodic on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  □  Now we give an example of application on dynamical systems of Corollary of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1 and  Corollary of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For the definition of a dynamical system, see above Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3 in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let (S(t))t≥0 be a dynamical system on a complete metric space (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) If u is a positive trajectory, then u is periodic on [t0, +∞) for some t0 ≥ 0 if and  only if u(R+) is a compact set and (S(t))t≥0 is equicontinuous on u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) If u is a complete trajectory, then u is periodic if and only if u(R) is a compact set  and (S(t))t≥0 is equicontinuous on u(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  iii) There exists a complete trajectory which is periodic if and only if there exists a  positive trajectory u such that u(R+) is a compact set and (S(t))t≥0 is equicontinuous on  u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Thus under the assumption of equicontinuity, a complete trajectory of a  dynamical system with a compact range is necessarily periodic, although there are almost  periodic functions with a compact range, which are not periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' An example of such  function is given by Haraux in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' i) Remark that if u is a positive trajectory which is periodic on  [t0, +∞) for some t0 ≥ 0, then first u(R+) is compact and second u ∈ AAP(R+, X)  (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' As consequence of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='5, the set (S(t))t≥0 is equicontinuous on  u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Reciprocally assume the positive trajectory u is such that (S(t))t≥0 is equicontinu-  ous on the compact set u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' It remains to prove that the positive trajectory u is periodic  on [t0, +∞) for some t0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For each x ∈ u(R+), the map S(·)x is continuous and satisfies  S(t)x ∈ u(R+) for each t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then the map S(·)x is bounded and continuous , so the map  Φ : u(R+) → BC(R+, X) with Φ(x) = S(·)x is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The continuity of Φ results  of the equicontinuity of (S(t))t≥0 on u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then the set Φ(u(R+)) = {Φ(u(τ)) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0}  is a compact of BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Moreover Φ(u(τ))(t) = S(t)u(τ) = u(t + τ) for t and τ ≥ 0,  12  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Cieutat  then Φ(u(τ)) = T +  τ u, so {T +  τ u ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ≥ 0} is a compact set of BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' According to  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2, the function u is periodic on [t0, +∞) for some t0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) The proof of ii) is similar to that of i) by using Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3 instead of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='5, Corollary  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='1 instead of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='2 and by replacing the map Φ : u(R+) → BC(R+, X) with  Φ(x) = S(·)x by the map Φ : u(R) → BC(R+, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' This permits to prove that the set  {Φ(u(τ)) = R(Tτu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' τ ∈ R} is a compact set of (BC(R+, X), d∞,+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  iii) If v is a complete trajectory which is periodic, then v(R) is compact and according  to ii), (S(t))t≥0 is equicontinuous on v(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' So the restriction u of v on R+ is a posi-  tive trajectory such that u(R+) = v(R+) = v(R) since v is periodic, then (S(t))t≥0 is  equicontinuous on the compact set u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Reciprocally, assume that u is a positive tra-  jectory such that (S(t))t≥0 is equicontinuous on the compact set u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' According to  i), u is ω-periodic on [t0, +∞) for some t0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let us denote by v the function defined  in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For t ≥ s, there exists n0 ∈ N such that s + n0ω ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The function  v is ω-periodic and u is a positive trajectory satisfying u(τ) = v(τ) for τ ≥ t0, then  v(t) = v(t + n0ω) = u(t + n0ω) = T(t − s)u(s + n0ω) = T(t − s)v(s + n0ω) = T(t − s)v(s)  for t ∈ R and n enough large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then v is a periodic complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  □  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Under i) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4, one can have t0 > 0, that is the positive trajectory  u is not the restriction of a periodic complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  For example, consider the  bounded dynamical system (S(t))t≥0 on L1(0, 1) defined by  (S(t)x)(s) =  \uf8f1  \uf8f2  \uf8f3  x(s − t)  if  t < s < 1  0  if  0 < s < t  for x ∈ L1(0, 1) and 0 < t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For t ≥ 1, we set S(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then all positive trajectories  have a compact range and the alone complete trajectory is the null function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Thus all  positive trajectories are not the restriction of a periodic complete trajectory except the  null function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Not all dynamical systems have this pathology, some systems are such that if two  positive trajectories have the same value at the same time, then they are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' If we  consider such systems, we get more refined results from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  A dynamical system (S(t))t≥0 has the backward uniqueness property if any two positive  trajectories having the same value at t = t0 ≥ 0 coincide for any other t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' This  property is equivalent to S(t) ∈ C(X, X) is injective for each t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' We say that a  positive trajectory u is extendable to a periodic complete trajectory, if there exists a periodic  complete trajectory such that its restriction on R+ is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let (S(t))t≥0 be a dynamical system on a complete metric space (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  Assume that (S(t))t≥0 has the backward uniqueness property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  i) If u is a positive trajectory, then u is periodic on R+ if and only if u(R+) is a  compact set and (S(t))t≥0 is equicontinuous on u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' In this case the positive trajectory  u is extendable to a periodic complete trajectory v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) If v is a complete trajectory, then v is periodic if and only if v(R+) is a compact set  and (S(t))t≥0 is equicontinuous on v(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  On Bochner’s almost-periodicity criterion  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' i) The direct implication results of i) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For the reciprocal implica-  tion we use i) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then the positive trajectory u is ω-periodic on [t0, +∞) for  some t0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let us denote by v ∈ C(R, X) the ω-periodic function satisfying u(t) = v(t)  for t ≥ t0 (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The restriction of v on R+ and u are two positive trajectories  having the same value at t = t0 (t0 ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' From the backward uniqueness property, we  have u(t) = v(t) for t ≥ 0, then u is periodic on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' By build, v is periodic and as in the  proof of iii) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4, we deduce that v is a complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  ii) The direct implication results of ii) Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4, since v(R+) = v(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' For the  reciprocal implication, we consider v a complete trajectory such that v(R+) is a compact  set and (S(t))t≥0 is equicontinuous on v(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Then the restriction u of the complete  trajectory v on R+ is a positive trajectory such that u(R+) is compact and (S(t))t≥0  is equicontinuous on u(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' According to i), u is ω-periodic on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Let us denote by  w ∈ C(R, X) the ω-periodic function satisfying u(t) = w(t) for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' As in proof of iii)  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='4, we deduce that w is a complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Fix T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' The two maps ˜v  and ˜w : R+ → X defined by ˜v = v(·, −T) and ˜w = w(·, −T) are two positive trajectories  having the same value at t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' From the backward uniqueness property, we have ˜v = ˜w,  that is v(t) = w(t) for t ≥ −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Since T is arbitrary, then v(t) = w(t) for each t ∈ R  where w is a periodic complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  This proves that v is a periodic complete  trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  □  References  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Amerio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Prouse, Almost periodic functions and functional equations, Van Nostrand Reinhold  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=', New York, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Bochner, A new approach to almost periodicity, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAyT4oBgHgl3EQfbfdc/content/2301.00263v1.pdf'}
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+Unlearnable Clusters: Towards Label-agnostic Unlearnable Examples
+Jiaming Zhang1 Xingjun Ma2* Qi Yi1 Jitao Sang1,4∗ Yugang Jiang2 Yaowei Wang4 Changsheng Xu3,4
+1Beijing Jiaotong University 2Fudan University 3Chinese Academy of Sciences 4Peng Cheng Lab
+Abstract
+There is a growing interest in developing unlearnable
+examples (UEs) against visual privacy leaks on the Internet.
+UEs are training samples added with invisible but unlearn-
+able noise, which have been found can prevent unauthorized
+training of machine learning models. UEs typically are gen-
+erated via a bilevel optimization framework with a surrogate
+model to remove (minimize) errors from the original samples,
+and then applied to protect the data against unknown target
+models. However, existing UE generation methods all rely
+on an ideal assumption called label-consistency, where the
+hackers and protectors are assumed to hold the same label
+for a given sample. In this work, we propose and promote
+a more practical label-agnostic setting, where the hackers
+may exploit the protected data quite differently from the
+protectors. E.g., a m-class unlearnable dataset held by the
+protector may be exploited by the hacker as a n-class dataset.
+Existing UE generation methods are rendered ineffective in
+this challenging setting. To tackle this challenge, we present
+a novel technique called Unlearnable Clusters (UCs) to gen-
+erate label-agnostic unlearnable examples with cluster-wise
+perturbations. Furthermore, we propose to leverage Vision-
+and-Language Pre-trained Models (VLPMs) like CLIP as the
+surrogate model to improve the transferability of the crafted
+UCs to diverse domains. We empirically verify the effective-
+ness of our proposed approach under a variety of settings
+with different datasets, target models, and even commercial
+platforms Microsoft Azure and Baidu PaddlePaddle.
+1. Introduction
+While the huge amount of “free” data available on the
+Internet has been key to the success of deep learning and
+computer vision, this has also raised public concerns on the
+unauthorized exploitation of personal data uploaded to the
+Internet to train commercial or even malicious models [16].
+For example, a company named Clearview AI has been
+found to have scraped billions of personal images from Face-
+book, YouTube, Venmo and millions of other websites to
+construct a commercial facial recognition application [44].
+*Corresponding authors
+Unconscious and unauthorized collection for training
+Label-consistency
+Label-agnostic
+Original images ������������
+Unlearnable image ������������′
+Hacker
+Protector
+������������-class surrogate 
+model ������������������������������������
+=
+car, dog, cat, 
+airplane, etc.
+...
+������������-class target 
+model ������������������������
+������������
+car, dog, cat, 
+airplane, etc.
+=
+������������-class target 
+model ������������������������
+������������
+=
+animal, 
+vehicle, etc.
+...
+Hacker
+Figure 1. An illustration of two different data protection assump-
+tions: label-consistency vs. label-agnostic, where the hacker ex-
+ploits the protected data in different manners.
+This has motivated the proposal of Unlearnable Examples
+(UEs) [17] to make data unlearnable (or unusable) to ma-
+chine learning models/services. Similar techniques are also
+known as availability attacks [2,41] or indiscriminate poi-
+soning attacks [14] in the literature. These techniques allow
+users to actively adding protective noise into their private
+data to avoid unauthorized exploitation, rather than putting
+our trust into the hands of large corporations.
+The original UE generation method generates error-
+minimizing noise via a bilevel min-min optimization frame-
+work with a surrogate model [17]. The noise can then be
+added to samples in a training set in either a sample-wise
+or class-wise manner to make the entire dataset unlearnable
+to different DNNs. It has been found that this method can-
+not survive adversarial training, which has been addressed
+by a recent method [11]. In this work, we identify one
+common assumption made by existing UE methods: label-
+consistency, where the hackers will exploit the protected
+dataset in the same way as the protector including the labels.
+This means that, for the same image, the hacker and protec-
+tor hold the same label. We argue that this assumption is
+too ideal, and it is possible that the hackers will collect the
+protected (unlearnable) samples into a dataset for a different
+task and label the dataset into different number of classes.
+As illustrated in Figure 1, an image can be labelled with
+1
+arXiv:2301.01217v1  [cs.CR]  31 Dec 2022
+
+盒盒000000different annotated labels (cat or animal), showing that a m-
+class (e.g., 10-class) unlearnable dataset may be exploited
+by the hacker as a n-class (e.g., 5-class or 20-class) dataset
+depending on its actual needs. We term this more generic
+assumption as label-agnostic and propose a novel method
+Unlearnable Clusters (UCs) to generate more effective and
+transferable unlearnable examples under this harsh setting.
+In Figure 2 (a), we show that this more generic label-
+agnostic setting poses a unique transferability challenge
+for the noise generated by existing methods like Error-
+Minimizing Noise (EMinN) [17], Adversarial Poisoning
+(AdvPoison) [10], Synthetic Perturbations (SynPer) [41] and
+DeepConfuse [9]. This indicates that the protective noise
+generated by these methods are label-dependent and are ren-
+dered ineffective when presented with different number of
+classes. As such, we need more fundamental approaches
+to make a dataset unlearnable regardless of the annotations.
+To this end, we start by analyzing the working mechanism
+of UEs generated by EMinN, AdvPoison as they are very
+representative under the label-consistency setting. Through
+a set of visual analyses, we find that the main reason why
+they could break supervised learners is that the generated
+noise tends to disrupts the distributional uniformity and dis-
+crepancy in the deep representation space. Uniformity refers
+to the property that the manifold of UEs in the deep rep-
+resentation space does not deviate much from that of the
+clean examples, while discrepancy refers to the property
+that examples belonging to the same class are richly diverse
+in the representation space. Inspired by the above obser-
+vation, we propose a novel approach called Unlearnable
+Clusters (UCs) to generate label-agnostic UEs using cluster-
+wise (rather than class-wise) perturbations. This allows us
+to achieve a simultaneous disruption of the uniformity and
+discrepancy without knowing the label information.
+Arguably, the choose of a proper surrogate model also
+plays an important role in generating effective UEs. Previ-
+ous methods generate UEs by directly attacking a surrogate
+model and then transfer the generated UEs to fight against
+a diverse set of target models [10,17]. This may be easily
+achievable under the label-consistency setting, but may fail
+badly under the label-agnostic setting. However, even un-
+der the label-consistency setting, few works have studied
+the impact of the surrogate model to the final unlearnable
+performance. To generate effective, and more importantly,
+transferable UEs under the label-agnostic setting, we need
+to explore more generic surrogate model selection strategies,
+especially those that can be tailored to a wider range of un-
+known target models. Intuitively, the surrogate model should
+be a classification DNN that contains as many classes as
+possible so as to facilitate the recognition and protection of
+billions of images on the Internet. In this paper, we propose
+to leverage the large-scale Vision-and-Language Pre-trained
+Models (VLPMs) [22,23,30] like CLIP [30] as the surrogate
+model. Pre-trained on over 400 million text-to-image pairs,
+CLIP has the power to extract the representation of extremely
+diverse semantics. Meanwhile, VLPMs are pre-trained with
+a textual description rather than a one-hot label to align with
+the image, making them less overfit to the actual class “la-
+bels”. In this work, we leverage the image encoder of CLIP
+to extract the embeddings of the input images and then use
+the embeddings to generate more transferable UCs.
+We evaluate our UC approach with different backbones
+and datasets, all in a black-box setting (the protector does
+not know the attacker’s network architecture or the class
+labels). Cluster-wise unlearnable noise can also prevent un-
+supervised exploitation against contrastive learning to certain
+extent, proving its superiority to existing UEs. We also com-
+pare UC with existing UE methods against two commercial
+machine learning platforms: Microsoft Azure1 and Baidu
+PaddlePaddle2. To the best of our knowledge, this is the
+first physical-world attack to commercial APIs in this line of
+work.
+Our main contributions are summarized as follows:
+• We promote a more generic data protection assumption
+called label-agnostic, which allows the hackers to ex-
+ploit the protected dataset differently (in terms of the
+annotated class labels) as the protector. This opens up
+a more practical and challenging setting against unau-
+thorized training of machine learning models.
+• We reveal the working mechanism of existing UE gener-
+ation methods: they all disrupt the distributional unifor-
+mity and discrepancy in the deep representation space.
+• We propose a novel approach called Unlearnable Clus-
+ters (UCs) to generate label-agnostic UEs with cluster-
+wise perturbations without knowing the label informa-
+tion. We also leverage VLPMs like CLIP as the surro-
+gate model to craft more transferable UCs.
+• We empirically verify the effectiveness of our proposed
+approach with different backbones on different datasets.
+We also show its effectiveness in protecting private data
+against commercial machine learning platforms Azure
+and PaddlePaddle.
+2. Related Work
+Unlearnable examples (UEs) can be viewed as one special
+type of data poisoning attacks [1,2] that aim to make model
+training fail completely on the poisoned (protected) dataset.
+UEs should be differentiated from the other two well-known
+attacks to deep learning models: backdoor attacks [5, 13,
+24] and adversarial attacks [12, 37]. Backdoor attacks are
+the other special type of data poisoning attacks that do not
+1https://portal.azure.com/
+2https://www.paddlepaddle.org.cn/en/
+2
+
+impact the model’s performance on clean data, which is in
+sharp contrast to UEs. Adversarial attacks are one type of
+test-time attacks that evade the model’s prediction by adding
+small imperceptible adversarial noise to the inputs.
+UEs can be generated via a min-min bilevel optimiza-
+tion framework with a surrogate model [17], similar to
+the generation of strong data poisons via bilevel optimiza-
+tion [18, 34, 36, 45]. The generated noise is termed Error-
+Minimizing Noise (EMinN) as it progressively eliminates
+errors from the training data to trick the target model to be-
+lieve there is nothing to learn [17]. We use EMinN to denote
+the original UE generation method. In addition to EMinN,
+there are also UE generation methods that utilize adversarial
+noise, such as Error-Maximizing Noise (EMaxN) [19], Deep-
+Confuse [9] and Adversarial Poisoning (AdvPoison) [10].
+Recently, Yu et al. [41] unveil a linear-separability property
+of unlearnable noise and propose the Synthetic Perturbations
+(SynPer) method to directly synthesize linearly-separable
+perturbations as effective unlearnable noise.
+The original UE method EMinN has a few limitations.
+First, the generated unlearnable noise can be removed to a
+large extent by adversarial training [26], although this will
+also decrease the model’s performance by a considerable
+amount [17]. This was later on solved by a recent work
+published at ICLR 2022 [11]. The idea is to optimize the
+adversarial training loss in place of the standard training loss
+to produce more robust error-minimizing noise. The other
+limitation is its transferability to different training schemes,
+target models (the models to protect against) or datasets.
+For example, it has been found that unlearnable noise gen-
+erated in a supervised manner fails to protect the dataset
+from unsupervised contrastive learning [14]. A unsupervised
+UE generation method was then proposed to craft UEs un-
+learnable to unsupervised contrastive learning. However, a
+very recent work by Ren et al. [32] demonstrates that, sur-
+prisingly, unsupervised UEs cannot protect the dataset from
+supervised exploitation. All above UE methods all rely on
+the ideal label-consistency assumption, i.e., the same (or
+no) labels for the protected data will be used by both the
+protectors and hackers. In this paper, we promote a more
+practical label-agnostic setting where different labels could
+be used by the hackers for their own purposes.
+Besides UEs, strong adversarial attacks have also been
+proposed to protect personal data from malicious face recog-
+nition systems, such as LowKey [6] and APF [44]. They
+differ from UEs by making a normally trained model unable
+to recognize the protected images, rather than preventing
+the proper training of any machine learning models on the
+protected images. In this work, we focus on UEs rather than
+other data protection techniques which we believe are of
+independent interest.
+3. Proposed Method
+Threat Model.
+We introduce two parties: the protector
+and the hacker. The protectors leverage a surrogate model
+to generate UEs for its private data before publishing it on
+the Internet. For example, online social network companies
+(or users) could convert their photos to their UE versions be-
+fore posting them online. These “protected” images are then
+collected, without the protectors’ consent, by a hacker into a
+dataset to train a commercial or malicious model. The pro-
+tectors’ goal is to make the collected dataset unlearnable, i.e.,
+cannot be used for model training, while the hackers’ goal
+is to train accurate models on the unlearnable (protected)
+dataset. Following prior works [11,17,25], we assume the
+released dataset is 100% protected, i.e., all the samples are
+perturbed to be unlearnable. While this assumption appears
+to be ideal, if the protection technique is reliable, there is no
+reason not to employ it to gain more protection and privacy.
+Therefore, in this work we choose to focus on the unlearn-
+able technique itself rather than changing the setting of the
+protectors. Following our label-agnostic setting, we also as-
+sume the hackers could exploit the unlearnable dataset with
+different labels. E.g., a m-class dataset could be exploited
+by the hacker as a n-class dataset.
+Here, we give an example of such label-agnostic scenario
+with a online social media company who strives to protect
+the contents created by all of its users. The company could
+leverage unlearnable techniques to develop systematic pro-
+tection scheme against unauthorized data explorers. In this
+case, we can assume all the images uploaded by the users are
+protected (by the company). Potential hackers like Clearview
+AI may crawl the images from the online platform without
+the users’ content into one or a set of datasets for its own
+purposes. Thus, the collected datasets cannot be guaranteed
+to have the same labels as their original versions. The pro-
+tector thus needs to craft more powerful and transferable
+unlearnable examples to make data unexploitable against
+different labeling strategies.
+3.1. Problem Formulation
+We focus on image classification tasks in this paper.
+Given a clean m-class training dataset Dm
+c = {(xi, yi)}k
+i=1
+consisting of k clean training images x ∈ X ⊂ Rd and their
+labels y ∈ Y, in a standard unlearnable setting [17], the
+protector trains an m-class surrogate model f m
+s on Dm
+c . The
+protector can then generate an unlearnable version of the
+dataset as Dm
+u = {(x′
+i, y′
+i)}k
+i=1, based on the clean dataset
+Dm
+c and the surrogate model f m
+s . The unlearnable images
+are denoted as x′ = x + δ (x ∈ Dm
+c ) with the same labels
+y ∈ Y as their original versions and δ ∈ ∆ ⊂ Rd are the
+generated unlearnable noise which is often regularized to be
+imperceptible. The unlearnable dataset Dm
+u is assumed to be
+the dataset collected by the hackers, and will be exploited
+to train a commercial or malicious m-class target model f m
+t
+3
+
+EMinN
+AdvPoison
+SynPer
+DeepConfuse
+0
+25
+50
+75
+100
+Accuracy (%)
+19.93
+6.25
+13.54
+28.77
+93.77
+93.6
+93
+93.31
+Label-consistency
+Label-agnostic
+(a)
+Clean
+EMinN
+AdvPoison
+(b)
+Figure 2. (a) Current UE methods become ineffective in the label-
+agnostic setting, even though they exhibit high effectiveness in the
+label-consistency setting (under noise constraint ϵ = 8/255). (b)
+A 3D feature visualization of clean CIFAR-10 examples and the
+UEs derived by EMinN and AdvPoison. Points in the same color
+denote samples of the same class.
+without the protectors’ consent.
+Label-consistency vs. Label-agnostic.
+The above formu-
+lation follows the standard label-consistency assumption of
+previous works [11,17], where the hackers collect, annotate
+and exploit the unlearnable dataset Dm
+u exactly the same
+as it was initially released by the protectors. Under a more
+general and practical label-agnostic assumption, the hackers
+could annotate the collected dataset Dm
+u differently, e.g., as-
+signing it with different number of classes. In this case, the
+hackers may exploit the dataset as a n-class (n ̸= m) classi-
+fication dataset Dn
+c = {(x′
+i, y′
+i)}k
+i=1 to train a n-class target
+model f n
+t . Note that the protectors have no knowledge of the
+target class number n nor the target labels y′
+i. Arguably, the
+hackers may even exploit the dataset as an object detection
+dataset rather than a classification dataset. We will explore
+such a more challenging task-agnostic assumption in our
+future work and focus on the label-agnostic in this work.
+3.2. The Label-agnostic Challenge
+Existing methods are not robust to label-agnostic ex-
+ploitation.
+We test the effectiveness of existing unlearn-
+able methods developed under the label-consistency set-
+ting against label-agnostic hackers. Here we consider cur-
+rent unlearnable method including Error-Minimizing Noise
+(EMinN) [17], Adversarial Poisoning (AdvPoison) [10], Syn-
+thetic Perturbations (SynPer) [41] and DeepConfuse [9], on
+the CIFAR-10 dataset [21]. The ResNet-18 [15] models are
+used for both the surrogate and target models. As shown
+in Figure 2 (a), these methods are extremely effective in
+preventing the training of machine learning models on the
+unlearnable dataset with the same labels. However, if the un-
+learnable dataset is crafted using ImageNet surrogate model
+with the predicted ImageNet labels (i.e., labels predicted by
+the surrogate model), it fails to prevent the model training
+with the original CIFAR-10 labels. This indicates one unique
+challenge of the label-agnostic setting: unlearnable noises
+generated to prevent one set of labels are not transferable to
+preventing other labeling strategies.
+The working mechanism of existing UEs under the label-
+consistency setting.
+Here, we investigate the representa-
+tions learned by the target model on clean vs. unlearnable
+examples, aiming to gain more understanding of the un-
+learnable mechanism. In Figure 2 (b), we visualize the
+3-dimensional PCA [39] projections of the original represen-
+tations learned by the ResNet-18 target model for a) clean
+CIFAR-10 training samples, b) unlearnable CIFAR-10 ex-
+amples crafted by EMinN method, and 3) unlearnable (poi-
+soned) CIFAR-10 examples crafted by AdvPoison. The rep-
+resentations are extracted at the last convolutional layer and
+projected using PCA from 512 to 3 dimensions. It shows in
+Figure 2 (b) that the unlearnable examples crafted by EMinN
+and AdvPoison tend to significantly reduce the variance at
+certain dimensions. There are also classes that collapse into
+smaller clusters, like the green class. This indicates that the
+noise disrupts the distributional discrepancy in the repre-
+sentation space to make the data “unlearnable”. The other
+key observation is that the noise greatly shifts the points
+away from the normal data manifold, causing an unneces-
+sary spread over a certain direction. This indicates that the
+noise also breaks the distributional uniformity of the data.
+Overall, it is evident the unlearnable noise crafted by EMinN
+and AdvPoison cripples the learning process by distorting
+both the discrepancy and uniformity of the data distribution
+in the deep representation space.
+Unlearnable examples can overfit to the labels.
+A closer
+look at the visualizations in Figure 2 (b), one may notice
+that the unlearning effects occur only within the classes. I.e.,
+the UEs have overfitted to the class labels. This is somewhat
+not surprising as the unlearnable noises are generated via
+a supervised loss function (i.e., cross-entropy) defined by
+the labels. The noise are thus optimized to thwart the most
+predictive information to the class labels. However, this
+causes the overfitting problem and fails to work if the labels
+are changed. Intuitively, if we could remove the dependency
+on the class labels and turn to exploit the clusters that natu-
+rally arise during the learning process, we could make the
+unlearnable noise more robust to different annotations.
+3.3. Unlearnable Clusters (UCs)
+Overview.
+Motivated by the above observations, in this
+work we propose to generate UEs by exploiting the clus-
+ters learned by a surrogate model and making the clusters
+unlearnable instead of the labeled classes. We term this
+approach as Unlearnable Clusters (UCs) and illustrate its
+working follow in Figure 3. The key components of UC are
+one generator model G and one surrogate model fs. At a
+high level, UC first employs a surrogate model fs to extract
+the representations E of all samples in the clean dataset Dc.
+It then utilizes the K-means [35] clustering method to derive
+p clusters from the representations E. Subsequently, for
+4
+
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+2兴
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.510.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+2
+2
+2Generator ������������(�; ������������������������)
+Uniform 
+Noise 
+������������
+Cluster-wise 
+Perturbation 
+������������������������
+Unlearnable
+Image
+������������′
+Surrogate Model ������������������������
+Origianl 
+Image
+������������
+������������1
+������������2
+������������3
+Feature Space
+������������1
+′
+Update ������������������������
+Feature Space
+K-means Initial
+Minimize ������������DDU
+×: clustering center ������������������������������������
+×: clustering center ������������������������������������
+������������2
+′
+������������3
+′
+Figure 3. The Unlearnable Clusters pipeline. The entire dataset is divided into p clusters via K-means clustering, where each cluster
+corresponds to a certain generator with parameters θi and a cluster-wise perturbation δi.
+each cluster, it generates a cluster-wise perturbation δi using
+the generator G. The noise will be generated and applied to
+craft the UE for each sample in Dc, with samples belonging
+to the same cluster are added with the same cluster-wise
+noise δi. UEs crafted in this manner can prevent the target
+model from learning meaningful clusters rather than class
+predictions, thus is more general to different types of label
+exploitations. Next, we will introduce the details of UCs.
+Cluster-wise Perturbations.
+In our UC framework, one
+encoder-decoder [29] generator network is used to gener-
+ate the cluster-wise perturbations, with each generator will
+be reinitialized for one cluster. As such, we need to ex-
+tract the clusters first. Here, we leverage the most classic
+clustering method K-means [35] to detect clusters from the
+deep representations. Particularly, the clean dataset Dc is
+fed into the surrogate model fs to extract the representation
+matrix before the classification layer E = [e1, · · · , ek]. K-
+means clustering is then applied on the representation matrix
+to detect p number of clusters C = {C1, · · · , Cp}, where
+Ci = {xij}τ(i)
+j=1 = {xi1, · · · , xiτ(i)} and �p
+i=1 τ(i) =
+k. The corresponding centers for the clusters are µC =
+{µC1, · · · , µCp}.
+With the detected clusters C, we can now propose the
+following method to generate the unlearnable noise for each
+cluster. Intuitively, for cluster Ci, we hope the unlearnable
+noise δi could move all samples in the cluster to a wrong
+cluster center, so as to force the model to forget the cor-
+rect clusters. This is done via the following minimization
+framework:
+θi = arg min
+θi
+LDDU(Ci, g(µCi), θi)
+= arg min
+θi
+�
+xij∈Ci
+d(fs(xij + G(σ; θi)), g(µCi)),
+(1)
+where, LDDU is our proposed Disrupting Discrepancy and
+Uniformity (DDU) loss that defines the distance (d(·)) of
+samples in Ci to a permuted (wrong) cluster center by a per-
+mutation function g(µCi); θi are the parameters of generator
+network G; G(σ; θi)) generates the unlearnable noise for all
+samples in Ci (i.e., xij ∈ Ci). Please note that the above
+problem needs to be solved for p times to obtain the cluster-
+wise unlearnable noise for all p clusters, and for each cluster,
+the generator G is reinitialized with new parameters θi. The
+complete procedure is described in Algorithm 1.
+Algorithm 1 Unlearnable Cluster Generation
+1: Input: surrogate model fs, distance metric d, uniform
+noise σ, number of clusters p, random permutation g,
+L∞-norm restriction ϵ, clean images x ∈ Dc, initialized
+generator G with parameters θ
+2: Output: cluster-wise perturbations δ = {δ1, · · · , δp}
+3: feature matrix E = fs(x)
+4: clusters and cluster centers {C, µC} = K-means(E, p)
+5: for i in 1 · · · p do
+6:
+Initialize θi
+7:
+δi = G(σ; θi)
+8:
+δi = Clamp(δi, −ϵ, ϵ)
+9:
+for xij in Ci do
+10:
+x′
+ij = Clamp(xij + δi, 0, 1)
+11:
+θi ← Optimize(x′
+ij, fs, g(µCi), d)
+12:
+end for
+13:
+δi = G(σ; θi)
+14:
+δi = Clamp(δi, −ϵ, ϵ)
+15: end for
+CLIP Surrogate Model.
+How to choose a surrogate
+model remains to be an independent challenge for gener-
+5
+
+X
+Xating effective cluster-wise unlearnable noise. As shown in
+prior works, it plays a central role in facilitating the trans-
+ferability of the generated UEs to different datasets or target
+models [17]. In the traditional label-consistency setting, the
+surrogate model can be a model that directly trained on the
+original (unprotected) dataset, which may of a different (and
+plausibly a better or more complex) model architecture. It
+could also be a model that trained on a larger dataset with
+more classes, e.g., ImageNet-trained models [10,17]. We
+thus adopt an ImageNet-pretrained ResNet-50 as the default
+surrogate model of our UC.
+Analogous to the classification surrogate models used
+for generating the traditional UEs, the ideal surrogate mod-
+els for unlearnable clusters could be those powerful fea-
+ture extractors that could lead to accurate detection of clus-
+ters from an image dataset. We thus propose to also lever-
+age one large-scale vision-and-language pre-trained model
+(VLPM) [22, 23] CLIP [30] as our surrogate model. Pre-
+trained on over 400 million text-to-image pairs, CLIP has
+the power to extract the representation of extremely diverse
+semantics. Moreover, CLIP was pre-trained with a textual de-
+scription rather than a one-hot label to align with the image,
+thus overfitting less to the actual class labels. Concretely,
+we employ the image encoder of CLIP to extract the feature
+matrix for the clean dataset, which is then used to compute
+the clusters and cluster centers. We denote the version of
+UC equipped with the CLIP surrogate model as UC-CLIP.
+4. Experiments
+In this section, we evaluate our UCs methods on different
+datasets against different target models, which is to simulate
+as many unknown cases as possible. We also examine the ro-
+bustness of UCs against several advanced defenses. Finally,
+we demonstrate its effectiveness in attacking commercial
+machine learning platforms Azure and PaddlePaddle.
+4.1. Experimental Settings
+Datasets and Models.
+We conduct our study on 6 high-
+resolution and industrial-scale vision datasets to simulate
+as diverse real-world applications as possible, including
+Pets [28], Cars [20], Flowers [27], Food [3], SUN397 [40]
+and ImageNet [33]. For ImageNet, we only use its first
+100 classes which is denoted as ImageNet⋆. For surrogate
+models, we consider ResNet-50 trained on ImageNet-1k as
+the default, unless otherwise explicitly stated. For target
+models, we employ randomly initialized ResNet-18 [15],
+EfficientNet-B1 [38] and RegNetX-1.6GF [31]. We train
+the target models with data augmentations (resizing, ran-
+dom crop, random horizontal flip and normalization) for 90
+epochs, using SGD with initial learning rate 0.1, Cosine
+annealing, momentum 0.9, and batch size 256.
+For generator G, we repeated p times to train the generator
+G for 10 epochs using SGD with initial learning rate 0.1,
+10
+20
+30
+Number of class
+0
+20
+40
+60
+Accuracy (%)
+UC
+UC-CLIP
+random guess
+clean
+(a) Different labelings
+Clean
+SynPer
+EMaxN
+EMinN
+AdvPoison
+DeepConfuse
+UC-CLIP
+UC
+0
+10
+20
+30
+40
+50
+Accuracy (%)
+48.3 46.8845.1948.1647.02
+24.37 26.11
+17.83
+(b) Unsupervised exploitation
+Figure 4. (a) The accuracy of ResNet-18 target models trained on
+the unlearnable Pets dataset but with its labels were re-labeled by
+the hacker into 5 to 35 classes. (b) Comparison of our approach
+with the baselines on Pets dataset against ResNet-18 target model
+trained via self-supervised SimCLR.
+Cosine annealing, momentum 0.9, and batch size 256, and
+10 epochs for ImageNet⋆ and 50 epochs for other datasets.
+For random permutation g(·), we simply chose i → i + 1 to
+build a closed loop. We consider L∞-norm restriction in this
+work, i.e., ∥δ∥∞ < ϵ = 16/255. The number of clusters p
+is set to 10, with an analysis is provided in Section 4.5.
+Baselines.
+We compare our UC and UC-CLIP with 5 base-
+line methods including DeepConfuse [9], Synthetic Perturba-
+tions (SynPer) [41], Error-minimizing Noise (EMinN) [17],
+Error-maximizing Noise(EMaxN) [19], and Adversarial Poi-
+soning (AdvPoison) [10]. We use their official implementa-
+tions and follow the suggested settings in the original papers
+to generate the UEs or poisons.
+Label-agnostic Setup.
+Please note that we conduct all of
+our experiments under the proposed label-agnostic setting.
+The UCs (and the UEs they serve) are all generated with the
+predicted labels by the surrogate models. The predicted la-
+bels may overlap with the ground truth labels to some extent,
+but are highly inconsistent with the original labels as the sur-
+rogate models are not trained on the particular datasets. The
+hackers train all target models on the unlearnable datasets
+with their ground truth labels. We report the test accuracy of
+the target models on the respective clean test sets.
+4.2. Main Results
+Effectiveness against different target models.
+We first
+compare our UC and UC-CLIP with the 5 baselines against
+different target models. Table 1 shows the results against
+ResNet-18, EfficientNet-B1, and RegNetX-1.6GF. We have
+the following main findings: (1) Our methods outperform
+the baselines by a huge margin consistently across different
+datasets and target models. This demonstrates the superi-
+ority of our methods over the baselines. (2) Our UC-CLIP
+achieves a better performance than UC, and in most of the
+cases, by a considerable margin. This proves the great poten-
+tial of using CLIP as the surrogate model to protect person
+data from unauthorized exploitations.
+6
+
+Table 1. The test accuracy (%) of different target models trained on the unlearnable datasets generated by our UC/UC-CLIP and the 5
+baseline methods, under the label-agnostic setting. The top-2 best results are highlighted in bold.
+RESNET-18
+EFFICIENTNET-B1
+REGNETX-1.6GF
+METHODS
+PETS CARS FLOWERS FOOD SUN397 IMAGENET⋆ PETS CARS FLOWERS FOOD SUN397 IMAGENET⋆ PETS CARS FLOWERS FOOD SUN397 IMAGENET⋆
+CLEAN
+62.31 67.18
+67.18
+78.97
+43.08
+77.76
+48.68 72.33
+52.46
+80.29
+42.84
+78.04
+44.86 63.84
+52.69
+84.02
+43.27
+80.78
+SYNPER
+52.60 53.50
+52.74
+74.80
+38.26
+74.69
+28.02 58.34
+42.93
+74.99
+35.92
+72.94
+34.51 45.54
+47.16
+77.65
+37.78
+60.38
+EMAXN
+54.70 52.95
+51.70
+73.77
+37.57
+73.82
+33.71 55.64
+42.66
+74.40
+37.30
+73.72
+34.26 43.40
+46.25
+78.76
+37.82
+76.72
+EMINN
+52.96 54.43
+50.58
+75.47
+38.48
+74.20
+36.88 54.23
+44.06
+75.54
+37.20
+72.20
+37.04 39.67
+47.34
+79.43
+36.82
+74.86
+ADVPOISON
+50.86 51.91
+50.64
+75.07
+38.51
+73.76
+37.99 50.08
+41.65
+74.88
+36.44
+72.54
+34.29 46.06
+47.41
+78.64
+36.42
+76.32
+DEEPCONFUSE
+53.72 51.11
+50.94
+73.13
+34.41
+55.12
+35.54 47.15
+43.28
+72.91
+35.22
+45.74
+33.71 41.15
+46.01
+77.26
+33.52
+49.88
+UC (OURS)
+12.21 33.57
+35.55
+55.29 20.38
+54.80
+17.06 13.92
+42.28
+53.45 22.97
+32.30
+4.28 29.46
+33.79
+64.48 22.28
+56.10
+UC-CLIP (OURS) 4.69
+4.74
+10.07
+19.07
+3.89
+39.78
+6.49 15.33
+14.13
+17.44 12.95
+31.82
+3.87
+4.18
+8.12
+26.76
+6.04
+41.66
+Effectiveness Against Different Labelings.
+An even
+more challenging label-agnostic setting is that the hacker
+may exploit the unlearnable dataset using different labeling
+strategies instead of one. So, a natural question is that what
+if the number of labeled classes of the unlearnable dataset
+is less than our cluster number p = 10? Here, we take the
+37-class Pets dataset as an example and explore the impact
+if the hacker re-labels the unlearnable version of the dataset
+as a 5 to 36 class dataset. One possible labeling strategy is
+that the hacker first extracts the embeddings of the original
+text labels using the BERT model [8], and then clusters the
+embeddings into 5-37 classes using K-means, so as to con-
+struct a mapping from the old labels to the new labels. As
+shown in Figure 4 (a), both our UC and UC-CLIP can bring
+the test accuracy of the target model down to a level that is
+close the random guess (the black curve). This verifies that
+our methods can craft more generic UEs against the most
+severe label-agnostic exploitations.
+Robustness to Unsupervised Exploitation.
+We also com-
+pare our methods with the baselines under an unsupervised
+contrastive learning setting against SimCLR [4]. Although
+our UC methods are not specifically designed for this un-
+supervised setting, Figure 4 (b) shows that cluster-wise un-
+learnable noise can also prevent unsupervised exploitation
+against SimCLR.
+4.3. Preventing Commercial Platforms
+Here, we apply our UC methods to prevent two com-
+mercial machine learning platforms: Microsoft Azure and
+Baidu PaddlePaddle. On both platforms, the training
+details are agnostic to us, including the model architecture,
+learning rate, batch size, epoch, data augmentation, splitting
+of the validation set, etc. Considering that ViT may be used
+on commercial platforms due to its recent popularity, we
+upgrade our UC-CLIP method by replacing the ResNet-50
+(RN50) surrogate model by a ViT-B-32 (ViTB32) surrogate
+model. The results are reported in Table 2, which are consis-
+Table 2. The test accuracy (%) of models trained by Azure and
+PaddlePaddle platforms on unlearnable Cars dataset crafted by
+different methods. The training configuration on the platform was
+set to “fastest training”.
+METHODS
+Azure
+PaddlePaddle
+CLEAN
+48.45
+83.74
+SYNPER
+42.38
+47.59
+EMAXN
+42.83
+42.99
+EMINN
+44.06
+44.40
+ADVPOISON
+43.97
+43.38
+DEEPCONFUSE
+39.47
+41.88
+UC (RN50)
+36.40
+30.96
+UC-CLIP (RN50)
+26.97
+25.79
+UC-CLIP (VITB32)
+22.47
+11.49
+tent with that in Table 1. I.e., both of our methods can protect
+the data uploaded to the two platforms against their training
+algorithms. Unsurprisingly, the ViTB32-powered UC-CLIP
+method achieves the best protection performance by causing
+the lowest test accuracy. This suggests the effectiveness of
+our methods even against commercial platforms.
+4.4. Resistance to Potential Defenses
+In this section, we test the robustness of our UC
+methods to several augmentation based defenses, includ-
+ing Mixup [43], Gaussian smoothing, Cutmix [42] and
+Cutout [7]. We did not consider adversarial training here is
+that the images involved in this paper are generally large in
+size, number and resolution, making adversarial training ex-
+tremely hard to converge without losing considerable clean
+accuracy on most of the datasets [26]. Compared with ad-
+versarial training, we believe that a more practical defense
+should be very efficient, such as image denoising or the con-
+sidered augmentation techniques. As can be observed in
+Table 3, the 4 data augmentation defenses have minimum
+impact on our UC and UC-CLIP methods. Particularly, Gaus-
+sian smoothing appears to be the most effective defense, but
+the accuracy is still below 25%.
+7
+
+Table 3. The test accuracy (%) of ResNet-18 trained using different
+defenses against our methods on Pets dataset.
+METHODS
+NO DEFENSE
+MIXUP
+GAUSSIAN
+CUTMIX
+CUTOUT
+UC
+12.21
+14.34
+24.26
+14.50
+12.35
+UC-CLIP
+4.69
+11.96
+18.59
+6.21
+12.29
+4.5. Ablation Study
+5
+10 15 20 25 30 35 40
+Number of clusters
+0
+5
+10
+15
+Accuracy (%)
+16.08
+12.21
+4.42 3.82 3.46 2.75
+4.51 3.92
+(a) Effect of p on UC
+5
+10 15 20 25 30 35 40
+Number of clusters
+0
+5
+10
+15
+Accuracy (%)
+6.38
+4.69
+3.11 3.33 3.33 2.73
+4.09
+2.67
+(b) Effect of p on UC-CLIP
+Figure 5. Analyzing the effect of cluster number p on Pets dataset.
+Here, we analyze the sensitivity of our methods to the
+number of clusters p, which has been set to p = 10 as a
+default. We take the 37-class Pets dataset as an example
+and evaluate our UC and UC-CLIP method under different
+values of p ∈ [5, 40]. As shown in Figure 5, our methods are
+quite stable to varying hyperparameter p for p ≥ 10. This
+indicates that, as long as the clusters can cover most of the
+concepts in a dataset, the generated unlearnable noise can
+effectively prevent the model from learning the real content
+from the dataset. As the number of clusters increases, the
+noise tends to become more effective, although there is a
+slight variation at 35. Note that, even in the worst case at
+p = 5, our methods still outperform the baselines.
+4.6. Mixture of Clean and Unlearnable Data
+0
+5
+10 15 20 25 30 35
+Number of classes
+0
+10
+20
+30
+40
+50
+60
+Accuracy (%)
+clean-only
+mixture
+(a) Mixture vs. Clean-only
+0
+25
+50
+75 100 125 150
+Epoch
+0
+20
+40
+60
+80
+100
+Accuracy (%)
+training (unlearnable)
+test (clean)
+test (unlearnable)
+(b) Accuracy trends
+Figure 6.
+(a) The test accuracy (%) of ResNet-18 trained on
+unlearnable-clean mixed vs. clean-only data; and (b) the accu-
+racy trends on clean vs. unlearnable examples. The unlearnable
+examples are crafted using our UC method on Pets dataset.
+All the above experiments are conducted under the as-
+sumption that all samples in the dataset are protected, a
+commonly adopted assumption in the literature [10,17,41].
+This setting is reasonable when the protectors have the access
+to the entire dataset, e.g., an online social media company
+adopts the technique to protect the contents created by all
+of its users. A more general case is that only a certain pro-
+portion of the users protect their data while others do not.
+This results in mixed dataset with both clean and unlearnable
+samples. Here we test our UC method under this setting
+and show the change in test accuracy with the number of
+clean classes in Figure 6 (a). I.e., for the mixture dataset,
+the rest of the classes are made unlearnable by UC. It can be
+inferred that the unlearnable classes almost do not contribute
+to the model training, a similar conclusion as in previous
+works [10,17,41]. This implies that only those who adopt
+the technique will get protected.
+4.7. More Understanding
+Why our UCs are more powerful than standard UEs
+against label-agnostic exploitation? As we explained in
+Section 3.1, the idea of UCs is inspired by the effectiveness
+of disrupting the uniformity and discrepancy in preventing
+the model from learning useful information. However, this
+also raises another question: what exactly does the target
+model learn? To answer these two questions, here we ana-
+lyze the learning curves of the target model on the clean vs.
+unlearnable examples separately. As shown in Figure 6 (b),
+as the training progresses, the training accuracy on the un-
+learnable training samples steadily improves until it reaches
+100%. But there is almost no improvement in the clean test
+accuracy on the clean test samples. This is consistent with
+the the above experimental results that the target model has
+not learned the capability to perceive normal samples. Sur-
+prisingly, however, the model’s accuracy on the perturbed
+test samples is fairly high (> 60%), considering that the
+normally trained ResNet-18 only achieves a test accuracy of
+62.31% on clean Pets dataset. This implies that the unlearn-
+able noise distribution contained in the UCs has effectively
+concealed the real data distribution.
+5. Conclusion
+Unlearnable examples (UEs) have shown great potential
+in preventing hackers from using users’ private data to train
+commercial or malicious models. A number of methods
+have been proposed to improve UEs’ transferability and ro-
+bustness to different datasets, target models and training
+paradigms. In this work, we identified one limitation of ex-
+isting UE methods, i.e., their label-consistency assumption.
+To overcome this limitation, we proposed a more general
+setting where the hackers could exploit the protected data
+with different sets of labels. We termed this more challeng-
+ing setting as label-agnostic, and proposed an Unlearnable
+Clusters (UCs) technique with conditioned generator models,
+K-means clustering, and large-scale vision-and-language pre-
+training model CLIP, to craft effective UEs against a wide
+range of datasets and target models. We also demonstrate
+its effectiveness against commercial platforms Microsoft
+Azure and Baidu PaddlePaddle.
+8
+
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diff --git a/1dAzT4oBgHgl3EQfRfuL/content/tmp_files/load_file.txt b/1dAzT4oBgHgl3EQfRfuL/content/tmp_files/load_file.txt
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@@ -0,0 +1,790 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf,len=789
+page_content='Unlearnable Clusters: Towards Label-agnostic Unlearnable Examples  Jiaming Zhang1 Xingjun Ma2* Qi Yi1 Jitao Sang1,4∗ Yugang Jiang2 Yaowei Wang4 Changsheng Xu3,4  1Beijing Jiaotong University 2Fudan University 3Chinese Academy of Sciences 4Peng Cheng Lab  Abstract  There is a growing interest in developing unlearnable  examples (UEs) against visual privacy leaks on the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  UEs are training samples added with invisible but unlearn-  able noise, which have been found can prevent unauthorized  training of machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' UEs typically are gen-  erated via a bilevel optimization framework with a surrogate  model to remove (minimize) errors from the original samples,  and then applied to protect the data against unknown target  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' However, existing UE generation methods all rely  on an ideal assumption called label-consistency, where the  hackers and protectors are assumed to hold the same label  for a given sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In this work, we propose and promote  a more practical label-agnostic setting, where the hackers  may exploit the protected data quite differently from the  protectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', a m-class unlearnable dataset held by the  protector may be exploited by the hacker as a n-class dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Existing UE generation methods are rendered ineffective in  this challenging setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' To tackle this challenge, we present  a novel technique called Unlearnable Clusters (UCs) to gen-  erate label-agnostic unlearnable examples with cluster-wise  perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Furthermore, we propose to leverage Vision-  and-Language Pre-trained Models (VLPMs) like CLIP as the  surrogate model to improve the transferability of the crafted  UCs to diverse domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We empirically verify the effective-  ness of our proposed approach under a variety of settings  with different datasets, target models, and even commercial  platforms Microsoft Azure and Baidu PaddlePaddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Introduction  While the huge amount of “free” data available on the  Internet has been key to the success of deep learning and  computer vision, this has also raised public concerns on the  unauthorized exploitation of personal data uploaded to the  Internet to train commercial or even malicious models [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  For example, a company named Clearview AI has been  found to have scraped billions of personal images from Face-  book, YouTube, Venmo and millions of other websites to  construct a commercial facial recognition application [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Corresponding authors  Unconscious and unauthorized collection for training  Label-consistency  Label-agnostic  Original images ������������  Unlearnable image ������������′  Hacker  Protector  ������������-class surrogate  model ������������������������������������  =  car, dog, cat,  airplane, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  ������������-class target  model ������������������������  ������������  car, dog, cat,  airplane, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  =  ������������-class target  model ������������������������  ������������  =  animal,  vehicle, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Hacker  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' An illustration of two different data protection assump-  tions: label-consistency vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' label-agnostic, where the hacker ex-  ploits the protected data in different manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  This has motivated the proposal of Unlearnable Examples  (UEs) [17] to make data unlearnable (or unusable) to ma-  chine learning models/services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Similar techniques are also  known as availability attacks [2,41] or indiscriminate poi-  soning attacks [14] in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' These techniques allow  users to actively adding protective noise into their private  data to avoid unauthorized exploitation, rather than putting  our trust into the hands of large corporations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  The original UE generation method generates error-  minimizing noise via a bilevel min-min optimization frame-  work with a surrogate model [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The noise can then be  added to samples in a training set in either a sample-wise  or class-wise manner to make the entire dataset unlearnable  to different DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' It has been found that this method can-  not survive adversarial training, which has been addressed  by a recent method [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In this work, we identify one  common assumption made by existing UE methods: label-  consistency, where the hackers will exploit the protected  dataset in the same way as the protector including the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  This means that, for the same image, the hacker and protec-  tor hold the same label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We argue that this assumption is  too ideal, and it is possible that the hackers will collect the  protected (unlearnable) samples into a dataset for a different  task and label the dataset into different number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  As illustrated in Figure 1, an image can be labelled with  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='01217v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='CR]  31 Dec 2022  盒盒000000different annotated labels (cat or animal), showing that a m-  class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', 10-class) unlearnable dataset may be exploited  by the hacker as a n-class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', 5-class or 20-class) dataset  depending on its actual needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We term this more generic  assumption as label-agnostic and propose a novel method  Unlearnable Clusters (UCs) to generate more effective and  transferable unlearnable examples under this harsh setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  In Figure 2 (a), we show that this more generic label-  agnostic setting poses a unique transferability challenge  for the noise generated by existing methods like Error-  Minimizing Noise (EMinN) [17], Adversarial Poisoning  (AdvPoison) [10], Synthetic Perturbations (SynPer) [41] and  DeepConfuse [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This indicates that the protective noise  generated by these methods are label-dependent and are ren-  dered ineffective when presented with different number of  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' As such, we need more fundamental approaches  to make a dataset unlearnable regardless of the annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  To this end, we start by analyzing the working mechanism  of UEs generated by EMinN, AdvPoison as they are very  representative under the label-consistency setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Through  a set of visual analyses, we find that the main reason why  they could break supervised learners is that the generated  noise tends to disrupts the distributional uniformity and dis-  crepancy in the deep representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Uniformity refers  to the property that the manifold of UEs in the deep rep-  resentation space does not deviate much from that of the  clean examples, while discrepancy refers to the property  that examples belonging to the same class are richly diverse  in the representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Inspired by the above obser-  vation, we propose a novel approach called Unlearnable  Clusters (UCs) to generate label-agnostic UEs using cluster-  wise (rather than class-wise) perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This allows us  to achieve a simultaneous disruption of the uniformity and  discrepancy without knowing the label information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Arguably, the choose of a proper surrogate model also  plays an important role in generating effective UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Previ-  ous methods generate UEs by directly attacking a surrogate  model and then transfer the generated UEs to fight against  a diverse set of target models [10,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This may be easily  achievable under the label-consistency setting, but may fail  badly under the label-agnostic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' However, even un-  der the label-consistency setting, few works have studied  the impact of the surrogate model to the final unlearnable  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' To generate effective, and more importantly,  transferable UEs under the label-agnostic setting, we need  to explore more generic surrogate model selection strategies,  especially those that can be tailored to a wider range of un-  known target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Intuitively, the surrogate model should  be a classification DNN that contains as many classes as  possible so as to facilitate the recognition and protection of  billions of images on the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In this paper, we propose  to leverage the large-scale Vision-and-Language Pre-trained  Models (VLPMs) [22,23,30] like CLIP [30] as the surrogate  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Pre-trained on over 400 million text-to-image pairs,  CLIP has the power to extract the representation of extremely  diverse semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Meanwhile, VLPMs are pre-trained with  a textual description rather than a one-hot label to align with  the image, making them less overfit to the actual class “la-  bels”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In this work, we leverage the image encoder of CLIP  to extract the embeddings of the input images and then use  the embeddings to generate more transferable UCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  We evaluate our UC approach with different backbones  and datasets, all in a black-box setting (the protector does  not know the attacker’s network architecture or the class  labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Cluster-wise unlearnable noise can also prevent un-  supervised exploitation against contrastive learning to certain  extent, proving its superiority to existing UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We also com-  pare UC with existing UE methods against two commercial  machine learning platforms: Microsoft Azure1 and Baidu  PaddlePaddle2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' To the best of our knowledge, this is the  first physical-world attack to commercial APIs in this line of  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Our main contributions are summarized as follows:  We promote a more generic data protection assumption  called label-agnostic, which allows the hackers to ex-  ploit the protected dataset differently (in terms of the  annotated class labels) as the protector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This opens up  a more practical and challenging setting against unau-  thorized training of machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  We reveal the working mechanism of existing UE gener-  ation methods: they all disrupt the distributional unifor-  mity and discrepancy in the deep representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  We propose a novel approach called Unlearnable Clus-  ters (UCs) to generate label-agnostic UEs with cluster-  wise perturbations without knowing the label informa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We also leverage VLPMs like CLIP as the surro-  gate model to craft more transferable UCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  We empirically verify the effectiveness of our proposed  approach with different backbones on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  We also show its effectiveness in protecting private data  against commercial machine learning platforms Azure  and PaddlePaddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Related Work  Unlearnable examples (UEs) can be viewed as one special  type of data poisoning attacks [1,2] that aim to make model  training fail completely on the poisoned (protected) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  UEs should be differentiated from the other two well-known  attacks to deep learning models: backdoor attacks [5, 13,  24] and adversarial attacks [12, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Backdoor attacks are  the other special type of data poisoning attacks that do not  1https://portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='azure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='com/  2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='paddlepaddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='cn/en/  2  impact the model’s performance on clean data, which is in  sharp contrast to UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Adversarial attacks are one type of  test-time attacks that evade the model’s prediction by adding  small imperceptible adversarial noise to the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  UEs can be generated via a min-min bilevel optimiza-  tion framework with a surrogate model [17], similar to  the generation of strong data poisons via bilevel optimiza-  tion [18, 34, 36, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The generated noise is termed Error-  Minimizing Noise (EMinN) as it progressively eliminates  errors from the training data to trick the target model to be-  lieve there is nothing to learn [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We use EMinN to denote  the original UE generation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In addition to EMinN,  there are also UE generation methods that utilize adversarial  noise, such as Error-Maximizing Noise (EMaxN) [19], Deep-  Confuse [9] and Adversarial Poisoning (AdvPoison) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Recently, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' [41] unveil a linear-separability property  of unlearnable noise and propose the Synthetic Perturbations  (SynPer) method to directly synthesize linearly-separable  perturbations as effective unlearnable noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  The original UE method EMinN has a few limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  First, the generated unlearnable noise can be removed to a  large extent by adversarial training [26], although this will  also decrease the model’s performance by a considerable  amount [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This was later on solved by a recent work  published at ICLR 2022 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The idea is to optimize the  adversarial training loss in place of the standard training loss  to produce more robust error-minimizing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The other  limitation is its transferability to different training schemes,  target models (the models to protect against) or datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  For example, it has been found that unlearnable noise gen-  erated in a supervised manner fails to protect the dataset  from unsupervised contrastive learning [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' A unsupervised  UE generation method was then proposed to craft UEs un-  learnable to unsupervised contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' However, a  very recent work by Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' [32] demonstrates that, sur-  prisingly, unsupervised UEs cannot protect the dataset from  supervised exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' All above UE methods all rely on  the ideal label-consistency assumption, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', the same (or  no) labels for the protected data will be used by both the  protectors and hackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In this paper, we promote a more  practical label-agnostic setting where different labels could  be used by the hackers for their own purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Besides UEs, strong adversarial attacks have also been  proposed to protect personal data from malicious face recog-  nition systems, such as LowKey [6] and APF [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' They  differ from UEs by making a normally trained model unable  to recognize the protected images, rather than preventing  the proper training of any machine learning models on the  protected images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In this work, we focus on UEs rather than  other data protection techniques which we believe are of  independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Proposed Method  Threat Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  We introduce two parties: the protector  and the hacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The protectors leverage a surrogate model  to generate UEs for its private data before publishing it on  the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' For example, online social network companies  (or users) could convert their photos to their UE versions be-  fore posting them online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' These “protected” images are then  collected, without the protectors’ consent, by a hacker into a  dataset to train a commercial or malicious model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The pro-  tectors’ goal is to make the collected dataset unlearnable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=',  cannot be used for model training, while the hackers’ goal  is to train accurate models on the unlearnable (protected)  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Following prior works [11,17,25], we assume the  released dataset is 100% protected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', all the samples are  perturbed to be unlearnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' While this assumption appears  to be ideal, if the protection technique is reliable, there is no  reason not to employ it to gain more protection and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Therefore, in this work we choose to focus on the unlearn-  able technique itself rather than changing the setting of the  protectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Following our label-agnostic setting, we also as-  sume the hackers could exploit the unlearnable dataset with  different labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', a m-class dataset could be exploited  by the hacker as a n-class dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Here, we give an example of such label-agnostic scenario  with a online social media company who strives to protect  the contents created by all of its users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The company could  leverage unlearnable techniques to develop systematic pro-  tection scheme against unauthorized data explorers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In this  case, we can assume all the images uploaded by the users are  protected (by the company).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Potential hackers like Clearview  AI may crawl the images from the online platform without  the users’ content into one or a set of datasets for its own  purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Thus, the collected datasets cannot be guaranteed  to have the same labels as their original versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The pro-  tector thus needs to craft more powerful and transferable  unlearnable examples to make data unexploitable against  different labeling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Problem Formulation  We focus on image classification tasks in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Given a clean m-class training dataset Dm  c = {(xi, yi)}k  i=1  consisting of k clean training images x ∈ X ⊂ Rd and their  labels y ∈ Y, in a standard unlearnable setting [17], the  protector trains an m-class surrogate model f m  s on Dm  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The  protector can then generate an unlearnable version of the  dataset as Dm  u = {(x′  i, y′  i)}k  i=1, based on the clean dataset  Dm  c and the surrogate model f m  s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The unlearnable images  are denoted as x′ = x + δ (x ∈ Dm  c ) with the same labels  y ∈ Y as their original versions and δ ∈ ∆ ⊂ Rd are the  generated unlearnable noise which is often regularized to be  imperceptible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The unlearnable dataset Dm  u is assumed to be  the dataset collected by the hackers, and will be exploited  to train a commercial or malicious m-class target model f m  t  3  EMinN  AdvPoison  SynPer  DeepConfuse  0  25  50  75  100  Accuracy (%)  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='93  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='25  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='54  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='77  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='77  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='6  93  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='31  Label-consistency  Label-agnostic  (a)  Clean  EMinN  AdvPoison  (b)  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' (a) Current UE methods become ineffective in the label-  agnostic setting, even though they exhibit high effectiveness in the  label-consistency setting (under noise constraint ϵ = 8/255).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' (b)  A 3D feature visualization of clean CIFAR-10 examples and the  UEs derived by EMinN and AdvPoison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Points in the same color  denote samples of the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  without the protectors’ consent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Label-consistency vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Label-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  The above formu-  lation follows the standard label-consistency assumption of  previous works [11,17], where the hackers collect, annotate  and exploit the unlearnable dataset Dm  u exactly the same  as it was initially released by the protectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Under a more  general and practical label-agnostic assumption, the hackers  could annotate the collected dataset Dm  u differently, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', as-  signing it with different number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In this case, the  hackers may exploit the dataset as a n-class (n ̸= m) classi-  fication dataset Dn  c = {(x′  i, y′  i)}k  i=1 to train a n-class target  model f n  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Note that the protectors have no knowledge of the  target class number n nor the target labels y′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Arguably, the  hackers may even exploit the dataset as an object detection  dataset rather than a classification dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We will explore  such a more challenging task-agnostic assumption in our  future work and focus on the label-agnostic in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The Label-agnostic Challenge  Existing methods are not robust to label-agnostic ex-  ploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  We test the effectiveness of existing unlearn-  able methods developed under the label-consistency set-  ting against label-agnostic hackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Here we consider cur-  rent unlearnable method including Error-Minimizing Noise  (EMinN) [17], Adversarial Poisoning (AdvPoison) [10], Syn-  thetic Perturbations (SynPer) [41] and DeepConfuse [9], on  the CIFAR-10 dataset [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The ResNet-18 [15] models are  used for both the surrogate and target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' As shown  in Figure 2 (a), these methods are extremely effective in  preventing the training of machine learning models on the  unlearnable dataset with the same labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' However, if the un-  learnable dataset is crafted using ImageNet surrogate model  with the predicted ImageNet labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', labels predicted by  the surrogate model), it fails to prevent the model training  with the original CIFAR-10 labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This indicates one unique  challenge of the label-agnostic setting: unlearnable noises  generated to prevent one set of labels are not transferable to  preventing other labeling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  The working mechanism of existing UEs under the label-  consistency setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Here, we investigate the representa-  tions learned by the target model on clean vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' unlearnable  examples, aiming to gain more understanding of the un-  learnable mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In Figure 2 (b), we visualize the  3-dimensional PCA [39] projections of the original represen-  tations learned by the ResNet-18 target model for a) clean  CIFAR-10 training samples, b) unlearnable CIFAR-10 ex-  amples crafted by EMinN method, and 3) unlearnable (poi-  soned) CIFAR-10 examples crafted by AdvPoison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The rep-  resentations are extracted at the last convolutional layer and  projected using PCA from 512 to 3 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' It shows in  Figure 2 (b) that the unlearnable examples crafted by EMinN  and AdvPoison tend to significantly reduce the variance at  certain dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' There are also classes that collapse into  smaller clusters, like the green class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This indicates that the  noise disrupts the distributional discrepancy in the repre-  sentation space to make the data “unlearnable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The other  key observation is that the noise greatly shifts the points  away from the normal data manifold, causing an unneces-  sary spread over a certain direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This indicates that the  noise also breaks the distributional uniformity of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Overall, it is evident the unlearnable noise crafted by EMinN  and AdvPoison cripples the learning process by distorting  both the discrepancy and uniformity of the data distribution  in the deep representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Unlearnable examples can overfit to the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  A closer  look at the visualizations in Figure 2 (b), one may notice  that the unlearning effects occur only within the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=',  the UEs have overfitted to the class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This is somewhat  not surprising as the unlearnable noises are generated via  a supervised loss function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', cross-entropy) defined by  the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The noise are thus optimized to thwart the most  predictive information to the class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' However, this  causes the overfitting problem and fails to work if the labels  are changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Intuitively, if we could remove the dependency  on the class labels and turn to exploit the clusters that natu-  rally arise during the learning process, we could make the  unlearnable noise more robust to different annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Unlearnable Clusters (UCs)  Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Motivated by the above observations, in this  work we propose to generate UEs by exploiting the clus-  ters learned by a surrogate model and making the clusters  unlearnable instead of the labeled classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We term this  approach as Unlearnable Clusters (UCs) and illustrate its  working follow in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The key components of UC are  one generator model G and one surrogate model fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' At a  high level, UC first employs a surrogate model fs to extract  the representations E of all samples in the clean dataset Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  It then utilizes the K-means [35] clustering method to derive  p clusters from the representations E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Subsequently, for  4  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
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+page_content='510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
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+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='5  2  2  2Generator ������������(�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' ������������������������)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Uniform  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Noise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Cluster-wise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Perturbation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Unlearnable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='������������′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Surrogate Model ������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Origianl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='������������1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='������������2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='������������3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Feature Space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='������������1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Update ������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Feature Space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='K-means Initial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Minimize ������������DDU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='×: clustering center ������������������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='×: clustering center ������������������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='������������2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='������������3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The Unlearnable Clusters pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The entire dataset is divided into p clusters via K-means clustering, where each cluster  corresponds to a certain generator with parameters θi and a cluster-wise perturbation δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  each cluster, it generates a cluster-wise perturbation δi using  the generator G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The noise will be generated and applied to  craft the UE for each sample in Dc, with samples belonging  to the same cluster are added with the same cluster-wise  noise δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' UEs crafted in this manner can prevent the target  model from learning meaningful clusters rather than class  predictions, thus is more general to different types of label  exploitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Next, we will introduce the details of UCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Cluster-wise Perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  In our UC framework, one  encoder-decoder [29] generator network is used to gener-  ate the cluster-wise perturbations, with each generator will  be reinitialized for one cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' As such, we need to ex-  tract the clusters first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Here, we leverage the most classic  clustering method K-means [35] to detect clusters from the  deep representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Particularly, the clean dataset Dc is  fed into the surrogate model fs to extract the representation  matrix before the classification layer E = [e1, · · · , ek].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' K-  means clustering is then applied on the representation matrix  to detect p number of clusters C = {C1, · · · , Cp}, where  Ci = {xij}τ(i)  j=1 = {xi1, · · · , xiτ(i)} and �p  i=1 τ(i) =  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The corresponding centers for the clusters are µC =  {µC1, · · · , µCp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  With the detected clusters C, we can now propose the  following method to generate the unlearnable noise for each  cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Intuitively, for cluster Ci, we hope the unlearnable  noise δi could move all samples in the cluster to a wrong  cluster center, so as to force the model to forget the cor-  rect clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This is done via the following minimization  framework:  θi = arg min  θi  LDDU(Ci, g(µCi), θi)  = arg min  θi  �  xij∈Ci  d(fs(xij + G(σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' θi)), g(µCi)),  (1)  where, LDDU is our proposed Disrupting Discrepancy and  Uniformity (DDU) loss that defines the distance (d(·)) of  samples in Ci to a permuted (wrong) cluster center by a per-  mutation function g(µCi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' θi are the parameters of generator  network G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' G(σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' θi)) generates the unlearnable noise for all  samples in Ci (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', xij ∈ Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Please note that the above  problem needs to be solved for p times to obtain the cluster-  wise unlearnable noise for all p clusters, and for each cluster,  the generator G is reinitialized with new parameters θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The  complete procedure is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Algorithm 1 Unlearnable Cluster Generation  1: Input: surrogate model fs, distance metric d, uniform  noise σ, number of clusters p, random permutation g,  L∞-norm restriction ϵ, clean images x ∈ Dc, initialized  generator G with parameters θ  2: Output: cluster-wise perturbations δ = {δ1, · · · , δp}  3: feature matrix E = fs(x)  4: clusters and cluster centers {C, µC} = K-means(E, p)  5: for i in 1 · · · p do  6:  Initialize θi  7:  δi = G(σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' θi)  8:  δi = Clamp(δi, −ϵ, ϵ)  9:  for xij in Ci do  10:  x′  ij = Clamp(xij + δi, 0, 1)  11:  θi ← Optimize(x′  ij, fs, g(µCi), d)  12:  end for  13:  δi = G(σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' θi)  14:  δi = Clamp(δi, −ϵ, ϵ)  15: end for  CLIP Surrogate Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  How to choose a surrogate  model remains to be an independent challenge for gener-  5  X  Xating effective cluster-wise unlearnable noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' As shown in  prior works, it plays a central role in facilitating the trans-  ferability of the generated UEs to different datasets or target  models [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In the traditional label-consistency setting, the  surrogate model can be a model that directly trained on the  original (unprotected) dataset, which may of a different (and  plausibly a better or more complex) model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' It  could also be a model that trained on a larger dataset with  more classes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', ImageNet-trained models [10,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We  thus adopt an ImageNet-pretrained ResNet-50 as the default  surrogate model of our UC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Analogous to the classification surrogate models used  for generating the traditional UEs, the ideal surrogate mod-  els for unlearnable clusters could be those powerful fea-  ture extractors that could lead to accurate detection of clus-  ters from an image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We thus propose to also lever-  age one large-scale vision-and-language pre-trained model  (VLPM) [22, 23] CLIP [30] as our surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Pre-  trained on over 400 million text-to-image pairs, CLIP has  the power to extract the representation of extremely diverse  semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Moreover, CLIP was pre-trained with a textual de-  scription rather than a one-hot label to align with the image,  thus overfitting less to the actual class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Concretely,  we employ the image encoder of CLIP to extract the feature  matrix for the clean dataset, which is then used to compute  the clusters and cluster centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We denote the version of  UC equipped with the CLIP surrogate model as UC-CLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Experiments  In this section, we evaluate our UCs methods on different  datasets against different target models, which is to simulate  as many unknown cases as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We also examine the ro-  bustness of UCs against several advanced defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Finally,  we demonstrate its effectiveness in attacking commercial  machine learning platforms Azure and PaddlePaddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Experimental Settings  Datasets and Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  We conduct our study on 6 high-  resolution and industrial-scale vision datasets to simulate  as diverse real-world applications as possible, including  Pets [28], Cars [20], Flowers [27], Food [3], SUN397 [40]  and ImageNet [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' For ImageNet, we only use its first  100 classes which is denoted as ImageNet⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' For surrogate  models, we consider ResNet-50 trained on ImageNet-1k as  the default, unless otherwise explicitly stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' For target  models, we employ randomly initialized ResNet-18 [15],  EfficientNet-B1 [38] and RegNetX-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='6GF [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We train  the target models with data augmentations (resizing, ran-  dom crop, random horizontal flip and normalization) for 90  epochs, using SGD with initial learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='1, Cosine  annealing, momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='9, and batch size 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  For generator G, we repeated p times to train the generator  G for 10 epochs using SGD with initial learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='1,  10  20  30  Number of class  0  20  40  60  Accuracy (%)  UC  UC-CLIP  random guess  clean  (a) Different labelings  Clean  SynPer  EMaxN  EMinN  AdvPoison  DeepConfuse  UC-CLIP  UC  0  10  20  30  40  50  Accuracy (%)  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='3 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='8845.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='1948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='1647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='02  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='37 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='11  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='83  (b) Unsupervised exploitation  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' (a) The accuracy of ResNet-18 target models trained on  the unlearnable Pets dataset but with its labels were re-labeled by  the hacker into 5 to 35 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' (b) Comparison of our approach  with the baselines on Pets dataset against ResNet-18 target model  trained via self-supervised SimCLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Cosine annealing, momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='9, and batch size 256, and  10 epochs for ImageNet⋆ and 50 epochs for other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  For random permutation g(·), we simply chose i → i + 1 to  build a closed loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We consider L∞-norm restriction in this  work, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', ∥δ∥∞ < ϵ = 16/255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The number of clusters p  is set to 10, with an analysis is provided in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  We compare our UC and UC-CLIP with 5 base-  line methods including DeepConfuse [9], Synthetic Perturba-  tions (SynPer) [41], Error-minimizing Noise (EMinN) [17],  Error-maximizing Noise(EMaxN) [19], and Adversarial Poi-  soning (AdvPoison) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We use their official implementa-  tions and follow the suggested settings in the original papers  to generate the UEs or poisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Label-agnostic Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Please note that we conduct all of  our experiments under the proposed label-agnostic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  The UCs (and the UEs they serve) are all generated with the  predicted labels by the surrogate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The predicted la-  bels may overlap with the ground truth labels to some extent,  but are highly inconsistent with the original labels as the sur-  rogate models are not trained on the particular datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The  hackers train all target models on the unlearnable datasets  with their ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We report the test accuracy of  the target models on the respective clean test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Main Results  Effectiveness against different target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  We first  compare our UC and UC-CLIP with the 5 baselines against  different target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Table 1 shows the results against  ResNet-18, EfficientNet-B1, and RegNetX-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='6GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We have  the following main findings: (1) Our methods outperform  the baselines by a huge margin consistently across different  datasets and target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This demonstrates the superi-  ority of our methods over the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' (2) Our UC-CLIP  achieves a better performance than UC, and in most of the  cases, by a considerable margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This proves the great poten-  tial of using CLIP as the surrogate model to protect person  data from unauthorized exploitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  6  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The test accuracy (%) of different target models trained on the unlearnable datasets generated by our UC/UC-CLIP and the 5  baseline methods, under the label-agnostic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The top-2 best results are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  RESNET-18  EFFICIENTNET-B1  REGNETX-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='6GF  METHODS  PETS CARS FLOWERS FOOD SUN397 IMAGENET⋆ PETS CARS FLOWERS FOOD SUN397 IMAGENET⋆ PETS CARS FLOWERS FOOD SUN397 IMAGENET⋆  CLEAN  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='31 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='18  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='18  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='97  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='08  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
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+page_content='71 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='15  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
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+page_content='21 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
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+page_content='06 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
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+page_content='28 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
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+page_content='10  UC-CLIP (OURS) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='69  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
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+page_content='07  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='07  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='89  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
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+page_content='49 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
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+page_content='44 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='95  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='82  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='87  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='18  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='12  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='76  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='04  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='66  Effectiveness Against Different Labelings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  An even  more challenging label-agnostic setting is that the hacker  may exploit the unlearnable dataset using different labeling  strategies instead of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' So, a natural question is that what  if the number of labeled classes of the unlearnable dataset  is less than our cluster number p = 10?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Here, we take the  37-class Pets dataset as an example and explore the impact  if the hacker re-labels the unlearnable version of the dataset  as a 5 to 36 class dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' One possible labeling strategy is  that the hacker first extracts the embeddings of the original  text labels using the BERT model [8], and then clusters the  embeddings into 5-37 classes using K-means, so as to con-  struct a mapping from the old labels to the new labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' As  shown in Figure 4 (a), both our UC and UC-CLIP can bring  the test accuracy of the target model down to a level that is  close the random guess (the black curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This verifies that  our methods can craft more generic UEs against the most  severe label-agnostic exploitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Robustness to Unsupervised Exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  We also com-  pare our methods with the baselines under an unsupervised  contrastive learning setting against SimCLR [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Although  our UC methods are not specifically designed for this un-  supervised setting, Figure 4 (b) shows that cluster-wise un-  learnable noise can also prevent unsupervised exploitation  against SimCLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Preventing Commercial Platforms  Here, we apply our UC methods to prevent two com-  mercial machine learning platforms: Microsoft Azure and  Baidu PaddlePaddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' On both platforms, the training  details are agnostic to us, including the model architecture,  learning rate, batch size, epoch, data augmentation, splitting  of the validation set, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Considering that ViT may be used  on commercial platforms due to its recent popularity, we  upgrade our UC-CLIP method by replacing the ResNet-50  (RN50) surrogate model by a ViT-B-32 (ViTB32) surrogate  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The results are reported in Table 2, which are consis-  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The test accuracy (%) of models trained by Azure and  PaddlePaddle platforms on unlearnable Cars dataset crafted by  different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The training configuration on the platform was  set to “fastest training”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  METHODS  Azure  PaddlePaddle  CLEAN  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='45  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='74  SYNPER  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='38  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='59  EMAXN  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='83  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='99  EMINN  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='06  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='40  ADVPOISON  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='97  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='38  DEEPCONFUSE  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='47  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='88  UC (RN50)  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='40  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='96  UC-CLIP (RN50)  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='97  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='79  UC-CLIP (VITB32)  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='47  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='49  tent with that in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', both of our methods can protect  the data uploaded to the two platforms against their training  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Unsurprisingly, the ViTB32-powered UC-CLIP  method achieves the best protection performance by causing  the lowest test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This suggests the effectiveness of  our methods even against commercial platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Resistance to Potential Defenses  In this section, we test the robustness of our UC  methods to several augmentation based defenses, includ-  ing Mixup [43], Gaussian smoothing, Cutmix [42] and  Cutout [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We did not consider adversarial training here is  that the images involved in this paper are generally large in  size, number and resolution, making adversarial training ex-  tremely hard to converge without losing considerable clean  accuracy on most of the datasets [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Compared with ad-  versarial training, we believe that a more practical defense  should be very efficient, such as image denoising or the con-  sidered augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' As can be observed in  Table 3, the 4 data augmentation defenses have minimum  impact on our UC and UC-CLIP methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Particularly, Gaus-  sian smoothing appears to be the most effective defense, but  the accuracy is still below 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  7  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The test accuracy (%) of ResNet-18 trained using different  defenses against our methods on Pets dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  METHODS  NO DEFENSE  MIXUP  GAUSSIAN  CUTMIX  CUTOUT  UC  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='21  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='34  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='26  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='50  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='35  UC-CLIP  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='69  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='96  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='59  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='21  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='29  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Ablation Study  5  10 15 20 25 30 35 40  Number of clusters  0  5  10  15  Accuracy (%)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='08  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='21  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='82 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='75  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='92  (a) Effect of p on UC  5  10 15 20 25 30 35 40  Number of clusters  0  5  10  15  Accuracy (%)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='38  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='69  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='33 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='73  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='09  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='67  (b) Effect of p on UC-CLIP  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Analyzing the effect of cluster number p on Pets dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Here, we analyze the sensitivity of our methods to the  number of clusters p, which has been set to p = 10 as a  default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We take the 37-class Pets dataset as an example  and evaluate our UC and UC-CLIP method under different  values of p ∈ [5, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' As shown in Figure 5, our methods are  quite stable to varying hyperparameter p for p ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This  indicates that, as long as the clusters can cover most of the  concepts in a dataset, the generated unlearnable noise can  effectively prevent the model from learning the real content  from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' As the number of clusters increases, the  noise tends to become more effective, although there is a  slight variation at 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Note that, even in the worst case at  p = 5, our methods still outperform the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Mixture of Clean and Unlearnable Data  0  5  10 15 20 25 30 35  Number of classes  0  10  20  30  40  50  60  Accuracy (%)  clean-only  mixture  (a) Mixture vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Clean-only  0  25  50  75 100 125 150  Epoch  0  20  40  60  80  100  Accuracy (%)  training (unlearnable)  test (clean)  test (unlearnable)  (b) Accuracy trends  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  (a) The test accuracy (%) of ResNet-18 trained on  unlearnable-clean mixed vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' clean-only data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' and (b) the accu-  racy trends on clean vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' unlearnable examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' The unlearnable  examples are crafted using our UC method on Pets dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  All the above experiments are conducted under the as-  sumption that all samples in the dataset are protected, a  commonly adopted assumption in the literature [10,17,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  This setting is reasonable when the protectors have the access  to the entire dataset, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', an online social media company  adopts the technique to protect the contents created by all  of its users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' A more general case is that only a certain pro-  portion of the users protect their data while others do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  This results in mixed dataset with both clean and unlearnable  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Here we test our UC method under this setting  and show the change in test accuracy with the number of  clean classes in Figure 6 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', for the mixture dataset,  the rest of the classes are made unlearnable by UC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' It can be  inferred that the unlearnable classes almost do not contribute  to the model training, a similar conclusion as in previous  works [10,17,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This implies that only those who adopt  the technique will get protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' More Understanding  Why our UCs are more powerful than standard UEs  against label-agnostic exploitation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' As we explained in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='1, the idea of UCs is inspired by the effectiveness  of disrupting the uniformity and discrepancy in preventing  the model from learning useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' However, this  also raises another question: what exactly does the target  model learn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' To answer these two questions, here we ana-  lyze the learning curves of the target model on the clean vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  unlearnable examples separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' As shown in Figure 6 (b),  as the training progresses, the training accuracy on the un-  learnable training samples steadily improves until it reaches  100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' But there is almost no improvement in the clean test  accuracy on the clean test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This is consistent with  the the above experimental results that the target model has  not learned the capability to perceive normal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Sur-  prisingly, however, the model’s accuracy on the perturbed  test samples is fairly high (> 60%), considering that the  normally trained ResNet-18 only achieves a test accuracy of  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='31% on clean Pets dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' This implies that the unlearn-  able noise distribution contained in the UCs has effectively  concealed the real data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Conclusion  Unlearnable examples (UEs) have shown great potential  in preventing hackers from using users’ private data to train  commercial or malicious models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' A number of methods  have been proposed to improve UEs’ transferability and ro-  bustness to different datasets, target models and training  paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In this work, we identified one limitation of ex-  isting UE methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=', their label-consistency assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  To overcome this limitation, we proposed a more general  setting where the hackers could exploit the protected data  with different sets of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We termed this more challeng-  ing setting as label-agnostic, and proposed an Unlearnable  Clusters (UCs) technique with conditioned generator models,  K-means clustering, and large-scale vision-and-language pre-  training model CLIP, to craft effective UEs against a wide  range of datasets and target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' We also demonstrate  its effectiveness against commercial platforms Microsoft  Azure and Baidu PaddlePaddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  8  References  [1] Battista Biggio, Blaine Nelson, and Pavel Laskov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Poison-  ing attacks against support vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' arXiv preprint  arXiv:1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='6389, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' 2  [2] Battista Biggio and Fabio Roli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Wild patterns: Ten years after  the rise of adversarial machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Pattern Recognition,  84:317–331, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' 1, 2  [3] Lukas Bossard, Matthieu Guillaumin, and Luc Van Gool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content='  Food-101–mining discriminative components with random  forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' In European conference on computer vision, pages  446–461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
+page_content=' Springer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfRfuL/content/2301.01217v1.pdf'}
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+Strong Collapse of Random Simplicial Complexes
+Jean-Daniel Boissonnat ∗ 1, Kunal Dutta † 2, Soumik Dutta † 3, and Siddharth Pritam ∗ 4
+1Universit´e Cˆote d’Azur, INRIA, Sophia Antipolis, France. email:
+Jean-Daniel.Boissonnat@inria.fr
+2Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland.
+email: K.dutta@mimuw.edu.pl
+3Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Poland.
+email: s.dutta2@mimuw.edu.pl
+4Universit´e Cˆote d’Azur, INRIA, Sophia Antipolis, France. email:
+siddharth.pritam@inria.fr
+Abstract
+The strong collapse of a simplicial complex, proposed by Barmak and Minian [6], is a
+combinatorial collapse of a complex onto its sub-complex. Recently, it has received attention
+from computational topology researchers [22, 7, 8], owing to its empirically observed useful-
+ness in simplification and size-reduction of the size of simplicial complexes while preserving
+the homotopy class. We consider the strong collapse process on random simplicial complexes.
+For the Erd˝os-R´enyi random clique complex X(n, c/n) on n vertices with edge probability
+c/n with c > 1, we show that after any maximal sequence of strong collapses the remaining
+subcomplex, or core must have (1 − γ)(1 − cγ)n + o(n) vertices asymptotically almost surely
+(a.a.s.), where γ is the least non-negative fixed point of the function f(x) = exp (−c(1 − x))
+in the range (0, 1). These are the first theoretical results proved for strong collapses on
+random (or non-random) simplicial complexes.
+1
+Introduction
+Motivation
+Simple collapse is a combinatorial notion which simplifies a simplicial complex
+without changing its topology. It can be expressed as a series of elementary moves of removals of
+pair of simplices σ and τ, such that σ is uniquely contained in τ. The notion of simple collapse
+was introduced by J.H.C Whitehead [21] to study homotopy types of cell complexes. Since then
+it has found usage in many different areas of topology, especially in computational topology.
+Recently new variants of simple collapses have been introduced, called strong collapses and more
+generally d-collapses [6, 8, 5]. In such collapses one removes special vertices (more generally
+d-simplices) called dominated vertices (simplices) whose link is a simplicial cone. It’s again
+expressed as a series of elementary moves of removals of dominated vertices (simplices). They
+have been shown to be very powerful tools to solve many problems in computational topology.
+In particular, the recent works of Pritam et. al. [9, 8, 23] has shown that strong collapses and
+edge collapses (d-collapse for d = 1) can be used for efficient computation of one parameter and
+∗J.-D. Boissonnat and S. Pritam received funding from the European Research Council (ERC) under the
+European Union’s Seventh Framework Programme (FP/2007- 2013) / ERC Grant Agreement No. 339025 GUDHI
+(Algorithmic Foundations of Geometry Understanding in Higher Dimensions).
+†K. Dutta and S. Dutta received funding from the Polish NCN SONATA Grant no. 2019/35/D/ST6/04525.
+1
+arXiv:2301.03514v1  [cs.CG]  9 Jan 2023
+
+multi-parameter persistence. Efficient computation of persistent homology is one of the central
+topic of research in topological data analysis.
+The computation of persistent homology involves computing homology groups of a nested
+sequence of simplicial complexes called filtrations. And to compute persistent homology requires
+O(n3) time and O(n2) space, here n is the total number of simplices in the filtration. The
+general technique developed in [9, 7, 8, 12, 23] is to reduce a filtration to a smaller filtration
+using strong or edge collapse such that the persistent homology is preserved. In [9, 7, 8, 12, 23],
+it has been established through experiments that in practice the reduced filtrations are very
+small and thereafter computation of persistent homology is extremely fast. The gain in efficiency
+is quite dramatic in the case of flag (clique) complexes, where the strong collapse and edge
+collapse can be computed using only the graph (1-skeleton) of the given complex [7, 12, 23].
+As mentioned above the efficiency reported in [9, 7, 12, 23] are through experiments and
+there is no theoretical guarantee over the reduction size. This is due to the fact that in general
+the amount of reduction depends on the individual complex and its combinatorial structure. In
+fact, the reduction is dependent even on the order of the collapses and a different order can
+result in a different core, except in the case of strong collapse. Which is even harder when we
+want study the reduction size in a filtered simplicial complexes. This motivates us to consider
+the case of random simplicial complexes and study the average reduction size by collapses.
+In this article, we study the problem of reduction size achieved by the strong collapses of a
+clique complex defined over an Erd˝os-R´enyi random graph.
+Previous
+The study of random simplicial complexes was initiated in the seminal paper of
+Linial and Meshulam [16]. Later Meshulam and Wallach [19] generalized the model of random
+complexes to obtain the Linial-Meshulam (LM) model of d-dimensional random complexes.
+Since then a large body of work from several authors has emerged on many different models
+of random simplicial complexes, studying various topological and geometric properties of such
+complexes [14]. The study of simple collapses for random simplicial complexes has also been of
+interest to researchers and there have been numerous works in this direction. In the d-dimensional
+LM model, Kozlov [15] proved bounds on the threshold for vanishing of the d-th homology.
+Simple collapses on random complexes were first studied by Aronshtam, Linial, �Luczak and
+Meshulam [4], who improved Kozlov’s bound to get a tight bound on the threshold and also gave
+a bound on the threshold for collapsibility in the d-dimensional LM model. Later Aronshtam
+and Linial [2, 3] extended this line of work, obtaining first the threshold for the vanishing of
+the d-th homology in [2] and then the threshold for non-collapsibility of the d-dimensional LM
+complex [3]. In [17] Linial and Peled obtained precise asymptotic bounds on the size of the core
+of such complexes. Very recently, Malen [18] has shown that the ER clique complex X(n, p) is
+(k + 1)-collapsible with high probability for p = n−α when α > 1/(k + 1).
+Thus, to the best of our knowledge, work on collapses in random complexes has so far
+considered only simple collapses.
+Throughout this paper, we shall use the notation asymptotically almost surely (a.a.s.) for a
+series of events (En)n≥1, when the probability of occurence of En goes to 1 as n → ∞.
+Models of Random Simplicial Complexes
+In this paper we shall consider two models of
+random simplicial complexes, which are described below. For a graph G, let cl(G) denotes the
+clique (flag) complex on G, i.e. the simplicial complex where each complete subgraph of G
+on d vertices is a d − 1-simplex in cl(G). The Erd˝os-R´enyi (ER) model X(n, p) on n vertices
+with probability parameter p ∈ [0, 1] is given by connecting each possible pair of elements of an
+n-element set by an edge randomly and independently with probability p to get the random
+graph G = G(n, p). Let X(n, p) := cl(G(n, p)).
+2
+
+Our Contribution
+We give bounds on the size of the core (i.e. the smallest sub complex
+without any dominated vertex) of a random simplicial complex after strong (vertex) collapses.
+Whereas previous works focused on computing the threshold for the appearance and disappearance
+of the k-th homology class for different k, here we are more interested in computing the size of
+the core. We show that for n-vertex ER clique complexes, the size of the core after a maximal
+series of strong collapses is a.a.s. a constant fraction of n, with the constant depending only on
+the edge probability, and bounded away from 1. Further, we also find a precise expression for
+the constant, as a fixed point of an implicit equation. Our first theorem is stated below.
+Let K be a simplicial complex. For i ≥ 1, we use fi(K) to denote the set of i−simplices of
+the complex. By a slight abuse of notation, we let f0(K) denote the set of non-isolated vertices
+of the complex and ˜f0(K) denote the set of non-isolated vertices of the complex. (A vertex
+v ∈ K is isolated the if v is not a face of any edge in K.) In a pruning phase run on K, all
+dominated (strong-collapsible) vertices of K are simultaneously collapsed. Let Rt(K) denote
+a complex obtained by running t pruning phases over K. Lastly, R∞(K) denotes a complex
+obtained from K after running a maximal series of strong collapses on K. i.e. the core of K.
+Theorem 1. Let c > 1 and X ∼ X(n, c/n). Then there exists a constant γ ≡ γ(c) given by
+the least non-negative fixed point of the function f(x) = exp (−c(1 − x)), x ∈ (0, 1), such that
+0 < γ < 1/c < 1 and a.a.s. the following holds
+|f0(R∞(X))| = (1 − γ)(1 − cγ)n + o(n).
+	0
+	5000
+	10000
+	15000
+	20000
+	25000
+	0
+	10000
+	20000
+	30000
+	40000
+	50000
+	60000
+	70000
+	80000
+	90000
+	100000
+size	of	the	CORE
+n
+experimental	value	for	c=1.5
+theoretical	value	for	c=1.5
+Figure 1: We ran experiments on ER complexes and plotted the size of the core (strong collapse)
+with n ∈ [0, 105] and c = 1.5. These experiments clearly validates our theoretical results.
+3
+
+	0
+	1000
+	2000
+	3000
+	4000
+	5000
+	6000
+	7000
+	8000
+	9000
+	10000
+	1
+	1.5
+	2
+	2.5
+	3
+	3.5
+	4
+	4.5
+	5
+size	of	the	CORE
+c
+experimental	value	for	n=10000
+theoretical	value	for	n=10000
+Figure 2: In a different set of experiments over ER complexes, we varied the constant c ∈ [1, 5]
+keeping the number of vertices fixes to n = 104.
+We then address a question of algorithmic interest: given an ε ∈ (0, 1), how many rounds
+do we need in the first epoch to get within an εn gap from the actual size of the core? The
+following theorem gives a bound on the number of rounds t as a function of ε.
+Theorem 2. Let X ∼ X(n, c/n) and ε > 0 (sufficiently small) be given. Then there exists
+t ∈ Z+ such that
+(c − cγ)ec
+1 − ce−c (ce−c)t ≤ ϵ ≤ (c + 1)ec
+1 − cγ (cγ)t,
+and |f0(Rt(X))| − |f0(R∞(X))| ≤ ϵn + o(n) a.a.s.
+1.1
+Overview of Proofs and Outline of Sections
+While our analysis shares the general flow of the analysis of the simple collapsibility of LM
+complexes in e.g. [4, 2, 3], there are several differences and difficulties. Firstly, in both cases
+(strong collapse of ER clique complex and simple collapse of LM model) our goal is to find the
+size of the core, rather than whether the complex is collapsible or not. Secondly, in the case of
+the random ER clique complex our analysis needs to take into account the non-homogeneity of
+the complex. That is, maximal simplices in this model can have different sizes. Further, unlike
+in the LM model, the existence of a maximal simplex is not independent of the existence of all
+other possible maximal simplices. Finally perhaps the most interesting difference of the random
+ER clique complex model in our context, is that the effect of removing a vertex is not necessarily
+localized – a fact which requires a fair bit of innovation to handle (in several places), especially
+in proving the concentration bounds in Section 6, and in the later stages of the analysis, in
+Section 7. We present a more detailed overview of the proof strategy used in our concentration
+bound, in the beginning of Section 6.
+With the above caveats in mind, we first briefly review the main ideas of the proof of
+Aronshtam and Linial [3]. The analysis was split into two epochs, each of which were further
+divided into several rounds (phases). In the first epoch, in each round, every simple-collapsible
+simplex was simultaneously collapsed, and this procedure was repeated for a constant number
+of rounds. Aronshtam and Linial [3] used a tree-like model of a random simplicial complex to
+approximate the local structure of the random LM complex and showed that the total number
+of collapsed simplices over all such rounds tended to a constant fraction of the number of initial
+4
+
+simplices, as the number of rounds increased. Moreover, this limit constant could be expressed
+as a fixed point of an implicit equation involving only the distribution parameter c.
+In the analysis of the second epoch, a simplex would be chosen randomly from the set of
+(non-neighbouring) simple-collapsible simplices, and collapsed in each round. The aim was to
+show that in this epoch, the number of simplices collapsed would be asymptotically negligible
+compared to the inital number of simplices present. Thus in summary, the final number of
+deleted simplices is determined by the first epoch itself, and the second epoch serves to show the
+tightness of this bound.
+Our proofs also split the analysis into two epochs. Similar to [3], we show that a certain tree-
+like model of random simplicial complexes provides a good approximation of local neighbourhoods,
+which is done in Section 3. This is followed by the analysis of the first epoch, in Section 4. The
+main theorem of this section gives an expression for the expected number of vertices remaining
+after t rounds (or pruning phases) of the first epoch. Bounds on t as a function of ε, γ and
+c, are given in Theorem 2, which is proved in Section 5. Before beginning the analysis of the
+second epoch however, we need bounds on the concentration of the size of the core itself, as well
+as several other random variables. These are proved in Section 6, where we use the notions of
+critical and precritical (sub)complexes – described in more detail in the beginning of Section 6.
+With these concentration bounds in place, we move to the analysis of the second epoch in
+Section 7.
+2
+Preliminaries
+In this section we briefly introduce some topological and probabilistic notions. Readers can refer
+to [13] for a comprehensive introduction to topics related to topology and [10] for topics related
+to probability theory and random structures.
+Simplicial complex.
+An abstract simplicial complex K is a collection of subsets of a
+non-empty finite set X, such that for every subset A in K, all the subsets of A are in K. An
+element of K is called a simplex. An element of cardinality d + 1 is called a d-simplex and d
+is called its dimension. Given a simplicial complex K, we denote its geometric realization as
+|K|. A simplex is called maximal if it is not a proper subset of any other simplex in K. A
+sub-collection L of K is called a subcomplex if it is a simplicial complex itself. A subcomplex
+K′ of K is called a d-skeleton of K if it contains all the simplices of K of dimension at most d.
+Erdos Renyi Graph Definition.
+This is the probability space G(n, p) consisting of
+all the graphs on n vertices. Probability of occurrence of a graph with m edges is pm(1 −
+p)(n)(n−1)/2−m. In other words, it is a random graph on n vertices where each edge can occur
+independently with probability p.
+Clique complex and Neighborhood.
+A complex K is a clique or a flag complex if,
+when a subset of its vertices form a clique (i.e. any pair of vertices is joined by an edge), they
+span a simplex. For a vertex v in G, the open neighborhood NG(v) of v in G is defined as
+NG(v) := {u ∈ G | [uv] ∈ E}, here E is the set of edges of G. The closed neighborhood
+NG[v] is NG[v] := NG(v) ∪ {v}. Similarly we define the closed and open neighborhood of an
+edge [xy] ∈ G, NG[xy] and NG(xy) as NG[xy] := N[x] ∩ N[y] and NG(xy) := N(x) ∩ N(y),
+respectively. The above definitions can be extended to any k-clique σ = [v1, v2, ..., vk] of G;
+NG[σ] := �
+vi∈σ N[vi] and NG(σ) := �
+vi∈σ N(vi).
+Star, Link and Simplicial Cone.
+Let σ be a simplex of a simplicial complex K, the
+closed star of σ in K, stK(σ) is a subcomplex of K which is defined as follows, stK(σ) := {τ ∈
+5
+
+K| τ ∪ σ ∈ K}. The link of σ in K, lkK(σ) is defined as the set of simplices in stK(σ) which
+do not intersect with σ, lkK(σ) := {τ ∈ stK(σ)|τ ∩ σ = ∅}. The open star of σ in K, sto
+K(σ) is
+defined as the set stK(σ) \ lkK(σ). Usually sto
+K(σ) is not a subcomplex of K.
+Let L be a simplicial complex and let a be a vertex not in L. Then the set aL defined as
+aL := {a, τ | τ ∈ L or τ = σ ∪ a; where σ ∈ L} is called a simplicial cone.
+Simple collapse.
+Given a complex K, a simplex σ ∈ K is called a free simplex if σ has
+a unique coface τ ∈ K. The pair {σ, τ} is called a free pair. The action of removing a free
+pair: K → K \ {σ, τ} is called an elementary simple collapse. A series of such elementary
+simple collapses is called a simple collapse. We denote it as K ↘ L. A subcomplex Kec of K
+is called an elementary core of K if K↘Kec and Kec has no free pair.
+Removal of a simplex.
+We denote by K \ σ the subcomplex of K obtained by removing
+σ, i.e. the complex that has all the simplices of K except the simplex σ and the cofaces of σ.
+Dominated simplex.
+A simplex σ in K is called a dominated simplex if the link lkK(σ)
+of σ in K is a simplicial cone, i.e. if there exists a vertex v′ /∈ σ and a subcomplex L of K, such
+that lkK(σ) = v′L. We say that the vertex v′ is dominating σ and that σ is dominated by v′,
+which we denote as σ ≺ v′.
+σ-algebra
+The reader can refer to [10] for the definition of σ-algebra.
+d-collapse.
+Given a complex K, the action of removing a dominated k-simplex σ from K is
+called an elementary d-collapse, denoted as K↘↘d{K \σ}. A series of elementary d-collapses
+is called a d-collapse, denoted as K ↘↘k L. We further call a complex K d-collapse minimal
+if it does not have any dominated d simplices. A subcomplex Kk of K is called a d-core if K
+↘↘k Kd and Kd is d-collapse minimal. A 0-core of a complex K is unique, however it is not
+true in general for k ≥ 1. Like simple collapses, d-collapses preserve the homotopy type of a
+simplicial complex.
+A 0-collapse is a strong collapse as introduced in [6] and 1-collapse is called an edge
+collapse [8]. The following lemma from [8] characterizes the domination of a simplex in the
+special case of a flag complex in terms of neighborhood.
+Lemma 3. Let σ be a simplex of a flag complex K. Then σ will be dominated by a vertex v′ if
+and only if NG[σ] ⊆ NG[v′].
+In this article, our main focus will be the case d = 0, i.e. when σ is a vertex. The next
+lemma from [7], though elementary, is of crucial significance.
+Lemma 4. Let K be a flag complex and let L be any subcomplex of K obtained by strong
+collapses. Then L is also a flag complex.
+Both lemmas (Lemma 3 and Lemma 4) show that strong collapse is well-suited to flag
+complexes. In the next sections we will investigate the reduction capabilities of strong of a clique
+complex of an Erd˝os-R´enyi random graph.
+3
+Tree process
+In this section, we describe the tree process which is used to simulate the collapse process in the
+first epoch of the (strong) collapse. A one-dimensional tree is built recursively as follows:
+1. Start with a single node(root).
+6
+
+2. In the nth iteration, add children to all the leaves at distance n − 1 from the root from
+Poisson distribution with parameter c(> 1).
+Let Tn denote the set of all possible trees after nth iteration for n > 1 and T0 being the root
+itself. Let T := �
+n∈N
+Tn.
+Let γt be the probability that a tree T ∈ Tt is pruned to the root in no more than t − 1 steps.
+Clearly, γ1 = e−c. Set γ0 = 0. Also, we have the following recursive relation which is true in
+general:
+γt+1 = e−c(1−γt).
+Note that, in this process we never prune the root itself even if its degree is 1. We call such
+a process root collapsing. Let γk
+t denote the probability that a tree T ∈ Tt has degree k after
+t − 1 root collapsing steps. Then,
+γk
+t = (c(1 − γt−1))k
+k!
+e−c(1−γt−1)
+Observe that γk
+1 gives the initial degree distribution. Also, let γ≥2
+t
+denote the probability that a
+vertex has degree atleast 2 after t − 1 root collapsing steps. Then we have
+γ≥2
+t
+=
+∞
+�
+k=2
+γk
+t = 1 − γt(1 + c(1 − γt−1)).
+Define βt := 1 − γt+1. Thus, βt is the probability that atleast t + 1 root collapsing steps are
+needed to isolate the root of a tree T ∈ Tt.
+Define f(x) := e−c(1−x) on the interval [0, 1]. We shall assume c > 1 for the rest of these
+paper unless specified otherwise. Note that f([0, 1]) ⊆ [0, 1], f′(x) = cf(x) and f(x) is strictly
+increasing on the interval [0, 1]. Let ft(x) denoted the function obtained by composing f t times.
+Then ft is also strictly increasing on [0, 1] for all t ≥ 1. As, γ1 > γ0, applying ft−1 on both sides,
+we get γt > γt−1 for all t ≥ 1.
+Also define γ to be the left most zero of the function g(x) := e−c(1−x) − x defined on the
+range [0, ∞). Note that 0 < γ ≤ 1 as g(1) = 0.
+Lemma 5. For c > 1 and γ = γ(c) defined as earlier we have cγ < 1.
+Proof of Lemma 5. Let g(x) be defined as above. So g(γ−) > 0 and g(γ) = 0. Thus, from
+the differentiability of g(x), cγ − 1 = g′(γ) ≤ 0. Now, if cγ − 1 = 0 then ce1−c = 1, which is
+impossible for c > 1. Thus cγ − 1 < 0.
+Now observe that, x ≤ γ =⇒ f(x) ≤ f(γ) = γ. Thus, by the fact that f′(x) = cf(x), f(x)
+restricted on [0, γ] becomes a contraction mapping. So, by Banach Fixed Point theorem, f|[0,γ]
+has an unique fixed point which, in our case, is γ.
+To summarize the above arguments we get the following remark.
+Remark 1. γt converges to 0 < γ < 1/c as an increasing sequence and βt converges to
+1 − 1/c < β < 1 as a decreasing sequence.
+4
+First Epoch
+In this section, we present the analysis of the first epoch of the collapse. The first epoch is executed
+in phases and in each phase we remove a maximal set of dominated vertices simultaneously.
+Our goal, in this section, is to prove the following theorem.
+7
+
+Theorem 6. Let X ∼ X(n, c/n). Let E(|f0(Rt(X))|) denote the expected number of non-isolated
+vertices in X after t strong collapse phases and γt be as defined in the last section. Then,
+E(|f0(Rt(X))|) = (1 − γt+1 − cγt + cγ2
+t )n.
+We start by proving some important lemmas about the local structure of the complex. For
+X ∼ X(n, c
+n), let us define the following event,
+D := {deg(v) ≤ log n ∀v ∈ ˜f0(X)}.
+Then, the following lemma can be proved using standard Chernoff bounds.
+Lemma 7. Pr{D} = 1 − on(1).
+Proof of Lemma 7. Note That for any v ∈ ˜f0(X), deg(v) ∼ Bin(n − 1, c/n). Let Dv be the
+event that deg(v) ≤ log(n). Then by the Chernoff Bound on Binomial Distribution
+Pr{¬Dv} = Pr{deg(v) > log(n)}
+≤ (ec(n − 1)
+n log(n) )log(n)
+≤ (
+ec
+log(n))log(n)
+= n− log(log(n/ec))
+Thus by the union bound Pr{�
+v∈f0(X) ¬Dv} ≤ n1−log(log(n/ec)). Hence,
+Pr{D} = Pr{
+�
+v∈f0(X)
+Dv} = 1 − Pr{
+�
+v∈f0(X)
+¬Dv} ≥ 1 − n1−log(log(n/ec)) = 1 − on(1)
+By Cl(S) we denote the simplicial closure of the set S. Fix v ∈ ˜f0(X). Define N0 := Cl(v)
+and N−1 := ∅. Also define Ni+1 := Cl({s ∈ X|∃u ∈ ˜f0(Ni)\ ˜f0(Ni−1)|u ⊂ s})∪Ni. Equivalently,
+this can also be defined in terms of the 1-skeleton of the complex.
+Define the event At = {Nt ∈ T }.
+Lemma 8. Let X ∼ X(n, p) and fix v ∈ ˜f0(X). Then Pr{At ∩ D} = 1 − o(1) .
+Proof. If X ∈ D then ˜f0(Nt) = O(logt+1n) and we want to avoid O(log2t+2n) edges to make Nt
+a one dimensional tree. Probability of that happening is
+(1 − c/n)O(log2t+2n) = 1 − o(1)
+Now the the degree of a node of this tree comes from Bin(n − 1, c/n). For large n this
+distribution can be approximated by Po(c).
+Proof of Theorem 6. Recall that for a simplicial complex X, Rt(X) denotes a complex obtained
+after t phases and f0(X) denotes the set of non-isolated vertices (0−simplices) of the complex.
+Note that if a vertex v ∈ f0(X) survived t pruning steps then it must have had degree deg(v) ≥ 2
+after t − 1 pruning steps. Thus, Pr{v ∈ f0(Rt(X))} ≤ γ≥2
+t
+= 1 − γt(1 + c(1 − γt−1)). This event
+counts both the isolated vertices and degree one, (i.e., collapsible) vertices. Thus this gives a
+slight over estimate. To get more precise estimate we observe that, in the spirit of [3], that a
+vertex v ∈ f0(X) survives t pruning steps if it is neither collapsed nor isolated after t pruning
+steps. Probability of such an event is 1 − γt+1 + cγt − cγ2
+t . The previous lemma asserts that it is
+indeed the survival probability of a vertex of the simplicial complex.
+8
+
+5
+Rate of Convergence
+In this section, we prove Theorem 2 thus giving bounds on the rate of convergence of the variable
+γt.
+Lemma 9. Let c, γt be as defined earlier. Then
+cγt ≤ γt+1 − γt
+γt − γt−1
+≤ cγt+1
+Proof. Let f(x) and g(x) be as defined in section 3. Clearly,
+γt+1 − γt
+γt − γt−1
+=
+g(γt)
+g(γt−1)
+= g(γt−1 + (γt − γt−1))
+g(γt−1)
+= g(γt−1) + g′(s)(γt − γt−1)
+g(γt−1)
+(for some s ∈ [γt−1, γt])
+= f′(s) = cf(s)
+(as γt − γt−1 = γt−1)
+As f(x) is an increasing function the result follows.
+In particular, ce−c ≤ γt+1−γt
+γt−γt−1 ≤ cγ for all t ≥ 1.
+Let δ(t) = (1 − γt+1 − cγt + cγ2
+t ) − (1 − γt+2 − cγt+1 + cγ2
+t+1), as defined in section 7. It can
+be shown that
+δ(t) = (1 − γt+1 − cγt + cγ2
+t ) − (1 − γt+2 − cγt+1 + cγ2
+t+1)
+= (γt+2 − γt+1) + c(γt+1 − γt) + c(γt + γt+1)(γt − γt+1)
+= (γt+2 − γt+1
+γt+1 − γt
++ c − c(γt+1 + γt))(γt+1 − γt)
+Hence,
+(c − cγ)ec(ce−c)t ≤ δ(t) ≤ (c + 1)ec(cγ)t
+Now define ϵ ≡ ϵ(t) := (1 − γt+1 − cγt + cγ2
+t ) − (1 − γ − cγ + cγ2) so that E[|f0(Rt(X))|] −
+(1 − γ)(1 − cγ] = ϵn. Consequently,
+ϵ(t) = (1 − γt+1 − cγt + cγ2
+t ) − (1 − γ − cγ + cγ2)
+= (γ − γt+1) + c(γ − γt) + c(γt + γ)(γt − γ)
+= (γ − γt+1) + (c − c(γt + γ))(γ − γt)
+So,
+(c − cγ)ec
+1 − ce−c (ce−c)t ≤ ϵ(t) ≤ (c + 1)ec
+1 − cγ (cγ)t.
+Thus we get the following corollary.
+Corollary 10. for any t ≥ 1
+ec(ce−c)t ≤ γt+1 − γt ≤ ec(cγ)t
+and
+ec
+1 − ce−c (ce−c)t ≤ γ − γt ≤
+ec
+1 − cγ (cγ)t.
+From the above corollary, we get the Theorem 2.
+9
+
+6
+Concentration of Size of the Complex after the First Epoch
+In this section, we shall prove a concentration bound on the size of the core. Unlike in the case
+of simple collapses in d-dimensional LM complexes [2, 3], concentration bounds in our case are
+less straightforward. Observe firstly, that deleting a single vertex v could potentially change
+the domination status of an arbitrary number of vertices, as for example when v dominates the
+entire complex. Thus the influence of a vertex can be n in the worst case. Therefore we shall
+need to use an edge exposure martingale inequality, in the form of a variant of an inequality of
+Freedman [11], given by Warnke [20], which allows us to consider the path variance of the effect
+of a single edge, rather than the worst case effect.
+In order to bound the path variance, we shall show that if the influence of a variable is large,
+there is a specific class of subcomplexes, which we call Critical Complexes, one of which must
+occur in the 1-skeleton of the complex. It is not hard to show (and we do) that the probability
+of occurence of these subgraphs is vanishingly low in the original random complex. However, the
+variance needs to be controlled at all steps in the edge exposure martingale, i.e. when we are
+computing expectations over arbitrarily small subcomplexes of the original complex. To handle
+this, we need to define a superset of critical complexes, which we call Precritical Complexes,
+and show that their probability of occurence will still be vanishingly small throughout the edge
+exposure process. We can then define a stopped martingale which stops if at any step of the edge
+exposure process, a precritical complex occurs, and prove concentration bounds using Warnke’s
+inequality for this martingale. The final concentration bound is then the bound obtained for
+the stopped martingale, together with the probability that the martingale ever encounters a
+precritical complex.
+Fix p = c
+n and m = n(n−1)
+2
+. For 1 ≤ i ≤ m, ei ∼ Bernoulli(p) be i.i.d. random variables
+corresponding to existance of edges. Clearly X(n, p) = e1 × · · · × em as probability spaces.
+Now we can define a filtration of σ-algebras {Fi}m
+i=0 on X(n, p) by setting Fi to the σ-algebra
+corresponding to e1, · · · , ei.
+Let X ∼ X(n, p). Now we construct an edge exposure martingale (see e.g. [1] for a definition
+of the edge exposure martingale) as follows: Clearly, Ym = |f0(Rt(X))| and Y0 = E(|f0(Rt(X))|).
+This section is devoted to prove the following concentration result, which says that the size
+of the complex after t pruning rounds of the first epoch is close to its expected value with high
+probability.
+Theorem 11. (Main Theorem) Let X ∼ X(n, p). Let |f0(Rt(X))| be number of vertices after t
+strong collapsing phases and Y0 = E(|f0(Rt(X))|) be its expected value. Then for any s ≥ 0 we
+have,
+Pr{||f0(Rt(X))| − Y0| ≥ s · n
+2
+3 }
+≤
+2 exp
+�
+−
+s2 · n
+1
+3
+(c4t+1 + (2/3)sn−1/32t+1) + O(1/n)
+�
++ O(1/n)
+=
+on(1).
+To prove this we begin by observing some combinatorial results. In the following lemmas, we
+show that the influence of deleting one vertex is bounded, with high probability.
+Lemma 12. Pr{deleting a vertex b gives birth to k newly generated dominated vertices} ≤
+O(
+1
+n3k−4 ).
+Proof of Lemma 12. We first claim that deleting a vertex b gives birth to k newly generated
+dominated vertices then b atleast have k neighbors one of which is the dominating vertex of
+b. In the following diagram, the solid arrow denotes domination and the white arrow denotes
+10
+
+future domination in the next phase only after deleting vertex b. The pointy head of the arrow is
+towards the dominated vertex. Vertices a, b, c may be connected to other vertices. The following
+diagrams exhibits some of the potential arrangements.
+a
+b
+c
+a
+b
+c
+a
+b
+c1
+ci
+cj
+. . .
+A careful inspection will show that these kind of arrangements are impossible. Indeed if
+it happens that will imply that the would-be-dominated vertices are already dominated. This
+is because we are only deleting b which is a common neighbor of all the would-be-dominated-
+dominating pairs. Thus neighbors of b can not have white arrows between themselves.
+Thus fig:1 gives the necessary minimal arrangements for the birth of k newly generated
+dominated vertices. In the following diagram, all the ci’s and their corresponding d’s are assumed
+to be connected to some non-neighbor of a which lies in the set {e1, · · · , el′}. We claim that in
+such case f1 − f0 ≥ (k − 2) + (k − 1) + (k − 1) ≥ 3k − 4. We shall prove our claim by induction
+on k ≥ 2. The case k = 2 is evident from the following diagram.
+a
+b
+c1
+d
+e
+We now prove the induction step. Consider the following figure again.
+a
+b
+c1
+ck−2 ck−1
+{d1, · · · , dl}
+{e1, · · · , el′}
+. . .
+Now assume that the claim holds for k − 1. Now just adding the kth vertex ck−1 increases
+f1 − f0 by 1. Also the corresponding d and e increases f1 − f0 by 1 each. This ends the
+induction step. So in the all the possible minimal arrangements f1 − f0 ≥ 3k − 4. Thus expected
+number of such arrangements is
+� n
+f0
+�
+· (c/n)f1 = O(
+1
+n3k−4 ). Thus the result follows from Markov’s
+inequality.
+Corollary 13. Pr{deleting a vertex b gives birth to 3 newly generated dominated vertices}
+≤ O( 1
+n5 ).
+Corollary 14. Let X ∼ X(n, p) and e ∈ f1(X), then
+Pr{|f0(Rt(X)) \ f0(Rt(X \ {e}))| ≥ 2t+1} ≤ O( 1
+n5 ).
+11
+
+Proof. If such an event happens then there must be a dominated vertex in the process whose
+deletion creates atleast 3 new dominating vertices. Thus the result follows from the previous
+corollary.
+Let Critical Complexes denote the minimal simplicial complexes corresponding to k = 3 (see
+the following diagrams). Let Precritical complexes be any of the Critical Complex without any
+four of the edges.. Let N denote the set of complexes from X(n, p) that contains a Critical
+Complex and N′ denote the set of complexes contains a Precritical Complex. . Clearly, N′ ⊇ N.
+a
+b
+c1
+c2
+d
+e
+a
+b
+c1
+c2
+d1
+d2
+e
+a
+b
+c1
+c2
+d
+e1
+e2
+a
+b
+c1
+c2
+d1
+d2
+e1
+e2
+The following result is immediate.
+Lemma 15. Let X ∼ X(n, p). Then,
+Pr{X ∈ N} ≤ O( 1
+n5 ),
+and
+Pr{X ∈ N′} ≤ O( 1
+n).
+Now define stopping time τ on {Fi}m
+i=0 such that τ = t if t = min(s≤t){Fs ∈ N′}. Define a
+stopped martingale with respect to {Fi}m
+i=0 by Mi := Yi∧τ.
+We first prove the following theorem.
+Theorem 16. (Stopped Martingale inequality) Let {Mi}m
+i=0 be the stopped martingale defined
+as above. Then for any s ≥ 0 we have,
+Pr{|Mm − M0| ≥ s · n2/3} ≤ 2 exp
+�
+−
+s2 · n1/3
+(c4t+1 + (2/3)sn−1/32t+1) + O(1/n)
+�
+.
+In order to prove the above theorem, we shall use the following lemma from Warnke [20].
+Assume that {FK}0≤k≤N is an increasing sequence of σ-algebras, and {MK}0≤k≤N is an
+{FK}0≤k≤N-adapted bounded martingale.
+Lemma 17. (2-sided version of Bounded Variance martingale Inequality) Let Uk be a Fk−1
+variable satisfying |Mk − Mk−1| ≤ Uk. Set Ck = maxi∈[k] Uk and Vk = �
+i∈[k] V (Mi−Mi−1|Fi−1).
+Let φ(x) = (1 + x) log(1 + x) − x. For every s ≥ 0 and V, C > 0 we have
+Pr{|MK − M0| ≥ s, Vk ≤ V, Ck ≤ C for some k ∈ [N]} ≤ 2e−s2/(2V +2Cs/3)
+Theorem 11 essentially follows from Theorem 16 and Lemma 15.
+12
+
+Proof. Proof of Theorem 16
+Let X ∈ X(n, p). We shall first try to calculate Pr{X ∈ N|ei, · · · , e1} where (e1, · · · , ei)
+does not form any precritical complex. Let M ⊂ N be the set of complexes that contains
+some critical complex not involving any of the edges from {ei, · · · , e1} and M′ ⊂ N be the set
+of complexes where all the critical complexes involves some edges from {ei, · · · , e1}. Clearly,
+N = M ⊔ M′. Thus,
+Pr{X ∈ N|ei, · · · , e1} = Pr{X ∈ M ⊔ M′|ei, · · · , e1}
+= Pr{X ∈ M|ei, · · · , e1} + Pr{X ∈ M′|ei, · · · , e1}
+= Pr{X ∈ M} + Pr{X ∈ M′|ei, · · · , e1}
+Pr{X ∈ M} is the probability that {ei+1, · · · , em} contains a critical complex. By reasoning
+similar to the proof of Lemma 12. We get Pr{X ∈ M} = O(1/n5). On the other hand, note
+that as (ei, · · · , e1) does not contain any precrtitical complex, atleast 5 more edges is needed
+for X to form a critical complex involving some edges {ei, · · · , e1}. Suppose, depending on
+(e1, · · · , ei), k more edges are needed to complete a critical complex. Clearly 5 ≤ k ≤ 13. Also
+note that, in a critical complex, there are atmost 2 vertices of degree two and rests have degree
+atleat 3. Thus even in the worst case one need to choose 3 vertices and construct 5 particular
+edges. Thus Pr{X ∈ M′|ei, · · · , e1} = O(1/n2). Hence, Pr{X ∈ N|ei, · · · , e1} = O(1/n2) given
+(e1, · · · , ei) does not form any precritical complex. In particular,
+Pr{|f0(Rt(X)) \ f0(Rt(X \ {ei+1}))| ≥ 2t+1|ei, · · · , e1} ≤ O( 1
+n2 )
+under the same assumption.
+Thus
+E(|f0(Rt(X)) \ f0(Rt(X \ {ei+1}))||ei, · · · , e1) ≤ 2t+1 + n · O(1/n2) ≤ 2t+1 + O(1/n)
+whenever (ei, · · · , e1) does not contain any precrtitical complex.
+We now claim that |Mi+1 − Mi| ≤ 2t+1 + O(1/n). If (ei, · · · , e1) contains a precritical
+complex then the martingale stops and the claim holds. Now suppose (ei, · · · , e1) does not
+contain any precritical complex. Then
+|[Mi+1 − Mi](1, ei, · · · , e1)| = |Mi+1(1, ei, · · · , e1) − Eei+1[Mi+1]|
+= |Mi+1(1, ei, · · · , e1) − p · (Mi+1(1, ei, · · · , e1)) − (1 − p) · (Mi+1(0, ei, · · · , e1))|
+= |(1 − p) · (Mi+1(1, ei, · · · , e1) − Mi+1(0, ei, · · · , e1))|
+≤ (1 − p)(2t+1 + O(1/n))
+Similarly,
+|[Mi+1 − Mi](0, ei, · · · , e1)| = |Mi+1(0, ei, · · · , e1) − Eei+1[Mi+1]|
+= |Mi+1(0, ei, · · · , e1) − p · (Mi+1(1, ei, · · · , e1)) − (1 − p) · (Mi+1(0, ei, · · · , e1))|
+= |p · (Mi+1(0, ei, · · · , e1) − Mi+1(1, ei, · · · , e1))|
+≤ p(2t+1 + O(1/n))
+Hence the claim follows.
+Next we claim that V ar(Mi+1 − Mi|Fi) = V ar(Mi+1|Fi) ≤ (c/n)(4t+1 + O(1/n)) ≤ O(1/n).
+Indeed if (ei, · · · , e1) contains a precritical complex then the martingale stops and the variance
+is zero. Otherwise
+13
+
+V ar(Mi+1 − Mi|ei, · · · , e1) = p · ([Mi+1 − Mi](1, ei, · · · , e1))2 + (1 − p) · ([Mi+1 − Mi](0, ei, · · · , e1))2
+≤ p(1 − p)2(2t+1 + O(1/n))2 + (1 − p)p2(2t+1 + O(1/n))2
+≤ p(1 − p)(4t+1 + O(1/n))
+≤ (c/n)(4t+1 + O(1/n))
+Thus by Lemma 17,
+Pr{|Mm − M0| ≥ s · n
+2
+3 } ≤ 2 exp(−
+s2 · n
+4
+3
+(n2 − n) · (c/n)(4t+1 + O(1/n)) + 2/3 · (2t+1 + O(1/n)) · s · n)
+≤ 2 exp(−
+s2 · n
+4
+3
+(n − 1)c4t+1 + (2/3)sn2/32t+1 + O(1))
+≤ 2 exp(−
+s2 · n
+4
+3
+nc4t+1 + (2/3)sn2/32t+1 + O(1))
+≤ 2 exp(−
+s2 · n
+4
+3
+n(c4t+1 + (2/3)sn−1/32t+1) + O(1))
+≤ 2 exp(−
+s2 · n
+1
+3
+(c4t+1 + (2/3)sn−1/32t+1) + O(1/n))
+Proof. Proof of Theorem 11 Theorem 11 follows from Theorem 16 and Lemma 15 via the
+following inequalities.
+Pr{|Ym − Y0| ≥ s} = Pr{|Ym − Y0| ≥ s and ¬N′} + Pr{|Ym − Y0| ≥ s and N′}
+≤ Pr{|Mm − M0| ≥ s} + Pr{N′}
+Thus
+Pr{|Ym − Y0| ≥ s · n
+2
+3 } ≤ 2 exp
+�
+−
+s2 · n
+1
+3
+(c4t+1 + (2/3)sn−1/32t+1) + O(1/n)
+�
++ O(1/n) = on(1).
+Now set X′
+0 := |f0(Rt(X))| − |f0(Rt+1(X))|.
+Lemma 18. For any s > 0, Pr{|X′
+0 − E[X′
+0]| > sn
+2
+3 } < on(1).
+Proof of Lemma 18. Observe that
+|X′
+0 − E[X′
+0]| > t =⇒ |(|f0(Rt−1(X))| − |f0(Rt(X))|) − (E[|f0(Rt−1(X))|] − E[|f0(Rt(X))|])| > t
+=⇒ ||f0(Rt(X)| − E[|f0(Rt(X))|]| + ||f0(Rt−1(X)| − E[|f0(Rt−1(X))|]| > t
+=⇒ ||f0(Rt(X)| − E[|f0(Rt(X))|]| > t/2
+or
+||f0(Rt−1(X)| − E[|f0(Rt−1(X))|]| > t/2
+Hence, by union bound,
+Pr{|X′
+0 − E[X′
+0]| > sn
+2
+3 }
+≤ Pr{||f0(Rt(X)| − E[|f0(Rt(X))|]| > (s/2)n
+2
+3 }+Pr{||f0(Rt−1(X)| − E[|f0(Rt−1(X))|]| > (s/2)n
+2
+3 } ≤ on(1)
+Let X0 be the random variable that denotes the number of dominated vertices at the end
+of the first epoch. Clearly 0 ≤ X0 ≤ |f0(Rt(X))| − |f0(Rt+1(X))| = X′
+0. As X′
+0 ≤ E[X′
+0] + o(n)
+a.a.s. we get that 0 ≤ X0 ≤ E[|f0(Rt(X))|] − E[|f0(Rt+1(X))|] + o(n) a.a.s.
+14
+
+7
+Second Epoch
+The second epoch will be a slower version of the first epoch. Here a dominated vertex is chosen
+uniformly randomly and is removed. The process continues until there is no more dominated
+vertices. Similar to the proof of [3], our strategy shall be to show that when a dominated vertex
+is deleted, the expected number of newly created dominated vertices is strictly less than 1, so
+that within o(n) steps, the strong collapse process comes to a halt. Thus the size of the core will
+be – up to a o(n)-factor – the number of vertices remaining after the first epoch.
+Let after t pruning phases the first epoch ends and the second epoch begins. Also, Let Yi be
+the random variable that denotes number of newly generated dominated vertices solely by the
+deletion of the dominated vertex at the i-th step of the second epoch. Note that Yi ∈ {0, · · · , n}.
+First we try to calculate Pr{Yi = 1}.
+Lemma 19. For any i ≤ 1
+12n(1 − γ)(1 − cγ),we have E[Yi] ≤ 1 − 3
+4(1 − cγ) + on(1) < 1 + on(1).
+Proof of Lemma 19. We shall say that a vertex is affected by ith collapse in the second epoch if
+its degree is changed by that collapsing step. Define Qi be the subset of the event {Yi = 1} that
+the newly generated vertex by the ith collapse is affected for the first time. Clearly the event
+{{Yi = 1} ∩ Qi} represents the fact that only one vertex, say v, is newly generated by the ith
+collapse (of vertex u) and v is affected for the first time. Thus that particular vertex retains the
+local structure since the first epoch. The idea here is that the edge {u, v} can be attached to
+any of the possible places after the first epoch ends. We are only calculating the probability
+that is is attached to a suitable vertex of Rt(X) \ {{u, v}, {u}}. To calculate Pr{{Yi = 1} ∩ Qi}
+we shall further partition it into two events. To this end, define P be the event that deg(v) = 2
+after i − 1 steps. Thus the probability Pr{{Yi = 1} ∩ Qi ∩ P} is the ratio of the numbers of
+degree one vertex in Rt(X) \ {{u, v}, {u}} to the number of non-isolated vertices after t − 1
+phase of the first epoch, as done in eq. 8 of [3]. It can be shown, by using similar arguments like
+section 6, that both these quantities are concentrated around their mean. These two quantities
+are, respectively, equal to c(1 − γt)γt+1n + o(n) and (1 − γt)n + o(n) a.a.s.
+For the event {Yi = 1} ∩ Qi ∩ P to occur v must be a part of a 2-simplex. Now we shall
+bound the number of 2-simplices remaining after the first epoch. Observe that during the
+collapsing phases number of 2-simplices can only decrease. Let us define the random variables
+T := |f2(X)| and T ′ := |f2(Rt(X))| for X ∼ X(n, c/n). Clearly T ′ ≤ T. From Markov’s
+inequality we get that Pr{T > log(n)} ≤ O(1/ log(n)). Thus, Pr{T ′ > log(n)} ≤ O(1/ log(n)).
+So Pr{{Yi = 1} ∩ Qi ∩ P} ≤ O(log(n))/(1 − γt)n a.a.s. Thus, a.a.s. Pr{{Yi = 1} ∩ Qi} ≤
+c(1−γt)γt+1n
+(1−γt)n
++ O(log(n))/(1 − γt)n ≤ c(1−γt)γt+1
+(1−γt)
++ on(1).
+Now we need to calculate Pr{{Yi = 1} ∩ Qi}. To do this we shall again partition this event
+into two disjoint events. Let Ai denote the number of affected vertices at ith step of the second
+epoch. Now define the event Bi := �i
+j=1{Aj ≤ 3}. So, Pr{{Yi = 1} ∩ Qi ∩ Bi−1} ≤
+3(i−1)
+n(1−γt).
+To calculate Pr{{Yi = 1} ∩ Qi ∩ Bi−1} first observe that Pr{{Yi = 1} ∩ Qi ∩ Bi−1} ≤
+Pr{Bi−1} ≤ Σi−1
+j=1{Aj ≥ 4} . But for {Ai ≥ 4} to happen the corresponding dominated vertex u
+must be a part of the following arrangement.
+a
+u
+ck
+ci
+cj
+Let S and S′ denote the number of such arrangements in X and Rt(X), respectively, for
+X ∼ X(n, c/n). Clearly, S′ ≤ S. From Markov’s inequality we get Pr{S ≥ 1} ≤ O(1/n2). Thus,
+Pr{S′ ≥ 1} ≤ O(1/n2). Hence, a.a.s. {Bi} never happens.
+15
+
+Similar argument combined with lemma 12 gives that Pr{Yi ∈ {2, · · · , n}} ≤ O(1/n2).
+By collecting all the terms we have the following inequality.
+E[Yi] = Σn
+j=0j · Pr{Yi = j}
+≤ Pr{Yi = 1} + n · O(1/n2)
+≤ Pr{{Yi = 1} ∩ Qi} + Pr{{Yi = 1} ∩ Qi} + n · O(1/n2)
+≤ Pr{{Yi = 1} ∩ Qi} + Pr{{Yi = 1} ∩ Qi ∩ Bi−1} + Pr{{Yi = 1} ∩ Qi ∩ Bi−1} + n · O(1/n2)
+≤ c(1 − γt)γt+1
+1 − γt
++ 3(i − 1)
+n(1 − γt) + O(1/n) + on(1)
+≤ cγt+1 + 3(i − 1)
+n(1 − γ) + on(1)
+≤ cγ + ϵi + on(1)
+≤ cγ + 1
+4(1 − cγ) + on(1)
+≤ 1 − 3
+4(1 − cγ) + on(1) < 1 + on(1), by lemma 5.
+Note that for any fixed i, ϵi = on(1) and for c > 1 we have cγ < 1.
+Let Xi be the number of dominated vertices at the end of the ith step of the second epoch.
+Then we have
+Xi = Xi−1 − 1 + Yi = X0 − n + Σn
+i=1Yi
+Untill the second epoch ends. If the second epoch stops at i′th step then Xj = 0
+∀j > i′.
+Thus we have
+E[X0] ≤ E[|f0(Rt(X))|] − E[|f0(Rt+1(X))|]
+= (1 − γt+1 − cγt + cγ2
+t )n − (1 − γt+2 − cγt+1 + cγ2
+t+1)n
+and,
+E[Xi] = E[Xi−1] − 1 + Yi = E[X0] − n + Σn
+i=1Yi
+as long as the second epoch continues.
+Now let us define δ ≡ δ(t) := (1 − γt+1 − cγt + cγ2
+t ) − (1 − γt+2 − cγt+1 + cγ2
+t+1) so that
+E[X0] ≤ nδ.
+Now we present the main lemmas of this section. The first lemma below shows that with
+high probability, for any sufficiently small ε > 0, we can choose a sufficiently large t, such that
+the number of vertices deleted in the second epoch is less than εn. The proof is by modelling
+the number of remaining dominated vertices after i steps of the epoch, as a biased random walk.
+Lemma 20. ∀0 < ϵ < min{ 1
+12(1 − γ)(1 − cγ),
+5
+192(1 − γ)(1 − cγ)2}
+∃T such that ∀t > T a.a.s.
+at most ϵn vertices will be deleted from Rt(X) before algorithm reaches the core.
+Proof of Lemma 20. Choose t such that δ ≡ δ(t) < 1
+8ϵ(1 − cγ). This can be done because as t
+increases γt − γt+1 and γt+1 − γt+2 approaches zero.
+Now suppose that the second epoch runs for ρ = ϵn steps. We shall show that E[Xρ] = 0
+a.a.s., i.e., there is no more dominated vertex left to be deleted. To this end we define a sequence
+of new random variable {Zi} as follows:
+Z0 := X0
+16
+
+and,
+Zi := Zi−1 − 1 + Yi = Z0 − n + Σn
+i=1Yi
+Note that Xi ≤ Zi and Zi ≤ 0 =⇒ Xi = 0.
+As ρ = ϵn ≤ 1
+4n(1 − γ)(1 − cγ), at the end of the ρ steps number of dominated vertices
+remaining is
+E[Xρ] ≤ E[Zρ] = E[Z0] − Σρ
+i 1 + Σρ
+i E[Yi]
+≤ nδ − 3
+4ρ(1 − cγ)
+by lemma 19
+≤ 1
+8nϵ(1 − cγ) − 3
+4nϵ(1 − cγ)
+≤ − 5
+96(1 − γ)(1 − cγ)2n ≤ 0
+Hence E[Xρ] = 0.
+Now we shall proving the concentration. Let us define l :=
+5
+96(1 − γ)(1 − cγ)2. Next we
+proceed to show
+Pr{Zρ > 0} ≤ Pr{Zρ − E[Zρ] > l · n} < on(1)
+Write Zρ as Zρ ≡ Z0 + Z′
+ρ(Y1, · · · , Yρ). First observe that, from lemma 18, Pr{Z0 − E[Z0] >
+(l/2)n} ≤ Pr{|Z0 − E[Z0]| > (l/2)n} ≤ on(1) for any s > 0.
+From the main geometric lemma we get
+E[et(Yi−E[Yi])] ≤ E[etYi] ≤ 1 + et + O(e2t
+n2 · 1 − (et/n3)n−1
+1 − et/n3
+)
+Fix t > 0. Then,
+Pr{Z′
+ρ − E[Z′
+ρ] > (l/2) · n} = Pr{Σρ
+i Yi − Σρ
+i E[Yi] > (l/2) · n}
+= Pr{et(Σρ
+i Yi−Σρ
+i E[Yi]) > etn(l/2)}
+≤ et(Σρ
+i Yi−Σρ
+i E[Yi])/etn(l/2)
+≤
+(1 + et + O( e2t
+n2 · 1−(et/n3)n−1
+1−et/n3
+))ρ
+etn(l/2)
+setting t = log(n) we get
+Pr{Z′
+ρ − E[Z′
+ρ] > (l/2) · n} ≤
+(1 + n + O( 1−(1/n2)n−1
+1−1/n2
+))ρ
+nn(l/2)
+≤
+(1 + n + O(
+1
+1−1/n2 ))ϵn
+nn(l/2)
+≤ ((O(1) + n)ϵ)n
+(n(l/2))n
+≤ ((O(1) + n)ϵ
+n(l/2)
+)n ≤ on(1)
+As ϵ < l/2 ,the quantity approaches zero as n increases, thus the last inequality follows. So,
+Pr{Zρ − E[Zρ] > l · n} ≤ Pr{Z0 − E[Z0] > (l/2) · n} + Pr{Z′
+ρ − E[Z′
+ρ] > (l/2) · n} ≤ on(1)
+17
+
+The next lemma follows from properties of γt and the concentration bounds presented in
+Section 6.
+Lemma 21. ∀0 < δ
+∃T such that ∀t > T a.a.s.
+(1 − γ)(1 − cγ)n + o(n) ≤ |f0(Rt(X))| ≤ (1 − γ)(1 − cγ)n + δn + o(n).
+Proof of Lemma 21. From theorem 6 and theorem 11 it can the shown that a.a.s.
+|f0(Rt(X))| = (1 − γt+1 − cγt + cγ2
+t )n + o(n)
+Define h(x) := 1 − e−c(1−x) − cx(1 − x) on the interval (0, 1). It can be checked that on this
+interval h′(x) < 0, i.e., h(x) is strictly decreasing. Thus we have that (1−γt+2 −cγt+1 +cγ2
+t+1) ≤
+(1 − γt+1 − cγt + cγ2
+t ). Hence the left inequality follows.
+Note that {γt}t is a monotonically increasing sequence that converges to γ. Therefore,
+{(1−γt+1−cγt+cγ2
+t )}t is a monotonically decreasing sequence that converges to (1−γ−cγ+cγ2).
+Thus, by choosing sufficiently large T, we can restrict {(1 − γt+1 − cγt + cγ2
+t )}t>T inside a δ-ball
+round (1 − γ − cγ + cγ2) for any δ > 0. Hence the right inequality follows.
+Using above two lemmas we have the proof of our first main result Theorem 1 about the size
+of the core (after strong collapse) of a ER complex.
+Proof of Theorem 1. By Lemma 20 and Lemma 21.
+8
+End Range Phase Transition
+Let P(X) denote the number all possible dominated-dominating pairs. It can be shown that
+for X ∼ X(n, p), E[P(X)] = (n(n − 1)/2)p(1 − p(1 − p))n−2. For p = λ log(n)
+n
+, where λ > 1,
+E[P(X)] = O( n log(n)
+nλ
+) = on(1). Thus, by Markov’s inequality, a.a.s. there is no dominated
+vertex to start the collapsing procedure.
+Now we shall focus on the behavior of X ∼ X(n, p) where p = 1 − λ log(n)
+n
+. For λ > 2,
+E[P(X)] = O( n2
+nλ ) = on(1). Thus a.a.s. there is no dominated vertex to start the collapsing
+procedure. But the situation is quite opposite when λ < 1 as the following lemma claims.
+Lemma 22. For X ∼ X(X, 1 − λ log n
+n
+) and λ < 1, a.a.s. X is collapsible.
+Proof of Lemma 22. We shall show that, in this range, a.a.s. there exits a vertex adjacent to
+every other vertices. Let us define the random variable V ∈ {0, · · · , n} that counts the number
+vertices that are adjacent to all other vertices. Clearly E[V ] = npn−1 = n(1− λ log(n)
+n
+)n−1 = Θ( n
+nλ ).
+Now we shall calculate V ar(V ). Let Ii denote the indicator random variable that vi is adjacent
+to all other vertices. Then,
+V ar(V ) = nV ar(I1) + n(n − 1)cov(I1, I2)
+= npn−1(1 − pn−1) + n(n − 1)(p2n−3 − p2n−2)
+Thus,
+Pr{V = 0} ≤ V ar(V )
+(E[V ])2
+≤ npn−1(1 − pn−1) + n(n − 1)(p2n−3 − p2n−2)
+n2p2n−2
+≤ 1/(npn−1) + (1/p − 1)
+≤ Θ(nλ
+n ) +
+λ log(n)
+n − λ log(n) = on(1)
+18
+
+Thus V ≥ 1 a.a.s.
+References
+[1] N. Alon and J. Spencer. The Probabilistic Method. Wiley, New York, 3rd edition, 2008.
+[2] Lior Aronshtam and Nathan Linial. When does the top homology of a random simplicial
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf,len=781
+page_content='Strong Collapse of Random Simplicial Complexes  Jean-Daniel Boissonnat ∗ 1, Kunal Dutta † 2, Soumik Dutta † 3, and Siddharth Pritam ∗ 4  1Universit´e Cˆote d’Azur, INRIA, Sophia Antipolis, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' email:  Jean-Daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='Boissonnat@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='fr  2Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  email: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='dutta@mimuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='pl  3Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Poland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  email: s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='dutta2@mimuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='pl  4Universit´e Cˆote d’Azur, INRIA, Sophia Antipolis, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' email:  siddharth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='pritam@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='fr  Abstract  The strong collapse of a simplicial complex, proposed by Barmak and Minian [6], is a  combinatorial collapse of a complex onto its sub-complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Recently, it has received attention  from computational topology researchers [22, 7, 8], owing to its empirically observed useful-  ness in simplification and size-reduction of the size of simplicial complexes while preserving  the homotopy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We consider the strong collapse process on random simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  For the Erd˝os-R´enyi random clique complex X(n, c/n) on n vertices with edge probability  c/n with c > 1, we show that after any maximal sequence of strong collapses the remaining  subcomplex, or core must have (1 − γ)(1 − cγ)n + o(n) vertices asymptotically almost surely  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ), where γ is the least non-negative fixed point of the function f(x) = exp (−c(1 − x))  in the range (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' These are the first theoretical results proved for strong collapses on  random (or non-random) simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  1  Introduction  Motivation  Simple collapse is a combinatorial notion which simplifies a simplicial complex  without changing its topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' It can be expressed as a series of elementary moves of removals of  pair of simplices σ and τ, such that σ is uniquely contained in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The notion of simple collapse  was introduced by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='C Whitehead [21] to study homotopy types of cell complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Since then  it has found usage in many different areas of topology, especially in computational topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Recently new variants of simple collapses have been introduced, called strong collapses and more  generally d-collapses [6, 8, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In such collapses one removes special vertices (more generally  d-simplices) called dominated vertices (simplices) whose link is a simplicial cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' It’s again  expressed as a series of elementary moves of removals of dominated vertices (simplices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' They  have been shown to be very powerful tools to solve many problems in computational topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  In particular, the recent works of Pritam et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' [9, 8, 23] has shown that strong collapses and  edge collapses (d-collapse for d = 1) can be used for efficient computation of one parameter and  ∗J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Boissonnat and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Pritam received funding from the European Research Council (ERC) under the  European Union’s Seventh Framework Programme (FP/2007- 2013) / ERC Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' 339025 GUDHI  (Algorithmic Foundations of Geometry Understanding in Higher Dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  †K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Dutta and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Dutta received funding from the Polish NCN SONATA Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' 2019/35/D/ST6/04525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='03514v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='CG]  9 Jan 2023  multi-parameter persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Efficient computation of persistent homology is one of the central  topic of research in topological data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  The computation of persistent homology involves computing homology groups of a nested  sequence of simplicial complexes called filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' And to compute persistent homology requires  O(n3) time and O(n2) space, here n is the total number of simplices in the filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The  general technique developed in [9, 7, 8, 12, 23] is to reduce a filtration to a smaller filtration  using strong or edge collapse such that the persistent homology is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In [9, 7, 8, 12, 23],  it has been established through experiments that in practice the reduced filtrations are very  small and thereafter computation of persistent homology is extremely fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The gain in efficiency  is quite dramatic in the case of flag (clique) complexes, where the strong collapse and edge  collapse can be computed using only the graph (1-skeleton) of the given complex [7, 12, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  As mentioned above the efficiency reported in [9, 7, 12, 23] are through experiments and  there is no theoretical guarantee over the reduction size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' This is due to the fact that in general  the amount of reduction depends on the individual complex and its combinatorial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In  fact, the reduction is dependent even on the order of the collapses and a different order can  result in a different core, except in the case of strong collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Which is even harder when we  want study the reduction size in a filtered simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' This motivates us to consider  the case of random simplicial complexes and study the average reduction size by collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  In this article, we study the problem of reduction size achieved by the strong collapses of a  clique complex defined over an Erd˝os-R´enyi random graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Previous  The study of random simplicial complexes was initiated in the seminal paper of  Linial and Meshulam [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Later Meshulam and Wallach [19] generalized the model of random  complexes to obtain the Linial-Meshulam (LM) model of d-dimensional random complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Since then a large body of work from several authors has emerged on many different models  of random simplicial complexes, studying various topological and geometric properties of such  complexes [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The study of simple collapses for random simplicial complexes has also been of  interest to researchers and there have been numerous works in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In the d-dimensional  LM model, Kozlov [15] proved bounds on the threshold for vanishing of the d-th homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Simple collapses on random complexes were first studied by Aronshtam, Linial, �Luczak and  Meshulam [4], who improved Kozlov’s bound to get a tight bound on the threshold and also gave  a bound on the threshold for collapsibility in the d-dimensional LM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Later Aronshtam  and Linial [2, 3] extended this line of work, obtaining first the threshold for the vanishing of  the d-th homology in [2] and then the threshold for non-collapsibility of the d-dimensional LM  complex [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In [17] Linial and Peled obtained precise asymptotic bounds on the size of the core  of such complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Very recently, Malen [18] has shown that the ER clique complex X(n, p) is  (k + 1)-collapsible with high probability for p = n−α when α > 1/(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Thus, to the best of our knowledge, work on collapses in random complexes has so far  considered only simple collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Throughout this paper, we shall use the notation asymptotically almost surely (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=') for a  series of events (En)n≥1, when the probability of occurence of En goes to 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Models of Random Simplicial Complexes  In this paper we shall consider two models of  random simplicial complexes, which are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For a graph G, let cl(G) denotes the  clique (flag) complex on G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' the simplicial complex where each complete subgraph of G  on d vertices is a d − 1-simplex in cl(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The Erd˝os-R´enyi (ER) model X(n, p) on n vertices  with probability parameter p ∈ [0, 1] is given by connecting each possible pair of elements of an  n-element set by an edge randomly and independently with probability p to get the random  graph G = G(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let X(n, p) := cl(G(n, p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  2  Our Contribution  We give bounds on the size of the core (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' the smallest sub complex  without any dominated vertex) of a random simplicial complex after strong (vertex) collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Whereas previous works focused on computing the threshold for the appearance and disappearance  of the k-th homology class for different k, here we are more interested in computing the size of  the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We show that for n-vertex ER clique complexes, the size of the core after a maximal  series of strong collapses is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' a constant fraction of n, with the constant depending only on  the edge probability, and bounded away from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Further, we also find a precise expression for  the constant, as a fixed point of an implicit equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Our first theorem is stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Let K be a simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For i ≥ 1, we use fi(K) to denote the set of i−simplices of  the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' By a slight abuse of notation, we let f0(K) denote the set of non-isolated vertices  of the complex and ˜f0(K) denote the set of non-isolated vertices of the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' (A vertex  v ∈ K is isolated the if v is not a face of any edge in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=') In a pruning phase run on K, all  dominated (strong-collapsible) vertices of K are simultaneously collapsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let Rt(K) denote  a complex obtained by running t pruning phases over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Lastly, R∞(K) denotes a complex  obtained from K after running a maximal series of strong collapses on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' the core of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let c > 1 and X ∼ X(n, c/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then there exists a constant γ ≡ γ(c) given by  the least non-negative fixed point of the function f(x) = exp (−c(1 − x)), x ∈ (0, 1), such that  0 < γ < 1/c < 1 and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' the following holds  |f0(R∞(X))| = (1 − γ)(1 − cγ)n + o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  0  5000  10000  15000  20000  25000  0  10000  20000  30000  40000  50000  60000  70000  80000  90000  100000  size of the CORE  n  experimental value for c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='5  theoretical value for c=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='5  Figure 1: We ran experiments on ER complexes and plotted the size of the core (strong collapse)  with n ∈ [0, 105] and c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' These experiments clearly validates our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  3  0  1000  2000  3000  4000  5000  6000  7000  8000  9000  10000  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='5  5  size of the CORE  c  experimental value for n=10000  theoretical value for n=10000  Figure 2: In a different set of experiments over ER complexes, we varied the constant c ∈ [1, 5]  keeping the number of vertices fixes to n = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  We then address a question of algorithmic interest: given an ε ∈ (0, 1), how many rounds  do we need in the first epoch to get within an εn gap from the actual size of the core?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The  following theorem gives a bound on the number of rounds t as a function of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let X ∼ X(n, c/n) and ε > 0 (sufficiently small) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then there exists  t ∈ Z+ such that  (c − cγ)ec  1 − ce−c (ce−c)t ≤ ϵ ≤ (c + 1)ec  1 − cγ (cγ)t,  and |f0(Rt(X))| − |f0(R∞(X))| ≤ ϵn + o(n) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='1  Overview of Proofs and Outline of Sections  While our analysis shares the general flow of the analysis of the simple collapsibility of LM  complexes in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' [4, 2, 3], there are several differences and difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Firstly, in both cases  (strong collapse of ER clique complex and simple collapse of LM model) our goal is to find the  size of the core, rather than whether the complex is collapsible or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Secondly, in the case of  the random ER clique complex our analysis needs to take into account the non-homogeneity of  the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' That is, maximal simplices in this model can have different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Further, unlike  in the LM model, the existence of a maximal simplex is not independent of the existence of all  other possible maximal simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Finally perhaps the most interesting difference of the random  ER clique complex model in our context, is that the effect of removing a vertex is not necessarily  localized – a fact which requires a fair bit of innovation to handle (in several places), especially  in proving the concentration bounds in Section 6, and in the later stages of the analysis, in  Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We present a more detailed overview of the proof strategy used in our concentration  bound, in the beginning of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  With the above caveats in mind, we first briefly review the main ideas of the proof of  Aronshtam and Linial [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The analysis was split into two epochs, each of which were further  divided into several rounds (phases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In the first epoch, in each round, every simple-collapsible  simplex was simultaneously collapsed, and this procedure was repeated for a constant number  of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Aronshtam and Linial [3] used a tree-like model of a random simplicial complex to  approximate the local structure of the random LM complex and showed that the total number  of collapsed simplices over all such rounds tended to a constant fraction of the number of initial  4  simplices, as the number of rounds increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Moreover, this limit constant could be expressed  as a fixed point of an implicit equation involving only the distribution parameter c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  In the analysis of the second epoch, a simplex would be chosen randomly from the set of  (non-neighbouring) simple-collapsible simplices, and collapsed in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The aim was to  show that in this epoch, the number of simplices collapsed would be asymptotically negligible  compared to the inital number of simplices present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus in summary, the final number of  deleted simplices is determined by the first epoch itself, and the second epoch serves to show the  tightness of this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Our proofs also split the analysis into two epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Similar to [3], we show that a certain tree-  like model of random simplicial complexes provides a good approximation of local neighbourhoods,  which is done in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' This is followed by the analysis of the first epoch, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The  main theorem of this section gives an expression for the expected number of vertices remaining  after t rounds (or pruning phases) of the first epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Bounds on t as a function of ε, γ and  c, are given in Theorem 2, which is proved in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Before beginning the analysis of the  second epoch however, we need bounds on the concentration of the size of the core itself, as well  as several other random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' These are proved in Section 6, where we use the notions of  critical and precritical (sub)complexes – described in more detail in the beginning of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  With these concentration bounds in place, we move to the analysis of the second epoch in  Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  2  Preliminaries  In this section we briefly introduce some topological and probabilistic notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Readers can refer  to [13] for a comprehensive introduction to topics related to topology and [10] for topics related  to probability theory and random structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  An abstract simplicial complex K is a collection of subsets of a  non-empty finite set X, such that for every subset A in K, all the subsets of A are in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' An  element of K is called a simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' An element of cardinality d + 1 is called a d-simplex and d  is called its dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Given a simplicial complex K, we denote its geometric realization as  |K|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' A simplex is called maximal if it is not a proper subset of any other simplex in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' A  sub-collection L of K is called a subcomplex if it is a simplicial complex itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' A subcomplex  K′ of K is called a d-skeleton of K if it contains all the simplices of K of dimension at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Erdos Renyi Graph Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  This is the probability space G(n, p) consisting of  all the graphs on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Probability of occurrence of a graph with m edges is pm(1 −  p)(n)(n−1)/2−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In other words, it is a random graph on n vertices where each edge can occur  independently with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Clique complex and Neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  A complex K is a clique or a flag complex if,  when a subset of its vertices form a clique (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' any pair of vertices is joined by an edge), they  span a simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For a vertex v in G, the open neighborhood NG(v) of v in G is defined as  NG(v) := {u ∈ G | [uv] ∈ E}, here E is the set of edges of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The closed neighborhood  NG[v] is NG[v] := NG(v) ∪ {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Similarly we define the closed and open neighborhood of an  edge [xy] ∈ G, NG[xy] and NG(xy) as NG[xy] := N[x] ∩ N[y] and NG(xy) := N(x) ∩ N(y),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The above definitions can be extended to any k-clique σ = [v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=', vk] of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  NG[σ] := �  vi∈σ N[vi] and NG(σ) := �  vi∈σ N(vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Star, Link and Simplicial Cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Let σ be a simplex of a simplicial complex K, the  closed star of σ in K, stK(σ) is a subcomplex of K which is defined as follows, stK(σ) := {τ ∈  5  K| τ ∪ σ ∈ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The link of σ in K, lkK(σ) is defined as the set of simplices in stK(σ) which  do not intersect with σ, lkK(σ) := {τ ∈ stK(σ)|τ ∩ σ = ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The open star of σ in K, sto  K(σ) is  defined as the set stK(σ) \\ lkK(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Usually sto  K(σ) is not a subcomplex of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Let L be a simplicial complex and let a be a vertex not in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then the set aL defined as  aL := {a, τ | τ ∈ L or τ = σ ∪ a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' where σ ∈ L} is called a simplicial cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Simple collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Given a complex K, a simplex σ ∈ K is called a free simplex if σ has  a unique coface τ ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The pair {σ, τ} is called a free pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The action of removing a free  pair: K → K \\ {σ, τ} is called an elementary simple collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' A series of such elementary  simple collapses is called a simple collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We denote it as K ↘ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' A subcomplex Kec of K  is called an elementary core of K if K↘Kec and Kec has no free pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Removal of a simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  We denote by K \\ σ the subcomplex of K obtained by removing  σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' the complex that has all the simplices of K except the simplex σ and the cofaces of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Dominated simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  A simplex σ in K is called a dominated simplex if the link lkK(σ)  of σ in K is a simplicial cone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' if there exists a vertex v′ /∈ σ and a subcomplex L of K, such  that lkK(σ) = v′L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We say that the vertex v′ is dominating σ and that σ is dominated by v′,  which we denote as σ ≺ v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  σ-algebra  The reader can refer to [10] for the definition of σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  d-collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Given a complex K, the action of removing a dominated k-simplex σ from K is  called an elementary d-collapse, denoted as K↘↘d{K \\σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' A series of elementary d-collapses  is called a d-collapse, denoted as K ↘↘k L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We further call a complex K d-collapse minimal  if it does not have any dominated d simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' A subcomplex Kk of K is called a d-core if K  ↘↘k Kd and Kd is d-collapse minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' A 0-core of a complex K is unique, however it is not  true in general for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Like simple collapses, d-collapses preserve the homotopy type of a  simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  A 0-collapse is a strong collapse as introduced in [6] and 1-collapse is called an edge  collapse [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The following lemma from [8] characterizes the domination of a simplex in the  special case of a flag complex in terms of neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let σ be a simplex of a flag complex K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then σ will be dominated by a vertex v′ if  and only if NG[σ] ⊆ NG[v′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  In this article, our main focus will be the case d = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' when σ is a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The next  lemma from [7], though elementary, is of crucial significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let K be a flag complex and let L be any subcomplex of K obtained by strong  collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then L is also a flag complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Both lemmas (Lemma 3 and Lemma 4) show that strong collapse is well-suited to flag  complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In the next sections we will investigate the reduction capabilities of strong of a clique  complex of an Erd˝os-R´enyi random graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  3  Tree process  In this section, we describe the tree process which is used to simulate the collapse process in the  first epoch of the (strong) collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' A one-dimensional tree is built recursively as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Start with a single node(root).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In the nth iteration, add children to all the leaves at distance n − 1 from the root from  Poisson distribution with parameter c(> 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Let Tn denote the set of all possible trees after nth iteration for n > 1 and T0 being the root  itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let T := �  n∈N  Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Let γt be the probability that a tree T ∈ Tt is pruned to the root in no more than t − 1 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Clearly, γ1 = e−c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Set γ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Also, we have the following recursive relation which is true in  general:  γt+1 = e−c(1−γt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Note that, in this process we never prune the root itself even if its degree is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We call such  a process root collapsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let γk  t denote the probability that a tree T ∈ Tt has degree k after  t − 1 root collapsing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then,  γk  t = (c(1 − γt−1))k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  e−c(1−γt−1)  Observe that γk  1 gives the initial degree distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Also, let γ≥2  t  denote the probability that a  vertex has degree atleast 2 after t − 1 root collapsing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then we have  γ≥2  t  =  ∞  �  k=2  γk  t = 1 − γt(1 + c(1 − γt−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Define βt := 1 − γt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus, βt is the probability that atleast t + 1 root collapsing steps are  needed to isolate the root of a tree T ∈ Tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Define f(x) := e−c(1−x) on the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We shall assume c > 1 for the rest of these  paper unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Note that f([0, 1]) ⊆ [0, 1], f′(x) = cf(x) and f(x) is strictly  increasing on the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let ft(x) denoted the function obtained by composing f t times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Then ft is also strictly increasing on [0, 1] for all t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' As, γ1 > γ0, applying ft−1 on both sides,  we get γt > γt−1 for all t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Also define γ to be the left most zero of the function g(x) := e−c(1−x) − x defined on the  range [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Note that 0 < γ ≤ 1 as g(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For c > 1 and γ = γ(c) defined as earlier we have cγ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let g(x) be defined as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' So g(γ−) > 0 and g(γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus, from  the differentiability of g(x), cγ − 1 = g′(γ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Now, if cγ − 1 = 0 then ce1−c = 1, which is  impossible for c > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus cγ − 1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Now observe that, x ≤ γ =⇒ f(x) ≤ f(γ) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus, by the fact that f′(x) = cf(x), f(x)  restricted on [0, γ] becomes a contraction mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' So, by Banach Fixed Point theorem, f|[0,γ]  has an unique fixed point which, in our case, is γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  To summarize the above arguments we get the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' γt converges to 0 < γ < 1/c as an increasing sequence and βt converges to  1 − 1/c < β < 1 as a decreasing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  4  First Epoch  In this section, we present the analysis of the first epoch of the collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The first epoch is executed  in phases and in each phase we remove a maximal set of dominated vertices simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Our goal, in this section, is to prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  7  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let X ∼ X(n, c/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let E(|f0(Rt(X))|) denote the expected number of non-isolated  vertices in X after t strong collapse phases and γt be as defined in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then,  E(|f0(Rt(X))|) = (1 − γt+1 − cγt + cγ2  t )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  We start by proving some important lemmas about the local structure of the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For  X ∼ X(n, c  n), let us define the following event,  D := {deg(v) ≤ log n ∀v ∈ ˜f0(X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Then, the following lemma can be proved using standard Chernoff bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Pr{D} = 1 − on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Note That for any v ∈ ˜f0(X), deg(v) ∼ Bin(n − 1, c/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let Dv be the  event that deg(v) ≤ log(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then by the Chernoff Bound on Binomial Distribution  Pr{¬Dv} = Pr{deg(v) > log(n)}  ≤ (ec(n − 1)  n log(n) )log(n)  ≤ (  ec  log(n))log(n)  = n− log(log(n/ec))  Thus by the union bound Pr{�  v∈f0(X) ¬Dv} ≤ n1−log(log(n/ec)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Hence,  Pr{D} = Pr{  �  v∈f0(X)  Dv} = 1 − Pr{  �  v∈f0(X)  ¬Dv} ≥ 1 − n1−log(log(n/ec)) = 1 − on(1)  By Cl(S) we denote the simplicial closure of the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Fix v ∈ ˜f0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Define N0 := Cl(v)  and N−1 := ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Also define Ni+1 := Cl({s ∈ X|∃u ∈ ˜f0(Ni)\\ ˜f0(Ni−1)|u ⊂ s})∪Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Equivalently,  this can also be defined in terms of the 1-skeleton of the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Define the event At = {Nt ∈ T }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let X ∼ X(n, p) and fix v ∈ ˜f0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then Pr{At ∩ D} = 1 − o(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' If X ∈ D then ˜f0(Nt) = O(logt+1n) and we want to avoid O(log2t+2n) edges to make Nt  a one dimensional tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Probability of that happening is  (1 − c/n)O(log2t+2n) = 1 − o(1)  Now the the degree of a node of this tree comes from Bin(n − 1, c/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For large n this  distribution can be approximated by Po(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Recall that for a simplicial complex X, Rt(X) denotes a complex obtained  after t phases and f0(X) denotes the set of non-isolated vertices (0−simplices) of the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Note that if a vertex v ∈ f0(X) survived t pruning steps then it must have had degree deg(v) ≥ 2  after t − 1 pruning steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus, Pr{v ∈ f0(Rt(X))} ≤ γ≥2  t  = 1 − γt(1 + c(1 − γt−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' This event  counts both the isolated vertices and degree one, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=', collapsible) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus this gives a  slight over estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' To get more precise estimate we observe that, in the spirit of [3], that a  vertex v ∈ f0(X) survives t pruning steps if it is neither collapsed nor isolated after t pruning  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Probability of such an event is 1 − γt+1 + cγt − cγ2  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The previous lemma asserts that it is  indeed the survival probability of a vertex of the simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  8  5  Rate of Convergence  In this section, we prove Theorem 2 thus giving bounds on the rate of convergence of the variable  γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let c, γt be as defined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then  cγt ≤ γt+1 − γt  γt − γt−1  ≤ cγt+1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let f(x) and g(x) be as defined in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Clearly,  γt+1 − γt  γt − γt−1  =  g(γt)  g(γt−1)  = g(γt−1 + (γt − γt−1))  g(γt−1)  = g(γt−1) + g′(s)(γt − γt−1)  g(γt−1)  (for some s ∈ [γt−1, γt])  = f′(s) = cf(s)  (as γt − γt−1 = γt−1)  As f(x) is an increasing function the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  In particular, ce−c ≤ γt+1−γt  γt−γt−1 ≤ cγ for all t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Let δ(t) = (1 − γt+1 − cγt + cγ2  t ) − (1 − γt+2 − cγt+1 + cγ2  t+1), as defined in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' It can  be shown that  δ(t) = (1 − γt+1 − cγt + cγ2  t ) − (1 − γt+2 − cγt+1 + cγ2  t+1)  = (γt+2 − γt+1) + c(γt+1 − γt) + c(γt + γt+1)(γt − γt+1)  = (γt+2 − γt+1  γt+1 − γt  + c − c(γt+1 + γt))(γt+1 − γt)  Hence,  (c − cγ)ec(ce−c)t ≤ δ(t) ≤ (c + 1)ec(cγ)t  Now define ϵ ≡ ϵ(t) := (1 − γt+1 − cγt + cγ2  t ) − (1 − γ − cγ + cγ2) so that E[|f0(Rt(X))|] −  (1 − γ)(1 − cγ] = ϵn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Consequently,  ϵ(t) = (1 − γt+1 − cγt + cγ2  t ) − (1 − γ − cγ + cγ2)  = (γ − γt+1) + c(γ − γt) + c(γt + γ)(γt − γ)  = (γ − γt+1) + (c − c(γt + γ))(γ − γt)  So,  (c − cγ)ec  1 − ce−c (ce−c)t ≤ ϵ(t) ≤ (c + 1)ec  1 − cγ (cγ)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Thus we get the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' for any t ≥ 1  ec(ce−c)t ≤ γt+1 − γt ≤ ec(cγ)t  and  ec  1 − ce−c (ce−c)t ≤ γ − γt ≤  ec  1 − cγ (cγ)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  From the above corollary, we get the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  9  6  Concentration of Size of the Complex after the First Epoch  In this section, we shall prove a concentration bound on the size of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Unlike in the case  of simple collapses in d-dimensional LM complexes [2, 3], concentration bounds in our case are  less straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Observe firstly, that deleting a single vertex v could potentially change  the domination status of an arbitrary number of vertices, as for example when v dominates the  entire complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus the influence of a vertex can be n in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Therefore we shall  need to use an edge exposure martingale inequality, in the form of a variant of an inequality of  Freedman [11], given by Warnke [20], which allows us to consider the path variance of the effect  of a single edge, rather than the worst case effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  In order to bound the path variance, we shall show that if the influence of a variable is large,  there is a specific class of subcomplexes, which we call Critical Complexes, one of which must  occur in the 1-skeleton of the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' It is not hard to show (and we do) that the probability  of occurence of these subgraphs is vanishingly low in the original random complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' However, the  variance needs to be controlled at all steps in the edge exposure martingale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' when we are  computing expectations over arbitrarily small subcomplexes of the original complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' To handle  this, we need to define a superset of critical complexes, which we call Precritical Complexes,  and show that their probability of occurence will still be vanishingly small throughout the edge  exposure process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We can then define a stopped martingale which stops if at any step of the edge  exposure process, a precritical complex occurs, and prove concentration bounds using Warnke’s  inequality for this martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The final concentration bound is then the bound obtained for  the stopped martingale, together with the probability that the martingale ever encounters a  precritical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Fix p = c  n and m = n(n−1)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For 1 ≤ i ≤ m, ei ∼ Bernoulli(p) be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' random variables  corresponding to existance of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Clearly X(n, p) = e1 × · · · × em as probability spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Now we can define a filtration of σ-algebras {Fi}m  i=0 on X(n, p) by setting Fi to the σ-algebra  corresponding to e1, · · · , ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Let X ∼ X(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Now we construct an edge exposure martingale (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' [1] for a definition  of the edge exposure martingale) as follows: Clearly, Ym = |f0(Rt(X))| and Y0 = E(|f0(Rt(X))|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  This section is devoted to prove the following concentration result, which says that the size  of the complex after t pruning rounds of the first epoch is close to its expected value with high  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' (Main Theorem) Let X ∼ X(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let |f0(Rt(X))| be number of vertices after t  strong collapsing phases and Y0 = E(|f0(Rt(X))|) be its expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then for any s ≥ 0 we  have,  Pr{||f0(Rt(X))| − Y0| ≥ s · n  2  3 }  ≤  2 exp  �  −  s2 · n  1  3  (c4t+1 + (2/3)sn−1/32t+1) + O(1/n)  �  + O(1/n)  =  on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  To prove this we begin by observing some combinatorial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In the following lemmas, we  show that the influence of deleting one vertex is bounded, with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Pr{deleting a vertex b gives birth to k newly generated dominated vertices} ≤  O(  1  n3k−4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We first claim that deleting a vertex b gives birth to k newly generated  dominated vertices then b atleast have k neighbors one of which is the dominating vertex of  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In the following diagram, the solid arrow denotes domination and the white arrow denotes  10  future domination in the next phase only after deleting vertex b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The pointy head of the arrow is  towards the dominated vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Vertices a, b, c may be connected to other vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The following  diagrams exhibits some of the potential arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  a  b  c  a  b  c  a  b  c1  ci  cj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  A careful inspection will show that these kind of arrangements are impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Indeed if  it happens that will imply that the would-be-dominated vertices are already dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' This  is because we are only deleting b which is a common neighbor of all the would-be-dominated-  dominating pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus neighbors of b can not have white arrows between themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Thus fig:1 gives the necessary minimal arrangements for the birth of k newly generated  dominated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In the following diagram, all the ci’s and their corresponding d’s are assumed  to be connected to some non-neighbor of a which lies in the set {e1, · · · , el′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We claim that in  such case f1 − f0 ≥ (k − 2) + (k − 1) + (k − 1) ≥ 3k − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We shall prove our claim by induction  on k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The case k = 2 is evident from the following diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  a  b  c1  d  e  We now prove the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Consider the following figure again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  a  b  c1  ck−2 ck−1  {d1, · · · , dl}  {e1, · · · , el′}  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Now assume that the claim holds for k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Now just adding the kth vertex ck−1 increases  f1 − f0 by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Also the corresponding d and e increases f1 − f0 by 1 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' This ends the  induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' So in the all the possible minimal arrangements f1 − f0 ≥ 3k − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus expected  number of such arrangements is  � n  f0  �  (c/n)f1 = O(  1  n3k−4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus the result follows from Markov’s  inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Pr{deleting a vertex b gives birth to 3 newly generated dominated vertices}  ≤ O( 1  n5 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let X ∼ X(n, p) and e ∈ f1(X), then  Pr{|f0(Rt(X)) \\ f0(Rt(X \\ {e}))| ≥ 2t+1} ≤ O( 1  n5 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  11  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' If such an event happens then there must be a dominated vertex in the process whose  deletion creates atleast 3 new dominating vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus the result follows from the previous  corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Let Critical Complexes denote the minimal simplicial complexes corresponding to k = 3 (see  the following diagrams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let Precritical complexes be any of the Critical Complex without any  four of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='. Let N denote the set of complexes from X(n, p) that contains a Critical  Complex and N′ denote the set of complexes contains a Precritical Complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Clearly, N′ ⊇ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  a  b  c1  c2  d  e  a  b  c1  c2  d1  d2  e  a  b  c1  c2  d  e1  e2  a  b  c1  c2  d1  d2  e1  e2  The following result is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let X ∼ X(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then,  Pr{X ∈ N} ≤ O( 1  n5 ),  and  Pr{X ∈ N′} ≤ O( 1  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Now define stopping time τ on {Fi}m  i=0 such that τ = t if t = min(s≤t){Fs ∈ N′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Define a  stopped martingale with respect to {Fi}m  i=0 by Mi := Yi∧τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  We first prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' (Stopped Martingale inequality) Let {Mi}m  i=0 be the stopped martingale defined  as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then for any s ≥ 0 we have,  Pr{|Mm − M0| ≥ s · n2/3} ≤ 2 exp  �  −  s2 · n1/3  (c4t+1 + (2/3)sn−1/32t+1) + O(1/n)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  In order to prove the above theorem, we shall use the following lemma from Warnke [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Assume that {FK}0≤k≤N is an increasing sequence of σ-algebras, and {MK}0≤k≤N is an  {FK}0≤k≤N-adapted bounded martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' (2-sided version of Bounded Variance martingale Inequality) Let Uk be a Fk−1  variable satisfying |Mk − Mk−1| ≤ Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Set Ck = maxi∈[k] Uk and Vk = �  i∈[k] V (Mi−Mi−1|Fi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Let φ(x) = (1 + x) log(1 + x) − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For every s ≥ 0 and V, C > 0 we have  Pr{|MK − M0| ≥ s, Vk ≤ V, Ck ≤ C for some k ∈ [N]} ≤ 2e−s2/(2V +2Cs/3)  Theorem 11 essentially follows from Theorem 16 and Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  12  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Proof of Theorem 16  Let X ∈ X(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We shall first try to calculate Pr{X ∈ N|ei, · · · , e1} where (e1, · · · , ei)  does not form any precritical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let M ⊂ N be the set of complexes that contains  some critical complex not involving any of the edges from {ei, · · · , e1} and M′ ⊂ N be the set  of complexes where all the critical complexes involves some edges from {ei, · · · , e1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Clearly,  N = M ⊔ M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus,  Pr{X ∈ N|ei, · · · , e1} = Pr{X ∈ M ⊔ M′|ei, · · · , e1}  = Pr{X ∈ M|ei, · · · , e1} + Pr{X ∈ M′|ei, · · · , e1}  = Pr{X ∈ M} + Pr{X ∈ M′|ei, · · · , e1}  Pr{X ∈ M} is the probability that {ei+1, · · · , em} contains a critical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' By reasoning  similar to the proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We get Pr{X ∈ M} = O(1/n5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' On the other hand, note  that as (ei, · · · , e1) does not contain any precrtitical complex, atleast 5 more edges is needed  for X to form a critical complex involving some edges {ei, · · · , e1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Suppose, depending on  (e1, · · · , ei), k more edges are needed to complete a critical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Clearly 5 ≤ k ≤ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Also  note that, in a critical complex, there are atmost 2 vertices of degree two and rests have degree  atleat 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus even in the worst case one need to choose 3 vertices and construct 5 particular  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus Pr{X ∈ M′|ei, · · · , e1} = O(1/n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Hence, Pr{X ∈ N|ei, · · · , e1} = O(1/n2) given  (e1, · · · , ei) does not form any precritical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In particular,  Pr{|f0(Rt(X)) \\ f0(Rt(X \\ {ei+1}))| ≥ 2t+1|ei, · · · , e1} ≤ O( 1  n2 )  under the same assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Thus  E(|f0(Rt(X)) \\ f0(Rt(X \\ {ei+1}))||ei, · · · , e1) ≤ 2t+1 + n · O(1/n2) ≤ 2t+1 + O(1/n)  whenever (ei, · · · , e1) does not contain any precrtitical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  We now claim that |Mi+1 − Mi| ≤ 2t+1 + O(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' If (ei, · · · , e1) contains a precritical  complex then the martingale stops and the claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Now suppose (ei, · · · , e1) does not  contain any precritical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then  |[Mi+1 − Mi](1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1)| = |Mi+1(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1) − Eei+1[Mi+1]|  = |Mi+1(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1) − p · (Mi+1(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1)) − (1 − p) · (Mi+1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1))|  = |(1 − p) · (Mi+1(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1) − Mi+1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1))|  ≤ (1 − p)(2t+1 + O(1/n))  Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  |[Mi+1 − Mi](0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1)| = |Mi+1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1) − Eei+1[Mi+1]|  = |Mi+1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1) − p · (Mi+1(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1)) − (1 − p) · (Mi+1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1))|  = |p · (Mi+1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1) − Mi+1(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1))|  ≤ p(2t+1 + O(1/n))  Hence the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Next we claim that V ar(Mi+1 − Mi|Fi) = V ar(Mi+1|Fi) ≤ (c/n)(4t+1 + O(1/n)) ≤ O(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Indeed if (ei, · · · , e1) contains a precritical complex then the martingale stops and the variance  is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Otherwise  13  V ar(Mi+1 − Mi|ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1) = p · ([Mi+1 − Mi](1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1))2 + (1 − p) · ([Mi+1 − Mi](0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' e1))2  ≤ p(1 − p)2(2t+1 + O(1/n))2 + (1 − p)p2(2t+1 + O(1/n))2  ≤ p(1 − p)(4t+1 + O(1/n))  ≤ (c/n)(4t+1 + O(1/n))  Thus by Lemma 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Pr{|Mm − M0| ≥ s · n  2  3 } ≤ 2 exp(−  s2 · n  4  3  (n2 − n) · (c/n)(4t+1 + O(1/n)) + 2/3 · (2t+1 + O(1/n)) · s · n)  ≤ 2 exp(−  s2 · n  4  3  (n − 1)c4t+1 + (2/3)sn2/32t+1 + O(1))  ≤ 2 exp(−  s2 · n  4  3  nc4t+1 + (2/3)sn2/32t+1 + O(1))  ≤ 2 exp(−  s2 · n  4  3  n(c4t+1 + (2/3)sn−1/32t+1) + O(1))  ≤ 2 exp(−  s2 · n  1  3  (c4t+1 + (2/3)sn−1/32t+1) + O(1/n))  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Proof of Theorem 11 Theorem 11 follows from Theorem 16 and Lemma 15 via the  following inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Pr{|Ym − Y0| ≥ s} = Pr{|Ym − Y0| ≥ s and ¬N′} + Pr{|Ym − Y0| ≥ s and N′}  ≤ Pr{|Mm − M0| ≥ s} + Pr{N′}  Thus  Pr{|Ym − Y0| ≥ s · n  2  3 } ≤ 2 exp  �  −  s2 · n  1  3  (c4t+1 + (2/3)sn−1/32t+1) + O(1/n)  �  + O(1/n) = on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Now set X′  0 := |f0(Rt(X))| − |f0(Rt+1(X))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For any s > 0, Pr{|X′  0 − E[X′  0]| > sn  2  3 } < on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Proof of Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Observe that  |X′  0 − E[X′  0]| > t =⇒ |(|f0(Rt−1(X))| − |f0(Rt(X))|) − (E[|f0(Rt−1(X))|] − E[|f0(Rt(X))|])| > t  =⇒ ||f0(Rt(X)| − E[|f0(Rt(X))|]| + ||f0(Rt−1(X)| − E[|f0(Rt−1(X))|]| > t  =⇒ ||f0(Rt(X)| − E[|f0(Rt(X))|]| > t/2  or  ||f0(Rt−1(X)| − E[|f0(Rt−1(X))|]| > t/2  Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' by union bound,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Pr{|X′  0 − E[X′  0]| > sn  2  3 }  ≤ Pr{||f0(Rt(X)| − E[|f0(Rt(X))|]| > (s/2)n  2  3 }+Pr{||f0(Rt−1(X)| − E[|f0(Rt−1(X))|]| > (s/2)n  2  3 } ≤ on(1)  Let X0 be the random variable that denotes the number of dominated vertices at the end  of the first epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Clearly 0 ≤ X0 ≤ |f0(Rt(X))| − |f0(Rt+1(X))| = X′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' As X′  0 ≤ E[X′  0] + o(n)  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' we get that 0 ≤ X0 ≤ E[|f0(Rt(X))|] − E[|f0(Rt+1(X))|] + o(n) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  14  7  Second Epoch  The second epoch will be a slower version of the first epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Here a dominated vertex is chosen  uniformly randomly and is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The process continues until there is no more dominated  vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Similar to the proof of [3], our strategy shall be to show that when a dominated vertex  is deleted, the expected number of newly created dominated vertices is strictly less than 1, so  that within o(n) steps, the strong collapse process comes to a halt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus the size of the core will  be – up to a o(n)-factor – the number of vertices remaining after the first epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Let after t pruning phases the first epoch ends and the second epoch begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Also, Let Yi be  the random variable that denotes number of newly generated dominated vertices solely by the  deletion of the dominated vertex at the i-th step of the second epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Note that Yi ∈ {0, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  First we try to calculate Pr{Yi = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For any i ≤ 1  12n(1 − γ)(1 − cγ),we have E[Yi] ≤ 1 − 3  4(1 − cγ) + on(1) < 1 + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Proof of Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We shall say that a vertex is affected by ith collapse in the second epoch if  its degree is changed by that collapsing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Define Qi be the subset of the event {Yi = 1} that  the newly generated vertex by the ith collapse is affected for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Clearly the event  {{Yi = 1} ∩ Qi} represents the fact that only one vertex, say v, is newly generated by the ith  collapse (of vertex u) and v is affected for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus that particular vertex retains the  local structure since the first epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The idea here is that the edge {u, v} can be attached to  any of the possible places after the first epoch ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We are only calculating the probability  that is is attached to a suitable vertex of Rt(X) \\ {{u, v}, {u}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' To calculate Pr{{Yi = 1} ∩ Qi}  we shall further partition it into two events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' To this end, define P be the event that deg(v) = 2  after i − 1 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus the probability Pr{{Yi = 1} ∩ Qi ∩ P} is the ratio of the numbers of  degree one vertex in Rt(X) \\ {{u, v}, {u}} to the number of non-isolated vertices after t − 1  phase of the first epoch, as done in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' 8 of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' It can be shown, by using similar arguments like  section 6, that both these quantities are concentrated around their mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' These two quantities  are, respectively, equal to c(1 − γt)γt+1n + o(n) and (1 − γt)n + o(n) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  For the event {Yi = 1} ∩ Qi ∩ P to occur v must be a part of a 2-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Now we shall  bound the number of 2-simplices remaining after the first epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Observe that during the  collapsing phases number of 2-simplices can only decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let us define the random variables  T := |f2(X)| and T ′ := |f2(Rt(X))| for X ∼ X(n, c/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Clearly T ′ ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' From Markov’s  inequality we get that Pr{T > log(n)} ≤ O(1/ log(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus, Pr{T ′ > log(n)} ≤ O(1/ log(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  So Pr{{Yi = 1} ∩ Qi ∩ P} ≤ O(log(n))/(1 − γt)n a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Pr{{Yi = 1} ∩ Qi} ≤  c(1−γt)γt+1n  (1−γt)n  + O(log(n))/(1 − γt)n ≤ c(1−γt)γt+1  (1−γt)  + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Now we need to calculate Pr{{Yi = 1} ∩ Qi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' To do this we shall again partition this event  into two disjoint events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let Ai denote the number of affected vertices at ith step of the second  epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Now define the event Bi := �i  j=1{Aj ≤ 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' So, Pr{{Yi = 1} ∩ Qi ∩ Bi−1} ≤  3(i−1)  n(1−γt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  To calculate Pr{{Yi = 1} ∩ Qi ∩ Bi−1} first observe that Pr{{Yi = 1} ∩ Qi ∩ Bi−1} ≤  Pr{Bi−1} ≤ Σi−1  j=1{Aj ≥ 4} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' But for {Ai ≥ 4} to happen the corresponding dominated vertex u  must be a part of the following arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  a  u  ck  ci  cj  Let S and S′ denote the number of such arrangements in X and Rt(X), respectively, for  X ∼ X(n, c/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Clearly, S′ ≤ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' From Markov’s inequality we get Pr{S ≥ 1} ≤ O(1/n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus,  Pr{S′ ≥ 1} ≤ O(1/n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Hence, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' {Bi} never happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  15  Similar argument combined with lemma 12 gives that Pr{Yi ∈ {2, · · · , n}} ≤ O(1/n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  By collecting all the terms we have the following inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  E[Yi] = Σn  j=0j · Pr{Yi = j}  ≤ Pr{Yi = 1} + n · O(1/n2)  ≤ Pr{{Yi = 1} ∩ Qi} + Pr{{Yi = 1} ∩ Qi} + n · O(1/n2)  ≤ Pr{{Yi = 1} ∩ Qi} + Pr{{Yi = 1} ∩ Qi ∩ Bi−1} + Pr{{Yi = 1} ∩ Qi ∩ Bi−1} + n · O(1/n2)  ≤ c(1 − γt)γt+1  1 − γt  + 3(i − 1)  n(1 − γt) + O(1/n) + on(1)  ≤ cγt+1 + 3(i − 1)  n(1 − γ) + on(1)  ≤ cγ + ϵi + on(1)  ≤ cγ + 1  4(1 − cγ) + on(1)  ≤ 1 − 3  4(1 − cγ) + on(1) < 1 + on(1), by lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Note that for any fixed i, ϵi = on(1) and for c > 1 we have cγ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Let Xi be the number of dominated vertices at the end of the ith step of the second epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Then we have  Xi = Xi−1 − 1 + Yi = X0 − n + Σn  i=1Yi  Untill the second epoch ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' If the second epoch stops at i′th step then Xj = 0  ∀j > i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Thus we have  E[X0] ≤ E[|f0(Rt(X))|] − E[|f0(Rt+1(X))|]  = (1 − γt+1 − cγt + cγ2  t )n − (1 − γt+2 − cγt+1 + cγ2  t+1)n  and,  E[Xi] = E[Xi−1] − 1 + Yi = E[X0] − n + Σn  i=1Yi  as long as the second epoch continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Now let us define δ ≡ δ(t) := (1 − γt+1 − cγt + cγ2  t ) − (1 − γt+2 − cγt+1 + cγ2  t+1) so that  E[X0] ≤ nδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Now we present the main lemmas of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The first lemma below shows that with  high probability, for any sufficiently small ε > 0, we can choose a sufficiently large t, such that  the number of vertices deleted in the second epoch is less than εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The proof is by modelling  the number of remaining dominated vertices after i steps of the epoch, as a biased random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ∀0 < ϵ < min{ 1  12(1 − γ)(1 − cγ),  5  192(1 − γ)(1 − cγ)2}  ∃T such that ∀t > T a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  at most ϵn vertices will be deleted from Rt(X) before algorithm reaches the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Proof of Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Choose t such that δ ≡ δ(t) < 1  8ϵ(1 − cγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' This can be done because as t  increases γt − γt+1 and γt+1 − γt+2 approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Now suppose that the second epoch runs for ρ = ϵn steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We shall show that E[Xρ] = 0  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=', there is no more dominated vertex left to be deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' To this end we define a sequence  of new random variable {Zi} as follows:  Z0 := X0  16  and,  Zi := Zi−1 − 1 + Yi = Z0 − n + Σn  i=1Yi  Note that Xi ≤ Zi and Zi ≤ 0 =⇒ Xi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  As ρ = ϵn ≤ 1  4n(1 − γ)(1 − cγ), at the end of the ρ steps number of dominated vertices  remaining is  E[Xρ] ≤ E[Zρ] = E[Z0] − Σρ  i 1 + Σρ  i E[Yi]  ≤ nδ − 3  4ρ(1 − cγ)  by lemma 19  ≤ 1  8nϵ(1 − cγ) − 3  4nϵ(1 − cγ)  ≤ − 5  96(1 − γ)(1 − cγ)2n ≤ 0  Hence E[Xρ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Now we shall proving the concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let us define l :=  5  96(1 − γ)(1 − cγ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Next we  proceed to show  Pr{Zρ > 0} ≤ Pr{Zρ − E[Zρ] > l · n} < on(1)  Write Zρ as Zρ ≡ Z0 + Z′  ρ(Y1, · · · , Yρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' First observe that, from lemma 18, Pr{Z0 − E[Z0] >  (l/2)n} ≤ Pr{|Z0 − E[Z0]| > (l/2)n} ≤ on(1) for any s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  From the main geometric lemma we get  E[et(Yi−E[Yi])] ≤ E[etYi] ≤ 1 + et + O(e2t  n2 · 1 − (et/n3)n−1  1 − et/n3  )  Fix t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Pr{Z′  ρ − E[Z′  ρ] > (l/2) · n} = Pr{Σρ  i Yi − Σρ  i E[Yi] > (l/2) · n}  = Pr{et(Σρ  i Yi−Σρ  i E[Yi]) > etn(l/2)}  ≤ et(Σρ  i Yi−Σρ  i E[Yi])/etn(l/2)  ≤  (1 + et + O( e2t  n2 · 1−(et/n3)n−1  1−et/n3  ))ρ  etn(l/2)  setting t = log(n) we get  Pr{Z′  ρ − E[Z′  ρ] > (l/2) · n} ≤  (1 + n + O( 1−(1/n2)n−1  1−1/n2  ))ρ  nn(l/2)  ≤  (1 + n + O(  1  1−1/n2 ))ϵn  nn(l/2)  ≤ ((O(1) + n)ϵ)n  (n(l/2))n  ≤ ((O(1) + n)ϵ  n(l/2)  )n ≤ on(1)  As ϵ < l/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='the quantity approaches zero as n increases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' thus the last inequality follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' So,  Pr{Zρ − E[Zρ] > l · n} ≤ Pr{Z0 − E[Z0] > (l/2) · n} + Pr{Z′  ρ − E[Z′  ρ] > (l/2) · n} ≤ on(1)  17  The next lemma follows from properties of γt and the concentration bounds presented in  Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' ∀0 < δ  ∃T such that ∀t > T a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  (1 − γ)(1 − cγ)n + o(n) ≤ |f0(Rt(X))| ≤ (1 − γ)(1 − cγ)n + δn + o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Proof of Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' From theorem 6 and theorem 11 it can the shown that a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  |f0(Rt(X))| = (1 − γt+1 − cγt + cγ2  t )n + o(n)  Define h(x) := 1 − e−c(1−x) − cx(1 − x) on the interval (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' It can be checked that on this  interval h′(x) < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=', h(x) is strictly decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus we have that (1−γt+2 −cγt+1 +cγ2  t+1) ≤  (1 − γt+1 − cγt + cγ2  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Hence the left inequality follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Note that {γt}t is a monotonically increasing sequence that converges to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Therefore,  {(1−γt+1−cγt+cγ2  t )}t is a monotonically decreasing sequence that converges to (1−γ−cγ+cγ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Thus, by choosing sufficiently large T, we can restrict {(1 − γt+1 − cγt + cγ2  t )}t>T inside a δ-ball  round (1 − γ − cγ + cγ2) for any δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Hence the right inequality follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Using above two lemmas we have the proof of our first main result Theorem 1 about the size  of the core (after strong collapse) of a ER complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' By Lemma 20 and Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  8  End Range Phase Transition  Let P(X) denote the number all possible dominated-dominating pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' It can be shown that  for X ∼ X(n, p), E[P(X)] = (n(n − 1)/2)p(1 − p(1 − p))n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For p = λ log(n)  n  , where λ > 1,  E[P(X)] = O( n log(n)  nλ  ) = on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus, by Markov’s inequality, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' there is no dominated  vertex to start the collapsing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Now we shall focus on the behavior of X ∼ X(n, p) where p = 1 − λ log(n)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For λ > 2,  E[P(X)] = O( n2  nλ ) = on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Thus a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' there is no dominated vertex to start the collapsing  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' But the situation is quite opposite when λ < 1 as the following lemma claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' For X ∼ X(X, 1 − λ log n  n  ) and λ < 1, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' X is collapsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Proof of Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' We shall show that, in this range, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' there exits a vertex adjacent to  every other vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let us define the random variable V ∈ {0, · · · , n} that counts the number  vertices that are adjacent to all other vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Clearly E[V ] = npn−1 = n(1− λ log(n)  n  )n−1 = Θ( n  nλ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Now we shall calculate V ar(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Let Ii denote the indicator random variable that vi is adjacent  to all other vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Then,  V ar(V ) = nV ar(I1) + n(n − 1)cov(I1, I2)  = npn−1(1 − pn−1) + n(n − 1)(p2n−3 − p2n−2)  Thus,  Pr{V = 0} ≤ V ar(V )  (E[V ])2  ≤ npn−1(1 − pn−1) + n(n − 1)(p2n−3 − p2n−2)  n2p2n−2  ≤ 1/(npn−1) + (1/p − 1)  ≤ Θ(nλ  n ) +  λ log(n)  n − λ log(n) = on(1)  18  Thus V ≥ 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  References  [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Alon and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Spencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The Probabilistic Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Wiley, New York, 3rd edition, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  [2] Lior Aronshtam and Nathan Linial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' When does the top homology of a random simplicial  complex vanish?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Random Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Algorithms, 46(1):26–35, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='1002/rsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='20495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  [3] Lior Aronshtam and Nathan Linial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' The threshold for d-collapsibility in random complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  Random Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
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+page_content='1002/rsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
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+page_content=' Vietoris-rips complexes also provide  topologically correct reconstructions of sampled shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Computational Geometry, 46(4):448–  465, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
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+page_content=' Strong homotopy types, nerves and collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Discrete  and Computational Geometry, 47:301–328, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
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+page_content=' In 26th Annual  European Symposium on Algorithms (ESA 2018), volume 112, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
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+page_content=' Computing persistent features in big data:  A distributed dimension reduction approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In International Conference on Acoustics,  Speech, and Signal Processing (ICASSP), pages 11–15, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content='  [23] Siddharth Pritam ´Angel Javier Alonso, Michael Kerber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' Filtration-Domination in Bifiltered  Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
+page_content=' In SIAM Symposium on Algorithm Engineering and Experiments (ALENEX23),  2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQf5wUG/content/2301.03514v1.pdf'}
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+Machine Translation for Accessible Multi-Language Text Analysis
+Edward W. Chew1, William D. Weisman1, Jingying Huang1, Seth Frey1,*
+1 Department of Communication, University of California Davis, Davis, CA, USA
+* Correspondence: Seth Frey (sethfrey@ucdavis.edu) 376 Kerr Hall, 1 Shields Dr., Davis,
+CA 95616, USA
+
+Machine Translation for Accessible Multi-Language Text Analysis
+Abstract
+English is the international standard of social research, but scholars are increasingly conscious
+of their responsibility to meet the need for scholarly insight into communication processes
+globally. This tension is as true in computational methods as any other area, with
+revolutionary advances in the tools for English language texts leaving most other languages
+far behind. In this paper, we aim to leverage those very advances to demonstrate that
+multi-language analysis is currently accessible to all computational scholars. We show that
+English-trained measures computed after translation to English have adequate-to-excellent
+accuracy compared to source-language measures computed on original texts. We show this for
+three major analytics—sentiment analysis, topic analysis, and word embeddings—over 16
+languages, including Spanish, Chinese, Hindi, and Arabic. We validate this claim by
+comparing predictions on original language tweets and their backtranslations: double
+translations from their source language to English and back to the source language. Overall,
+our results suggest that Google Translate, a simple and widely accessible tool, is effective in
+preserving semantic content across languages and methods. Modern machine translation can
+thus help computational scholars make more inclusive and general claims about human
+communication.
+Keywords: multi-language text analysis, multi-lingual text analysis, computational text
+analysis, natural language processing, backtranslation, topic modeling, word embedding,
+sentiment analysis.
+
+Introduction
+Humans communicate in thousands of languages, and yet a single language, English,
+attracts the bulk of communication research. This not only has the effect of depriving other
+languages of adequate attention, but depriving English-focused scholars of any sense of where
+the language stands relative to others. The general use of English-trained tools for
+English-focused analyses in the social science community is particularly notable given the
+ubiquity of multi-lingual data and the power of modern computational natural language
+processing. For example, social media researchers on Twitter typically begin with raw data
+that is highly multilingual, before filtering out all tweets except for English, or some other
+single language. With such practices scholars miss a tremendous opportunity to test the
+generalizability of social media-observed big data claims. But bringing standard text analysis
+tools to the level of training and refinement that English-trained tools receive is a forbidding
+prospect that few multi-lingual scholars have the training, resources, and language
+background to pursue. We propose a simple alternative approach that makes texts from over
+100 languages accessible to the full variety of analyses that are typically available to only
+English-focused scholars. Specifically we demonstrate that modern machine translation has
+reached a level of refinement necessary to preserve sentiment, lexical topics, and semantic
+distance, making multi-language datasets legible to state of the art English-trained tools. By
+providing a validation of state-of-the-art machine translation, along with easily adaptable
+demonstration code, we aim to broaden the horizon of computational research and support
+Communication scholars in increasing the relevance and generality of their work.
+Google Translate, the most popular, accurate, and accessible multilingual neural
+machine translation service, offers translations for over 133 languages (Caswell, 2022). In this
+
+paper, we demonstrate the efficacy of Google Translate in retaining sentiment valence across
+translations of large hand-coded and machine-coded Twitter datasets composed of tweets in
+16 global non-English languages from four language families, being of Indo-European,
+Uralic, Semitic, and Sinitic origin. With our findings that Google Translate preserves the
+sentiment of tweets, as well as other dimensions of semantics, scholars may be emboldened to
+utilize Google Translate and other multilingual neural machine translation services to expand
+the generalizability of their research. In so doing, non-English languages can benefit from
+advanced English-trained natural language processing tools, and computational findings
+normally restricted to the English language can be expanded upon to broaden scholars’
+knowledge of global social phenomena.
+Academics use Twitter datasets for a wide range of scholarship, including sentiment
+analysis (e.g. Gautam & Yadav, 2014), algorithmic training (e.g. Braithwaite et al., 2016), and
+even coronavirus disease 2019 detection (e.g. Gharavi et al., 2020). English-language corpora
+have been used to predict election results (Nausheen & Begum, 2018), analyze consumer
+preferences (Ahmed & Danti, 2016), and explore pro- and anti-childhood vaccine
+communities’ influence on Twitter (Featherstone et al., 2020).
+As valuable as this work is, it can only be more valuable extended across languages.
+Frey et al. (2018) use a corpus of six languages to document general ripple effects of
+emotional influence through others and back around to the self.  Mocanu et al. (2013) use data
+on 78 languages to characterize inter-linguistic diversity and intra-linguistic drift across
+municipalities and regions. And Alshaabi et al. (2021) compare the dynamics of social
+influence on Twitter over 150 languages. In other disciplines, large-scale multi-language
+comparisons in other disciplines have identified universal patterns in the cross-language
+
+naming of colors (Lindsey & Brown, 2009), a well as a universal preference for the shortening
+of dependency length in human sentence production (Futrell et al., 2015).
+Our research demonstrates the effectiveness of Google Translate on the maintenance
+of sentiments, topic clusters, and semantic distance for tweets in all languages we examine.
+We validate the approach using “backtranslation,” a classic validation method of machine
+translation in which scholars compare an original text to a version of that text that has been
+translated from its original language to another language (in our case English) and then back
+again to the original (Figure 1). This makes it possible to directly compare the accuracy of
+English-trained tools on English translations to original-language-trained tools on
+original-language texts, while controlling for semantic drift introduced by the translation
+process itself. We first test the preservation of sentiments using two large public multi-lingual
+Twitter datasets, one with hand-coded sentiments (Mozetič et al., 2016) and another with
+machine-coded sentiments (Imran et al., 2022) for this analysis. The second portion of our
+research applies the same two datasets to show that Google Translate preserves topic clusters
+after backtranslation, thereby demonstrating a similar level of semantic conservation for this
+second common text analysis task. In the third and final portion of our present study we
+demonstrate the effectiveness of out-of-the-box machine translation on a third common text
+analysis approach: neural word embeddings. Here, after backtranslation, 10 of 16 languages
+examined performed better than chance in maintaining a minimal embedding distance. Our
+findings provide strong support for the use of modern machine translation for expanding
+academic attention to the languages spoken by most humans.
+Method
+
+Datasets
+We utilized two large, multilingual Twitter datasets. First, we examine the Mozetič et
+al. (2016) dataset, which contains over 1.6 million general Twitter posts hand-labeled as
+containing “positive”, “negative”, or “neutral” labels for 15 European languages: Albanian,
+Bosnian, Bulgarian, Croatian, English, German, Hungarian, Polish, Portuguese, Russian,
+Serbian, Slovak, Slovenian, Spanish, and Swedish. To expand the scope of our research
+beyond European languages, we added tweets from the Imran et al. (2022) COVID-19 dataset,
+a larger (70 million tweet) corpus including tweets in Chinese, Hindi, and Arabic. While these
+two datasets are comparable (both include sentiment labels), they differ in subject and date, as
+well as in how they determined sentiment scores. Those in Mozetič et al. (2016) were applied
+by human language-domain experts, while tweets from Imran et al. (2022) dataset were
+determined by algorithms (all trained within-language).
+Data cleaning
+Before translation and subsequent analysis, we preprocessed all Twitter data to remove
+Twitter handles, return handles, URLs, numbers, empty tweets, and converting all content to
+lowercase. We dropped Serbian from our analysis halfway through the study, after discovering
+that the Mozetič dataset contains Cyrillic Serbian, but Google Translate only supports
+Latin-character Serbian. We obviously excluded all English language tweets from validation
+by backtranslation through English.
+To reduce our dataset to a more manageable, and affordable, size (the Google
+Translate API is paid), we randomly sampled 30,000 tweets from each of the 13 applicable
+
+European languages from Mozetič et al. (2016) dataset, and 10,000 tweets from Chinese,
+Hindi, and Arabic from the Imran et al. (2022) dataset, for a total of 16 languages.
+Translation process
+Utilizing the Google Translate API, we translate all tweets from their “original
+language” datasets into English, saving the results as our “English translated” dataset. We then
+translate all the English translated tweets back to their original language, saving it as our
+“backtranslated” dataset (Figure 1). Our results are based only partly on three-way
+comparisons between these datasets. Where there is not a meaningful correspondence between
+English- and original-language analyses we use only two-way comparisons between the
+original and backtranslated datasets.
+With this manuscript we share the scripts and instructions, to enable researchers to
+easily extend their single-language corpus research to multiple languages. The code is
+available at https://osf.io/jx476/.
+Sentiment analysis
+We conduct our sentiment analysis task with the free open-source software Polyglot
+(Al-Rfou, 2015). Polyglot allows the generation of sentiment labels in more than 100
+languages, with “-1” indicating negative sentiment, “0” indicating neutral sentiment, and “1”
+indicating positive sentiment for each word in each original-language tweet. Based on the
+difference between the number of positive sentiment words and negative sentiment words, we
+generate an overall polarity for each tweet. Polyglot’s lexicon-based sentiment analysis relies
+on a valence dictionary of positive and negative words, computing the sentiment of a text as
+
+the simple sum of the valences of its words, normalized back down to the [-1, 1] interval. Our
+pipeline excluded neutrally labeled tweets: as a result of Polyglot’s lexicon-based sentiments,
+short texts like Twitter posts are overwhelmingly labeled as neutral which makes it difficult to
+distinguish the performance of sentiment analyses across translations.
+We computed confidence intervals around the accuracy of each language’s sentiments
+with bootstrapping (1000x). The final sentiment accuracies are the bootstrapped medians.
+Topic clustering
+While sentiment analysis is a common application for natural language tools, it only
+serves to answer a small range of questions. We expand our investigation of Google
+Translate’s ability to preserve the content of translated text through topic analysis. Compared
+to sentiment analysis, topic analysis provides a more technical, but much more flexible
+approach to computationally representing the meanings of text. Although the process of topic
+analysis is language agnostic, common computational tools are typically built to only support
+the English language, from stopwords to supported character sets.
+We model our topic clustering approach after Yin and Wang (2014) who present the
+open-source software GSDMM (“Gibbs Sampling algorithm of the Dirichlet Multinomial
+Mixture). GSDMM is trained upon short text posts such as those found within social media
+environments such as Twitter (Yin & Wang, 2014). We follow the original work’s data
+cleaning steps of removing both emojis and stopwords. We excluded Albanian and Bosnian
+due to their incompatibility with our data cleaning dependencies.
+
+Our cluster analysis process was as follows. For each language, we used a total of five
+iterations of the clustering algorithm. We then classified the backtranslated tweets to the
+clusters generated on the original language tweets. To estimate the success of machine
+translation at semantic preservation under topic analysis, we computed the proportion of
+backtranslated tweet that were correctly assigned to the cluster of their original-language
+version. We compare these proportions to the baseline “null” proportions expected by chance,
+as derived from random permutations of original cluster assignments. Like many clustering
+algorithms, GSDMM requires researchers to impose a desired number of clusters, rather than
+identifying the number of clusters through the same emergent process as cluster assignments.
+But the ability of backtranslation to preserve topic clusters depends on the number of clusters.
+Therefore, we observe the effectiveness of topic preservation across a range of clusterings by
+training models on each original language dataset for 2, 5, 10, 15, 20, 50, 100, 150, and 200
+clusters.
+Unlike with our evaluation of sentiment analysis, the analysis of topics is only able to
+compare original language and backtranslated analytics: it is not able to compare either to
+English. While the framework of sentiment analysis imposes the same meaning to the idea of
+“positive” and “negative” sentiments across languages, topics emerge from a narrow
+understanding of a word as the sequence of characters that constitutes it. To the extent that
+original language and backtranslated tweets use the same characters (as in languages
+borrowings from each other), they can be assigned to the same set of clusters. But lexicons in
+English and each original language are mostly non-overlapping, and there is ultimately no
+basis to map English translations to original-language-derived topics.
+Word embedding
+
+Polyglot (Al-Rfou, 2015) also supports semantic word embeddings across its
+languages. We determine semantic preservation under word embedding by embedding
+original and backtranslated tweets (as the normalized sum of the embeddings of their words)
+and calculating their (cosine) distance in the embedding’s high-dimensional semantic space.
+Under this formalism, a distance of zero indicates perfect preservation of semantics after
+translation. Because Polyglot has a different semantic space for each language, it was not
+possible to compare the distance of the intermediate English texts to the original and
+backtranslated texts.
+To measure how well machine translation preserves semantics under word embedding,
+we compared the embedding distances after backtranslation to two baseline distances. We
+computed the average minimum distance of tweets from 5,000 other tweets in that language
+and their average average distances. Our rationale is that meaning is preserved despite
+semantic drift imposed by the process of (double) machine translation if the average distance
+of backtranslated tweets from their originals is smaller than the average distances of different
+original language tweets from each other.
+Results
+Our primary finding is that Google Translate is faithful enough to preserve the
+semantics of multilingual texts under three common text analysis tasks: sentiment analysis,
+topic analysis, and word embeddings.
+Application 1: Preservation of sentiment
+
+We find that overall accuracy of sentiment scores decreases less than 1% after
+backtranslation, from a median 65.17% accuracy (with a very tight 99% high-confidence
+interval (HCI) of [65.16, 65.18]) to 64.76% [64.75, 64.77]. While small, this decline was
+statistically significant, as measured by the separation of the 99% HCI bars. We display this
+result in Figure 2, below.
+We did have one surprise from this process. We expected that the accuracy of
+English-trained sentiment on the English-translated tweets would be between or below the
+accuracy of the original or backtranslated tweets, whose “ground truth” sentiments were
+computed with models trained specifically for those languages. Instead median sentiment
+accuracy increases 4.95% following original languages’ translation into English (original
+language median accuracy: 65.17%, HCI [65.16, 65.18]; English translated median accuracy:
+70.12%, HCI [70.11, 70.13]). Somehow sentiments extracted from English translations are
+more accurate than sentiments of original language tweets, despite the process of translation
+in between (Figure 2). We speculate on this result in the Discussion section.
+Looking specifically at how different languages performed, we found the expected
+decrease in accuracy rates between the original language datasets and the backtranslated
+datasets for Albanian, Arabic, Chinese, Slovak, and Spanish (Figure 3). Unexpectedly, the
+remaining language datasets, belonging to the languages of Bosnian, Bulgarian, Croatian,
+German, Hindi, Hungarian, Polish, Portuguese, Russian, Slovenian, and Swedish, experienced
+an increase in sentiment accuracy from original language to backtranslated form. Although
+languages on average showed higher accuracy in English translation, the original language
+datasets of Portuguese, Russian, Slovenian, and Swedish show a drop in sentiment accuracy
+
+when translated to English (while the remaining others, Albanian, Bosnian, Bulgarian,
+Croatian, English, German, Hungarian, Polish, Slovak, and Spanish all improve).
+Application 2: Preservation of lexical topic assignments
+Our primary finding from Application 2 is that machine translation also preserves
+topic structure (Figure 4). Specifically, backtranslated tweets are assigned to the same cluster
+and their original language version at a rate well above chance. As expected, the ability of
+backtranslation to preserve topic structure declines as the number of topics increases (Figure
+5). Performance accuracy of topic cluster preservation is 78% when there are two topic
+clusters, and declines to about 60% when there are 10 or more clusters. However, chance
+accuracy declines much faster, from 52% to 10% over the same span. In other words, the
+relative accuracy of topic recovery actually improves with the number of clusters, even as
+absolute efficacy declines.
+It may seem peculiar that chance performance of topic assignment remains at 10%
+even with 200 clusters. This is probably due to an unequal distribution of cluster sizes. For
+200 equally sized clusters, the baseline, null probability of a backtranslated tweet being
+randomly assigned to its correct topic is 0.05%, one half of a percent.  Chance performance
+higher than this is easy to arrange in a system with a few large clusters and a large number of
+very small ones.
+Overall, we find that Google Translate preserves topic clusters across languages, with
+accuracy ranging from 60% to 80% depending on the number of topics we set the GSDMM
+algorithm to impose.
+
+Application 3: Preservation of semantics in embedding space
+In the final application of this work, we examine the multilingual preservation of
+semantic vectors in high-dimensional neural embeddings after machine translation and
+backtranslation.
+On average, original language tweets are significantly closer to their backtranslations
+than to other original language tweets in the same collection (Figure 6). Across languages,
+average distances of original language tweets from each other are 0.041–0.184 units, their
+minimum distances from each other are 0.028–0.132, and their distances from their
+backtranslations are 0.203–0.492. Being less than half of the average distance (except
+Chinese) and below or slightly above the minimum distance, we can conclude that the
+semantic change introduced by the translation algorithm is enough to change the meaning of a
+backtranslated tweet to be mistakable for a different closely related tweet, but not the typical
+more distantly related tweet.
+Of the 16 languages involved in the analysis, 6 languages (Albanian, Arabic, Chinese,
+German, Hindi, and Portuguese) failed the minimum baseline test, with backtranslated tweets
+having greater semantic distance from their originals than the average closest outside tweet.
+The Albanian, German, and Portuguese corpora failed by small margins (mean distance of
+0.065 compared to minimum baseline distance 0.059 in Albanian; distance 0.110 compared to
+baseline 0.094 in German; 0.089 against 0.085 in Portuguese). But in Arabic, Chinese, and
+Hindi, embeddings of translations were even further from their original (distance 0.146
+against 0.101 in Arabic; 0.184 against 0.053 in Chinese; 0.041 against 0.028 in Hindi). It
+should be noted that Arabic, Chinese, and Hindi were drawn from the Imran et al. (2022)
+
+dataset focused on COVID-19 related tweets, included as part of our effort to expand this
+project’s analysis beyond languages of European origin. As baseline measures were calculated
+on the distance between random tweets relative to their distance with all other tweets, and
+these tweets were semantically more closely related, these languages’ baseline measures may
+have been especially narrow relative to those of the other languages as a result of their shared
+topic. Although they failed the rigorous minimum distance test, even Arabic, Chinese, and
+Hindi passed the mean distance test: they were closer in meaning to their original than the
+average tweet (mean baseline distances 0.416, 0.352, and 0.203, respectively).
+Discussion
+As the global academic world becomes increasingly interconnected, Communication
+scholars must meet the challenge to make claims about communication processes more
+globally relevant. Fortunately, with recent advances in natural language processing,
+quantitative Communication research has an opportunity to be multilingual by default.
+Advances that bring equal attention to more of the world’s languages will not only provide
+greater generality of results, but greater attention to the work of Communication scholars from
+all parts of the world. Standard approaches to large multilingual corpora will also allow the
+rapid transfer of groundbreaking knowledge to and from the international Communication
+community.
+Of course, these advances have downsides. When multi-language analyses are
+conducted by scholars who can’t speak all of those languages, it becomes harder for them to
+“gut check” or “sanity check” their results, culturally contextualize those results, and interpret
+whatever valid cross-linguistic differences that do appear,. By encouraging researchers to
+
+conduct multilingual studies by default, we are almost necessarily advocating for a
+circumstance in which scholars are making conclusions about languages that they do not
+know. Although this approach has some acceptance in other fields, such as large-scale
+comparative linguistics, it would be understandable to see it as controversial. As novel as this
+problem may be, the way forward may not be novel at all. Quantitative and qualitative
+methods have a fundamental complementarity, with the former bringing generality as the
+latter brings depth and sensitivity to context. By supporting the summary quantitative claims
+of non-speakers with citations to other work by native speakers and other domain experts,
+scholars may be able to justify not knowing the languages they are engaging with. This
+complementary approach will be particularly valuable for understanding outliers.  In the case
+of our research, Chinese, Arabic, and Hindi all perform worse than the other languages.
+Having ruled out explanations that go to the phylogeny, character set, and geography of these
+languages, domain experts become the best candidates for understanding how and why
+specific languages deviate from the majority of their peers. This illustrates the importance of
+framing our contribution as a complement to expert-based multi-language communication
+research, rather than a substitute.
+In this work we are able to validate the performance of Google Translate by leveraging
+source-language versions of our three methods: sentiment analysis, topic analysis, and word
+embeddings.  However, as others make use of machine translation, they will not have the
+comfort of source-language tools, and may feel that they are “flying blind”. Although we
+succeed in showing that translation introduces negligible drift, it may still be uncomfortable to
+apply it to a new dataset, particularly with text analysis methods beyond the three that we
+validate here (such as custom classifiers). To address this concern, researchers can take not
+
+only our conclusions, but the backtranslation method itself, to perform partial validations for
+their own case. Most likely, it should be possible to find “home language” tools for at least a
+handful of languages in a larger corpus. If an author can show satisfactory and stable
+performance across this subset, by comparing original and backtranslated texts, they can
+assure their audience that the method is probably working for other languages as well. Even
+lacking such “ground truth,” there may be ways of using our method to instill confidence in a
+multilingual result. For example, a scholar could perform iterative backtranslations to
+calculate how many cycles must be introduced for the statistical significance of their result to
+degrade below threshold. If it takes a large number of backtranslations to degrade a result,
+readers can have confidence that artifacts introduced by the method are not sufficient to
+explain those results. Conversely, if machine translation is artificially amplifying a result,
+scholars can measure this effect with iterated backtranslation to suggest an appropriate amount
+of caution.
+Another design choice of this work ensures the generality of the method we introduce.
+All three applications of this work were performed with tweets. Tweets are short, making
+them challenging for text analysis methods like sentiment and topic analysis. That our method
+is very effective on challenging text is encouraging for scholars who would extend this
+method to more typical (longer) texts.
+A limitation of our approach is its accessibility. We have argued that Google Translate
+is very accessible, and this is true in that it requires a small amount of code (that we provide)
+to translate large quantities of text to English. However, our approach is not as financially
+accessible. The Google’s Translation API costs $20 USD per million characters. In practice,
+this translates to roughly $100 USD per 130,000 tweets. Fortunately, free translation tools of
+
+comparable quality are increasingly common, and can also be validated in practice using
+backtranslation.
+One surprising result from this work was that the accuracy of sentiment detection after
+translation into English, and in some cases after backtranslation, was higher than in the
+original texts. This finding is easier to understand with an appreciation of how sentiment
+analysis works in libraries like Polyglot. Polyglot uses the “dictionary” method, in which
+hundreds to thousands of high-frequency words are given sentiment scores, and the score of a
+statement is calculated from the sum of scores of the subset of words that are in the detector’s
+sentiment dictionary. If the dictionary is large, or the text is long, then its assigned sentiment
+score will be based on a lot of signal. Consequently, this method is less suitable for rarer
+languages and shorter texts (like tweets), which are less likely to contain scored words. It is
+also more suitable to texts with more common words, since uncommon words are less likely
+to appear in a language’s sentiment dictionary.
+Why would translation to English, or backtranslation from English, improve task
+performance? Translation to English may be improving dictionary-based sentiment detection
+because English-language sentiment dictionaries tend to be longer. And subsequent
+backtranslation may be improving detection performance if it results in uncommon unscored
+words from the source text being backtranslated into more common words that are scored. We
+believe that this finding can be explained by Polyglots’ apparent relative greater capacity to
+detect sentiment in English language content, relative to the content of other languages. This
+result underscores the need to validate Google Translate for each natural language task that it
+is being used to support.
+Conclusion
+
+There is an unmet need to extend Communication scholars’ applications of text
+analysis to more languages, particularly in the data-rich context of social media studies.
+Translation tools such as Google Translate can be immensely helpful in meeting this need. We
+have quantified Google Translate’s effectiveness in maintaining sentence meaning in
+translations to and from English. Across 16 non-English languages, sentiment analysis scores
+were shown to improve when translated to English, and only diminish marginally when
+translated back to their original languages. Similarly, both topic and semantic distances are
+preserved during backtranslation. Our findings demonstrate that machine translation is able to
+preserve semantic content and make non-English datasets legible to English-trained
+computational tools. We hope this analysis gives researchers the confidence to use machine
+translation to simply and economically increase the number of languages involved in their
+research, and thereby the generality of their findings.
+Acknowledgements
+We would like to thank Arti Thakur, Communication Ph.D. Candidate at the
+University of California, Davis, for her assistance with the analysis. All code is available at
+https://osf.io/jx476/. The authors report there are no competing interests to declare.
+
+Figure 1
+We compare analytics computed on texts in their original languages to translated English
+language analytics and texts translated back to the original language. Differences between
+the original and translated texts are typically difficult to attribute to semantic differences
+between the languages and “semantic” imposed by poor translation. Comparing original and
+backtranslated texts enables us to control for the effect of drift and focus on semantics.
+
+Original
+Intermediate
+Backtranslated
+source
+English
+source
+language
+language
+language
+text
+text
+textFigure 2
+Sentiment analysis overall retains accuracy after backtranslation by machine methods.
+Median sentiment detection accuracy increases 4.9% from original language to English
+translated language datasets, and falls less than 1% from original language datasets to
+backtranslated language datasets. Note that 99% error bars are too narrow to be displayed.
+
+0.701
+0.652
+0.648Figure 3
+Comparison of sentiment labeling accuracy across languages, before, during, and after
+backtranslation. Seventeen language sentiment detection accuracy from original language >
+English translated > backtranslated datasets. Note that 99% error bars are too narrow to be
+displayed.
+Figure 4
+
+Albanian
+Arabic
+Bosnian
+Bulgarian
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+English Translated w/o Neutrals
+0.8 -
+0.594
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+Original Language w/o Neutrals
+0.2 -
+0.2 -
+Backtranslated w/o Neutrals
+0.0
+0.0Machine translation preserves topic clusters across languages, regardless of number of
+topics. Percentage of backtranslated tweets assigned to the same cluster as original language
+tweets by language. White text denotes permutation test accuracy. The 14 languages examined
+demonstrate topic clustering accuracy preserved above chance.
+2 Clusters
+100 Clusters
+
+0.930
+0.882
+0.876
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+0.058
+0.019
+0.037
+0.024Figure 5
+Recovery of topics after backtranslation declines in an absolute sense with number of
+topics, but increases relative to baseline. Average topic cluster accuracy by size of topic
+cluster. Note that topic cluster accuracy decreases from two to ten clusters, whereby it is
+approximately stable to 200 clusters.
+
+Arabic
+Bulgarian
+Chinese
+Croatian
+German
+Hindi
+Hungarian
+Polish
+Portuguese
+Russian
+Slovak
+Slovenian
+Spanish
+Swedish
+Average
+BaselineFigure 6
+Tweets are closer to their backtranslations on average than to other tweets.
+Average distance between original language and backtranslated sentence embeddings by
+language. Black lines denote the mean baseline distance and blue lines denote the minimum
+baseline distance. All 16 languages have mean distances below their mean baseline. All
+languages but Albanian, Arabic, Chinese, German, Hindi, and Portuguese have mean
+distances below their minimum baseline. In these languages, tweets backtranslated tweets are
+further from their source tweet in meaning than tweets that are very semantically similar to
+the source, but tweets in these languages are still consistently closer to their source than the
+average tweet.
+
+0.492
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+0.085
+0.065
+0.067
+0.073
+0.059
+0.028
+0.041
+0.044References
+Ahmed, S., & Danti, A. (2015). Effective sentimental analysis and opinion mining of web
+reviews using rule based classifiers. Advances in Intelligent Systems and Computing,
+171–179. https://doi.org/10.1007/978-81-322-2734-2_18
+Al-Rfou, R., Kulkarni, V., Perozzi, B., & Skiena, S. (2015). Polyglot-NER: Massive
+Multilingual Named Entity Recognition. Proceedings of the 2015 SIAM International
+Conference on Data Mining, Vancouver, British Columbia, Canada, April 30- May 2,
+2015. https://doi.org/10.48550/arXiv.1307.1662
+Alshaabi, T., Dewhurst, D. R., Minot, J. R., Arnold, M. V., Adams, J. L., Danforth, C. M., &
+Dodds, P. S. (2021). The growing amplification of social media: Measuring temporal
+and social contagion dynamics for over 150 languages on Twitter for 2009–2020. EPJ
+Data Science, 10(1), Art. 1. https://doi.org/10/gjq4qq
+Braithwaite, S. R., Giraud-Carrier, C., West, J., Barnes, M. D., & Hanson, C. L. (2016).
+Validating machine learning algorithms for Twitter data against established measures
+of Suicidality. JMIR Mental Health, 3(2). https://doi.org/10.2196/mental.4822
+Caswell, I. (2022, May 11). Google Translate learns 24 new languages. Google. Retrieved
+December 15, 2022, from https://blog.google/products/translate/24-new-languages/
+Chen, Y., & Skiena, S. (2014). Building sentiment lexicons for all major languages.
+Proceedings of the 52nd Annual Meeting of the Association for Computational
+Linguistics (Volume 2: Short Papers), 383–389. https://doi.org/10.3115/v1/p14-2063
+
+Featherstone, J. D., & Barnett, G. A. (2020). Validating sentiment analysis on opinion mining
+using self-reported attitude scores. 2020 Seventh International Conference on Social
+Networks Analysis, Management and Security (SNAMS).
+https://doi.org/10.1109/snams52053.2020.9336540
+Featherstone, J. D., Barnett, G. A., Ruiz, J. B., Zhuang, Y., & Millam, B. J. (2020). Exploring
+childhood anti-vaccine and pro-vaccine communities on Twitter – a perspective from
+influential users. Online Social Networks and Media, 20, 100105.
+https://doi.org/10.1016/j.osnem.2020.100105
+Frey, S., Donnay, K., Helbing, D., Sumner, R. W., Bos, M. W. (2018) The rippling dynamics
+of valenced messages in naturalistic youth chat. Behavior Research Methods
+http://doi.org/cwbz
+Futrell, R., Mahowald, K., & Gibson, E. (2015). Large-scale evidence of dependency length
+minimization in 37 languages. Proceedings of the National Academy of Sciences,
+112(33), 10336–10341. https://doi.org/10.1073/pnas.1502134112
+Gautam, G., & Yadav, D. (2014). Sentiment analysis of Twitter data using machine learning
+approaches and semantic analysis. 2014 Seventh International Conference on
+Contemporary Computing (IC3). https://doi.org/10.1109/ic3.2014.6897213
+Gharavi, E., Nazemi, N., & Dadgostari, F. (2020). Early Outbreak Detection for Proactive
+Crisis Management Using Twitter Data: COVID-19 a Case Study in the US. ArXiv.
+https://doi.org/arXiv:2005.00475
+
+Imran, M., Qazi, U., & Ofli, F. (2022). TBCOV: Two Billion multilingual COVID-19 tweets
+with sentiment, entity, GEO, and gender labels. Data, 7(1), 8.
+https://doi.org/10.3390/data7010008
+Lindsey, D. T., & Brown, A. M. (2009). World color survey color naming reveals universal
+motifs and their within-language diversity. Proceedings of the National Academy of
+Sciences, 106(47), 19785–19790. https://doi.org/10.1073/pnas.0910981106
+Mocanu, D., Baronchelli, A., Perra, N., Gonçalves, B., Zhang, Q., & Vespignani, A. (2013).
+The Twitter of Babel: Mapping World Languages through Microblogging Platforms.
+PLOS ONE, 8(4), e61981. https://doi.org/10/f4vdv4
+Mozetič, I., Grčar, M., & Smailović, J. (2016). Multilingual twitter sentiment classification:
+The role of human annotators. PLOS ONE, 11(5).
+https://doi.org/10.1371/journal.pone.0155036
+Nausheen, F., & Begum, S. H. (2018). Sentiment analysis to predict election results using
+Python. 2018 2nd International Conference on Inventive Systems and Control (ICISC).
+https://doi.org/10.1109/icisc.2018.8399007
+Yin, J., & Wang, J. (2014). A Dirichlet multinomial mixture model-based approach for short
+text clustering. Proceedings of the 20th ACM SIGKDD International Conference on
+Knowledge Discovery and Data Mining, 233–242.
+https://doi.org/10.1145/2623330.2623715
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf,len=677
+page_content='Machine Translation for Accessible Multi-Language Text Analysis  Edward W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Chew1, William D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Weisman1, Jingying Huang1, Seth Frey1,*  1 Department of Communication, University of California Davis, Davis, CA, USA  Correspondence: Seth Frey (sethfrey@ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='edu) 376 Kerr Hall, 1 Shields Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=', Davis,  CA 95616, USA  Machine Translation for Accessible Multi-Language Text Analysis  Abstract  English is the international standard of social research, but scholars are increasingly conscious  of their responsibility to meet the need for scholarly insight into communication processes  globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' This tension is as true in computational methods as any other area, with  revolutionary advances in the tools for English language texts leaving most other languages  far behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' In this paper, we aim to leverage those very advances to demonstrate that  multi-language analysis is currently accessible to all computational scholars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We show that  English-trained measures computed after translation to English have adequate-to-excellent  accuracy compared to source-language measures computed on original texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We show this for  three major analytics—sentiment analysis, topic analysis, and word embeddings—over 16  languages, including Spanish, Chinese, Hindi, and Arabic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We validate this claim by  comparing predictions on original language tweets and their backtranslations: double  translations from their source language to English and back to the source language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Overall,  our results suggest that Google Translate, a simple and widely accessible tool, is effective in  preserving semantic content across languages and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Modern machine translation can  thus help computational scholars make more inclusive and general claims about human  communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Keywords: multi-language text analysis, multi-lingual text analysis, computational text  analysis, natural language processing, backtranslation, topic modeling, word embedding,  sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Introduction  Humans communicate in thousands of languages, and yet a single language, English,  attracts the bulk of communication research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' This not only has the effect of depriving other  languages of adequate attention, but depriving English-focused scholars of any sense of where  the language stands relative to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' The general use of English-trained tools for  English-focused analyses in the social science community is particularly notable given the  ubiquity of multi-lingual data and the power of modern computational natural language  processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' For example, social media researchers on Twitter typically begin with raw data  that is highly multilingual, before filtering out all tweets except for English, or some other  single language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' With such practices scholars miss a tremendous opportunity to test the  generalizability of social media-observed big data claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' But bringing standard text analysis  tools to the level of training and refinement that English-trained tools receive is a forbidding  prospect that few multi-lingual scholars have the training, resources, and language  background to pursue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We propose a simple alternative approach that makes texts from over  100 languages accessible to the full variety of analyses that are typically available to only  English-focused scholars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Specifically we demonstrate that modern machine translation has  reached a level of refinement necessary to preserve sentiment, lexical topics, and semantic  distance, making multi-language datasets legible to state of the art English-trained tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' By  providing a validation of state-of-the-art machine translation, along with easily adaptable  demonstration code, we aim to broaden the horizon of computational research and support  Communication scholars in increasing the relevance and generality of their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Google Translate, the most popular, accurate, and accessible multilingual neural  machine translation service, offers translations for over 133 languages (Caswell, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' In this  paper, we demonstrate the efficacy of Google Translate in retaining sentiment valence across  translations of large hand-coded and machine-coded Twitter datasets composed of tweets in  16 global non-English languages from four language families, being of Indo-European,  Uralic, Semitic, and Sinitic origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' With our findings that Google Translate preserves the  sentiment of tweets, as well as other dimensions of semantics, scholars may be emboldened to  utilize Google Translate and other multilingual neural machine translation services to expand  the generalizability of their research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' In so doing, non-English languages can benefit from  advanced English-trained natural language processing tools, and computational findings  normally restricted to the English language can be expanded upon to broaden scholars’  knowledge of global social phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Academics use Twitter datasets for a wide range of scholarship, including sentiment  analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Gautam & Yadav, 2014), algorithmic training (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Braithwaite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=', 2016), and  even coronavirus disease 2019 detection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Gharavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' English-language corpora  have been used to predict election results (Nausheen & Begum, 2018), analyze consumer  preferences (Ahmed & Danti, 2016), and explore pro- and anti-childhood vaccine  communities’ influence on Twitter (Featherstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  As valuable as this work is, it can only be more valuable extended across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Frey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' (2018) use a corpus of six languages to document general ripple effects of  emotional influence through others and back around to the self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Mocanu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' (2013) use data  on 78 languages to characterize inter-linguistic diversity and intra-linguistic drift across  municipalities and regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' And Alshaabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' (2021) compare the dynamics of social  influence on Twitter over 150 languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' In other disciplines, large-scale multi-language  comparisons in other disciplines have identified universal patterns in the cross-language  naming of colors (Lindsey & Brown, 2009), a well as a universal preference for the shortening  of dependency length in human sentence production (Futrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Our research demonstrates the effectiveness of Google Translate on the maintenance  of sentiments, topic clusters, and semantic distance for tweets in all languages we examine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  We validate the approach using “backtranslation,” a classic validation method of machine  translation in which scholars compare an original text to a version of that text that has been  translated from its original language to another language (in our case English) and then back  again to the original (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' This makes it possible to directly compare the accuracy of  English-trained tools on English translations to original-language-trained tools on  original-language texts, while controlling for semantic drift introduced by the translation  process itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We first test the preservation of sentiments using two large public multi-lingual  Twitter datasets, one with hand-coded sentiments (Mozetič et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=', 2016) and another with  machine-coded sentiments (Imran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=', 2022) for this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' The second portion of our  research applies the same two datasets to show that Google Translate preserves topic clusters  after backtranslation, thereby demonstrating a similar level of semantic conservation for this  second common text analysis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' In the third and final portion of our present study we  demonstrate the effectiveness of out-of-the-box machine translation on a third common text  analysis approach: neural word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Here, after backtranslation, 10 of 16 languages  examined performed better than chance in maintaining a minimal embedding distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Our  findings provide strong support for the use of modern machine translation for expanding  academic attention to the languages spoken by most humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Method  Datasets  We utilized two large, multilingual Twitter datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' First, we examine the Mozetič et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' (2016) dataset, which contains over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='6 million general Twitter posts hand-labeled as  containing “positive”, “negative”, or “neutral” labels for 15 European languages: Albanian,  Bosnian, Bulgarian, Croatian, English, German, Hungarian, Polish, Portuguese, Russian,  Serbian, Slovak, Slovenian, Spanish, and Swedish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' To expand the scope of our research  beyond European languages, we added tweets from the Imran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' (2022) COVID-19 dataset,  a larger (70 million tweet) corpus including tweets in Chinese, Hindi, and Arabic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' While these  two datasets are comparable (both include sentiment labels), they differ in subject and date, as  well as in how they determined sentiment scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Those in Mozetič et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' (2016) were applied  by human language-domain experts, while tweets from Imran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' (2022) dataset were  determined by algorithms (all trained within-language).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Data cleaning  Before translation and subsequent analysis, we preprocessed all Twitter data to remove  Twitter handles, return handles, URLs, numbers, empty tweets, and converting all content to  lowercase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We dropped Serbian from our analysis halfway through the study, after discovering  that the Mozetič dataset contains Cyrillic Serbian, but Google Translate only supports  Latin-character Serbian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We obviously excluded all English language tweets from validation  by backtranslation through English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  To reduce our dataset to a more manageable, and affordable, size (the Google  Translate API is paid), we randomly sampled 30,000 tweets from each of the 13 applicable  European languages from Mozetič et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' (2016) dataset, and 10,000 tweets from Chinese,  Hindi, and Arabic from the Imran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' (2022) dataset, for a total of 16 languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Translation process  Utilizing the Google Translate API, we translate all tweets from their “original  language” datasets into English, saving the results as our “English translated” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We then  translate all the English translated tweets back to their original language, saving it as our  “backtranslated” dataset (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Our results are based only partly on three-way  comparisons between these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Where there is not a meaningful correspondence between  English- and original-language analyses we use only two-way comparisons between the  original and backtranslated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  With this manuscript we share the scripts and instructions, to enable researchers to  easily extend their single-language corpus research to multiple languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' The code is  available at https://osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='io/jx476/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Sentiment analysis  We conduct our sentiment analysis task with the free open-source software Polyglot  (Al-Rfou, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Polyglot allows the generation of sentiment labels in more than 100  languages, with “-1” indicating negative sentiment, “0” indicating neutral sentiment, and “1”  indicating positive sentiment for each word in each original-language tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Based on the  difference between the number of positive sentiment words and negative sentiment words, we  generate an overall polarity for each tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Polyglot’s lexicon-based sentiment analysis relies  on a valence dictionary of positive and negative words, computing the sentiment of a text as  the simple sum of the valences of its words, normalized back down to the [-1, 1] interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Our  pipeline excluded neutrally labeled tweets: as a result of Polyglot’s lexicon-based sentiments,  short texts like Twitter posts are overwhelmingly labeled as neutral which makes it difficult to  distinguish the performance of sentiment analyses across translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  We computed confidence intervals around the accuracy of each language’s sentiments  with bootstrapping (1000x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' The final sentiment accuracies are the bootstrapped medians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Topic clustering  While sentiment analysis is a common application for natural language tools, it only  serves to answer a small range of questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We expand our investigation of Google  Translate’s ability to preserve the content of translated text through topic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Compared  to sentiment analysis, topic analysis provides a more technical, but much more flexible  approach to computationally representing the meanings of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Although the process of topic  analysis is language agnostic, common computational tools are typically built to only support  the English language, from stopwords to supported character sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  We model our topic clustering approach after Yin and Wang (2014) who present the  open-source software GSDMM (“Gibbs Sampling algorithm of the Dirichlet Multinomial  Mixture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' GSDMM is trained upon short text posts such as those found within social media  environments such as Twitter (Yin & Wang, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We follow the original work’s data  cleaning steps of removing both emojis and stopwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We excluded Albanian and Bosnian  due to their incompatibility with our data cleaning dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Our cluster analysis process was as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' For each language, we used a total of five  iterations of the clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We then classified the backtranslated tweets to the  clusters generated on the original language tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' To estimate the success of machine  translation at semantic preservation under topic analysis, we computed the proportion of  backtranslated tweet that were correctly assigned to the cluster of their original-language  version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We compare these proportions to the baseline “null” proportions expected by chance,  as derived from random permutations of original cluster assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Like many clustering  algorithms, GSDMM requires researchers to impose a desired number of clusters, rather than  identifying the number of clusters through the same emergent process as cluster assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  But the ability of backtranslation to preserve topic clusters depends on the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Therefore, we observe the effectiveness of topic preservation across a range of clusterings by  training models on each original language dataset for 2, 5, 10, 15, 20, 50, 100, 150, and 200  clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Unlike with our evaluation of sentiment analysis, the analysis of topics is only able to  compare original language and backtranslated analytics: it is not able to compare either to  English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' While the framework of sentiment analysis imposes the same meaning to the idea of  “positive” and “negative” sentiments across languages, topics emerge from a narrow  understanding of a word as the sequence of characters that constitutes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' To the extent that  original language and backtranslated tweets use the same characters (as in languages  borrowings from each other), they can be assigned to the same set of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' But lexicons in  English and each original language are mostly non-overlapping, and there is ultimately no  basis to map English translations to original-language-derived topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Word embedding  Polyglot (Al-Rfou, 2015) also supports semantic word embeddings across its  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We determine semantic preservation under word embedding by embedding  original and backtranslated tweets (as the normalized sum of the embeddings of their words)  and calculating their (cosine) distance in the embedding’s high-dimensional semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Under this formalism, a distance of zero indicates perfect preservation of semantics after  translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Because Polyglot has a different semantic space for each language, it was not  possible to compare the distance of the intermediate English texts to the original and  backtranslated texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  To measure how well machine translation preserves semantics under word embedding,  we compared the embedding distances after backtranslation to two baseline distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We  computed the average minimum distance of tweets from 5,000 other tweets in that language  and their average average distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Our rationale is that meaning is preserved despite  semantic drift imposed by the process of (double) machine translation if the average distance  of backtranslated tweets from their originals is smaller than the average distances of different  original language tweets from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Results  Our primary finding is that Google Translate is faithful enough to preserve the  semantics of multilingual texts under three common text analysis tasks: sentiment analysis,  topic analysis, and word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Application 1: Preservation of sentiment  We find that overall accuracy of sentiment scores decreases less than 1% after  backtranslation, from a median 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='17% accuracy (with a very tight 99% high-confidence  interval (HCI) of [65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='16, 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='18]) to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='76% [64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='75, 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' While small, this decline was  statistically significant, as measured by the separation of the 99% HCI bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We display this  result in Figure 2, below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  We did have one surprise from this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We expected that the accuracy of  English-trained sentiment on the English-translated tweets would be between or below the  accuracy of the original or backtranslated tweets, whose “ground truth” sentiments were  computed with models trained specifically for those languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Instead median sentiment  accuracy increases 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='95% following original languages’ translation into English (original  language median accuracy: 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='17%, HCI [65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='16, 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' English translated median accuracy:  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='12%, HCI [70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='11, 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Somehow sentiments extracted from English translations are  more accurate than sentiments of original language tweets, despite the process of translation  in between (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We speculate on this result in the Discussion section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Looking specifically at how different languages performed, we found the expected  decrease in accuracy rates between the original language datasets and the backtranslated  datasets for Albanian, Arabic, Chinese, Slovak, and Spanish (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Unexpectedly, the  remaining language datasets, belonging to the languages of Bosnian, Bulgarian, Croatian,  German, Hindi, Hungarian, Polish, Portuguese, Russian, Slovenian, and Swedish, experienced  an increase in sentiment accuracy from original language to backtranslated form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Although  languages on average showed higher accuracy in English translation, the original language  datasets of Portuguese, Russian, Slovenian, and Swedish show a drop in sentiment accuracy  when translated to English (while the remaining others, Albanian, Bosnian, Bulgarian,  Croatian, English, German, Hungarian, Polish, Slovak, and Spanish all improve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Application 2: Preservation of lexical topic assignments  Our primary finding from Application 2 is that machine translation also preserves  topic structure (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Specifically, backtranslated tweets are assigned to the same cluster  and their original language version at a rate well above chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' As expected, the ability of  backtranslation to preserve topic structure declines as the number of topics increases (Figure  5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Performance accuracy of topic cluster preservation is 78% when there are two topic  clusters, and declines to about 60% when there are 10 or more clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' However, chance  accuracy declines much faster, from 52% to 10% over the same span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' In other words, the  relative accuracy of topic recovery actually improves with the number of clusters, even as  absolute efficacy declines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  It may seem peculiar that chance performance of topic assignment remains at 10%  even with 200 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' This is probably due to an unequal distribution of cluster sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' For  200 equally sized clusters, the baseline, null probability of a backtranslated tweet being  randomly assigned to its correct topic is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='05%, one half of a percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Chance performance  higher than this is easy to arrange in a system with a few large clusters and a large number of  very small ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Overall, we find that Google Translate preserves topic clusters across languages, with  accuracy ranging from 60% to 80% depending on the number of topics we set the GSDMM  algorithm to impose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Application 3: Preservation of semantics in embedding space  In the final application of this work, we examine the multilingual preservation of  semantic vectors in high-dimensional neural embeddings after machine translation and  backtranslation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  On average, original language tweets are significantly closer to their backtranslations  than to other original language tweets in the same collection (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Across languages,  average distances of original language tweets from each other are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='041–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='184 units, their  minimum distances from each other are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='028–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='132, and their distances from their  backtranslations are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='203–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Being less than half of the average distance (except  Chinese) and below or slightly above the minimum distance, we can conclude that the  semantic change introduced by the translation algorithm is enough to change the meaning of a  backtranslated tweet to be mistakable for a different closely related tweet, but not the typical  more distantly related tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Of the 16 languages involved in the analysis, 6 languages (Albanian, Arabic, Chinese,  German, Hindi, and Portuguese) failed the minimum baseline test, with backtranslated tweets  having greater semantic distance from their originals than the average closest outside tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  The Albanian, German, and Portuguese corpora failed by small margins (mean distance of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='065 compared to minimum baseline distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='059 in Albanian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='110 compared to  baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='094 in German;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='089 against 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='085 in Portuguese).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' But in Arabic, Chinese, and  Hindi, embeddings of translations were even further from their original (distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='146  against 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='101 in Arabic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='184 against 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='053 in Chinese;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='041 against 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='028 in Hindi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' It  should be noted that Arabic, Chinese, and Hindi were drawn from the Imran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' (2022)  dataset focused on COVID-19 related tweets, included as part of our effort to expand this  project’s analysis beyond languages of European origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' As baseline measures were calculated  on the distance between random tweets relative to their distance with all other tweets, and  these tweets were semantically more closely related, these languages’ baseline measures may  have been especially narrow relative to those of the other languages as a result of their shared  topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Although they failed the rigorous minimum distance test, even Arabic, Chinese, and  Hindi passed the mean distance test: they were closer in meaning to their original than the  average tweet (mean baseline distances 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='416, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='352, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='203, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Discussion  As the global academic world becomes increasingly interconnected, Communication  scholars must meet the challenge to make claims about communication processes more  globally relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Fortunately, with recent advances in natural language processing,  quantitative Communication research has an opportunity to be multilingual by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Advances that bring equal attention to more of the world’s languages will not only provide  greater generality of results, but greater attention to the work of Communication scholars from  all parts of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Standard approaches to large multilingual corpora will also allow the  rapid transfer of groundbreaking knowledge to and from the international Communication  community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Of course, these advances have downsides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' When multi-language analyses are  conducted by scholars who can’t speak all of those languages, it becomes harder for them to  “gut check” or “sanity check” their results, culturally contextualize those results, and interpret  whatever valid cross-linguistic differences that do appear,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' By encouraging researchers to  conduct multilingual studies by default, we are almost necessarily advocating for a  circumstance in which scholars are making conclusions about languages that they do not  know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Although this approach has some acceptance in other fields, such as large-scale  comparative linguistics, it would be understandable to see it as controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' As novel as this  problem may be, the way forward may not be novel at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Quantitative and qualitative  methods have a fundamental complementarity, with the former bringing generality as the  latter brings depth and sensitivity to context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' By supporting the summary quantitative claims  of non-speakers with citations to other work by native speakers and other domain experts,  scholars may be able to justify not knowing the languages they are engaging with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' This  complementary approach will be particularly valuable for understanding outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' In the case  of our research, Chinese, Arabic, and Hindi all perform worse than the other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Having ruled out explanations that go to the phylogeny, character set, and geography of these  languages, domain experts become the best candidates for understanding how and why  specific languages deviate from the majority of their peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' This illustrates the importance of  framing our contribution as a complement to expert-based multi-language communication  research, rather than a substitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  In this work we are able to validate the performance of Google Translate by leveraging  source-language versions of our three methods: sentiment analysis, topic analysis, and word  embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' However, as others make use of machine translation, they will not have the  comfort of source-language tools, and may feel that they are “flying blind”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Although we  succeed in showing that translation introduces negligible drift, it may still be uncomfortable to  apply it to a new dataset, particularly with text analysis methods beyond the three that we  validate here (such as custom classifiers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' To address this concern, researchers can take not  only our conclusions, but the backtranslation method itself, to perform partial validations for  their own case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Most likely, it should be possible to find “home language” tools for at least a  handful of languages in a larger corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' If an author can show satisfactory and stable  performance across this subset, by comparing original and backtranslated texts, they can  assure their audience that the method is probably working for other languages as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Even  lacking such “ground truth,” there may be ways of using our method to instill confidence in a  multilingual result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' For example, a scholar could perform iterative backtranslations to  calculate how many cycles must be introduced for the statistical significance of their result to  degrade below threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' If it takes a large number of backtranslations to degrade a result,  readers can have confidence that artifacts introduced by the method are not sufficient to  explain those results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Conversely, if machine translation is artificially amplifying a result,  scholars can measure this effect with iterated backtranslation to suggest an appropriate amount  of caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Another design choice of this work ensures the generality of the method we introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  All three applications of this work were performed with tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Tweets are short, making  them challenging for text analysis methods like sentiment and topic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' That our method  is very effective on challenging text is encouraging for scholars who would extend this  method to more typical (longer) texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  A limitation of our approach is its accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We have argued that Google Translate  is very accessible, and this is true in that it requires a small amount of code (that we provide)  to translate large quantities of text to English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' However, our approach is not as financially  accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' The Google’s Translation API costs $20 USD per million characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' In practice,  this translates to roughly $100 USD per 130,000 tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Fortunately, free translation tools of  comparable quality are increasingly common, and can also be validated in practice using  backtranslation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  One surprising result from this work was that the accuracy of sentiment detection after  translation into English, and in some cases after backtranslation, was higher than in the  original texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' This finding is easier to understand with an appreciation of how sentiment  analysis works in libraries like Polyglot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Polyglot uses the “dictionary” method, in which  hundreds to thousands of high-frequency words are given sentiment scores, and the score of a  statement is calculated from the sum of scores of the subset of words that are in the detector’s  sentiment dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' If the dictionary is large, or the text is long, then its assigned sentiment  score will be based on a lot of signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Consequently, this method is less suitable for rarer  languages and shorter texts (like tweets), which are less likely to contain scored words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' It is  also more suitable to texts with more common words, since uncommon words are less likely  to appear in a language’s sentiment dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Why would translation to English, or backtranslation from English, improve task  performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Translation to English may be improving dictionary-based sentiment detection  because English-language sentiment dictionaries tend to be longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' And subsequent  backtranslation may be improving detection performance if it results in uncommon unscored  words from the source text being backtranslated into more common words that are scored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We  believe that this finding can be explained by Polyglots’ apparent relative greater capacity to  detect sentiment in English language content, relative to the content of other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' This  result underscores the need to validate Google Translate for each natural language task that it  is being used to support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Conclusion  There is an unmet need to extend Communication scholars’ applications of text  analysis to more languages, particularly in the data-rich context of social media studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Translation tools such as Google Translate can be immensely helpful in meeting this need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We  have quantified Google Translate’s effectiveness in maintaining sentence meaning in  translations to and from English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Across 16 non-English languages, sentiment analysis scores  were shown to improve when translated to English, and only diminish marginally when  translated back to their original languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Similarly, both topic and semantic distances are  preserved during backtranslation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Our findings demonstrate that machine translation is able to  preserve semantic content and make non-English datasets legible to English-trained  computational tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' We hope this analysis gives researchers the confidence to use machine  translation to simply and economically increase the number of languages involved in their  research, and thereby the generality of their findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Acknowledgements  We would like to thank Arti Thakur, Communication Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Candidate at the  University of California, Davis, for her assistance with the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' All code is available at  https://osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
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+page_content=' The authors report there are no competing interests to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  Figure 1  We compare analytics computed on texts in their original languages to translated English  language analytics and texts translated back to the original language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Differences between  the original and translated texts are typically difficult to attribute to semantic differences  between the languages and “semantic” imposed by poor translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
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+page_content='  Original  Intermediate  Backtranslated  source  English  source  language  language  language  text  text  textFigure 2  Sentiment analysis overall retains accuracy after backtranslation by machine methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
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+page_content=' Percentage of backtranslated tweets assigned to the same cluster as original language  tweets by language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
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+page_content=' Average topic cluster accuracy by size of topic  cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
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+page_content='  Arabic  Bulgarian  Chinese  Croatian  German  Hindi  Hungarian  Polish  Portuguese  Russian  Slovak  Slovenian  Spanish  Swedish  Average  BaselineFigure 6  Tweets are closer to their backtranslations on average than to other tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
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+page_content=' Effective sentimental analysis and opinion mining of web  reviews using rule based classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
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+page_content=' Polyglot-NER: Massive  Multilingual Named Entity Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
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+page_content=' Sentiment analysis to predict election results using  Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' 2018 2nd International Conference on Inventive Systems and Control (ICISC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
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+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
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+page_content=' A Dirichlet multinomial mixture model-based approach for short  text clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
+page_content=' Proceedings of the 20th ACM SIGKDD International Conference on  Knowledge Discovery and Data Mining, 233–242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dFAT4oBgHgl3EQfDhxB/content/2301.08416v1.pdf'}
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+1
+A deep local attention network for pre-operative
+lymph node metastasis prediction in pancreatic
+cancer via multiphase CT imaging
+Zhilin Zheng, Xu Fang, Jiawen Yao, Mengmeng Zhu, Le Lu Fellow, IEEE, Lingyun Huang, Jing Xiao, Yu Shi,
+Hong Lu, Jianping Lu, Ling Zhang, Chengwei Shao*, Yun Bian*
+Abstract—Lymph node (LN) metastasis status is one of the
+most critical prognostic and cancer staging factors for patients
+with resectable pancreatic ductal adenocarcinoma (PDAC), or in
+general, for any types of solid malignant tumors. Preoperative
+prediction of LN metastasis from non-invasive CT imaging
+is highly desired, as it might be straightforwardly used to
+guide the following neoadjuvant treatment decision and surgical
+planning. Most studies only capture the tumor characteristics
+in CT imaging to implicitly infer LN metastasis and very few
+work exploit direct LN’s CT imaging information. LN staging
+is usually confirmed from pathological images acquired after
+invasive procedures of biopsy or surgery. To the best of our
+knowledge, this is the first work to propose a fully-automated
+LN segmentation and identification network to directly facilitate
+the LN metastasis status prediction task. Nevertheless LN seg-
+mentation/detection is very challenging since LN can be easily
+confused with other hard negative anatomic structures (e.g.,
+vessels) from radiological images. 1) We explore the anatomical
+spatial context priors of pancreatic LN locations by generating a
+guiding attention map from related organs and vessels to assist
+segmentation and infer LN status. As such, LN segmentation
+is impelled to focus on regions that are anatomically adjacent
+or plausible with respect to the specific organs and vessels
+(thus hard negative samples with certain distance ranges can
+be ignored). 2) The metastasized LN identification network is
+trained to classify the segmented LN instances into positives
+or negatives by reusing the segmentation network as a pre-
+trained backbone and padding a new classification head. 3)
+More importantly, we develop a LN metastasis status prediction
+network that combines the patient-wise aggregation results of LN
+segmentation/identification and deep imaging features extracted
+from the tumor region. 4) Extensive quantitative nested five-
+fold cross-validation is conducted on a discovery dataset of 749
+patients with PDAC. External multi-center clinical evaluation is
+further performed on two other hospitals of 191 patients in total.
+Our final multi-staged LN status prediction network statistically
+significantly outperforms the strong baseline of nnUNet and
+several other compared methods, including CT-reported LN
+status, radiomics, and deep learning models.
+Index Terms—Pancreatic ductal adenocarcinoma (PDAC),
+Z. Zheng, L. Huang, J. Xiao are with Ping An Technology (Shanghai &
+Shenzhen), People’s Republic of China, (e-mail: zhilin.zheng95@gmail.com).
+X. Fang, M. Zhu and J. Lu are with Changhai Hospital, Shanghai, People’s
+Republic of China. J. Yao, L. Lu and L. Zhang were with PAII Inc., Bethesda,
+MD 20817, USA. X. Fang and J. Yao contributed equally.
+Y. Shi is with Department of Radiology, Shengjing Hospital of China
+Medical University, Shenyang, China.
+H. Lu is with Department of Radiology, Tianjin Medical University Cancer
+Institute and Hospital, National Clinical Research Center of Cancer, Key
+Laboratory of Cancer Prevention and Therapy, Tianjin, China.
+*s indicate joint corresponding authors. Y. Bian and C. Shao are
+with Changhai Hospital, Shanghai, People’s Republic of China, (email:
+bianyun2012@foxmail.com, cwshao@sina.com).
+Lymph node metastasis, Lymph node segmentation, Contrast-
+enhanced computed tomography
+I. INTRODUCTION
+P
+ANCREATIC cancer is the third leading cause of overall
+cancer death in the United States [1], of which approx-
+imately 95% is pancreatic ductal adenocarcinoma (PDAC)
+[2]. With the poorest prognosis (i.e., 5-year overall survival
+(OS) of 10% approximately), surgical resection is the most
+effective way to achieve long-term survival for patients with
+PDAC [2]. However, not all patients can benefit from the
+margin-negative (R0) resection and comprehensive treatment
+protocol is usually established for pancreatic cancer. The
+patients’ treatment selections can be determined by whether
+their peripancreatic lymph nodes (LNs) have metastasized with
+the options of adjuvant radiotherapy (RT) or chemotherapy.
+It was found that neoadjuvant therapy before surgery was
+associated with improved survival and time to recurrence in
+patients with LN metastasis, since neoadjuvant therapy can
+not only treat lymphovascular invasion but also benefit tumor
+downstaging [3], [4]. The accurate preoperative detection of
+LN metastasis becomes vital and would aid in treatment
+planning and management.
+Contrast-enhanced CT is used as the typical imaging pro-
+tocol for identifying the presence of peripancreatic metastatic
+disease to LNs, but it is a very challenging task for radiologists
+to determine whether a patient has LN metastasis by using only
+CT scans. To this end, poor diagnostic accuracy of CT with a
+pooled sensitivity of 25% and positive predictive value (PPV)
+of 28% was reported in a meta-analysis [5] on assessing extra-
+regional LN metastasis in pancreatic and peri-ampullary can-
+cer. Recently, several radiomics based approaches have been
+proposed to tackle the LN metastasis differentiation problem
+of various cancer types
+[6]–[12]. However, these methods
+require hand-crafted feature design which can bring concerns
+of reproducibility and human bias is often introduced due to
+manual selection of 2D tumor slice with limited representation
+power. Although there are some deep learning work that report
+promising performance on predicting LN metastasis status in
+gastric cancer [13], [14], those models assume that the risk of
+metastases is fundamentally driven by the primary tumor. They
+rely on LN CT report information for the integration model
+without using any LNs detection or segmentation. The model
+that takes both tumor morphology and lymphatic anatomy into
+arXiv:2301.01448v1  [eess.IV]  4 Jan 2023
+
+2
+Arterial
+Arterial
+Venous
+Patient with metastasis
+Patient without metastasis
+Tumor
+LN
+Tumor-LN anatomy
+Negative LN
+Positive LN
+Tumor
+Tumor & LN
+Spleen
+Esophagus
+Stomach
+Aorta
+Pancreas
+Duodenum
+SMA
+TC
+LGA
+CHA&PHA
+Organs & Vessels
+Venous
+LN
+LN
+LN
+LN
+LN
+Tumor
+LN
+LN
+LN
+LN
+LN
+Tumor
+LN
+LN
+LN
+LN
+Tumor
+LN
+LN
+LN
+LN
+Tumor
+Fig. 1. A visualization of pancreatic tumor (in dark red) and LNs (in pink red for positives or green for negatives) in multi-phase CT images
+and their spatial distributions corresponding to key anatomical structures as follows. SMA: superior mesenteric artery. TC&SA: truncus
+coeliacus and splenic artery. LGA: left gastric artery; CHA&PHA: common hepatic artery and proper hepatic artery.
+account could be of more clinically usefulness on addressing
+these aforementioned issues, similarly as in the diagnostic
+processes performed by radiologist readers. PET/CT is another
+imaging modality worth exploring. PET/CT based approaches
+[15]–[17] generally use maximum standardized uptake value
+(SUVmax) of manually-drawn LN RoIs as the prediction
+element, but it comes with challenges of numerous false
+positives from inflammatory LNs and false negatives from
+small-sized metastatic LNs [18], [19]. Also, it is relatively
+not as common as CT, which is less affordable, available and
+accessible, hence we opt for CT for our research purpose.
+In this paper, we tackle the LN metastasis status prediction
+problem in patients with PDAC by first segmenting and
+identifying instances of LNs and then classifying the patients
+into metastasis-positive or -negative group. LNs are tiny struc-
+tures that anatomically locate surrounding organs and vessels.
+Their locations have been mapped into 18 stations that are
+relevant for pancreatic cancer tumor according to their relative
+positions against adjacent anatomical structures, as defined
+by Japan Pancreas Society (JPS) [20] (see Supplementary
+Table 1 in the supplementary material for details). Examples
+of their spatial distribution are shown in Fig. 1. Metastasis
+happens when cancer cells spread from the primary tumor
+to LNs, causing enlargement of LNs and other underlying
+changes. Response Evaluation Criteria in Solid Tumors (RE-
+CIST) criteria [21] defines the criteria for LN metastasis
+suspicion, i.e., nodes with short axis greater than 10mm,
+heterogeneity and central necrosis. However, these criteria are
+not pathognomonic since there exist false negatives associated
+with small node micrometastases and false positives with
+inflammatory nodes larger than 10mm in short axis. Hence,
+finding LNs in CT images is quite time-consuming and can be
+inconsistent depending on radiologists’ subjective experiences.
+It is ambiguous for radiologists to identify nodal positivity
+accurately from CT without referring to pathology reports.
+The gold standard for determination of metastasis is based on
+post-operative pathological evaluation of pancreatectomy spec-
+imens. Automated yet reliable pre-operative LN segmentation
+and identification are highly desirable for patient surgical or
+RT treatment planing.
+LN segmentation is inherently challenging due to two
+reasons: 1) small to tiny sizes of LN cause extreme foreground-
+background class imbalance problem; 2) LNs have CT atten-
+uation values similar to vessels and other soft-tissue struc-
+tures, resulting in visual confusions. Existing work [22]–[24]
+mainly adopt U-Net based deep networks [25]–[27] as strong
+backbones, in which skip connections aggregate multi-level
+semantic representation and alleviate vanishing gradient prob-
+lem. They incorporate anatomical context by directly taking
+organs&vessels segmentation masks as either supervision tar-
+gets [22] or additional inputs [24], [28]. Concerns are remained
+that the relationship between lymphatic anatomy and adjacent
+anatomical structures is not well explored. We address it by
+introducing a distance-guided attention map to fully utilize the
+spatial priors. In our segmentation framework, the LN attention
+map is obtained via a pre-defined mapping from distance
+maps of related organs/vessels that have been integrated into
+UNet-based backbone to control the segmentation network’s
+
+TO:
+B
+TVN00SL0003TO五
+TO
+TVNO0O0003TO:
+BTVN001O0003TO:
+B
+TVN00SL0003TO:
+B3
+spatial focus. It simultaneously assists in improving sample
+selection strategy that filters out non-informative negative
+samples (called ”informative negative selection”) to tackle the
+class imbalance problem. The segmented LNs are labeled as
+positive/negative using radiologist’s judgement as the standard
+that combines information from pathological results and CT
+intensities. A classification network is subsequently derived by
+sharing the same backbone with segmentation and initialized
+with the trained segmentation parameters. This strategy ben-
+efits the classification task from densely structured prediction
+in segmentation. By predicting LN metastasis in patients
+with PDAC, we employ a modified ResNet [29] classification
+model. Tumor characteristics are proven to be important cues
+for metastasis [8], [9], [11], so we integrate both tumor and
+LN cues by taking as inputs the image patches of tumor
+and the patient-wise aggregation of LN segmentation and
+identification.
+Our main contribution is four folds: 1) To the best of
+our knowledge, this work is the first to directly incorporate
+automated LN segmentation and identification for assisting
+preoperative LN metastasis status prediction for patients with
+PDAC. 2) We propose an attention-based LN segmentation
+network with the guidance of distances to nearby anatomical
+structures, explicitly exploiting the spatial priors and simul-
+taneously addressing the foreground-background imbalance
+issue. 3) We design a compositive LN metastasis status
+prediction network combining both tumor and positive LN
+characteristics, showing the potential of tumor-LN guided
+cues. 4) Extensive quantitative experiments are conducted to
+evidently validate the effectiveness of our deep local attention
+network in both tasks of LN segmentation and LN metastasis
+status prediction, and external multi-center clinical evaluation
+is performed to demonstrate the generalization ability. Without
+loss of generality, our proposed method is applicable for
+finding the preoperative LN metastasis status of other types
+of solid tumor or cancers, such as liver or gastric cancers.
+II. RELATED WORK
+A. Lymph Node Segmentation
+Automated LN segmentation in CT images is an essential
+yet challenging task in medical image analysis. Traditional
+approaches tackle this problem by the means of atlas based
+search space restriction [30], spatial prior features combination
+[31], [32], supervoxel clustering [33], etc. In recent years, U-
+Net based deep networks have shown remarkable performance
+in numerous organ or tumor segmentation tasks [34]–[38].
+nnUNet [27] further proposes a self-configuring approach,
+with automatic configurations including preprocessing, net-
+work architecture, training and post-processing, that achieves
+robust performance and general applicability. To address the
+strong class imbalance issues in LN segmentation, four other
+anatomical structures are included as training targets [22]
+using 3D U-Net [26] framework. [23] utilizes parallel net-
+works of 2D U-Net [25] and Mask R-CNN [39] with the
+supervision of all considered anatomical structures and LNs,
+benefiting from both semantic and instance segmentation.
+Another strategy to incorporate anatomical context is to take
+organ segmentation masks as additional channels of the input.
+[28] proposes an ensemble approach for a slab-wise and
+a downsampled full volume based LN segmentation, taking
+the concatenation of CT image and segmented anatomical
+structure mask as input. DeepStationing [24] presents a key
+referencing organ auto-search strategy and combines selected
+organs into the network via input concatentation for LN station
+parsing. All above methods implicitly exploit spatial priors
+of LNs by injecting the anatomical structure masks either as
+inputs or supervisions, hence the prior knowledge has not been
+fully exploited. More importantly, there is a lack of studies on
+how LN segmentation could be used for predicting metastasis.
+B. Lymph Node Metastasis Prediction
+Radiomics Methods. Radiomics is a powerful technique
+for extracting quantitative image features with the purpose
+of clinical decision support, and thus widely used in can-
+cer research [11], [12], [40]–[42]. It converts imaging data
+into different types of hand-crafted features, including shape,
+intensity, texture and filter-based (e.g., wavelet, Laplacian of
+Gaussian) features. Applications of radiomics on predicting
+LN metastasis from primary tumor have been explored in
+many recent works [6]–[10]. Radiomics features are first
+extracted from manually delineated tumor regions in any
+contrast-enhanced CT images. Feature selection and classifi-
+cation model construction (e.g., logistic regression, random
+forest) are then performed to give LN metastasis prediction
+for various cancer types like gastric cancer [7], [12], biliary
+tract cancer [6] and PDAC [8], [9], [11]. Relying only on
+primary tumor radiomics without considering LNs character-
+istics may limit the prediction performance, thus [10] uses
+manual annotations of the largest LN visible in the gastric
+region and combines LN radiomics into the prediction model
+for gastric cancer. However, problem still remains because it
+simply involves the largest LN without identifying the nodal
+positivity.
+Deep Learning based Methods. Recent advances in deep
+learning have made it a mainstream method of addressing the
+entire workflow of diagnosis and treatment for various cancer
+types on medical imaging, such as orapharageal cancer [43],
+lung cancer [44], as well as pancreatic cancer [45]–[47]. Deep
+neural networks are applied to LN metastasis in many studies
+[13], [14], [48], [49]. In [48], deep features are extracted
+from tumor ROIs in bimodal image (i.e., US and SWE) using
+ResNet [29], and then fed into a SVM model for predicting
+axillary LN status in breast cancer. For gastric cancer, [14]
+combines DenseNet [50] features with some hand-crafted
+features, extracted from the 2D tumor ROI with the largest
+area in multi-phase CT images. To investigate metastasis in
+individual LN stations for gastric cancer, [13] develops a
+system of multiple independent ResNets with tumor ROIs and
+corresponding annotation masks as inputs where each ResNet
+is responsible to predict metastasis at one specific nodal sta-
+tion. Most existing studies capture only tumor characteristics
+for LN metastasis prediction, while the one leveraging LN
+radiomics requires manual delineation and considers simply
+the LN with the largest size [10]. An automated and accurate
+
+4
+b)   LN metastasis status prediction network
+Volume 
+Image
+PDAC
+Mask
+Metastasis
+-positive
+Metastasis-
+Negative
+√
+Cropping
+Cropping
+c
+Lymph node 
+segmentation & 
+identification
+Lymph node 
+segmentation & 
+identification
+Volume 
+Image
+PDAC
+Mask
+Metastasis
+-positive
+Metastasis-
+Negative
+√
+Cropping
+Cropping
+c
+Lymph node 
+segmentation & 
+identification
+Image
+Lymph Node 
+Segmentation 
+Instance-wise 
+Lymph Node 
+Identification
+Organ& 
+Vessel
+Distance
+Transform
+Non-linear
+Mapping
+Attention
+Mechanism
+GAP
+a)  Lymph node segmentation & identification network
+c
+concatenation
+element-wise 
+multiplication
+Downsampling
+Fig. 2. The proposed framework for (a) two-stage LN segmentation and identification and (b) LN metastasis status prediction .
+process of LN segmentation and nodal positivity identification
+is hence of high importance for assisting metastasis prediction.
+III. METHODOLOGY
+The overall framework is illustrated in Fig. 2, which is com-
+posed of (a) distance-guided attention-based LN segmentation
+and identification network, and (b) tumor and LN combined
+metastasis status prediction network.
+A. Distance-guided Attention-based Lymph Node Segmenta-
+tion and Identification Network
+We perform LN detection from any input CT scan by a two-
+stage strategy: segmenting the image into two classes of LN
+and background voxels, followed by identifying segmented LN
+instances as positive or negative.
+1) Stage 1: Class-agnostic Lymph Node Segmentation:
+Based on the spatial prior that LN stations are geometrically
+distributed or constrained around certain anatomical structures,
+we propose an attention based LN segmentation network by
+taking the distances to nearby organs/vessels into account.
+Our LN segmentation network differs from the strong baseline
+(i.e., nnUNet [27]) in that attention mechanism is applied to
+guide possible LN locations, with the advantage of reducing
+false positive predictions outside those locations. The intuition
+behind the attention module is that the attention map can cover
+regions adjacently constrained to those organs and vessels.
+Attention Map Generation. To explicitly capture and
+model the lymphatic anatomy, attention computation is im-
+plemented as a pre-defined geometric mapping function from
+organ&vessel distance maps. An example of attention map
+generation process is shown in Fig. 3. Specifically, given a
+multi-phase input CT volume X ∈ RN×W ×H×D, we first
+obtain organ&vessel segmentation mask using nnUNet [27]
+model trained with 19 classes of annotations. Ten classes
+among them involved with 17 LN stations are used (see
+Supplementary Table 1 in the supplementary material for the
+definition of LN stations), i.e., spleen, esophagus, stomach,
+aorta, pancreas, duodenum, superior mesenteric artery (SMA),
+truncus coeliacus and splenic artery (TC&SA), left gastric
+artery (LGA), common hepatic artery and proper hepatic
+artery (CHA&PHA). Note that station 15# (LNs along middle
+Spleen
+Aorta
+SMA
+LGA
+Image
+Organ&Vessel
+Organ/Vessel 
+Distance Map
+Organ/Vessel 
+Attention Map
+Integrated 
+Attention Map
+CHA&PHA
+Esophagus
+Duodenum
+Non-linear 
+Mapping
+0
+1
+d imin
+min
+d imin
+dmax
+imax
+i
+dmax
+i
+d imin
+min
+d imin-3
+d imin-3
+dmax
+imax
+i
+dmax
+i
++3
+dmax
+i
++3 d
+fifi
+Stomach
+Pancreas
+TC&SA
+Fig. 3. An illustration of attention map generation process.
+colic artery) is left aside here since it is related to distant
+metastasis that rarely happens in our patient population. A
+SDT is applied to each class of the segmentation mask M ∈
+{0, 1, 2, ..., 10}W ×H×D, generating a total of 10 organ/vessel
+distance maps Di where i ∈ {1, 2, ..., 10} is the index of
+organ/vessel class. Di has positive values at the voxels outside
+the i-th organ/vessel and negative scores inside it. Intuitively,
+LNs are likely to appear within a certain range of distance
+to each organ/vessel, which requires paying attention to. To
+obtain the distance-guided attention maps, Di is passed to an
+isosceles trapezium-shaped non-linear mapping function (see
+
+TO
+B
+c000.r100.ocbqT0
+B
+E000_100_3sbq5
+Fig. 3), formulated as
+f i(d) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1,
+di
+min ≤ d ≤ di
+max
+− (d − di
+max − 3)
+3
+,
+di
+max<d<di
+max + 3
+(d − di
+min + 3)
+3
+,
+di
+min − 3<d<di
+min
+0,
+Otherwise
+.
+(1)
+where d is the individual element in Di; di
+min and di
+max
+determine the distance range in mm; the smooth border 3mm
+is chosen empirically. This mapping function converts the
+distance maps to the attention scores ranging from 0 to 1, with
+1 indicating possible locations of LNs, 0 indicating impossible
+locations, and decimals lying in between. The i-th attention
+map Ai is obtained by Ai = f i(Di).
+The final attention map Aall is produced by integrating all
+of the organ/vessel-specific attention maps, thus Aall can cover
+the whole areas that need attending to. In specific, Aall takes
+the element-wise maximum of all Ai except for the voxels
+inside an organ/vessel, illustrated as
+aall
+v
+=
+�
+max
+i=1,2,...,10 ai
+v,
+mv = 0
+ai
+v,
+mv = i
+.
+(2)
+where a∗
+v and mv are the values of A∗ and M at the voxel v.
+Attention based Lymph Node Segmentation. After ob-
+taining the distance-guided attention map, we incorporate it
+to the segmentation network with 3D nnUNet [27] as the
+backbone. The attention mechanism emphasizes LN-related
+regions by spatially scaling the features with the attention map.
+Given the input image X and the one-hot segmentation label
+Y , the deep features at the penultimate layer are extracted
+and multiplied element-wisely with Aall. It is finally passed
+through a convolution block with a softmax layer to produce
+the segmentation output P ∈ R2×W ×H×D.
+Due to GPU memory limitation, patch-based training strat-
+egy is employed. nnUNet randomly samples 3D image patches
+from the whole CT scan and enforces that more than a third
+of samples in a batch contain at least one foreground class to
+control the foreground-to-background ratio. Considering the
+extreme class imbalance problem caused by the small LN
+targets, we improve it with the “informative negative selection”
+scheme. Note that our proposed attention mechanism helps
+block out features at the voxels with a certain distance to
+or inside all organs and vessels, resulting in lots of non-
+informative negative patches filled with 0 by applying zero
+attention scores. Thus we can naturally throw out those non-
+informative patches, and select patches containing at least one
+attention score > 0 (called informative patches) for training.
+This sampling strategy further boosts the network’s concen-
+tration on targeted regions of interest (ROIs) surrounding
+organs/vessels.
+For training objectives to better balance precision and recall
+, we modify the Dice loss in nnUNet with Tversky loss [51]:
+LT = − 2
+|V |
+�
+v p1,vy1,v
+2 �
+v p1,vy1,v + α �
+v p1,vy0,v + β �
+v p0,vy1,v
+(3)
+where |V | is the number of voxels. p1,v is the probability
+of voxel v being a LN, and p0,v is the probability being a
+non-LN. Also, y1,v is 1 for a LN voxel and 0 for a non-LN
+voxel, and vice versa for the y0,v. In practice, we set α = 0.5
+and β = 1.5 to emphasis on false negatives and boost recall.
+The whole network is trained by the combination of cross
+entropy loss LCE and Tversky loss LT with equal weights as
+in nnUNet.
+LCE = − 1
+|V |
+�
+v
+�
+k=0,1
+yk,v log(pk,v)
+(4)
+LSEG = LCE + LT
+(5)
+Following nnUNet, the network is trained with deep supervi-
+sion, i.e., losses are calculated over multi-resolution outputs
+given by final and intermediate layers of the decoder, and
+the corresponding downsampled ground-truth (GT) masks are
+used as targets. Here attention mechanism is applied in a multi-
+scale manner. That is, the attention map, after downsampled
+to match the resolution, is injected to the intermediate decoder
+feature for each deep supervision output.
+2) Stage 2: Instance-wise Lymph Node Identification: After
+segmenting LN instances from the whole CT image, we then
+classify them into either positive or negative class. To benefit
+from the already trained dense segmentation network of stage
+1, the task of LN instance identification reuses 3D nnUNet
+backbone and is initialized using the trained segmentation
+parameters, with a new classification head added upon it.
+Cross entropy loss is adopted to finetune the whole network
+for classifying the instance as positive/negative. To generate
+LN instances, we crop patches centered at the connected
+components of the segmentation mask. GT LN instances are
+cropped and employed in the training phase. While at inference
+time, we can apply the classification network to identify
+each segmented LN of stage 1, and obtain a class-aware LN
+segmentation mask.
+B. Tumor and Lymph Node Combined LN Metastasis Status
+Prediction Network
+Besides LNs themselves, imaging characteristics in the
+primary tumor play an important role in predicting the status
+of LN metastasis. To further boost the performance, we build
+a combined classification network, integrating both PDAC and
+LNs related information. In contrast to previous work that
+only consider tumor characteristics [8], [9], [11], our method
+benefits from directly observing the status of LN instances by
+automated LN segmentation and identification.
+Given a CT image and the corresponding PDAC mask,
+2D slices with the top three largest PDAC areas in each of
+axial, sagittal, and coronal planes are cropped, resulting in
+nine image patches in total. Each image patch is fed into
+a ResNet [29] pre-trained on ImageNet [52] for metastasis
+prediction. Inspired by [53], a side branch with the PDAC
+mask as input is added to encourage the network to concentrate
+on the PDAC region. Specifically, the side branch consists of
+a Conv-ReLU block and maps the input mask to a feature
+map with the same shape as the output of “Conv1” layer in
+ResNet. It is then integrated into the backbone by element-wise
+
+6
+multiplication with the “Conv1” feature. Our initial experiment
+empirically shows that such incorporation produces better per-
+formance than direct input-level fusion, as the convolution in
+the side branch learns which region to focus on in each channel
+of “Conv1” feature (e.g., regions inside the mask, around the
+mask border or outside the mask). To be better aligned with
+the pre-trained backbone and eliminate the initial effect of the
+side branch, the weights and biases in the convolution layer
+are initialized to 0 and 1 respectively. Before classification, we
+additionally employ a Texture Encoding Layer (TEL) [54] on
+top of the “Layer4” feature FL4 to extract respective texture
+representation. TEL serves as an orderless feature pooling
+layer that encodes spatially invariant representation describing
+the feature distributions, which benefits texture recognition of
+the PDAC region. The original deep feature is merged with
+the texture feature to form an enhanced representation F:
+F = Concat(GAP(FL4), TEL(FL4))
+(6)
+where Concat and GAP denote feature concatenation and
+global average layer, respectively.
+We further integrate LN-related cues into the network given
+the LN segmentation and identification results described in
+Section III-A. A patient is considered as metastasis-positive if
+there exists at least one positive LN, thus it is very sensitive
+to the false positives in LN identification. Therefore, we
+employ the volume of positive LN as the feature instead
+of its binary status of presence/absence, based on the fact
+that positive LNs tend to have larger volume than negative
+ones. The volume of the largest positive LN in each patient
+Vmax
+pLN =
+max
+ln∈{positive LNs} Vln (in mm3) is mapped to a vector-
+shaped feature, and fused with F by element-wise addition,
+formulated as follows:
+Fcomb = FC(BN(Vmax
+pLN)) + F
+(7)
+where FC and BN denote the full-connected layer and batch
+normalization layer, respectively. Other LN features, such as
+the average or total volume of positive LNs, are also evaluated,
+with the current setting giving the best result. Finally, the
+classification probabilities generated from nine image patches
+are averaged to given an ensembled prediction for a patient.
+IV. EXPERIMENTS
+In this section, we first demonstrate the multicenter datasets
+(i.e. the discovery dataset and two external datasets) and
+implementation details, and then elaborate the strategy we
+use to generate PDAC segmentation masks. Subsequently we
+present results on the discovery dataset in each step of our
+method, including organ&vessel segmentation, attention map
+generation, LN segmentation and identification, and LN metas-
+tasis status prediction. Finally, external validation is conducted
+to evaluate the generalization performance of LN metastasis
+status prediction, with only pathology reports accessible in two
+external datasets.
+A. Experimental Settings
+1) Dataset:
+We conduct a multicenter study on three
+independent datasets with a total of 940 patients collected
+from Changhai Hospital in Shanghai, Shengjing Hospital in
+Liaoning Province, and Tianjin Cancer Hospital in Tianjin. All
+patients had a pathologically confirmed diagnosis of PDAC,
+and contrast-enhanced CT scans of arterial (A) and venous
+(V) phases acquired before treatment were included in this
+study. We labeled LNs on the dataset from Changhai Hospital
+(denoted as Discovery dataset), and developed our model on
+it using nested cross-validation (CV). The rest two datasets
+from Shengjing Hospital and Tianjin Cancer Hospital (denoted
+as Ext-validation dataset 1 and Ext-validation dataset 2) were
+used as external validation sets with only pathologically diag-
+nosed LN metastasis status provided. This study was reviewed
+and approved by the Biomedical Research Ethics Committee
+of the institution (No. CHEC2021164), and was performed in
+accordance with the ethical standards of the 1964 Declaration
+of Helsinki. The requirement for patient informed consent
+was waived by the Institutional Review Board due to the
+retrospective nature of the study and because all procedures
+performed were part of routine care.
+Discovery dataset contains CT scans of 749 patients,
+among which there are 351 positive samples (patients with
+LN metastasis) and 398 negative samples (patients without
+LN metastasis). The annotation of LNs was performed by
+two board-certified radiologists (XF with 7 and MZ with 5
+years of experiences in pancreatic imaging) with referring to
+pathology report under supervision of a senior radiologist (YB)
+with 17 years of experiences in pancreatic imaging. There are
+totally 2,467 labeled LNs, of which 476 are positive and the
+rest are negative. In specific, 351 metastasis-positive patients
+contain 476 labeled positive and 322 labeled negative LNs, and
+398 metastasis-negative patients have the rest 1,669 labeled
+negative LNs.
+This dataset was split using nested five-fold CV, with 64%,
+16% and 20% as training, validation and testing sets in each
+CV round. As for the primary tumor, 163 patients among the
+whole dataset were annotated with 3D tumor masks by two
+radiologists (XF and MZ). We use these 163 patients as the
+testing set and the remaining unlabeled 586 patients as the
+training set for an annotation-efficient PDAC segmentation.
+Additionally, we generate pseudo annotations of 17 classes
+of organs and vessels using the self-learning segmentation
+model described in [55], and manually annotate extra two
+vessels (LGA, CHA&PHA) and extend other two vessels
+(SMA and TC&SA) under the supervision of a radiologist
+(XF) for 50 patients randomly sampled from our dataset. 40/10
+of these patients are used as training and validation sets for
+organ&vessel segmentation, respectively.
+Ext-validation dataset 1 contains CT scans of 132 patients
+with 39 positive and 93 negative patients; Ext-validation
+dataset 2 contains 59 patients with 37 positive and 22 negative
+patients. More detailed information on three datasets can be
+seen in Table I.
+2) Implementation Details: In our experiments, CT images
+of arterial phase are registered to venous phase using DEEDS
+[56], and they are all resampled to a median spacing of 0.68
+× 0.68 × 0.80 mm. For LN segmentation and organ&vessel
+segmentation, sub-volumes of 160 × 192 × 80 are randomly
+cropped as training patches. In the non-linear mapping from
+
+7
+TABLE I
+DEMOGRAPHIC DISTRIBUTIONS AND TUMOR CHARACTERISTICS IN THE THREE DATASETS (DISCOVERY DATASET, EXT-VALIDATION
+DATASET 1 AND EXT-VALIDATION DATASET 2). MEDIAN [INTERQUARTILE RANGE, 25TH–75TH PERCENTILE] VALUES ARE REPORTED
+FOR CONTINUOUS VARIABLES.
+Characteristics
+Discovery dataset
+Ext-validation dataset 1
+Ext-validation dataset 2
+(n=749)
+(n=132)
+(n=59)
+Gender, n (%)
+Female
+282 (38%)
+60 (45%)
+28 (47%)
+Male
+467 (62%)
+72 (55 %)
+31 (53%)
+Age at Diagnosis, yrs
+63 [56-69]
+60 [53-65]
+58 [51-62]
+pT Stage, n (%)
+pT1 / pT2
+92 (12%) / 314 (42%)
+24 (18%)/ 80 (61%)
+10 (17%) / 31 (53%)
+pT3 / pT4
+316 (42%) / 13 (2%)
+15 (11%)/ 13 (10%)
+5 (8%)/ 13 (22%)
+Missing
+14 (2%)
+0 (0 %)
+0 (0 %)
+pN Stage, n (%)
+pN0
+398 (53%)
+93 (70%)
+22 (37%)
+pN1
+242 (32%)
+32 (24%)
+22 (37%)
+pN2
+109 (15%)
+7 (5%)
+15 (25%)
+Tumor Size, cm
+3.0 [2.5-4.1]
+2.7 [2.2-3.0]
+2.9 [2.2-3.4]
+Tumor Location, n (%)
+Head / Uncinate
+475 (63%)
+56 (42%)/ 52 (39%)
+35 (59%)/ 22 (37%)
+Body / Tail
+274 (37%)
+2 (2%)/ 22 (17%)
+2 (3%)/ 0 (0%)
+Positive LN Volume, mm3
+665[210-804]
+-
+-
+Negative LN Volume, mm3
+300[106-377]
+-
+-
+distance maps to attention maps for our LN segmentation,
+parameters of the mapping function are determined by group-
+ing GT LN voxels according to which organ/vessel is closest
+to, and calculating the minimum and maximum distances to
+organ/vessel boundaries in each group. Parameters are listed
+in Table II, in which negative values indicate voxels inside
+organ/vessel.
+TABLE II
+PARAMETERS (I.E. dmin AND dmax) OF NON-LINEAR MAPPING FUNCTION
+FOR EACH ORGAN OR VESSEL. SMA: SUPERIOR MESENTERIC ARTERY.
+TC&SA: TRUNCUS COELIACUS AND SPLENIC ARTERY. LGA: LEFT
+GASTRIC ARTERY; CHA&PHA: COMMON HEPATIC ARTERY AND PROPER
+HEPATIC ARTERY.
+Organ/Vessel
+dmin (mm)
+dmax (mm)
+Spleen
+0
+16
+Esophagus
+0
+25
+Stomach
+-2
+18
+Aorta
+0
+28
+Pancreas
+-5
+20
+Duodenum
+-5
+22
+SMA
+-1
+20
+TC&SA
+-2
+18
+LGA
+0
+21
+CHA&PHA
+0
+20
+As for instance-wise LN identification, 3D image training
+samples are generated by cropping a 96 × 96 × 80 sub-volume
+centered per each GT LN. SGD optimizer with Nesterov
+momentum (µ = 0.95) is adopted to train the network, whose
+initial learning rate and weight decay are 5 × 10−4 and
+1 × 10−4, respectively. Furthermore, the final LN metastasis
+status prediction model takes 2D inputs of 224 × 224 cen-
+tered at PDAC, and is trained using the same optimizer as
+above. Details of the network architecture are presented in the
+supplementary material.
+B. PDAC Segmentation Mask Acquisition/Harvesting
+We employ an annotation-efficient strategy to generate
+3D masks of tumors for the labor cost reduction purpose.
+Specifically, we start with the PDAC segmentation model
+trained with arterial-late phase described in [46] to generate
+pseudo annotations. Next, the model is fine-tuned under the
+supervision of pseudo annotations and then applied to produce
+segmentation masks on our dataset. To obtain the PDAC
+segmentation model on venous phase, those segmentation
+masks are registered to venous phase and are then used to
+train a nnUNet model from scratch to generate the final 3D
+masks of tumors. We evaluate the final PDAC segmentation
+model on the labeled testing set. Median Dice score, average
+surface distance (ASD, mm), and Hausdorff distance (HD, mm)
+are 0.683, 2.186, and 12.805 respectively.
+C. Evaluation of Organ&Vessel Segmentation and Attention
+Maps
+To evaluate the performance of organ&vessel segmentation,
+a testing set of 19 randomly selected CT volumes with ten
+classes of organ/vessel is manually annotated by a radiologist
+(XF). To reduce the annotation burden, all dense CT volumes
+are downsampled to 5mm in the slice thickness dimension. We
+compare our self-learning model with the pseudo annotation
+generator [55], which is able to segment eight of ten classes
+(except for LGA and CHA&PHA) on single-phase CT. Dice
+score, ASD (mm), and HD (mm) are adopted as the evaluation
+metrics and the results are provided in Table III. Our model
+that is trained on two phases outperforms [55] on seven
+of eight organs/vessels. Note that SMA and TC&SA masks
+segmented by [55] contain shorter parts compared with those
+segmented by our model, therefore, resulting in significantly
+lower performance than ours (0.331 lower Dice score in SMA,
+and 0.171 lower in TC&SA).
+
+8
+TABLE III
+QUANTITATIVE PERFORMANCE OF ORGAN&VESSEL
+SEGMENTATION. A: ARTERIAL. V: VENOUS. SMA: SUPERIOR
+MESENTERIC ARTERY. TC&SA: TRUNCUS COELIACUS AND
+SPLENIC ARTERY. LGA: LEFT GASTRIC ARTERY; CHA&PHA:
+COMMON HEPATIC ARTERY AND PROPER HEPATIC ARTERY.
+Organ/Vessel
+Methods
+CT Phases
+Dice
+ASD (mm)
+HD (mm)
+Spleen
+[55]
+A
+0.938
+0.643
+14.252
+[55]
+V
+0.954
+0.422
+11.107
+Ours
+A+V
+0.959
+0.384
+8.129
+Esophagus
+[55]
+A
+0.557
+0.936
+13.897
+[55]
+V
+0.598
+0.854
+11.003
+Ours
+A+V
+0.745
+0.641
+8.125
+Stomach
+[55]
+A
+0.846
+2.223
+35.338
+[55]
+V
+0.813
+3.765
+43.114
+Ours
+A+V
+0.905
+1.519
+19.183
+Aorta
+[55]
+A
+0.893
+0.519
+8.130
+[55]
+V
+0.924
+0.417
+6.158
+Ours
+A+V
+0.920
+0.359
+5.863
+Pancreas
+[55]
+A
+0.712
+2.905
+25.880
+[55]
+V
+0.756
+1.897
+19.258
+Ours
+A+V
+0.847
+0.975
+12.859
+Duodenum
+[55]
+A
+0.613
+2.976
+34.187
+[55]
+V
+0.665
+3.366
+32.174
+Ours
+A+V
+0.764
+1.892
+29.131
+SMA
+[55]
+A
+0.387
+0.663
+68.869
+[55]
+V
+0.415
+0.710
+68.098
+Ours
+A+V
+0.746
+0.860
+28.840
+TC&SA
+[55]
+A
+0.563
+0.780
+43.974
+[55]
+V
+0.407
+1.245
+51.432
+Ours
+A+V
+0.734
+0.305
+22.224
+LGA
+[55]
+A
+-
+-
+-
+[55]
+V
+-
+-
+-
+Ours
+A+V
+0.651
+0.371
+10.420
+CHA&PHA
+[55]
+A
+-
+-
+-
+[55]
+V
+-
+-
+-
+Ours
+A+V
+0.715
+1.424
+24.239
+Qualitative evaluation of organ&vessel segmentation exam-
+ples as well as the corresponding attention maps are visualized
+in Fig. 4 (b).
+D. Evaluation of Lymph Node Instance Segmentation and
+Identification
+TABLE IV
+AVERAGE INSTANCE-WISE LN CLASSIFICATION PERFORMANCE
+ACROSS 5 FOLDS. THE RESULTS ARE REPORTED ON GT
+INSTANCES.
+Metric
+Performance
+AUC
+0.854
+Accuracy
+0.789
+Balanced accuracy
+0.771
+Sensitivity
+0.742
+Specificity
+0.800
+Quantitative Evaluation. LNs are first detected by the
+class-agnostic segmentation model, and then identified as
+positive/negative by applying the classification model on the
+cropped instances. For positive/negative LN identification, our
+classification model is trained with Ground-Truth (GT) LNs,
+yielding an average AUC of 0.854 across 5 folds (in Table IV).
+At inference, the automatically segmented LNs are cropped
+and then identified by the classification model. To evaluate the
+segmentation performance before and after identification, we
+compare our method with a strong baseline, nnUNet [27]. The
+segmentation accuracy is measured by voxel-wise metrics (i.e.,
+Dice, Recall, Precision) and instance-wise metrics (F-measure,
+Recall, Precision). To achieve the statistical analysis, we apply
+1,000 iterations of Wilcoxon signed rank test to voxel-wise
+Dice and instance-wise F-measure. Results are provided in
+Table V. An instance is considered successfully detected if
+its (intersect-over-union) IoU score between the segmentation
+mask and GT mask is ≥ 30 %. Before identification, our
+segmentation model significantly outperforms nnUNet on both
+voxel-wise and instance-wise metrics, with the voxel-wise
+Dice increasing from 45.9% to 47.7% and the instance-wise
+F-measure increasing from 36.1% to 40.6%, as shown in Table
+V. In addition, our model also yields superior performance in
+terms of both positive and negative LNs after identification,
+achieving 1.8% higher voxel-wise Dice and 1.8% higher
+instance-wise F-measure in terms of positive LNs, and 0.2%
+higher voxel-wise Dice and 1.2% higher instance-wise F-
+measure in terms of negative LNs. In total five out of six
+comparisons, our method is statistically significantly better or
+more accurate (i.e., with p-value < 0.05) in LN segmentation
+than the nnUNet baseline (implemented without the attention
+maps).
+Qualitative Evaluation. Examples of LN segmentation
+and identification results are shown in Fig. 4 (a) for qual-
+itative comparison. Our segmentation model leverages prior
+knowledge of LNs’ position distribution by incorporating the
+attention mechanism to remove false positives that are far
+from anatomically plausible LN areas. In Fig. 4 (a), we can
+observe that nnUNet tends to falsely detect an instance inside
+some organs or located very far, while our method provides
+noticeably less false positives.
+E. Evaluation of Patient-wise Lymph Node Metastasis Status
+Prediction
+Metrics. In this section, we evaluate various performance
+metrics of LN metastasis status prediction. For this binary
+classification problem, AUC, accuracy, balanced accuracy,
+sensitivity and specificity are adopted as evaluation metrics
+and the average results across 5 folds are reported. Statistical
+analysis is also carried out to verify the significance of
+performance improvement. We collect the predictions of all
+5 folds, repeat 1,000 times of bootstrapping for calculating
+balanced accuracy, and apply Wilcoxon signed rank test to
+balanced accuracy distributoins to compare our method with
+several other configurations. For comparing ROC curves,
+DeLong test is performed. P-values < 0.05 are considered
+as statistically significant. To compute 95% CI, the 2.5th
+percentile and 97.5th percentile are estimated after 1,000 times
+of bootstrapping.
+Ablation Study. We first investigate the impact of each
+component in our framework. To evaluate the metastasis
+prediction performance of LN segmentation and identification,
+the results can be aggregated into patient-level prediction,
+based on the definition that a patient with at least one positive
+LN is metastasis-positive. However, due to a large number of
+false positives produced by segmentation (LN segmentation
+in CT images is challenging after all), it will lead to a poor
+performance if predicting metastasis simply based on the
+
+9
+TABLE V
+PERFORMANCE COMPARISON ON LN SEGMENTATION BEFORE AND AFTER INSTANCE-WISE IDENTIFICATION (DENOTED AS
+Class-agnostic Seg AND Class-aware Seg). POS AND NEG DENOTE POSITIVE AND NEGATIVE LNS. RESULTS ARE AVERAGED ACROSS 5
+FOLDS. WILCOXON SIGNED RANK TEST IS CONDUCTED ON VOXEL-WISE DICE AND INSTANCE-WISE F-MEASURE. * INDICATES
+p-VALUE < 0.05. NS INDICATES NO SIGNIFICANCE.
+Stage
+Class
+Method
+Voxel-wise Metrics (%)
+Instance-wise Metrics (%)
+Dice
+Recall
+Precision
+F-measure
+Recall
+Precision
+Class-agnostic Seg
+-
+nnUNet
+45.9∗
+75.4
+36.2
+36.1∗
+81.0
+25.3
+(before identification)
+Ours
+47.7ref
+77.7
+37.5
+40.6ref
+80.9
+29.9
+Pos
+nnUNet
+10.2∗
+32.3
+11.3
+11.7∗
+36.1
+12.0
+Class-aware Seg
+Ours
+12.0ref
+38.9
+11.7
+13.5ref
+41.5
+13.3
+(after identification)
+Neg
+nnUNet
+27.5NS
+51.1
+25.4
+27.7∗
+60.0
+22.9
+Ours
+27.7ref
+49.0
+27.0
+28.9ref
+56.2
+25.8
+Image
+Label
+nnUNet
+Ours
+Positive Lymph Node
+Negative Lymph Node
+Attmap
+Organ&Vessel
+Organ&Vessel Anatomy
+(a) Lymph Node Segmentation and Identification Results
+(b)  Organ&Vessel Segmentation and  Attention Map Results
+Spleen
+RightKidney
+LeftKidney
+Gallbladder
+Esophagus
+Liver
+Stomach
+Aorta
+IVC
+PV&SV
+Pancreas
+RAG
+LAG
+Duodenum
+SMV
+SMA
+TC
+LGA
+CHA&PHA
+Spleen
+RightKidney
+LeftKidney
+Gallbladder
+Esophagus
+Liver
+Stomach
+Aorta
+IVC
+PV&SV
+Pancreas
+RAG
+LAG
+Duodenum
+SMV
+SMA
+TC
+LGA
+CHA&PHA
+Organs & Vessels
+Lymph Nodes
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+Fig. 4. Examples of (a) LN segmentation and identification results, and (b) Organ&Vessel segmentation and attention map results.
+presence of positive LN in the segmentation results. We instead
+conduct ROC analysis on the volume of the largest positive
+LN in each case, and find an optimal threshold with the best
+balanced accuracy in the validation set. Then the threshold
+is applied to the testing set. A patients with positive LNs
+larger than the threshold are classified into metastasis-positive;
+otherwise, it is classified into metastasis-negative. The ablation
+models for consideration/comparison are listed as follows:
+• ClsfromPDAC: The straightforward strategy combining
+ResNet2D [29] feature and DeepTEN [54] feature, extracted
+
+N980-2021103010
+TABLE VI
+PERFORMANCE COMPARISON AND ABLATION STUDY ON LN METASTASIS STATUS PREDICTION OF DISCOVERY DATASET. RESULTS ARE
+AVERAGED ACROSS 5 FOLDS. WILCOXON SIGNED RANK TEST IS CONDUCTED ON BALANCED ACCURACY. * INDICATES p-VALUE <
+0.05. NS INDICATES NO SIGNIFICANCE.
+Method
+Balanced Accuracy
+AUC
+Accuracy
+Sensitivity
+Specificity
+[95% CI]
+[95% CI]
+[95% CI]
+[95% CI]
+[95% CI]
+CT-reported LN status
+0.599∗
+-
+0.599
+0.588
+0.609
+[0.564-0.634]
+[0.565-0.634]
+[0.538-0.635]
+[0.558-0.657]
+Radiomics
+0.597∗
+0.648
+0.603
+0.508
+0.686
+[0.563-0.633]
+[0.598-0.681]
+[0.569-0.637]
+[0.456-0.561]
+[0.638-0.734]
+Radiomics +
+0.604∗
+0.654
+0.610
+0.524
+0.684
+CT-reported LN status
+[0.572-0.641]
+[0.612-0.692]
+[0.575-0.644]
+[0.470-0.581]
+[0.641-0.731]
+ResNet3D
+0.562∗
+0.609
+0.554
+0.599
+0.524
+[0.521-0.593]
+[0.550-0.631]
+[0.519-0.587]
+[0.538-0.644]
+[0.475-0.568]
+ResNet2D
+0.571∗
+0.631
+0.574
+0.568
+0.574
+[0.540-0.609]
+[0.590-0.667]
+[0.537-0.607]
+[0.519-0.624]
+[0.530-0.628]
+DeepTEN
+0.588∗
+0.634
+0.593
+0.609
+0.566
+[0.560-0.628]
+[0.599-0.679]
+[0.559-0.628]
+[0.564-0.667]
+[0.520-0.621]
+ClsfromPDAC
+0.599∗
+0.654
+0.597
+0.600
+0.597
+[0.558-0.634]
+[0.608-0.685]
+[0.561-0.633]
+[0.547-0.647]
+[0.550-0.646]
+ClsbyLNSeg w/o Attn
+0.545∗
+0.590
+0.566
+0.433
+0.657
+[0.525-0.593]
+[0.548-0.625]
+[0.534-0.601]
+[0.393-0.499]
+[0.623-0.716]
+ClsbyLNSeg w/ Attn
+0.563∗
+0.603
+0.572
+0.351
+0.775
+[0.530-0.594]
+[0.564-0.642]
+[0.539-0.605]
+[0.299-0.396]
+[0.731-0.814]
+Ours (ClsfromPDAC +
+0.633ref
+0.682
+0.635
+0.618
+0.649
+ClsbyLNSeg w/ Attn)
+[0.599-0.669]
+[0.640-0.717]
+[0.601-0.669]
+[0.567-0.664]
+[0.603-696]
+(a)
+(b)
+Fig. 5. ROC curve comparison of (a) ablation study and (b) baseline models and our method using nested five-fold cross-validation in Discovery dataset.
+from PDAC slices, in the input of the classification layer.
+• ClsbyLNSeg w/o Attn: Patient-level metastasis aggregation
+from the results of LN segmentation without attention (i.e.
+nnUNet).
+• ClsbyLNSeg w/ Attn: Patient-level metastasis aggregation
+from the results of our proposed LN segmentation with
+attention.
+• ClsfromPDAC + ClsbyLNSeg w/ Attn: Combined model
+incorporating the volume of the largest positive LN given
+by ClsbyLNSeg w/ Attn into the classification layer of
+ClsfromPDAC.
+The results of the ablation experiments are summarized in
+Table VI, and ROC analysis is illustrated in Fig. 5(a). By
+using only information about LNs, ClsbyLNSeg w/ Attn gives
+better aggregation results compared with ClsbyLNSeg w/o
+Attn (balanced accuracy 0.563 versus 0.545). Our final model
+(ClsfromPDAC + ClsbyLNSeg w/ Attn) significantly outper-
+forms the other three models with a balanced accuracy of 0.633
+(p-value < 0.05), which reveals the success of integrating both
+tumor and LNs imaging information for metastasis prediction.
+Comparison with Baselines. To validate the effective-
+ness of our method, radiomics model [8] and 2D/3D deep
+classification models are taken for comparison. To build the
+radiomics model, 1688 radiomics features of PDAC for each
+
+Ablation study: ROC curve on testing set
+1.0
+0.8
+True Positive Rate
+0.6
+0.4
+0.2
+ClsfromPDAC,AUC=0.654,p=0.038
+ClsbyLNSeg w/o Attn, AUC=0.590, p<0.001
+ClsbyLNSeg w/Attn,AUC=0.603,p<0.001
+0.0
+ClsfromPDAC+ClsbyLNSeg w/ Attn, AUC=0.682, ref
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive RateComparison with baselines: ROC curve on testing set
+1.0
+0.8
+True Positive Rate
+0.6
+0.4
+Radiomics.AUC=0.648.p=0.052
+Radiomics+Radiologists,AUC=0.654,p=0.16
+0.2
+Resnet3D,AUC=0.609,p<0.0001
+Resnet2D,AUC=0.631,p=0.0014
+DeepTEN,AUC=0.634,p=0.0099
+0.0
+Ours, AUC=0.682,ref
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate11
+CT phase are extracted using Pyradiomics package [57]1, and
+then shrunk using the least absolute shrinkage and selection
+operator (LASSO) method. Then a logistic regression model
+is applied to the selected features. The combined model of
+radiomics and CT-reported LN status is implemented with a
+logistic regression model on radiomics signature and radiolo-
+gists’ diagnosis. For 2D deep networks, ResNet2D [29] and
+DeepTEN [54], we use ResNet-18 backbone pre-trained on
+ImageNet [52]; while for 3D deep networks, we adopt 3D-
+ResNet-18 [58] backbone pre-trained on Kinetics-700 [59]
+and Moments in Time [60]. In all of 2D/3D deep networks,
+a side branch with the PDAC mask as input is added to
+the backbone, as we implemented in our method, for fair
+comparison. Table VI and Fig. 5(b) present the quantitative
+results of different models. More importantly, our method
+yields the best balanced accuracy (0.633) among all compared
+methods, and is significantly better than the radiomics method
+and all of 2D/3D deep networks.
+F. External Validation
+In this section, we demonstrate the generalization ability
+of our LN metastasis status prediction in two external multi-
+center datasets (i.e., Ext-validation dataset 1 and Ext-validation
+dataset 2). After training the model on Discovery dataset
+using cross validation, we apply the model to external datasets
+for inference. For each patient, the ensemble prediction is
+generated by averaging the model predictions from five folds.
+We first evaluate the performance of ablation variants, and
+then compare our method with baseline models. Metrics are
+used the same as Section IV-E.
+Ablation Study. We conduct ablation study on two external
+datasets, and results are shown in Table VII. With respect
+to LN metastasis status prediction using only LN-related
+information, our method (ClsbyLNSeg w/ Attn) outperforms
+nnUNet (ClsbyLNSeg w/o Attn) on both two datasets (bal-
+anced accuracy 0.589 versus 0.579 on Ext-validation dataset 1,
+0.639 versus 0.607 on Ext-validation dataset 2). By integrating
+PDAC characteristics, our final model (ClsfromPDAC + Cls-
+byLNSeg w/ Attn) gives the best results among all ablation
+models (balanced accuracy 0.620 on Ext-validation dataset 1
+and 0.684 on Ext-validation dataset 2).
+Comparison with Baselines. Table VII validates the gener-
+alization performance of our method compared with radiomics
+and 2D/3D deep learning models. Note that we skip methods
+involved with CT-reported LN status since there is no CT
+report available in two external datasets. The radiomics model
+shows poor generalization ability with a large drop in perfor-
+mance compared with that in Table VI, while deep learning
+models are relatively more robust. Our method significantly
+surpasses all of 2D/2D deep learning models with p-value <
+0.05 on both external datasets (balanced accuracy 0.620 and
+0.684 respectively), demonstrating the power of our model to
+generalize across different data sites.
+1https://pyradiomics.readthedocs.io/
+V. DISCUSSION
+Pre-operative LN metastasis status prediction is of vital sig-
+nificance for PDAC patients in the following aspects. Firstly,
+if diagnosed with LN metastasis, patients with resectable
+PDAC are recommended to receive neoadjuvant therapy first
+before surgery, according to NCCN guidelines [61]. Secondly,
+pancreatectomy could be guided by whether and where their
+LNs have metastasized, that is, whether or not a standard or an
+extended lymphadenectomy should be performed. This could
+make the surgical procedure being more targeted beforehand
+which could lead to better patient outcome and avoid over-
+treatment. Thirdly, LN metastasis is highly associated with pa-
+tients’ survival, which can evidently assist with good prognosis
+prediction value [55]. Note that it is very time consuming and
+highly dependent on (board-certified radiologist) observer’s
+experience and energy level to manually determine whether
+a patients has LN metastasis primarily from CT scans (even it
+is a very desirable task for patient care). CT-reported LN status
+in this study shows limited performance with an accuracy of
+0.599, thus accurate LN metastasis status prediction is highly
+desired.
+In the literature, LN metastasis status prediction has pre-
+dominantly been studied through tumor feature extraction,
+combined with CT report information, using radiomics [6]–
+[12] or deep learning approaches [13], [14], while the one
+leveraging LN radiomics requires manual delineation and
+considers simply the LN with the largest size [10]. An
+automated and accurate process of LN segmentation and
+nodal positivity identification is hence of high importance for
+assisting metastasis prediction. Predicting the metastasis status
+from automated segmented LNs is formulated by detecting
+metastatic LNs with Faster R-CNN [62], however the spatial
+context priors towards LNs are not exploited. This work
+proposes an automated geometric attention mechanism using
+LN segmentation and identification to predict the patient-level
+status of LN metastasis.
+To demonstrate the effectiveness of our method, we pro-
+vide extensive quantitative experiments on LN segmenta-
+tion/identification and LN metastasis status prediction. Our
+LN segmentation model statistically significantly outperforms
+the strong baseline nnUNet in voxel-wise and instance-wise
+metrics. For LN instance-wise detection, our model achieves
+considerable quantitative improvements (4.6%) in precision
+(with respect to a similar recall level) as compared to nnUNet
+(see Table V). This observation clearly validates that the
+proposed distance-guided attention mechanism is beneficial
+to remove false positives as we expect. The success of our
+model can be attributed to its attention map design and
+informative negative selection scheme. The former defines the
+LN-plausible regions that deserve network’s focus, and the
+latter helps to throw out non-informative negative training
+patches accordingly. As such, it becomes more efficient to
+train and force the model to learn discriminative features from
+possible LN locations. To verify the effect of LN detection
+improvements on patient-level metastasis status prediction, we
+perform instance-wise positivity identification and patient-wise
+aggregation on the LN instances to classify the patients into
+
+12
+TABLE VII
+PERFORMANCE COMPARISON AND ABLATION STUDY ON LN METASTASIS STATUS PREDICTION OF TWO EXTERNAL DATASETS.
+PREDICTIONS ARE AVERAGED ACROSS 5 FOLDS. WILCOXON SIGNED RANK TEST IS CONDUCTED ON BALANCED ACCURACY. *
+INDICATES p-VALUE < 0.05. NS INDICATES NO SIGNIFICANCE.
+Dataset
+Method
+Balanced
+AUC
+Accuracy
+Sensitivity
+Specificity
+Accuracy
+[95% CI]
+[95% CI]
+[95% CI]
+[95% CI]
+[95% CI]
+Radiomics
+0.493∗
+0.511
+0.672
+0.051
+0.935
+[0.451-0.537]
+[0.400-0.620]
+[0.626-0.710]
+[0.000-0.128]
+[0.880-0.978]
+ResNet3D
+0.508∗
+0.509
+0.527
+0.462
+0.554
+[0.415-0.612]
+[0.409-0.617]
+[0.450-0.611]
+[0.308-0.615]
+[0.457-0.663]
+ResNet2D
+0.563∗
+0.564
+0.542
+0.615
+0.511
+[0.470-0.656]
+[0.460-0.676]
+[0.466-0.626]
+[0.462-0.769]
+[0.413-0.609]
+DeepTEN
+0.556∗
+0.557
+0.511
+0.667
+0.446
+Ext-validation
+[0.467-0.647]
+[0.454-0.666]
+[0.427-0.595]
+[0.513-0.795]
+[0.348-0.544]
+dataset 1
+ClsfromPDAC
+0.515∗
+0.554
+0.485
+0.590
+0.441
+[0.423-0.609]
+[0.450-0.661]
+[0.402-0.568]
+[0.436-0.744]
+[0.344-0.548]
+ClsbyLNSeg w/o Attn
+0.579∗
+0.555
+0.689
+0.308
+0.849
+[0.498-0.662]
+[0.454-0.641]
+[0.621-0.750]
+[0.179-0.462]
+[0.774-0.914]
+ClsbyLNSeg w/ Attn
+0.589∗
+0.580
+0.705
+0.308
+0.871
+[0.511-0.672]
+[0.474-0.694]
+[0.644-0.765]
+[0.154-0.462]
+[0.796-0.935]
+Ours (ClsfromPDAC +
+0.620ref
+0.603
+0.674
+0.487
+0.753
+ClsbyLNSeg w/ Attn)
+[0.538-0.713]
+[0.498-0.712]
+[0.598-0.742]
+[0.333-0.641]
+[0.667-0.839]
+Radiomics
+0.508∗
+0.609
+0.441
+0.243
+0.773
+[0.391-0.626]
+[0.452-0.757]
+[0.339-0.542]
+[0.108-0.378]
+[0.591-0.909]
+ResNet3D
+0.461∗
+0.442
+0.475
+0.514
+0.409
+[0.334-0.584]
+[0.300-0.593]
+[0.356-0.594]
+[0.351-0.676]
+[0.182-0.591]
+ResNet2D
+0.681∗
+0.687
+0.650
+0.577
+0.786
+[0.536-0.810]
+[0.508-0.849]
+[0.500-0.800]
+[0.385-0.731]
+[0.571-0.930]
+DeepTEN
+0.613∗
+0.647
+0.640
+0.697
+0.529
+Ext-validation
+[0.465-0.747]
+[0.474-0.806]
+[0.520-0.760]
+[0.545-0.848]
+[0.294-0.765]
+dataset 2
+ClsfromPDAC
+0.620∗
+0.639
+0.593
+0.514
+0.727
+[0.493-0.734]
+[0.490-0.781]
+[0.475-0.712]
+[0.378-0.676]
+[0.545-0.909]
+ClsbyLNSeg w/o Attn
+0.607∗
+0.690
+0.542
+0.351
+0.864
+[0.503-0.716]
+[0.554-0.818]
+[0.441-0.661]
+[0.216-0.487]
+[0.682-1.000]
+ClsbyLNSeg w/ Attn
+0.639∗
+0.695
+0.593
+0.459
+0.818
+[0.525-0.752]
+[0.552-0.833]
+[0.475-0.695]
+[0.297-0.622]
+[0.636-0.955]
+Ours (ClsfromPDAC +
+0.684ref
+0.703
+0.661
+0.595
+0.773
+ClsbyLNSeg w/ Attn)
+[0.570-0.797]
+[0.554-0.846]
+[0.542-0.780]
+[0.432-0.757]
+[0.591-0.909]
+metastasis-positive/-negative, and our model presents better
+prediction performance than nnUNet (balanced accuracy 0.563
+versus 0.545, Table VI). We further combine the results
+with tumor CT imaging characteristics and our final pre-
+diction model achieves statistically significant performance
+gains compared to radiomics methods and other deep 2D/3D
+models (see Table VI), which demonstrates the success and
+effectiveness of integrating tumor morphology and lymphatic
+anatomy. It is worth noting that our method achieves sta-
+tistically significant improvement (balanced accuracy 0.633
+versus 0.604) compared to the approach even with radiologists
+involved in “Radiomics + CT-reported LN status” in Table
+VI. Nevertheless, using our method, this time-consuming,
+subjective and highly challenging manual process of CT-
+reported LN status can be fully automated. External multi-
+center clinical validation is further conducted on extra two
+datasets from different hospitals, and the results evidently
+exhibit our superior performance accuracy and generalization
+ability with the best results (balanced accuracy 0.620 and
+0.684 on the two external datasets) among several compared
+models (see Table VII). With all above-mentioned experi-
+ments, our model reports highly generalized prediction per-
+formance (0.620∼0.684) on multi-center datasets and robust
+improvements over CT-reported LN status (0.599) as well as
+radiomics and deep learning models, which clearly clarifies
+the advantage and stability of our model.
+Although recent work [8], [11] report exceedingly high
+accuracy (AUC > 0.9), they use small datasets of < 200
+patients, which would be subject to overfitting. Another recent
+progress in gastric cancer [12] enrolls over 500 patients from
+multiple hospitals, and yields noticeably lower but probably
+more reliable AUC score of 0.615∼0.712 in validation using
+2D/2.5D/3D radiomics features and under different patient
+splits. [12] is probably more suitable to serve as reference
+baseline for our work. We employs 940 patients in total in this
+study, in which 749 patients are from a high-volume pancreatic
+cancer clinical center and 191 are from two external centers.
+The studied patient population is arguably much closer and
+more realistic to the real-world patient data distributions com-
+paring to [8], [11], similar to [12]. We present a very promising
+approach that explicitly explores the role of automated LN
+segmentation in promoting LN metastasis status prediction
+to facilitate future clinical adoption as a fully-automated and
+generalizable clinical tool. One limitation of our framework
+
+13
+lies in the intuitive but simple solution that extracts tumor
+and LNs imaging information separately and then integrates
+them by feature concatenation, which does not fully exploit
+the nature of interactions between tumor and cancer cells
+in LNs. This work could be further improved in the future
+by designing an enriched deep learning geometric network
+representation to encode the tumor-LN topology information
+and spatial anatomical interactions, by modeling the pathways
+of nodal metastasis explicitly.
+Last, without loss of generality, our proposed method is
+applicable for finding the preoperative LN metastasis status of
+other types of solid tumor or cancers, such as liver or gastric
+cancers. We leave this as future work.
+VI. CONCLUSION
+We present an attention based LN segmentation network and
+utilize it on predicting LN metastasis in patients with PDAC.
+The proposed LN segmentation network involves an attention
+mechanism that encourages the network to focus on regions
+around certain anatomical organs/vessels. It outperforms the
+strong baseline nnUNet [27] by leveraging the context infor-
+mation of surrounding anatomical structures. Our segmenta-
+tion model, followed by a nodal positivity identification model,
+can serve as a single predictor for LN metastasis. Combined
+with tumor imaging characteristics, we further build a compos-
+itive LN metastasis status prediction model that is validated
+to surpass the CT-reported results, radiomics based method,
+and other 2D/3D deep learning models. Further investigations
+include conceiving a more complicated way to encode tumor-
+LN relationship and exploring its applications to prognosis and
+treatment planning in cancer patient management.
+ACKNOWLEDGMENTS
+This
+work
+was
+supported
+in
+part
+by
+the
+National
+Science Foundation for Scientists of China (81871352,
+82171915, and 82171930), Clinical Research Plan of SHDC
+(SHDC2020CR4073), 234 Platform Discipline Consolidation
+Foundation Project (2019YPT001, 2020YPT001), and The
+Natural Science Foundation of Shanghai Science and Technol-
+ogy Innovation Action Plan (21ZR1478500, 21Y11910300).
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf,len=2016
+page_content='1  A deep local attention network for pre-operative  lymph node metastasis prediction in pancreatic  cancer via multiphase CT imaging  Zhilin Zheng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Xu Fang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Jiawen Yao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Mengmeng Zhu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Le Lu Fellow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Lingyun Huang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Jing Xiao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Yu Shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Hong Lu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Jianping Lu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Ling Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Chengwei Shao*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Yun Bian*  Abstract—Lymph node (LN) metastasis status is one of the  most critical prognostic and cancer staging factors for patients  with resectable pancreatic ductal adenocarcinoma (PDAC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' or in  general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' for any types of solid malignant tumors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Preoperative  prediction of LN metastasis from non-invasive CT imaging  is highly desired, as it might be straightforwardly used to  guide the following neoadjuvant treatment decision and surgical  planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Most studies only capture the tumor characteristics  in CT imaging to implicitly infer LN metastasis and very few  work exploit direct LN’s CT imaging information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' LN staging  is usually confirmed from pathological images acquired after  invasive procedures of biopsy or surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To the best of our  knowledge, this is the first work to propose a fully-automated  LN segmentation and identification network to directly facilitate  the LN metastasis status prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Nevertheless LN seg-  mentation/detection is very challenging since LN can be easily  confused with other hard negative anatomic structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=',  vessels) from radiological images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 1) We explore the anatomical  spatial context priors of pancreatic LN locations by generating a  guiding attention map from related organs and vessels to assist  segmentation and infer LN status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' As such, LN segmentation  is impelled to focus on regions that are anatomically adjacent  or plausible with respect to the specific organs and vessels  (thus hard negative samples with certain distance ranges can  be ignored).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 2) The metastasized LN identification network is  trained to classify the segmented LN instances into positives  or negatives by reusing the segmentation network as a pre-  trained backbone and padding a new classification head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 3)  More importantly, we develop a LN metastasis status prediction  network that combines the patient-wise aggregation results of LN  segmentation/identification and deep imaging features extracted  from the tumor region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 4) Extensive quantitative nested five-  fold cross-validation is conducted on a discovery dataset of 749  patients with PDAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' External multi-center clinical evaluation is  further performed on two other hospitals of 191 patients in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Our final multi-staged LN status prediction network statistically  significantly outperforms the strong baseline of nnUNet and  several other compared methods, including CT-reported LN  status, radiomics, and deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Index Terms—Pancreatic ductal adenocarcinoma (PDAC),  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Zheng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Xiao are with Ping An Technology (Shanghai &  Shenzhen), People’s Republic of China, (e-mail: zhilin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='zheng95@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Fang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Zhu and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Lu are with Changhai Hospital, Shanghai, People’s  Republic of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Yao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Lu and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Zhang were with PAII Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', Bethesda,  MD 20817, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Fang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Yao contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Shi is with Department of Radiology, Shengjing Hospital of China  Medical University, Shenyang, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Lu is with Department of Radiology, Tianjin Medical University Cancer  Institute and Hospital, National Clinical Research Center of Cancer, Key  Laboratory of Cancer Prevention and Therapy, Tianjin, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  s indicate joint corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Bian and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Shao are  with Changhai Hospital, Shanghai, People’s Republic of China, (email:  bianyun2012@foxmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='com, cwshao@sina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Lymph node metastasis, Lymph node segmentation, Contrast-  enhanced computed tomography  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' INTRODUCTION  P  ANCREATIC cancer is the third leading cause of overall  cancer death in the United States [1], of which approx-  imately 95% is pancreatic ductal adenocarcinoma (PDAC)  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' With the poorest prognosis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', 5-year overall survival  (OS) of 10% approximately), surgical resection is the most  effective way to achieve long-term survival for patients with  PDAC [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' However, not all patients can benefit from the  margin-negative (R0) resection and comprehensive treatment  protocol is usually established for pancreatic cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The  patients’ treatment selections can be determined by whether  their peripancreatic lymph nodes (LNs) have metastasized with  the options of adjuvant radiotherapy (RT) or chemotherapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  It was found that neoadjuvant therapy before surgery was  associated with improved survival and time to recurrence in  patients with LN metastasis, since neoadjuvant therapy can  not only treat lymphovascular invasion but also benefit tumor  downstaging [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The accurate preoperative detection of  LN metastasis becomes vital and would aid in treatment  planning and management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Contrast-enhanced CT is used as the typical imaging pro-  tocol for identifying the presence of peripancreatic metastatic  disease to LNs, but it is a very challenging task for radiologists  to determine whether a patient has LN metastasis by using only  CT scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To this end, poor diagnostic accuracy of CT with a  pooled sensitivity of 25% and positive predictive value (PPV)  of 28% was reported in a meta-analysis [5] on assessing extra-  regional LN metastasis in pancreatic and peri-ampullary can-  cer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Recently, several radiomics based approaches have been  proposed to tackle the LN metastasis differentiation problem  of various cancer types  [6]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' However, these methods  require hand-crafted feature design which can bring concerns  of reproducibility and human bias is often introduced due to  manual selection of 2D tumor slice with limited representation  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Although there are some deep learning work that report  promising performance on predicting LN metastasis status in  gastric cancer [13], [14], those models assume that the risk of  metastases is fundamentally driven by the primary tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' They  rely on LN CT report information for the integration model  without using any LNs detection or segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The model  that takes both tumor morphology and lymphatic anatomy into  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='01448v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='IV]  4 Jan 2023  2  Arterial  Arterial  Venous  Patient with metastasis  Patient without metastasis  Tumor  LN  Tumor-LN anatomy  Negative LN  Positive LN  Tumor  Tumor & LN  Spleen  Esophagus  Stomach  Aorta  Pancreas  Duodenum  SMA  TC  LGA  CHA&PHA  Organs & Vessels  Venous  LN  LN  LN  LN  LN  Tumor  LN  LN  LN  LN  LN  Tumor  LN  LN  LN  LN  Tumor  LN  LN  LN  LN  Tumor  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' A visualization of pancreatic tumor (in dark red) and LNs (in pink red for positives or green for negatives) in multi-phase CT images  and their spatial distributions corresponding to key anatomical structures as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' SMA: superior mesenteric artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' TC&SA: truncus  coeliacus and splenic artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' LGA: left gastric artery;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' CHA&PHA: common hepatic artery and proper hepatic artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  account could be of more clinically usefulness on addressing  these aforementioned issues, similarly as in the diagnostic  processes performed by radiologist readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' PET/CT is another  imaging modality worth exploring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' PET/CT based approaches  [15]–[17] generally use maximum standardized uptake value  (SUVmax) of manually-drawn LN RoIs as the prediction  element, but it comes with challenges of numerous false  positives from inflammatory LNs and false negatives from  small-sized metastatic LNs [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Also, it is relatively  not as common as CT, which is less affordable, available and  accessible, hence we opt for CT for our research purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  In this paper, we tackle the LN metastasis status prediction  problem in patients with PDAC by first segmenting and  identifying instances of LNs and then classifying the patients  into metastasis-positive or -negative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' LNs are tiny struc-  tures that anatomically locate surrounding organs and vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Their locations have been mapped into 18 stations that are  relevant for pancreatic cancer tumor according to their relative  positions against adjacent anatomical structures, as defined  by Japan Pancreas Society (JPS) [20] (see Supplementary  Table 1 in the supplementary material for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Examples  of their spatial distribution are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Metastasis  happens when cancer cells spread from the primary tumor  to LNs, causing enlargement of LNs and other underlying  changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Response Evaluation Criteria in Solid Tumors (RE-  CIST) criteria [21] defines the criteria for LN metastasis  suspicion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', nodes with short axis greater than 10mm,  heterogeneity and central necrosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' However, these criteria are  not pathognomonic since there exist false negatives associated  with small node micrometastases and false positives with  inflammatory nodes larger than 10mm in short axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Hence,  finding LNs in CT images is quite time-consuming and can be  inconsistent depending on radiologists’ subjective experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  It is ambiguous for radiologists to identify nodal positivity  accurately from CT without referring to pathology reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  The gold standard for determination of metastasis is based on  post-operative pathological evaluation of pancreatectomy spec-  imens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Automated yet reliable pre-operative LN segmentation  and identification are highly desirable for patient surgical or  RT treatment planing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  LN segmentation is inherently challenging due to two  reasons: 1) small to tiny sizes of LN cause extreme foreground-  background class imbalance problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 2) LNs have CT atten-  uation values similar to vessels and other soft-tissue struc-  tures, resulting in visual confusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Existing work [22]–[24]  mainly adopt U-Net based deep networks [25]–[27] as strong  backbones, in which skip connections aggregate multi-level  semantic representation and alleviate vanishing gradient prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' They incorporate anatomical context by directly taking  organs&vessels segmentation masks as either supervision tar-  gets [22] or additional inputs [24], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Concerns are remained  that the relationship between lymphatic anatomy and adjacent  anatomical structures is not well explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' We address it by  introducing a distance-guided attention map to fully utilize the  spatial priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In our segmentation framework, the LN attention  map is obtained via a pre-defined mapping from distance  maps of related organs/vessels that have been integrated into  UNet-based backbone to control the segmentation network’s  TO:  B  TVN00SL0003TO五  TO  TVNO0O0003TO:  BTVN001O0003TO:  B  TVN00SL0003TO:  B3  spatial focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' It simultaneously assists in improving sample  selection strategy that filters out non-informative negative  samples (called ”informative negative selection”) to tackle the  class imbalance problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The segmented LNs are labeled as  positive/negative using radiologist’s judgement as the standard  that combines information from pathological results and CT  intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' A classification network is subsequently derived by  sharing the same backbone with segmentation and initialized  with the trained segmentation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' This strategy ben-  efits the classification task from densely structured prediction  in segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' By predicting LN metastasis in patients  with PDAC, we employ a modified ResNet [29] classification  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Tumor characteristics are proven to be important cues  for metastasis [8], [9], [11], so we integrate both tumor and  LN cues by taking as inputs the image patches of tumor  and the patient-wise aggregation of LN segmentation and  identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Our main contribution is four folds: 1) To the best of  our knowledge, this work is the first to directly incorporate  automated LN segmentation and identification for assisting  preoperative LN metastasis status prediction for patients with  PDAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 2) We propose an attention-based LN segmentation  network with the guidance of distances to nearby anatomical  structures, explicitly exploiting the spatial priors and simul-  taneously addressing the foreground-background imbalance  issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 3) We design a compositive LN metastasis status  prediction network combining both tumor and positive LN  characteristics, showing the potential of tumor-LN guided  cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 4) Extensive quantitative experiments are conducted to  evidently validate the effectiveness of our deep local attention  network in both tasks of LN segmentation and LN metastasis  status prediction, and external multi-center clinical evaluation  is performed to demonstrate the generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Without  loss of generality, our proposed method is applicable for  finding the preoperative LN metastasis status of other types  of solid tumor or cancers, such as liver or gastric cancers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' RELATED WORK  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Lymph Node Segmentation  Automated LN segmentation in CT images is an essential  yet challenging task in medical image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Traditional  approaches tackle this problem by the means of atlas based  search space restriction [30], spatial prior features combination  [31], [32], supervoxel clustering [33], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In recent years, U-  Net based deep networks have shown remarkable performance  in numerous organ or tumor segmentation tasks [34]–[38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  nnUNet [27] further proposes a self-configuring approach,  with automatic configurations including preprocessing, net-  work architecture, training and post-processing, that achieves  robust performance and general applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To address the  strong class imbalance issues in LN segmentation, four other  anatomical structures are included as training targets [22]  using 3D U-Net [26] framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' [23] utilizes parallel net-  works of 2D U-Net [25] and Mask R-CNN [39] with the  supervision of all considered anatomical structures and LNs,  benefiting from both semantic and instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Another strategy to incorporate anatomical context is to take  organ segmentation masks as additional channels of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  [28] proposes an ensemble approach for a slab-wise and  a downsampled full volume based LN segmentation, taking  the concatenation of CT image and segmented anatomical  structure mask as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' DeepStationing [24] presents a key  referencing organ auto-search strategy and combines selected  organs into the network via input concatentation for LN station  parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' All above methods implicitly exploit spatial priors  of LNs by injecting the anatomical structure masks either as  inputs or supervisions, hence the prior knowledge has not been  fully exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' More importantly, there is a lack of studies on  how LN segmentation could be used for predicting metastasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Lymph Node Metastasis Prediction  Radiomics Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Radiomics is a powerful technique  for extracting quantitative image features with the purpose  of clinical decision support, and thus widely used in can-  cer research [11], [12], [40]–[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' It converts imaging data  into different types of hand-crafted features, including shape,  intensity, texture and filter-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', wavelet, Laplacian of  Gaussian) features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Applications of radiomics on predicting  LN metastasis from primary tumor have been explored in  many recent works [6]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Radiomics features are first  extracted from manually delineated tumor regions in any  contrast-enhanced CT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Feature selection and classifi-  cation model construction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', logistic regression, random  forest) are then performed to give LN metastasis prediction  for various cancer types like gastric cancer [7], [12], biliary  tract cancer [6] and PDAC [8], [9], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Relying only on  primary tumor radiomics without considering LNs character-  istics may limit the prediction performance, thus [10] uses  manual annotations of the largest LN visible in the gastric  region and combines LN radiomics into the prediction model  for gastric cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' However, problem still remains because it  simply involves the largest LN without identifying the nodal  positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Deep Learning based Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Recent advances in deep  learning have made it a mainstream method of addressing the  entire workflow of diagnosis and treatment for various cancer  types on medical imaging, such as orapharageal cancer [43],  lung cancer [44], as well as pancreatic cancer [45]–[47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Deep  neural networks are applied to LN metastasis in many studies  [13], [14], [48], [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In [48], deep features are extracted  from tumor ROIs in bimodal image (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', US and SWE) using  ResNet [29], and then fed into a SVM model for predicting  axillary LN status in breast cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' For gastric cancer, [14]  combines DenseNet [50] features with some hand-crafted  features, extracted from the 2D tumor ROI with the largest  area in multi-phase CT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To investigate metastasis in  individual LN stations for gastric cancer, [13] develops a  system of multiple independent ResNets with tumor ROIs and  corresponding annotation masks as inputs where each ResNet  is responsible to predict metastasis at one specific nodal sta-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Most existing studies capture only tumor characteristics  for LN metastasis prediction, while the one leveraging LN  radiomics requires manual delineation and considers simply  the LN with the largest size [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' An automated and accurate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='b)   ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='LN metastasis status prediction network  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Volume  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='PDAC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Mask  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Metastasis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='positive  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Metastasis-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Negative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Cropping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Cropping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='c  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Lymph node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='segmentation &  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='identification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Lymph node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='segmentation &  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='identification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Volume  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='PDAC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Mask  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Metastasis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='positive  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Metastasis-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Negative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Cropping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Cropping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='c  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Lymph node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='segmentation &  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='identification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Lymph Node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Segmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Instance-wise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Lymph Node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Identification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Organ&  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Vessel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Distance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Transform  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Non-linear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Mapping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Attention  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Mechanism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='GAP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Lymph node segmentation & identification network  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='c  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='concatenation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='element-wise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='multiplication  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Downsampling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The proposed framework for (a) two-stage LN segmentation and identification and (b) LN metastasis status prediction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  process of LN segmentation and nodal positivity identification  is hence of high importance for assisting metastasis prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' METHODOLOGY  The overall framework is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 2, which is com-  posed of (a) distance-guided attention-based LN segmentation  and identification network, and (b) tumor and LN combined  metastasis status prediction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Distance-guided Attention-based Lymph Node Segmenta-  tion and Identification Network  We perform LN detection from any input CT scan by a two-  stage strategy: segmenting the image into two classes of LN  and background voxels, followed by identifying segmented LN  instances as positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  1) Stage 1: Class-agnostic Lymph Node Segmentation:  Based on the spatial prior that LN stations are geometrically  distributed or constrained around certain anatomical structures,  we propose an attention based LN segmentation network by  taking the distances to nearby organs/vessels into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Our LN segmentation network differs from the strong baseline  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', nnUNet [27]) in that attention mechanism is applied to  guide possible LN locations, with the advantage of reducing  false positive predictions outside those locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The intuition  behind the attention module is that the attention map can cover  regions adjacently constrained to those organs and vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Attention Map Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To explicitly capture and  model the lymphatic anatomy, attention computation is im-  plemented as a pre-defined geometric mapping function from  organ&vessel distance maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' An example of attention map  generation process is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Specifically, given a  multi-phase input CT volume X ∈ RN×W ×H×D, we first  obtain organ&vessel segmentation mask using nnUNet [27]  model trained with 19 classes of annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Ten classes  among them involved with 17 LN stations are used (see  Supplementary Table 1 in the supplementary material for the  definition of LN stations), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', spleen, esophagus, stomach,  aorta, pancreas, duodenum, superior mesenteric artery (SMA),  truncus coeliacus and splenic artery (TC&SA), left gastric  artery (LGA), common hepatic artery and proper hepatic  artery (CHA&PHA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Note that station 15# (LNs along middle  Spleen  Aorta  SMA  LGA  Image  Organ&Vessel  Organ/Vessel  Distance Map  Organ/Vessel  Attention Map  Integrated  Attention Map  CHA&PHA  Esophagus  Duodenum  Non-linear  Mapping  0  1  d imin  min  d imin  dmax  imax  i  dmax  i  d imin  min  d imin-3  d imin-3  dmax  imax  i  dmax  i  +3  dmax  i  +3 d  fifi  Stomach  Pancreas  TC&SA  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' An illustration of attention map generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  colic artery) is left aside here since it is related to distant  metastasis that rarely happens in our patient population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' A  SDT is applied to each class of the segmentation mask M ∈  {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', 10}W ×H×D, generating a total of 10 organ/vessel  distance maps Di where i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', 10} is the index of  organ/vessel class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Di has positive values at the voxels outside  the i-th organ/vessel and negative scores inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Intuitively,  LNs are likely to appear within a certain range of distance  to each organ/vessel, which requires paying attention to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To  obtain the distance-guided attention maps, Di is passed to an  isosceles trapezium-shaped non-linear mapping function (see  TO  B  c000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='r100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='ocbqT0  B  E000_100_3sbq5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 3), formulated as  f i(d) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  1,  di  min ≤ d ≤ di  max  − (d − di  max − 3)  3  ,  di  max<d<di  max + 3  (d − di  min + 3)  3  ,  di  min − 3<d<di  min  0,  Otherwise  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  (1)  where d is the individual element in Di;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' di  min and di  max  determine the distance range in mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' the smooth border 3mm  is chosen empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' This mapping function converts the  distance maps to the attention scores ranging from 0 to 1, with  1 indicating possible locations of LNs, 0 indicating impossible  locations, and decimals lying in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The i-th attention  map Ai is obtained by Ai = f i(Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  The final attention map Aall is produced by integrating all  of the organ/vessel-specific attention maps, thus Aall can cover  the whole areas that need attending to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In specific, Aall takes  the element-wise maximum of all Ai except for the voxels  inside an organ/vessel, illustrated as  aall  v  =  �  max  i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=',10 ai  v,  mv = 0  ai  v,  mv = i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  (2)  where a∗  v and mv are the values of A∗ and M at the voxel v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Attention based Lymph Node Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' After ob-  taining the distance-guided attention map, we incorporate it  to the segmentation network with 3D nnUNet [27] as the  backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The attention mechanism emphasizes LN-related  regions by spatially scaling the features with the attention map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Given the input image X and the one-hot segmentation label  Y , the deep features at the penultimate layer are extracted  and multiplied element-wisely with Aall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' It is finally passed  through a convolution block with a softmax layer to produce  the segmentation output P ∈ R2×W ×H×D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Due to GPU memory limitation, patch-based training strat-  egy is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' nnUNet randomly samples 3D image patches  from the whole CT scan and enforces that more than a third  of samples in a batch contain at least one foreground class to  control the foreground-to-background ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Considering the  extreme class imbalance problem caused by the small LN  targets, we improve it with the “informative negative selection”  scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Note that our proposed attention mechanism helps  block out features at the voxels with a certain distance to  or inside all organs and vessels, resulting in lots of non-  informative negative patches filled with 0 by applying zero  attention scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Thus we can naturally throw out those non-  informative patches, and select patches containing at least one  attention score > 0 (called informative patches) for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  This sampling strategy further boosts the network’s concen-  tration on targeted regions of interest (ROIs) surrounding  organs/vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  For training objectives to better balance precision and recall  , we modify the Dice loss in nnUNet with Tversky loss [51]:  LT = − 2  |V |  �  v p1,vy1,v  2 �  v p1,vy1,v + α �  v p1,vy0,v + β �  v p0,vy1,v  (3)  where |V | is the number of voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' p1,v is the probability  of voxel v being a LN, and p0,v is the probability being a  non-LN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Also, y1,v is 1 for a LN voxel and 0 for a non-LN  voxel, and vice versa for the y0,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In practice, we set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='5  and β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='5 to emphasis on false negatives and boost recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  The whole network is trained by the combination of cross  entropy loss LCE and Tversky loss LT with equal weights as  in nnUNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  LCE = − 1  |V |  �  v  �  k=0,1  yk,v log(pk,v)  (4)  LSEG = LCE + LT  (5)  Following nnUNet, the network is trained with deep supervi-  sion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', losses are calculated over multi-resolution outputs  given by final and intermediate layers of the decoder, and  the corresponding downsampled ground-truth (GT) masks are  used as targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Here attention mechanism is applied in a multi-  scale manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' That is, the attention map, after downsampled  to match the resolution, is injected to the intermediate decoder  feature for each deep supervision output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  2) Stage 2: Instance-wise Lymph Node Identification: After  segmenting LN instances from the whole CT image, we then  classify them into either positive or negative class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To benefit  from the already trained dense segmentation network of stage  1, the task of LN instance identification reuses 3D nnUNet  backbone and is initialized using the trained segmentation  parameters, with a new classification head added upon it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Cross entropy loss is adopted to finetune the whole network  for classifying the instance as positive/negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To generate  LN instances, we crop patches centered at the connected  components of the segmentation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' GT LN instances are  cropped and employed in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' While at inference  time, we can apply the classification network to identify  each segmented LN of stage 1, and obtain a class-aware LN  segmentation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Tumor and Lymph Node Combined LN Metastasis Status  Prediction Network  Besides LNs themselves, imaging characteristics in the  primary tumor play an important role in predicting the status  of LN metastasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To further boost the performance, we build  a combined classification network, integrating both PDAC and  LNs related information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In contrast to previous work that  only consider tumor characteristics [8], [9], [11], our method  benefits from directly observing the status of LN instances by  automated LN segmentation and identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Given a CT image and the corresponding PDAC mask,  2D slices with the top three largest PDAC areas in each of  axial, sagittal, and coronal planes are cropped, resulting in  nine image patches in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Each image patch is fed into  a ResNet [29] pre-trained on ImageNet [52] for metastasis  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Inspired by [53], a side branch with the PDAC  mask as input is added to encourage the network to concentrate  on the PDAC region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Specifically, the side branch consists of  a Conv-ReLU block and maps the input mask to a feature  map with the same shape as the output of “Conv1” layer in  ResNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' It is then integrated into the backbone by element-wise  6  multiplication with the “Conv1” feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Our initial experiment  empirically shows that such incorporation produces better per-  formance than direct input-level fusion, as the convolution in  the side branch learns which region to focus on in each channel  of “Conv1” feature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', regions inside the mask, around the  mask border or outside the mask).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To be better aligned with  the pre-trained backbone and eliminate the initial effect of the  side branch, the weights and biases in the convolution layer  are initialized to 0 and 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Before classification, we  additionally employ a Texture Encoding Layer (TEL) [54] on  top of the “Layer4” feature FL4 to extract respective texture  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' TEL serves as an orderless feature pooling  layer that encodes spatially invariant representation describing  the feature distributions, which benefits texture recognition of  the PDAC region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The original deep feature is merged with  the texture feature to form an enhanced representation F:  F = Concat(GAP(FL4), TEL(FL4))  (6)  where Concat and GAP denote feature concatenation and  global average layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  We further integrate LN-related cues into the network given  the LN segmentation and identification results described in  Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' A patient is considered as metastasis-positive if  there exists at least one positive LN, thus it is very sensitive  to the false positives in LN identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Therefore, we  employ the volume of positive LN as the feature instead  of its binary status of presence/absence, based on the fact  that positive LNs tend to have larger volume than negative  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The volume of the largest positive LN in each patient  Vmax  pLN =  max  ln∈{positive LNs} Vln (in mm3) is mapped to a vector-  shaped feature, and fused with F by element-wise addition,  formulated as follows:  Fcomb = FC(BN(Vmax  pLN)) + F  (7)  where FC and BN denote the full-connected layer and batch  normalization layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Other LN features, such as  the average or total volume of positive LNs, are also evaluated,  with the current setting giving the best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Finally, the  classification probabilities generated from nine image patches  are averaged to given an ensembled prediction for a patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' EXPERIMENTS  In this section, we first demonstrate the multicenter datasets  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' the discovery dataset and two external datasets) and  implementation details, and then elaborate the strategy we  use to generate PDAC segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Subsequently we  present results on the discovery dataset in each step of our  method, including organ&vessel segmentation, attention map  generation, LN segmentation and identification, and LN metas-  tasis status prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Finally, external validation is conducted  to evaluate the generalization performance of LN metastasis  status prediction, with only pathology reports accessible in two  external datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Experimental Settings  1) Dataset:  We conduct a multicenter study on three  independent datasets with a total of 940 patients collected  from Changhai Hospital in Shanghai, Shengjing Hospital in  Liaoning Province, and Tianjin Cancer Hospital in Tianjin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' All  patients had a pathologically confirmed diagnosis of PDAC,  and contrast-enhanced CT scans of arterial (A) and venous  (V) phases acquired before treatment were included in this  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' We labeled LNs on the dataset from Changhai Hospital  (denoted as Discovery dataset), and developed our model on  it using nested cross-validation (CV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The rest two datasets  from Shengjing Hospital and Tianjin Cancer Hospital (denoted  as Ext-validation dataset 1 and Ext-validation dataset 2) were  used as external validation sets with only pathologically diag-  nosed LN metastasis status provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' This study was reviewed  and approved by the Biomedical Research Ethics Committee  of the institution (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' CHEC2021164), and was performed in  accordance with the ethical standards of the 1964 Declaration  of Helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The requirement for patient informed consent  was waived by the Institutional Review Board due to the  retrospective nature of the study and because all procedures  performed were part of routine care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Discovery dataset contains CT scans of 749 patients,  among which there are 351 positive samples (patients with  LN metastasis) and 398 negative samples (patients without  LN metastasis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The annotation of LNs was performed by  two board-certified radiologists (XF with 7 and MZ with 5  years of experiences in pancreatic imaging) with referring to  pathology report under supervision of a senior radiologist (YB)  with 17 years of experiences in pancreatic imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' There are  totally 2,467 labeled LNs, of which 476 are positive and the  rest are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In specific, 351 metastasis-positive patients  contain 476 labeled positive and 322 labeled negative LNs, and  398 metastasis-negative patients have the rest 1,669 labeled  negative LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  This dataset was split using nested five-fold CV, with 64%,  16% and 20% as training, validation and testing sets in each  CV round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' As for the primary tumor, 163 patients among the  whole dataset were annotated with 3D tumor masks by two  radiologists (XF and MZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' We use these 163 patients as the  testing set and the remaining unlabeled 586 patients as the  training set for an annotation-efficient PDAC segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Additionally, we generate pseudo annotations of 17 classes  of organs and vessels using the self-learning segmentation  model described in [55], and manually annotate extra two  vessels (LGA, CHA&PHA) and extend other two vessels  (SMA and TC&SA) under the supervision of a radiologist  (XF) for 50 patients randomly sampled from our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 40/10  of these patients are used as training and validation sets for  organ&vessel segmentation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Ext-validation dataset 1 contains CT scans of 132 patients  with 39 positive and 93 negative patients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Ext-validation  dataset 2 contains 59 patients with 37 positive and 22 negative  patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' More detailed information on three datasets can be  seen in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  2) Implementation Details: In our experiments, CT images  of arterial phase are registered to venous phase using DEEDS  [56], and they are all resampled to a median spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='68  × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='68 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='80 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' For LN segmentation and organ&vessel  segmentation, sub-volumes of 160 × 192 × 80 are randomly  cropped as training patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In the non-linear mapping from  7  TABLE I  DEMOGRAPHIC DISTRIBUTIONS AND TUMOR CHARACTERISTICS IN THE THREE DATASETS (DISCOVERY DATASET, EXT-VALIDATION  DATASET 1 AND EXT-VALIDATION DATASET 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' MEDIAN [INTERQUARTILE RANGE, 25TH–75TH PERCENTILE] VALUES ARE REPORTED  FOR CONTINUOUS VARIABLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Characteristics  Discovery dataset  Ext-validation dataset 1  Ext-validation dataset 2  (n=749)  (n=132)  (n=59)  Gender,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' n (%)  Female  282 (38%)  60 (45%)  28 (47%)  Male  467 (62%)  72 (55 %)  31 (53%)  Age at Diagnosis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' yrs  63 [56-69]  60 [53-65]  58 [51-62]  pT Stage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' n (%)  pT1 / pT2  92 (12%) / 314 (42%)  24 (18%)/ 80 (61%)  10 (17%) / 31 (53%)  pT3 / pT4  316 (42%) / 13 (2%)  15 (11%)/ 13 (10%)  5 (8%)/ 13 (22%)  Missing  14 (2%)  0 (0 %)  0 (0 %)  pN Stage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' n (%)  pN0  398 (53%)  93 (70%)  22 (37%)  pN1  242 (32%)  32 (24%)  22 (37%)  pN2  109 (15%)  7 (5%)  15 (25%)  Tumor Size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' cm  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='0 [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='5-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='1]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='7 [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='0]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='9 [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='4]  Tumor Location,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' n (%)  Head / Uncinate  475 (63%)  56 (42%)/ 52 (39%)  35 (59%)/ 22 (37%)  Body / Tail  274 (37%)  2 (2%)/ 22 (17%)  2 (3%)/ 0 (0%)  Positive LN Volume,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' mm3  665[210-804] Negative LN Volume,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' mm3  300[106-377] distance maps to attention maps for our LN segmentation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  parameters of the mapping function are determined by group-  ing GT LN voxels according to which organ/vessel is closest  to,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' and calculating the minimum and maximum distances to  organ/vessel boundaries in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Parameters are listed  in Table II, in which negative values indicate voxels inside  organ/vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  TABLE II  PARAMETERS (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' dmin AND dmax) OF NON-LINEAR MAPPING FUNCTION  FOR EACH ORGAN OR VESSEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' SMA: SUPERIOR MESENTERIC ARTERY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  TC&SA: TRUNCUS COELIACUS AND SPLENIC ARTERY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' LGA: LEFT  GASTRIC ARTERY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' CHA&PHA: COMMON HEPATIC ARTERY AND PROPER  HEPATIC ARTERY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Organ/Vessel  dmin (mm)  dmax (mm)  Spleen  0  16  Esophagus  0  25  Stomach  2  18  Aorta  0  28  Pancreas  5  20  Duodenum  5  22  SMA  1  20  TC&SA  2  18  LGA  0  21  CHA&PHA  0  20  As for instance-wise LN identification, 3D image training  samples are generated by cropping a 96 × 96 × 80 sub-volume  centered per each GT LN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' SGD optimizer with Nesterov  momentum (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='95) is adopted to train the network, whose  initial learning rate and weight decay are 5 × 10−4 and  1 × 10−4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Furthermore, the final LN metastasis  status prediction model takes 2D inputs of 224 × 224 cen-  tered at PDAC, and is trained using the same optimizer as  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Details of the network architecture are presented in the  supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' PDAC Segmentation Mask Acquisition/Harvesting  We employ an annotation-efficient strategy to generate  3D masks of tumors for the labor cost reduction purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Specifically, we start with the PDAC segmentation model  trained with arterial-late phase described in [46] to generate  pseudo annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Next, the model is fine-tuned under the  supervision of pseudo annotations and then applied to produce  segmentation masks on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To obtain the PDAC  segmentation model on venous phase, those segmentation  masks are registered to venous phase and are then used to  train a nnUNet model from scratch to generate the final 3D  masks of tumors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' We evaluate the final PDAC segmentation  model on the labeled testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Median Dice score, average  surface distance (ASD, mm), and Hausdorff distance (HD, mm)  are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='683, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='186, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='805 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Evaluation of Organ&Vessel Segmentation and Attention  Maps  To evaluate the performance of organ&vessel segmentation,  a testing set of 19 randomly selected CT volumes with ten  classes of organ/vessel is manually annotated by a radiologist  (XF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To reduce the annotation burden, all dense CT volumes  are downsampled to 5mm in the slice thickness dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' We  compare our self-learning model with the pseudo annotation  generator [55], which is able to segment eight of ten classes  (except for LGA and CHA&PHA) on single-phase CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Dice  score, ASD (mm), and HD (mm) are adopted as the evaluation  metrics and the results are provided in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Our model  that is trained on two phases outperforms [55] on seven  of eight organs/vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Note that SMA and TC&SA masks  segmented by [55] contain shorter parts compared with those  segmented by our model, therefore, resulting in significantly  lower performance than ours (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='331 lower Dice score in SMA,  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='171 lower in TC&SA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  8  TABLE III  QUANTITATIVE PERFORMANCE OF ORGAN&VESSEL  SEGMENTATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' A: ARTERIAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' V: VENOUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' SMA: SUPERIOR  MESENTERIC ARTERY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' TC&SA: TRUNCUS COELIACUS AND  SPLENIC ARTERY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' LGA: LEFT GASTRIC ARTERY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' CHA&PHA:  COMMON HEPATIC ARTERY AND PROPER HEPATIC ARTERY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Organ/Vessel  Methods  CT Phases  Dice  ASD (mm)  HD (mm)  Spleen  [55]  A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='239  Qualitative evaluation of organ&vessel segmentation exam-  ples as well as the corresponding attention maps are visualized  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Evaluation of Lymph Node Instance Segmentation and  Identification  TABLE IV  AVERAGE INSTANCE-WISE LN CLASSIFICATION PERFORMANCE  ACROSS 5 FOLDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' THE RESULTS ARE REPORTED ON GT  INSTANCES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Metric  Performance  AUC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='854  Accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='789  Balanced accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='771  Sensitivity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='742  Specificity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='800  Quantitative Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' LNs are first detected by the  class-agnostic segmentation model, and then identified as  positive/negative by applying the classification model on the  cropped instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' For positive/negative LN identification, our  classification model is trained with Ground-Truth (GT) LNs,  yielding an average AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='854 across 5 folds (in Table IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  At inference, the automatically segmented LNs are cropped  and then identified by the classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To evaluate the  segmentation performance before and after identification, we  compare our method with a strong baseline, nnUNet [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The  segmentation accuracy is measured by voxel-wise metrics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=',  Dice, Recall, Precision) and instance-wise metrics (F-measure,  Recall, Precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To achieve the statistical analysis, we apply  1,000 iterations of Wilcoxon signed rank test to voxel-wise  Dice and instance-wise F-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Results are provided in  Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' An instance is considered successfully detected if  its (intersect-over-union) IoU score between the segmentation  mask and GT mask is ≥ 30 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Before identification, our  segmentation model significantly outperforms nnUNet on both  voxel-wise and instance-wise metrics, with the voxel-wise  Dice increasing from 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='9% to 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='7% and the instance-wise  F-measure increasing from 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='1% to 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='6%, as shown in Table  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In addition, our model also yields superior performance in  terms of both positive and negative LNs after identification,  achieving 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='8% higher voxel-wise Dice and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='8% higher  instance-wise F-measure in terms of positive LNs, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='2%  higher voxel-wise Dice and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='2% higher instance-wise F-  measure in terms of negative LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In total five out of six  comparisons, our method is statistically significantly better or  more accurate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', with p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='05) in LN segmentation  than the nnUNet baseline (implemented without the attention  maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Qualitative Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Examples of LN segmentation  and identification results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 4 (a) for qual-  itative comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Our segmentation model leverages prior  knowledge of LNs’ position distribution by incorporating the  attention mechanism to remove false positives that are far  from anatomically plausible LN areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 4 (a), we can  observe that nnUNet tends to falsely detect an instance inside  some organs or located very far, while our method provides  noticeably less false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Evaluation of Patient-wise Lymph Node Metastasis Status  Prediction  Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In this section, we evaluate various performance  metrics of LN metastasis status prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' For this binary  classification problem, AUC, accuracy, balanced accuracy,  sensitivity and specificity are adopted as evaluation metrics  and the average results across 5 folds are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Statistical  analysis is also carried out to verify the significance of  performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' We collect the predictions of all  5 folds, repeat 1,000 times of bootstrapping for calculating  balanced accuracy, and apply Wilcoxon signed rank test to  balanced accuracy distributoins to compare our method with  several other configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' For comparing ROC curves,  DeLong test is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' P-values < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='05 are considered  as statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To compute 95% CI, the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='5th  percentile and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='5th percentile are estimated after 1,000 times  of bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Ablation Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' We first investigate the impact of each  component in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To evaluate the metastasis  prediction performance of LN segmentation and identification,  the results can be aggregated into patient-level prediction,  based on the definition that a patient with at least one positive  LN is metastasis-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' However, due to a large number of  false positives produced by segmentation (LN segmentation  in CT images is challenging after all), it will lead to a poor  performance if predicting metastasis simply based on the  9  TABLE V  PERFORMANCE COMPARISON ON LN SEGMENTATION BEFORE AND AFTER INSTANCE-WISE IDENTIFICATION (DENOTED AS  Class-agnostic Seg AND Class-aware Seg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' POS AND NEG DENOTE POSITIVE AND NEGATIVE LNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' RESULTS ARE AVERAGED ACROSS 5  FOLDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' WILCOXON SIGNED RANK TEST IS CONDUCTED ON VOXEL-WISE DICE AND INSTANCE-WISE F-MEASURE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' * INDICATES  p-VALUE < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' NS INDICATES NO SIGNIFICANCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Stage  Class  Method  Voxel-wise Metrics (%)  Instance-wise Metrics (%)  Dice  Recall  Precision  F-measure  Recall  Precision  Class-agnostic Seg nnUNet  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='9∗  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='4  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='2  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='1∗  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='0  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='3  (before identification)  Ours  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='7ref  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='7  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='5  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='6ref  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='9  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='9  Pos  nnUNet  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='2∗  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='3  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='3  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='7∗  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='1  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='0  Class-aware Seg  Ours  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='0ref  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='9  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='7  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='5ref  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='5  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='3  (after identification)  Neg  nnUNet  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='5NS  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='1  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='4  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='7∗  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='9  Ours  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='7ref  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='0  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='0  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='9ref  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='2  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Label  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='nnUNet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Ours  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Positive Lymph Node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Negative Lymph Node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Attmap  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Organ&Vessel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Organ&Vessel Anatomy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='(a) Lymph Node Segmentation and Identification Results  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='(b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Organ&Vessel Segmentation and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Attention Map Results  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Spleen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='RightKidney  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='LeftKidney  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Gallbladder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Esophagus  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Liver  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Stomach  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Aorta  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='IVC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='PV&SV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='Pancreas  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='RAG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='LAG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Examples of (a) LN segmentation and identification results, and (b) Organ&Vessel segmentation and attention map results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  presence of positive LN in the segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' We instead  conduct ROC analysis on the volume of the largest positive  LN in each case, and find an optimal threshold with the best  balanced accuracy in the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Then the threshold  is applied to the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' A patients with positive LNs  larger than the threshold are classified into metastasis-positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  otherwise, it is classified into metastasis-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The ablation  models for consideration/comparison are listed as follows:  ClsfromPDAC: The straightforward strategy combining  ResNet2D [29] feature and DeepTEN [54] feature, extracted  N980-2021103010  TABLE VI  PERFORMANCE COMPARISON AND ABLATION STUDY ON LN METASTASIS STATUS PREDICTION OF DISCOVERY DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' RESULTS ARE  AVERAGED ACROSS 5 FOLDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' WILCOXON SIGNED RANK TEST IS CONDUCTED ON BALANCED ACCURACY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' * INDICATES p-VALUE <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' NS INDICATES NO SIGNIFICANCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Method  Balanced Accuracy  AUC  Accuracy  Sensitivity  Specificity  [95% CI]  [95% CI]  [95% CI]  [95% CI]  [95% CI]  CT-reported LN status  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='603-696]  (a)  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' ROC curve comparison of (a) ablation study and (b) baseline models and our method using nested five-fold cross-validation in Discovery dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  from PDAC slices, in the input of the classification layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  nnUNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='  ClsfromPDAC + ClsbyLNSeg w/ Attn: Combined model  incorporating the volume of the largest positive LN given  by ClsbyLNSeg w/ Attn into the classification layer of  ClsfromPDAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content=' By  using only information about LNs, ClsbyLNSeg w/ Attn gives  better aggregation results compared with ClsbyLNSeg w/o  Attn (balanced accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='  Comparison with Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To validate the effective-  ness of our method, radiomics model [8] and 2D/3D deep  classification models are taken for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='052  Radiomics+Radiologists,AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='2  Resnet3D,AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='609,p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='0001  Resnet2D,AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='631,p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='0014  DeepTEN,AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='0  Ours, AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='682,ref  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='0  False Positive Rate11  CT phase are extracted using Pyradiomics package [57]1, and  then shrunk using the least absolute shrinkage and selection  operator (LASSO) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Then a logistic regression model  is applied to the selected features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The combined model of  radiomics and CT-reported LN status is implemented with a  logistic regression model on radiomics signature and radiolo-  gists’ diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' For 2D deep networks, ResNet2D [29] and  DeepTEN [54], we use ResNet-18 backbone pre-trained on  ImageNet [52];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' while for 3D deep networks, we adopt 3D-  ResNet-18 [58] backbone pre-trained on Kinetics-700 [59]  and Moments in Time [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' In all of 2D/3D deep networks,  a side branch with the PDAC mask as input is added to  the backbone, as we implemented in our method, for fair  comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Table VI and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' 5(b) present the quantitative  results of different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' More importantly, our method  yields the best balanced accuracy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='633) among all compared  methods, and is significantly better than the radiomics method  and all of 2D/3D deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' External Validation  In this section, we demonstrate the generalization ability  of our LN metastasis status prediction in two external multi-  center datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=', Ext-validation dataset 1 and Ext-validation  dataset 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' After training the model on Discovery dataset  using cross validation, we apply the model to external datasets  for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' For each patient, the ensemble prediction is  generated by averaging the model predictions from five folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  We first evaluate the performance of ablation variants, and  then compare our method with baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Metrics are  used the same as Section IV-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Ablation Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' We conduct ablation study on two external  datasets, and results are shown in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' With respect  to LN metastasis status prediction using only LN-related  information, our method (ClsbyLNSeg w/ Attn) outperforms  nnUNet (ClsbyLNSeg w/o Attn) on both two datasets (bal-  anced accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='589 versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='579 on Ext-validation dataset 1,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='639 versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='607 on Ext-validation dataset 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' By integrating  PDAC characteristics, our final model (ClsfromPDAC + Cls-  byLNSeg w/ Attn) gives the best results among all ablation  models (balanced accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='620 on Ext-validation dataset 1  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='684 on Ext-validation dataset 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Comparison with Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Table VII validates the gener-  alization performance of our method compared with radiomics  and 2D/3D deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Note that we skip methods  involved with CT-reported LN status since there is no CT  report available in two external datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The radiomics model  shows poor generalization ability with a large drop in perfor-  mance compared with that in Table VI, while deep learning  models are relatively more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Our method significantly  surpasses all of 2D/2D deep learning models with p-value <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='05 on both external datasets (balanced accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='620 and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='684 respectively), demonstrating the power of our model to  generalize across different data sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  1https://pyradiomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='io/  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' DISCUSSION  Pre-operative LN metastasis status prediction is of vital sig-  nificance for PDAC patients in the following aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Firstly,  if diagnosed with LN metastasis, patients with resectable  PDAC are recommended to receive neoadjuvant therapy first  before surgery, according to NCCN guidelines [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Secondly,  pancreatectomy could be guided by whether and where their  LNs have metastasized, that is, whether or not a standard or an  extended lymphadenectomy should be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' This could  make the surgical procedure being more targeted beforehand  which could lead to better patient outcome and avoid over-  treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Thirdly, LN metastasis is highly associated with pa-  tients’ survival, which can evidently assist with good prognosis  prediction value [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Note that it is very time consuming and  highly dependent on (board-certified radiologist) observer’s  experience and energy level to manually determine whether  a patients has LN metastasis primarily from CT scans (even it  is a very desirable task for patient care).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' CT-reported LN status  in this study shows limited performance with an accuracy of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='599, thus accurate LN metastasis status prediction is highly  desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  In the literature, LN metastasis status prediction has pre-  dominantly been studied through tumor feature extraction,  combined with CT report information, using radiomics [6]–  [12] or deep learning approaches [13], [14], while the one  leveraging LN radiomics requires manual delineation and  considers simply the LN with the largest size [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' An  automated and accurate process of LN segmentation and  nodal positivity identification is hence of high importance for  assisting metastasis prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Predicting the metastasis status  from automated segmented LNs is formulated by detecting  metastatic LNs with Faster R-CNN [62], however the spatial  context priors towards LNs are not exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' This work  proposes an automated geometric attention mechanism using  LN segmentation and identification to predict the patient-level  status of LN metastasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  To demonstrate the effectiveness of our method, we pro-  vide extensive quantitative experiments on LN segmenta-  tion/identification and LN metastasis status prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Our  LN segmentation model statistically significantly outperforms  the strong baseline nnUNet in voxel-wise and instance-wise  metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' For LN instance-wise detection, our model achieves  considerable quantitative improvements (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='6%) in precision  (with respect to a similar recall level) as compared to nnUNet  (see Table V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' This observation clearly validates that the  proposed distance-guided attention mechanism is beneficial  to remove false positives as we expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The success of our  model can be attributed to its attention map design and  informative negative selection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' The former defines the  LN-plausible regions that deserve network’s focus, and the  latter helps to throw out non-informative negative training  patches accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' As such, it becomes more efficient to  train and force the model to learn discriminative features from  possible LN locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' To verify the effect of LN detection  improvements on patient-level metastasis status prediction, we  perform instance-wise positivity identification and patient-wise  aggregation on the LN instances to classify the patients into  12  TABLE VII  PERFORMANCE COMPARISON AND ABLATION STUDY ON LN METASTASIS STATUS PREDICTION OF TWO EXTERNAL DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  PREDICTIONS ARE AVERAGED ACROSS 5 FOLDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' WILCOXON SIGNED RANK TEST IS CONDUCTED ON BALANCED ACCURACY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' *  INDICATES p-VALUE < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' NS INDICATES NO SIGNIFICANCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Dataset  Method  Balanced  AUC  Accuracy  Sensitivity  Specificity  Accuracy  [95% CI]  [95% CI]  [95% CI]  [95% CI]  [95% CI]  Radiomics  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='909]  metastasis-positive/-negative, and our model presents better  prediction performance than nnUNet (balanced accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content=' We further combine the results  with tumor CT imaging characteristics and our final pre-  diction model achieves statistically significant performance  gains compared to radiomics methods and other deep 2D/3D  models (see Table VI), which demonstrates the success and  effectiveness of integrating tumor morphology and lymphatic  anatomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' It is worth noting that our method achieves sta-  tistically significant improvement (balanced accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='604) compared to the approach even with radiologists  involved in “Radiomics + CT-reported LN status” in Table  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Nevertheless, using our method, this time-consuming,  subjective and highly challenging manual process of CT-  reported LN status can be fully automated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' External multi-  center clinical validation is further conducted on extra two  datasets from different hospitals, and the results evidently  exhibit our superior performance accuracy and generalization  ability with the best results (balanced accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='684 on the two external datasets) among several compared  models (see Table VII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' With all above-mentioned experi-  ments, our model reports highly generalized prediction per-  formance (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content='684) on multi-center datasets and robust  improvements over CT-reported LN status (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='599) as well as  radiomics and deep learning models, which clearly clarifies  the advantage and stability of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Although recent work [8], [11] report exceedingly high  accuracy (AUC > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+page_content=' Another recent  progress in gastric cancer [12] enrolls over 500 patients from  multiple hospitals, and yields noticeably lower but probably  more reliable AUC score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='615∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='712 in validation using  2D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='5D/3D radiomics features and under different patient  splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' [12] is probably more suitable to serve as reference  baseline for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' We employs 940 patients in total in this  study, in which 749 patients are from a high-volume pancreatic  cancer clinical center and 191 are from two external centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  The studied patient population is arguably much closer and  more realistic to the real-world patient data distributions com-  paring to [8], [11], similar to [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' We present a very promising  approach that explicitly explores the role of automated LN  segmentation in promoting LN metastasis status prediction  to facilitate future clinical adoption as a fully-automated and  generalizable clinical tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' One limitation of our framework  13  lies in the intuitive but simple solution that extracts tumor  and LNs imaging information separately and then integrates  them by feature concatenation, which does not fully exploit  the nature of interactions between tumor and cancer cells  in LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' This work could be further improved in the future  by designing an enriched deep learning geometric network  representation to encode the tumor-LN topology information  and spatial anatomical interactions, by modeling the pathways  of nodal metastasis explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  Last, without loss of generality, our proposed method is  applicable for finding the preoperative LN metastasis status of  other types of solid tumor or cancers, such as liver or gastric  cancers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' We leave this as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' CONCLUSION  We present an attention based LN segmentation network and  utilize it on predicting LN metastasis in patients with PDAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  The proposed LN segmentation network involves an attention  mechanism that encourages the network to focus on regions  around certain anatomical organs/vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' It outperforms the  strong baseline nnUNet [27] by leveraging the context infor-  mation of surrounding anatomical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Our segmenta-  tion model, followed by a nodal positivity identification model,  can serve as a single predictor for LN metastasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Combined  with tumor imaging characteristics, we further build a compos-  itive LN metastasis status prediction model that is validated  to surpass the CT-reported results, radiomics based method,  and other 2D/3D deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content=' Further investigations  include conceiving a more complicated way to encode tumor-  LN relationship and exploring its applications to prognosis and  treatment planning in cancer patient management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This  work  was  supported  in  part  by  the  National  Science Foundation for Scientists of China (81871352,  82171915, and 82171930), Clinical Research Plan of SHDC  (SHDC2020CR4073), 234 Platform Discipline Consolidation  Foundation Project (2019YPT001, 2020YPT001), and The  Natural Science Foundation of Shanghai Science and Technol-  ogy Innovation Action Plan (21ZR1478500, 21Y11910300).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQffPw3/content/2301.01448v1.pdf'}
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+Cubic Double Perovskites Host Noncoplanar Spin Textures
+Joseph A. M. Paddison,1, ∗ Hao Zhang,2 Jiaqiang Yan,1 Matthew J. Cliffe,3 Seung-Hwan Do,1 Shang Gao,1, 4
+Matthew B. Stone,4 David Dahlbom,2 Kipton Barros,5 Cristian D. Batista,6 and Andrew D. Christianson1, †
+1Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
+2Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
+3School of Chemistry, University of Nottingham, Nottingham NG7 2RD, UK
+4Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
+5Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
+6Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA
+Magnetic materials with noncoplanar magnetic structures can show unusual physical properties driven by
+nontrivial topology. Topologically-active states are often multi-q structures, which are challenging to stabilize
+in models and to identify in materials. Here, we use inelastic neutron-scattering experiments to show that the
+insulating double perovskites Ba2YRuO6 and Ba2LuRuO6 host a noncoplanar 3-q structure on the face-centered
+cubic lattice. Quantitative analysis of our neutron-scattering data reveals that these 3-q states are stabilized by
+biquadratic interactions. Our study identifies double perovskites as a highly promising class of materials to
+realize topological magnetism, elucidates the stabilization mechanism of the 3-q state in these materials, and
+establishes neutron spectroscopy on powder samples as a valuable technique to distinguish multi-q from single-q
+states, facilitating the discovery of topologically-nontrivial magnetic materials.
+Most magnetic materials order with simple magnetic struc-
+tures in which spins are collinear or coplanar.
+Noncopla-
+nar magnetic structures are relatively rare, but are of great
+current interest, because they can exhibit topological charac-
+ter and exotic physical properties [1, 2]. For example, the
+finite scalar spin chirality of noncoplanar spin textures can
+generate a topological magneto-optical effect [3] and anoma-
+lous quantum Hall effect [4, 5], even in the absence of spin-
+orbit coupling. Topologically-nontrivial spin textures are typ-
+ically multi-q structures, which superpose magnetic modula-
+tions with symmetry-related wavevectors q [2]. Multi-q spin
+textures with long-wavelength modulations, such as skyrmion
+and hedgehog crystals, are well-studied as hosts of topology-
+driven phenomena [6–8]. In this context, multi-q antiferro-
+magnets are increasingly important [9], because they offer
+higher densities of topological objects with the potential to
+generate stronger physical responses [10].
+To probe the relationships between spin structure, interac-
+tions, topology, and physical response, it is crucial to identify
+real materials that host noncoplanar spin textures. This has
+proved a challenging task, for three main reasons. First, it
+is necessary to identify noncoplanar spin textures that are ro-
+bust to subleading effects such as magnetic anisotropies, spin-
+lattice coupling [11, 12], fluctuations [13–16], and anisotropic
+interactions [17], which usually favor collinear states. Sec-
+ond, most noncoplanar states are found in metals, such as
+USb [18, 19] and γ-Mn alloys [20–25], and are often stable
+only under an applied magnetic field [6, 26].
+On the one
+hand, itinerant electrons can support the generation of phys-
+ical responses; on the other hand, modeling the magnetic in-
+teractions of metals presents fundamental challenges [27–32],
+such that insulators are often more suitable as model materi-
+als. Third, powder neutron-diffraction measurements play a
+central role in solving magnetic structures, but suffer from a
+“multi-q problem”: Such measurements are generally unable
+to distinguish 1-q from multi-q structures [33]. Therefore,
+multi-q spin textures are challenging to stabilize in models,
+and to identify in real materials.
+Here, we identify the Mott-insulating double perovskites
+Ba2YRuO6 and Ba2LuRuO6 [34–37] as prototypical exam-
+ples of noncoplanar 3-q magnetism on the face-centered cu-
+bic (FCC) lattice in zero magnetic field. We obtain evidence
+for 3-q magnetism from a spin-wave analysis of neutron spec-
+troscopy data. By optimizing the magnetic structure and inter-
+actions simultaneously against our data, we show that the 3-
+q structure is stabilized by biquadratic interactions within an
+antiferromagnetic Heisenberg-Kitaev model. Our study ex-
+perimentally establishes that noncoplanar multi-q states are
+stabilized in frustrated FCC antiferromagnets, identifies cubic
+double perovskites as model materials to realize this behavior,
+and identifies guiding principles to facilitate design of materi-
+als with noncoplanar states.
+Our study is motivated by theoretical results for the
+FCC antiferromagnet [13, 38–40].
+The nearest-neighbor
+Heisenberg-Kitaev spin Hamiltonian on the FCC lattice can
+be written as
+H = J ∑
+⟨i, j⟩
+Si ·Sj +K ∑
+⟨i, j⟩γ
+Sγ
+i Sγ
+j,
+(1)
+where Si is a Ru5+ spin with quantum number S = 3/2, J
+and K denote the Heisenberg and Kitaev interactions, respec-
+tively, and γ ∈ {x,y,z} is perpendicular to the cubic plane con-
+taining the bond between neighbors ⟨i, j⟩. For antiferromag-
+netic J > 0 only, the model is frustrated, and orderings with
+q ∈ [1,q,0] are degenerate [13, 39, 40]. The degenerate mani-
+fold includes q = [1,0,0] (“Type I”) ordering, which is favored
+by fluctuations [13, 14, 41] and is observed in Ba2YRuO6
+and Ba2LuRuO6 [34]. Henceforth, we therefore restrict our
+discussion to q = [1,0,0] ordering. For a collinear structure,
+spins may be either parallel or perpendicular to q; the former
+is favored by K < 0 and the latter by K > 0 [38–40].
+Figure 1(a) shows the collinear (1-q) and noncollinear
+arXiv:2301.11395v1  [cond-mat.str-el]  26 Jan 2023
+
+2
+1-q
+tetragonal
+2-q
+tetragonal
+3-q
+cubic
+3-q
+cubic
+2-q
+tetragonal
+1-q
+orthorhombic
+(a)
+��������
+����������������
+�������
+�
+���
+�
+���
+�
+���
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+�������
+�
+���
+�
+���
+�
+���
+��������
+�������
+�
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+�
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+�
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+���������
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+�
+���
+�
+���
+�
+���
+K ≥ 0, mX5
++
+K ≤ 0, mX3
++
+(b)
+Figure 1.
+(a) Symmetry-allowed magnetic structures with propagation vector q = [1,0,0] on the FCC lattice for Ba2MRuO6 (space group
+Fm¯3m; a = 8.29 and 8.24 Å for M = Y and Lu, respectively). The 1-q, 2-q, and 3-q structures are shown for the mX+
+3 irrep (left) and the
+mX+
+5 irrep (right). Spins along different directions are colored differently; note that 1-q, 2-q, and 3-q structures have [100], ⟨110⟩, and ⟨111⟩
+spin directions, respectively. (b) Elastic scattering data (−1.3 ≤ E ≤ 1.3 meV) measured at T = 5 K with Ei = 11.8 meV for Ba2YRuO6 and
+Ba2LuRuO6 (black circles), Rietveld refinements (red lines), and data – fit (blue lines). Tick marks show (top to bottom): nuclear, impurity
+M2O3, and magnetic phases. The mX+
+3 irrep (left) does not reproduce our data, whereas the mX+
+5 irrep (right) agrees well with our data.
+(multi-q) structures associated with Type I antiferromag-
+netism. A remarkable property of the FCC lattice is that 1-q,
+2-q, and 3-q structures are energetically degenerate for all bi-
+linear interactions for which Type I ordering is stable [39, 40].
+Moreover, uniaxial anisotropy (∼S2
+z) and antisymmetric ex-
+change terms are forbidden by Fm¯3m symmetries, and quartic
+anisotropy (∼S4
+x +S4
+y +S4
+z) is forbidden for S = 3/2 operators
+in a cubic environment. Consequently, interactions that would
+usually favor collinear magnetic structures are inactive in the
+S = 3/2 FCC antiferromagnet. This remarkable property po-
+tentially allows noncollinear structures to appear.
+To identify candidate systems for 3-q spin textures among
+the diverse magnetic ground states of double perovskites [42–
+50], we consider two criteria: Type I antiferromagnetic or-
+dering, and strictly cubic symmetry below the magnetic or-
+dering temperature, TN. The second criterion is key because
+3-q structures have cubic symmetry, while 1-q and 2-q struc-
+tures have tetragonal or orthorhombic symmetry that could
+drive a crystallographic distortion via spin-lattice coupling
+[Figure 1(a)].
+We investigate Ba2YRuO6 and Ba2LuRuO6
+because they are chemically well-ordered and show no evi-
+dence for low-temperature deviations from cubic symmetry
+[34, 36]. Moreover, recent first-principles calculations predict
+that their magnetic structures might not be collinear [51], in
+apparent contradiction with interpretations of previous exper-
+iments [34].
+We prepared ∼8 g polycrystalline samples of Ba2YRuO6
+and Ba2LuRuO6 by solid-state reaction [52].
+Rietveld re-
+finement revealed stoichiometric samples with minor Lu2O3
+(1.94 wt.%) or Y2O3 (0.65 wt.%) impurities. The magnetic
+ordering temperature TN ≈ 37 K is the same for both sam-
+ples, and is suppressed compared to the Weiss temperature
+θ ∼ −500 K, indicating strong magnetic frustration [36]. We
+performed inelastic neutron-scattering measurements on the
+SEQUOIA instrument at ORNL [53] using incident neutron
+energies Ei = 62 and 11.8 meV, yielding elastic energy reso-
+lutions δ ins ≈ 1.68 and 0.27 meV, respectively.
+Figure 1(b) shows magnetic Rietveld refinements to our
+elastic neutron-scattering data, measured with Ei = 11.8 meV
+at T ≈ 5 K. Applying the q = [1,0,0] propagation vector to
+Fm¯3m crystal symmetry generates two magnetic irreducible
+representations (irreps), notated mX+
+3 and mX+
+5 [54, 55].
+These irreps can be distinguished by their magnetic Bragg
+profiles. The mX+
+5 irrep agrees well with our elastic-scattering
+data for both materials; we obtain ordered magnetic moment
+lengths of 2.56(2) and 2.43(2) µB per Ru for Ba2YRuO6 and
+Ba2LuRuO6, respectively, from Rietveld refinement. Since
+the magnetic form factor for Ru5+ is not known, we tested
+several 4d magnetic form factors [56]; while this choice does
+not qualitatively affect our results, the form factor for Zr+
+(isoelectronic with Ru5+) yields optimal agreement with our
+data and is used throughout. In contrast to the mX+
+5 irrep, the
+mX+
+3 irrep strongly disagrees with our data, as it yields zero
+intensity for the strong (100) magnetic Bragg peak. This can
+be understood intuitively for a collinear 1-q structure, because
+neutrons are only sensitive to spin components perpendicular
+to the scattering wavevector, and the mX3+ irrep has S ∥ q
+while the mX5+ irrep has S ⊥ q [Figure 1(a)]. A previous elas-
+
+3
+Figure 2. Broadband inelastic neutron-scattering data (Ei = 62 meV)
+measured at T = 5 K for Ba2YRuO6 (upper panels) and Ba2LuRuO6
+(lower panels), showing (a) intensity as a color plot, and (b) energy
+dependence integrated over 4.0 ≤ Q ≤ 4.5 Å−1, where experimental
+data are shown as black circles, and Gaussian fits to the ∼14 meV
+phonon band as red lines.
+tic neutron-scattering study of Ba2YRuO6 and Ba2LuRuO6
+considered only collinear 1-q structures [34], but could not
+rule out multi-q structures, due to the multi-q problem.
+To overcome the multi-q problem, we consider the en-
+ergy dependence of our neutron-scattering data [57].
+Fig-
+ure 2(a) shows our inelastic data measured with Ei = 62 meV
+at T ≈ 5 K. A structured inelastic signal appears at T < TN
+for small scattering wavevectors, Q ≲ 2 Å−1, which we iden-
+tify as magnon scattering.
+The top of the magnetic band
+overlaps with an intense phonon signal for Q ≳ 2 Å−1. Fig-
+ure 2(b) shows the scattering intensity integrated over 4.0 ≤
+Q ≤ 4.5 Å−1, from which we extract the average energy Eph
+and width σph of this phonon band via Gaussian fits for each
+material. The energy overlap of magnon and phonon modes
+suggests that spin-lattice coupling may be significant, which
+we consider further below.
+Our starting point for modeling the magnetic scattering is
+the Heisenberg-Kitaev model, Eq. (1). For all models, we re-
+quire J > 0 and K > 0 to stabilize mX+
+5 ordering. We con-
+sider three additional interactions in turn. First, the symmet-
+ric off-diagonal interaction HΓ = Γ∑⟨i, j⟩γ
+�
+Sα
+i Sβ
+j +Sβ
+i Sα
+j
+�
+is
+the only additional bilinear nearest-neighbor interaction al-
+lowed by symmetry.
+Second, the Heisenberg next-nearest
+neighbor interaction H2 = J2 ∑⟨⟨i, j⟩⟩ Si · Sj has been invoked
+for Ba2YRuO6 [37]; we require J2 ≤ 0 to stabilize Type
+I ordering.
+Third, the nearest-neighbor biquadratic cou-
+pling Hbq = Jbq ∑⟨i, j⟩(Si · Sj)2 has been invoked in density-
+functional-theory calculations for 4d double perovskites due
+to their increased electron hopping relative to 3d analogs [51].
+For Jbq = 0, the classical energy of 1-q, 2-q, and 3-q struc-
+tures is equal for all K, Γ, and J2 that stabilize Type I or-
+dering. Nonzero Jbq removes this degeneracy, and stabilizes
+����
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+�
+�
+��
+��
+��
+����
+����
+����
+����
+����
+����
+����
+����
+��
+�
+��
+��
+��
+Rwp (%)
+1-q
+2-q
+3-q
+(a)
+(b)
+Jbq
+K
+3-q
+mX5
++
+1-q
+mX5
++
+3-q
+mX3
++
+1-q
+mX3
++
+Figure 3. (a) Schematic phase diagram showing the magnetic ground
+states of the J-K-Jbq model. (b) Goodness-of-fit metric Rwp for can-
+didate magnetic structures and interaction models of Ba2YRuO6 (up-
+per graph) and Ba2LuRuO6 (lower graph). The graphs show Rwp for
+refinements of the Heisenberg-Kitaev (J-K) model including a third
+refined parameter Γ (red bars), J2 (blue bars), or Jbq (green bars);
+note that the 2-q structure is stable only for Jbq = 0.
+J (K)
+K (K)
+Jbq (K) A (meV)
+Ba2YRuO6 21.85(3) 0.39(1) 1.32(2) 0.97(3)
+Ba2LuRuO6 22.27(4) 0.36(2) 1.17(3) 2.25(5)
+Table I. Refined values of magnetic interaction parameters for the J-
+K-Jbq model and 3-q structure. Uncertainties indicate 1σ statistical
+confidence intervals.
+1-q ordering for Jbq < 0 and 3-q ordering for Jbq > 0 [Fig-
+ure 3(a)]. Importantly, since single-ion anisotropies are for-
+bidden for S = 3/2 in a cubic environment, biquadratic ex-
+change is the only physically-plausible mechanism that can
+remove the degeneracy of 1-q and 3-q structures.
+We performed extensive fits to our inelastic neutron-
+scattering data to optimize the magnetic structure and inter-
+actions simultaneously. For each structure associated with the
+mX+
+5 irrep (1-q, 2-q, or 3-q), we optimized three spin Hamil-
+tonian parameters (J, K, and either Γ, J2, or Jbq) against the
+broadband inelastic data shown in Figure 4(a) and the energy
+dependence near the (100) magnetic Bragg position shown
+in Figure 4(b). The powder-averaged magnon spectrum was
+calculated within a renormalized linear spin-wave theory [58]
+using the SpinW program [59]. The renormalization factor,
+which takes into account higher-order corrections in the 1/S
+expansion, is strictly necessary to extract a correct value of
+Jbq, since the unrenormalized spin-wave theory would lead to
+a value of Jbq that is 2.25 times smaller than the correct value
+[60]. The parameter values were optimized to minimize the
+sum of squared residuals using nonlinear least-squares refine-
+ment [52]. We calculated the energy-dependent broadening of
+the magnon spectrum as δ(E) = δins(E) + Ae−(E−Eph)2/2δ 2
+ph,
+where δ(E) is the overall Gaussian energy width, δins(E) is
+the instrumental resolution, and A is a refined parameter that
+phenomenologically accounts for magnon broadening due to
+coupling with phonons at E ∼ Eph.
+Figure 4(a) compares our broadband inelastic data (Ei =
+62 meV) with the best fit for each of the 1-q, 2-q, and 3-q
+
+4
+Figure 4. (a) Broadband inelastic neutron-scattering data (Ei = 62 meV) and optimal spin-wave fits for different magnetic structures, showing
+(left to right) experimental data, 1-q fit, 2-q fit, and 3-q fit. (b) Low-energy inelastic neutron-scattering data (Ei = 11.8 meV) and 3-q model
+calculations, showing (left to right) a cut at Q = 0.7450±0.0175 Å−1 comparing experimental data (black circles) and spin-wave fit (red lines),
+experimental data as a Q-E slice, and spin-wave calculation.
+structures. The data show two V-shaped features centered at
+≈ 0.85 and ≈ 1.70 Å−1, with a sharp cutoff of magnetic signal
+for energies above ∼14 meV. For both materials, these char-
+acteristics are best reproduced by the 3-q structure, while the
+1-q structure disagrees with our experimental data. These ob-
+servations are confirmed by the goodness-of-fit metric Rwp,
+shown in Figure 3(b). For both materials and for every in-
+teraction model we considered, the 3-q structure yields better
+agreement with our data than the 1-q or 2-q structures. No-
+tably, the goodness-of-fit is more sensitive to the structure than
+the precise magnetic interactions; indeed, the main differences
+between 1-q and 3-q spectra are apparent for Heisenberg ex-
+change only [52]. The global best fit is for the 3-q structure
+and J, K, and Jbq interactions with the refined values given in
+Table I. The refined values of A indicate significant magnon
+broadening, which is larger for Ba2LuRuO6 and is likely due
+to magnon-phonon coupling. Importantly, for both materi-
+als, the biquadratic term is significant, with Jbq/J ∼ 0.06.
+Hence, our key results are that only the 3-q spin texture agrees
+well with our neutron data, and this state is stabilized by bi-
+quadratic interactions in Ba2YRuO6 and Ba2LuRuO6.
+Our model provides insight into the mechanism of gap
+opening [35]. Figure 4(b) compares our low-energy inelastic
+data (Ei = 11.8 meV) with the 3-q magnon spectrum for the
+optimal J-K-Jbq model [Table I]. This calculation reproduces
+the observed ≈ 2.8 meV gap, unlike the J-K-J2 model that
+yields the next-best Rwp [52]. Since single-ion anisotropies
+are forbidden here, the mechanism of gap opening is subtle.
+If K = 0, there is no gap, because the energy of the Heisenberg
+and biquadratic terms is unchanged by global spin rotations.
+For K > 0, whether a gap opens depends on both structure
+and interactions. If the structure is 1-q with Jbq < 0, the clas-
+sical energy is unchanged by global spin rotations in the plane
+perpendicular to q. In this case, there is no gap at the linear
+spin-wave level; a gap is generated only by magnon interac-
+tions in the quantum (S = 1/2) limit [61]. By contrast, if the
+structure is 3-q with Jbq > 0, a gap is present at the linear spin-
+wave level, because Jbq > 0 and K > 0 together favor ⟨111⟩
+spin alignment. Since Ba2YRuO6 and Ba2LuRuO6 are not in
+the quantum limit, the experimental observation of a gap sup-
+ports the presence of biquadratic and Kitaev interactions in a
+3-q structure.
+We have shown that the magnetic ground states of
+Ba2YRuO6 and Ba2LuRuO6 are noncoplanar 3-q structures
+stabilized by biquadratic interactions. Macroscopic topolog-
+ical physical responses may be generated synthesizing thin
+films of these materials with [111] strain [62]. Our exper-
+imental results strikingly confirm recent first-principles pre-
+dictions [51]. The positive sign of Jbq suggests that the ef-
+fect of inter-site electron hopping outweighs spin-lattice cou-
+pling, since the latter would give a negative contribution to
+Jbq [11, 12]. Crucially, we quantify the magnetic interactions
+that stabilize the noncoplanar state, in contrast to other pro-
+posed 3-q structures in NiS2 [63–65], MnTe2 [66], and UO2
+[67–70], where the relevant interactions are not yet well un-
+derstood. Our work provides several guiding principles to fa-
+cilitate the identification of multi-q spin textures. First, the
+near-degeneracy of 1-q and multi-q structures on the FCC lat-
+tice makes double perovskites enticing systems. In candidate
+materials, the crystal symmetry should be higher than a 1-q
+model would imply. Second, magnets that are not deep in-
+side the Mott-insulating regime are expected to have larger
+Jbq and, consequently, more robust 3-q orderings. This cri-
+terion hints that cubic Ba2YOsO6 [71, 72] may also host a
+3-q state, due to its extended Os 5d orbitals, potentially offer-
+ing a route to investigate the effect of increased electron hop-
+ping. For small Jbq, we anticipate a thermally-induced tran-
+sition from 3-q to 1-q ordering, since thermal fluctuations fa-
+
+5
+vor collinear states. Third, quartic single-ion anisotropy may
+play a role in FCC magnets with S > 3/2; in particular, easy-
+⟨111⟩ axis anisotropy should favor 3-q ordering. Finally, our
+key methodological insight is that refining the magnetic struc-
+ture and interactions simultaneously enables 1-q and multi-q
+structures to be distinguished on the FCC lattice, even when
+single-crystal samples are not available.
+This work was supported by the U.S. Department of En-
+ergy, Office of Science, Basic Energy Sciences, Materials Sci-
+ences and Engineering Division. This research used resources
+at the Spallation Neutron Source, a DOE Office of Science
+User Facility operated by the Oak Ridge National Laboratory.
+∗ paddisonja@ornl.gov
+† christiansad@ornl.gov
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+
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new file mode 100644
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+page_content='Cubic Double Perovskites Host Noncoplanar Spin Textures  Joseph A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Paddison,1, ∗ Hao Zhang,2 Jiaqiang Yan,1 Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Cliffe,3 Seung-Hwan Do,1 Shang Gao,1, 4  Matthew B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Stone,4 David Dahlbom,2 Kipton Barros,5 Cristian D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Batista,6 and Andrew D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Christianson1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' †  1Materials Science and Technology Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Oak Ridge National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Oak Ridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' TN 37831,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' USA  2Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' University of Tennessee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Knoxville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Tennessee 37996,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' USA  3School of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' University of Nottingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Nottingham NG7 2RD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' UK  4Neutron Scattering Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Oak Ridge National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Oak Ridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Tennessee 37831,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' USA  5Theoretical Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Los Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Los Alamos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' New Mexico 87545,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' USA  6Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The University of Tennessee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Knoxville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Tennessee 37996,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' USA  Magnetic materials with noncoplanar magnetic structures can show unusual physical properties driven by  nontrivial topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Topologically-active states are often multi-q structures, which are challenging to stabilize  in models and to identify in materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Here, we use inelastic neutron-scattering experiments to show that the  insulating double perovskites Ba2YRuO6 and Ba2LuRuO6 host a noncoplanar 3-q structure on the face-centered  cubic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Quantitative analysis of our neutron-scattering data reveals that these 3-q states are stabilized by  biquadratic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Our study identifies double perovskites as a highly promising class of materials to  realize topological magnetism, elucidates the stabilization mechanism of the 3-q state in these materials, and  establishes neutron spectroscopy on powder samples as a valuable technique to distinguish multi-q from single-q  states, facilitating the discovery of topologically-nontrivial magnetic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Most magnetic materials order with simple magnetic struc-  tures in which spins are collinear or coplanar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Noncopla-  nar magnetic structures are relatively rare, but are of great  current interest, because they can exhibit topological charac-  ter and exotic physical properties [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' For example, the  finite scalar spin chirality of noncoplanar spin textures can  generate a topological magneto-optical effect [3] and anoma-  lous quantum Hall effect [4, 5], even in the absence of spin-  orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Topologically-nontrivial spin textures are typ-  ically multi-q structures, which superpose magnetic modula-  tions with symmetry-related wavevectors q [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Multi-q spin  textures with long-wavelength modulations, such as skyrmion  and hedgehog crystals, are well-studied as hosts of topology-  driven phenomena [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' In this context, multi-q antiferro-  magnets are increasingly important [9], because they offer  higher densities of topological objects with the potential to  generate stronger physical responses [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  To probe the relationships between spin structure, interac-  tions, topology, and physical response, it is crucial to identify  real materials that host noncoplanar spin textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' This has  proved a challenging task, for three main reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' First, it  is necessary to identify noncoplanar spin textures that are ro-  bust to subleading effects such as magnetic anisotropies, spin-  lattice coupling [11, 12], fluctuations [13–16], and anisotropic  interactions [17], which usually favor collinear states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Sec-  ond, most noncoplanar states are found in metals, such as  USb [18, 19] and γ-Mn alloys [20–25], and are often stable  only under an applied magnetic field [6, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  On the one  hand, itinerant electrons can support the generation of phys-  ical responses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' on the other hand, modeling the magnetic in-  teractions of metals presents fundamental challenges [27–32],  such that insulators are often more suitable as model materi-  als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Third, powder neutron-diffraction measurements play a  central role in solving magnetic structures, but suffer from a  “multi-q problem”: Such measurements are generally unable  to distinguish 1-q from multi-q structures [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Therefore,  multi-q spin textures are challenging to stabilize in models,  and to identify in real materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Here, we identify the Mott-insulating double perovskites  Ba2YRuO6 and Ba2LuRuO6 [34–37] as prototypical exam-  ples of noncoplanar 3-q magnetism on the face-centered cu-  bic (FCC) lattice in zero magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' We obtain evidence  for 3-q magnetism from a spin-wave analysis of neutron spec-  troscopy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' By optimizing the magnetic structure and inter-  actions simultaneously against our data, we show that the 3-  q structure is stabilized by biquadratic interactions within an  antiferromagnetic Heisenberg-Kitaev model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Our study ex-  perimentally establishes that noncoplanar multi-q states are  stabilized in frustrated FCC antiferromagnets, identifies cubic  double perovskites as model materials to realize this behavior,  and identifies guiding principles to facilitate design of materi-  als with noncoplanar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Our study is motivated by theoretical results for the  FCC antiferromagnet [13, 38–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  The nearest-neighbor  Heisenberg-Kitaev spin Hamiltonian on the FCC lattice can  be written as  H = J ∑  ⟨i, j⟩  Si ·Sj +K ∑  ⟨i, j⟩γ  Sγ  i Sγ  j,  (1)  where Si is a Ru5+ spin with quantum number S = 3/2, J  and K denote the Heisenberg and Kitaev interactions, respec-  tively, and γ ∈ {x,y,z} is perpendicular to the cubic plane con-  taining the bond between neighbors ⟨i, j⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' For antiferromag-  netic J > 0 only, the model is frustrated, and orderings with  q ∈ [1,q,0] are degenerate [13, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The degenerate mani-  fold includes q = [1,0,0] (“Type I”) ordering, which is favored  by fluctuations [13, 14, 41] and is observed in Ba2YRuO6  and Ba2LuRuO6 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Henceforth, we therefore restrict our  discussion to q = [1,0,0] ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' For a collinear structure,  spins may be either parallel or perpendicular to q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' the former  is favored by K < 0 and the latter by K > 0 [38–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Figure 1(a) shows the collinear (1-q) and noncollinear  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='11395v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='str-el]  26 Jan 2023  2  1-q  tetragonal  2-q  tetragonal  3-q  cubic  3-q  cubic  2-q  tetragonal  1-q  orthorhombic  (a)  ��������  ����������������  �������  �  ���  �  ���  �  ���  ���������  �������  �  ���  �  ���  �  ���  ��������  �������  �  ���  �  ���  �  ���  ���������  �������  �  ���  �  ���  �  ���  K ≥ 0, mX5  +  K ≤ 0, mX3  +  (b)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  (a) Symmetry-allowed magnetic structures with propagation vector q = [1,0,0] on the FCC lattice for Ba2MRuO6 (space group  Fm¯3m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' a = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='29 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='24 Å for M = Y and Lu, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The 1-q, 2-q, and 3-q structures are shown for the mX+  3 irrep (left) and the  mX+  5 irrep (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Spins along different directions are colored differently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' note that 1-q, 2-q, and 3-q structures have [100], ⟨110⟩, and ⟨111⟩  spin directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' (b) Elastic scattering data (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='3 ≤ E ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='3 meV) measured at T = 5 K with Ei = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='8 meV for Ba2YRuO6 and  Ba2LuRuO6 (black circles), Rietveld refinements (red lines), and data – fit (blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Tick marks show (top to bottom): nuclear, impurity  M2O3, and magnetic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The mX+  3 irrep (left) does not reproduce our data, whereas the mX+  5 irrep (right) agrees well with our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  (multi-q) structures associated with Type I antiferromag-  netism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' A remarkable property of the FCC lattice is that 1-q,  2-q, and 3-q structures are energetically degenerate for all bi-  linear interactions for which Type I ordering is stable [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Moreover, uniaxial anisotropy (∼S2  z) and antisymmetric ex-  change terms are forbidden by Fm¯3m symmetries, and quartic  anisotropy (∼S4  x +S4  y +S4  z) is forbidden for S = 3/2 operators  in a cubic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Consequently, interactions that would  usually favor collinear magnetic structures are inactive in the  S = 3/2 FCC antiferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' This remarkable property po-  tentially allows noncollinear structures to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  To identify candidate systems for 3-q spin textures among  the diverse magnetic ground states of double perovskites [42–  50], we consider two criteria: Type I antiferromagnetic or-  dering, and strictly cubic symmetry below the magnetic or-  dering temperature, TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The second criterion is key because  3-q structures have cubic symmetry, while 1-q and 2-q struc-  tures have tetragonal or orthorhombic symmetry that could  drive a crystallographic distortion via spin-lattice coupling  [Figure 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  We investigate Ba2YRuO6 and Ba2LuRuO6  because they are chemically well-ordered and show no evi-  dence for low-temperature deviations from cubic symmetry  [34, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Moreover, recent first-principles calculations predict  that their magnetic structures might not be collinear [51], in  apparent contradiction with interpretations of previous exper-  iments [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  We prepared ∼8 g polycrystalline samples of Ba2YRuO6  and Ba2LuRuO6 by solid-state reaction [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Rietveld re-  finement revealed stoichiometric samples with minor Lu2O3  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='94 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='%) or Y2O3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='65 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='%) impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The magnetic  ordering temperature TN ≈ 37 K is the same for both sam-  ples, and is suppressed compared to the Weiss temperature  θ ∼ −500 K, indicating strong magnetic frustration [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' We  performed inelastic neutron-scattering measurements on the  SEQUOIA instrument at ORNL [53] using incident neutron  energies Ei = 62 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='8 meV, yielding elastic energy reso-  lutions δ ins ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='68 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='27 meV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Figure 1(b) shows magnetic Rietveld refinements to our  elastic neutron-scattering data, measured with Ei = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='8 meV  at T ≈ 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Applying the q = [1,0,0] propagation vector to  Fm¯3m crystal symmetry generates two magnetic irreducible  representations (irreps), notated mX+  3 and mX+  5 [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  These irreps can be distinguished by their magnetic Bragg  profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The mX+  5 irrep agrees well with our elastic-scattering  data for both materials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' we obtain ordered magnetic moment  lengths of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='56(2) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='43(2) µB per Ru for Ba2YRuO6 and  Ba2LuRuO6, respectively, from Rietveld refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Since  the magnetic form factor for Ru5+ is not known, we tested  several 4d magnetic form factors [56];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' while this choice does  not qualitatively affect our results, the form factor for Zr+  (isoelectronic with Ru5+) yields optimal agreement with our  data and is used throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' In contrast to the mX+  5 irrep, the  mX+  3 irrep strongly disagrees with our data, as it yields zero  intensity for the strong (100) magnetic Bragg peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' This can  be understood intuitively for a collinear 1-q structure, because  neutrons are only sensitive to spin components perpendicular  to the scattering wavevector, and the mX3+ irrep has S ∥ q  while the mX5+ irrep has S ⊥ q [Figure 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' A previous elas-  3  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Broadband inelastic neutron-scattering data (Ei = 62 meV)  measured at T = 5 K for Ba2YRuO6 (upper panels) and Ba2LuRuO6  (lower panels), showing (a) intensity as a color plot, and (b) energy  dependence integrated over 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='0 ≤ Q ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='5 Å−1, where experimental  data are shown as black circles, and Gaussian fits to the ∼14 meV  phonon band as red lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  tic neutron-scattering study of Ba2YRuO6 and Ba2LuRuO6  considered only collinear 1-q structures [34], but could not  rule out multi-q structures, due to the multi-q problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  To overcome the multi-q problem, we consider the en-  ergy dependence of our neutron-scattering data [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Fig-  ure 2(a) shows our inelastic data measured with Ei = 62 meV  at T ≈ 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' A structured inelastic signal appears at T < TN  for small scattering wavevectors, Q ≲ 2 Å−1, which we iden-  tify as magnon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  The top of the magnetic band  overlaps with an intense phonon signal for Q ≳ 2 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Fig-  ure 2(b) shows the scattering intensity integrated over 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='0 ≤  Q ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='5 Å−1, from which we extract the average energy Eph  and width σph of this phonon band via Gaussian fits for each  material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The energy overlap of magnon and phonon modes  suggests that spin-lattice coupling may be significant, which  we consider further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Our starting point for modeling the magnetic scattering is  the Heisenberg-Kitaev model, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' For all models, we re-  quire J > 0 and K > 0 to stabilize mX+  5 ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' We con-  sider three additional interactions in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' First, the symmet-  ric off-diagonal interaction HΓ = Γ∑⟨i, j⟩γ  �  Sα  i Sβ  j +Sβ  i Sα  j  �  is  the only additional bilinear nearest-neighbor interaction al-  lowed by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Second, the Heisenberg next-nearest  neighbor interaction H2 = J2 ∑⟨⟨i, j⟩⟩ Si · Sj has been invoked  for Ba2YRuO6 [37];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' we require J2 ≤ 0 to stabilize Type  I ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Third, the nearest-neighbor biquadratic cou-  pling Hbq = Jbq ∑⟨i, j⟩(Si · Sj)2 has been invoked in density-  functional-theory calculations for 4d double perovskites due  to their increased electron hopping relative to 3d analogs [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  For Jbq = 0, the classical energy of 1-q, 2-q, and 3-q struc-  tures is equal for all K, Γ, and J2 that stabilize Type I or-  dering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Nonzero Jbq removes this degeneracy, and stabilizes  ����  ����  ����  ����  ����  ����  ����  ����  �  �  ��  ��  ��  ����  ����  ����  ����  ����  ����  ����  ����  ��  �  ��  ��  ��  Rwp (%)  1-q  2-q  3-q  (a)  (b)  Jbq  K  3-q  mX5  +  1-q  mX5  +  3-q  mX3  +  1-q  mX3  +  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' (a) Schematic phase diagram showing the magnetic ground  states of the J-K-Jbq model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' (b) Goodness-of-fit metric Rwp for can-  didate magnetic structures and interaction models of Ba2YRuO6 (up-  per graph) and Ba2LuRuO6 (lower graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The graphs show Rwp for  refinements of the Heisenberg-Kitaev (J-K) model including a third  refined parameter Γ (red bars), J2 (blue bars), or Jbq (green bars);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  note that the 2-q structure is stable only for Jbq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  J (K)  K (K)  Jbq (K) A (meV)  Ba2YRuO6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='85(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='39(1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='32(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='97(3)  Ba2LuRuO6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='27(4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='36(2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='17(3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='25(5)  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Refined values of magnetic interaction parameters for the J-  K-Jbq model and 3-q structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Uncertainties indicate 1σ statistical  confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  1-q ordering for Jbq < 0 and 3-q ordering for Jbq > 0 [Fig-  ure 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Importantly, since single-ion anisotropies are for-  bidden for S = 3/2 in a cubic environment, biquadratic ex-  change is the only physically-plausible mechanism that can  remove the degeneracy of 1-q and 3-q structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  We performed extensive fits to our inelastic neutron-  scattering data to optimize the magnetic structure and inter-  actions simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' For each structure associated with the  mX+  5 irrep (1-q, 2-q, or 3-q), we optimized three spin Hamil-  tonian parameters (J, K, and either Γ, J2, or Jbq) against the  broadband inelastic data shown in Figure 4(a) and the energy  dependence near the (100) magnetic Bragg position shown  in Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The powder-averaged magnon spectrum was  calculated within a renormalized linear spin-wave theory [58]  using the SpinW program [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The renormalization factor,  which takes into account higher-order corrections in the 1/S  expansion, is strictly necessary to extract a correct value of  Jbq, since the unrenormalized spin-wave theory would lead to  a value of Jbq that is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='25 times smaller than the correct value  [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The parameter values were optimized to minimize the  sum of squared residuals using nonlinear least-squares refine-  ment [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' We calculated the energy-dependent broadening of  the magnon spectrum as δ(E) = δins(E) + Ae−(E−Eph)2/2δ 2  ph,  where δ(E) is the overall Gaussian energy width, δins(E) is  the instrumental resolution, and A is a refined parameter that  phenomenologically accounts for magnon broadening due to  coupling with phonons at E ∼ Eph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Figure 4(a) compares our broadband inelastic data (Ei =  62 meV) with the best fit for each of the 1-q, 2-q, and 3-q  4  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' (a) Broadband inelastic neutron-scattering data (Ei = 62 meV) and optimal spin-wave fits for different magnetic structures, showing  (left to right) experimental data, 1-q fit, 2-q fit, and 3-q fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' (b) Low-energy inelastic neutron-scattering data (Ei = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='8 meV) and 3-q model  calculations, showing (left to right) a cut at Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='7450±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='0175 Å−1 comparing experimental data (black circles) and spin-wave fit (red lines),  experimental data as a Q-E slice, and spin-wave calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The data show two V-shaped features centered at  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='85 and ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='70 Å−1, with a sharp cutoff of magnetic signal  for energies above ∼14 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' For both materials, these char-  acteristics are best reproduced by the 3-q structure, while the  1-q structure disagrees with our experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' These ob-  servations are confirmed by the goodness-of-fit metric Rwp,  shown in Figure 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' For both materials and for every in-  teraction model we considered, the 3-q structure yields better  agreement with our data than the 1-q or 2-q structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' No-  tably, the goodness-of-fit is more sensitive to the structure than  the precise magnetic interactions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' indeed, the main differences  between 1-q and 3-q spectra are apparent for Heisenberg ex-  change only [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The global best fit is for the 3-q structure  and J, K, and Jbq interactions with the refined values given in  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The refined values of A indicate significant magnon  broadening, which is larger for Ba2LuRuO6 and is likely due  to magnon-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Importantly, for both materi-  als, the biquadratic term is significant, with Jbq/J ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Hence, our key results are that only the 3-q spin texture agrees  well with our neutron data, and this state is stabilized by bi-  quadratic interactions in Ba2YRuO6 and Ba2LuRuO6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  Our model provides insight into the mechanism of gap  opening [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Figure 4(b) compares our low-energy inelastic  data (Ei = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='8 meV) with the 3-q magnon spectrum for the  optimal J-K-Jbq model [Table I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' This calculation reproduces  the observed ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='8 meV gap, unlike the J-K-J2 model that  yields the next-best Rwp [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Since single-ion anisotropies  are forbidden here, the mechanism of gap opening is subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  If K = 0, there is no gap, because the energy of the Heisenberg  and biquadratic terms is unchanged by global spin rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  For K > 0, whether a gap opens depends on both structure  and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' If the structure is 1-q with Jbq < 0, the clas-  sical energy is unchanged by global spin rotations in the plane  perpendicular to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' In this case, there is no gap at the linear  spin-wave level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' a gap is generated only by magnon interac-  tions in the quantum (S = 1/2) limit [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' By contrast, if the  structure is 3-q with Jbq > 0, a gap is present at the linear spin-  wave level, because Jbq > 0 and K > 0 together favor ⟨111⟩  spin alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Since Ba2YRuO6 and Ba2LuRuO6 are not in  the quantum limit, the experimental observation of a gap sup-  ports the presence of biquadratic and Kitaev interactions in a  3-q structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  We have shown that the magnetic ground states of  Ba2YRuO6 and Ba2LuRuO6 are noncoplanar 3-q structures  stabilized by biquadratic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Macroscopic topolog-  ical physical responses may be generated synthesizing thin  films of these materials with [111] strain [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Our exper-  imental results strikingly confirm recent first-principles pre-  dictions [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' The positive sign of Jbq suggests that the ef-  fect of inter-site electron hopping outweighs spin-lattice cou-  pling, since the latter would give a negative contribution to  Jbq [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Crucially, we quantify the magnetic interactions  that stabilize the noncoplanar state, in contrast to other pro-  posed 3-q structures in NiS2 [63–65], MnTe2 [66], and UO2  [67–70], where the relevant interactions are not yet well un-  derstood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Our work provides several guiding principles to fa-  cilitate the identification of multi-q spin textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' First, the  near-degeneracy of 1-q and multi-q structures on the FCC lat-  tice makes double perovskites enticing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' In candidate  materials, the crystal symmetry should be higher than a 1-q  model would imply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Second, magnets that are not deep in-  side the Mott-insulating regime are expected to have larger  Jbq and, consequently, more robust 3-q orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' This cri-  terion hints that cubic Ba2YOsO6 [71, 72] may also host a  3-q state, due to its extended Os 5d orbitals, potentially offer-  ing a route to investigate the effect of increased electron hop-  ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' For small Jbq, we anticipate a thermally-induced tran-  sition from 3-q to 1-q ordering, since thermal fluctuations fa-  5  vor collinear states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Third, quartic single-ion anisotropy may  play a role in FCC magnets with S > 3/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' in particular, easy-  ⟨111⟩ axis anisotropy should favor 3-q ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Finally, our  key methodological insight is that refining the magnetic struc-  ture and interactions simultaneously enables 1-q and multi-q  structures to be distinguished on the FCC lattice, even when  single-crystal samples are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  This work was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Department of En-  ergy, Office of Science, Basic Energy Sciences, Materials Sci-  ences and Engineering Division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' This research used resources  at the Spallation Neutron Source, a DOE Office of Science  User Facility operated by the Oak Ridge National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='  ∗ paddisonja@ornl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content='gov  † christiansad@ornl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
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+page_content=' Shannon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Shiba, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
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+page_content=' Vishwanath, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
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+page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
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+page_content=' Zhitomirsky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFIT4oBgHgl3EQf6yuB/content/2301.11395v1.pdf'}
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+arXiv:2301.04867v1  [physics.chem-ph]  12 Jan 2023
+Model for vibrationally enhanced tunneling of
+proton transfer in hydrogen bond
+A.E. Sitnitsky,
+Kazan Institute of Biochemistry and Biophysics, FRC Kazan Scientific Center of
+RAS, P.O.B. 30, 420111, Russian Federation. e-mail: sitnitsky@kibb.knc.ru
+Abstract
+Theoretical analysis of the effect of an external vibration on proton transfer (PT) in
+a hydrogen bond (HB) is carried out. It is based on the two-dimensional Schr¨odinger
+equation with trigonometric double-well potential. Its solution obtained within the
+framework of the standard adiabatic approximation is available. An analytic for-
+mula is derived that provides the calculation of PT rate with the help of elements
+implemented in Mathematica. We exemplify the general theory by calculating PT
+rate constant for the intermolecular HB in the Zundel ion H5O+
+2 (oxonium hydrate).
+This object enables one to explore a wide range of the HB lengths. Below some crit-
+ical value of the frequency of the external vibration the calculated PT rate yields
+extremely rich resonant behavior (multiple manifestations of bell-shaped peaks). It
+takes place at symmetric coupling of the external vibration to the proton coordinate.
+This phenomenon is absent for anti-symmetric and squeezed mode couplings.
+Key words: Schr¨odinger equation, double-well potential, quantum tunneling,
+spheroidal function, Zundel ion.
+1
+Introduction
+Proton transfer (PT) in hydrogen bonds (HB) is one of the main processes
+in the reaction rate theory. It takes place in the most important biological
+molecules such as proteins (participating in some enzymatic reactions) and
+DNA (arguably participating in the occurrence of mutations). In particular
+the phenomenon of vibrationally enhanced (or assisted or promoted) tunnel-
+ing at PT (i.e., resonant acceleration of the process by a coupled oscillation
+in some frequency range) [1], [2] is of great interest especially in regard of its
+Email address: sitnitsky@kibb.knc.ru ( A.E. Sitnitsky).
+Preprint submitted to Chemical Physics Letters
+13 January 2023
+
+possible role in a mechanism for enzymatic hydrogen transfer [3-6]. Within
+the context of enzyme catalysis it is a specific case of the more general trend
+named ”rate-promoting vibration” [6-11]. There are several cases in which a
+vibration can be coupled to the proton coordinate in HB. First of all there
+is the heavy atoms stretching mode which is an intrinsic degree of freedom
+in HB. It is thoroughly studied theoretically since the pioneer articles [12],
+[13], [1] dealing with vibrationally promoted PT in solids. Unfortunately it
+is an internal vibration and its fixed frequency is not an experimentally con-
+trollable parameter. Then there are external vibrations exerted on HB and
+provided either by protein scaffold for HB in enzymes or by some means from
+the researcher’s toolkit for HB in model compounds. One of the most effi-
+cient ways for such purpose is the usage of the IR electromagnetic field of an
+optical cavity. The phenomenon of resonant activation (or, in contrast, sup-
+pression) of reaction rates is widely discussed for modifying chemical kinetics
+by optical cavities (for recent articles in this field which is sometimes called
+vibrational polariton chemistry see, e.g., [14-16] and refs. therein). The reso-
+nance, i.e., maximal cavity induced enhancement of the reaction rate under
+the vibrational resonance condition is produced in this case by mixing the
+electromagnetic field with quantum states of molecular systems. The cavity is
+equivalent to a harmonic oscillator of a given frequency coupled to the molec-
+ular system. The Hamiltonian of the molecule degree of freedom coupled to
+the field oscillator in the electric dipole approximation of light-matter interac-
+tion has the same structure as those used for PT coupled to the heavy atoms
+stretching mode in HB. In this regard constructing reliable theoretical models
+of PT which take into account the possibility of varying the frequency of the
+external vibration exerted on HB is a long-standing problem for the reaction
+rate theory and seems to be of interest for perspectives of various application.
+A proton in HB is known to be sufficiently light to exhibit full-fledged quan-
+tum behavior leading to tunneling effect, energy levels splitting, etc (see, e.g.,
+[12,13,17-28] and refs. therein). Physical models of PT based on simplified
+Hamiltonians take their peculiar place in the enormous amount of literature on
+HB including also ab initio calculations by the methods of quantum chemistry,
+DFT, their combination with molecular dynamics simulations (considering nu-
+clei as classical Newtonian particles), QM/MM, chemical physics approaches
+within the framework of modern trends in TST, quantum-classical Liouville
+dynamics, etc. In physical models of PT the reaction coordinate is singled out
+and studied separately from the environment for which various approxima-
+tions are assumed. The problem of PT rate estimate is inevitably reduced to
+a one-dimensional path on the potential energy surface (PES) and as a result
+to a one-dimensional cross-section of PES for HB which usually has the form
+of a double-well potential (DWP). Quantum mechanical models of PT are
+motivated by the necessity to take into account other (than the reaction co-
+ordinate) internal dynamic modes, e.g., the heavy atoms stretching mode and
+to account for vibrationally and/or thermally assisted tunneling. The modern
+2
+
+physical approach to taking into account dissipative effects at tunneling is
+based on the Lindblad master equation (describing the dynamics of Marko-
+vian open quantum systems) for the time evolution of the density matrix and
+Caldeira-Leggett model of the thermal bath. For PT such scheme was initiated
+in [12,13] and by now it has been thoroughly studied within the framework of
+the general context for the reaction rate theory (see, e.g., [17] and refs. therein).
+The problem of the rate-promoting vibration for PT is considered with the
+help of this theory in [28]. The authors obtained the desired increase of the PT
+rate at adding the vibrational mode. However they came to a conclusion that
+the lower its frequency the stronger the enhancement of PT rate. Our aim is to
+find out the conceptual possibility of resonant activation (bell-shaped peaks)
+with frequency. In the present article we avoid the complications of the above
+mentioned theory (ensuing from the necessity to deal with numerous evolution
+equations for the density matrix elements) which seem to be unimportant for
+our aims. For calculating PT rate we make use of the Weiner’s theory [18,19].
+Our model of HB is maximally simple and constructed in an ad hoc manner
+for studying the effect of PT resonant activation. It corresponds to HB in a
+gas phase and does not touch upon the effects of environment taking place
+in solution. It deals only with two salient degrees of freedom, i.e., the proton
+coordinate and that of an oscillator (e.g., the heavy atoms stretching mode or
+an external vibration) with symmetric coupling between them. We treat both
+degrees of freedom quantum-mechanically by solving the corresponding two-
+dimensional Schr¨odinger equation (SE). We make use of literature data for
+the one-dimensional cross-section of PES from quantum-chemical calculations
+and model it by a suitable phenomenological DWP. For the case of the heavy
+atoms stretching mode we use literature data of IR-spectroscopy for HB to
+determine its frequency and the strength of proton coordinate coupling to it.
+The calculation of PT rate in HB requires the knowledge of the energy levels
+which are the eigenvalues of the corresponding SE with DWP. PT is a typical
+example of a quantum particle in DWP which is an omnipresent problem in
+physics and chemistry [23,26,29-41]. Most DWPs used for the analysis of HB
+and composed of polynomials, exponentials (e.g., the double Morse potential)
+or their combinations are amenable only to numerical solutions (even in one-
+dimensional case let alone its two-dimensional generalization) or approximate
+analytic approaches like the quasi-classical (WKB) method. This restriction
+was inevitable until 2010-s because of the lack of a convenient DWP for which
+SE would have an exact analytic solution (see [29] and refs. therein). Since then
+a number of exactly solvable DWPs suitable for chemical problems (taking
+infinite values at the boundaries of the spatial variable interval) appeared. For
+them analytic solutions of SE are feasible via the confluent Heun’s function
+[23], [31-38] or the spheroidal function [39,40]. The latter is a well-studied
+special function of mathematical physics [42] implemented in Mathematica.
+The case which is amenable to the treatment by both functions [23,31,39]
+makes use of the so-called trigonometric DWP (TDWP). In the previous years
+3
+
+TDWP was applied to numerous objects [23,24,25,31,39,40,43,44]. The aim
+of the present article is to show that TDWP enables one to construct an
+analytically tractable model for PT resonant activation in HB. We exemplify
+the general theory by the analysis of PT rate for intermolecular HB in the
+Zundel ion H5O+
+2 (oxonium hydrate H2O · · · H · · · OH2 in which the proton is
+equally shared between two water molecules). For the Zundel ion the detailed
+data of IR spectroscopy [20-22] along with the quantum chemical ab initio
+calculations [45,46] are available. As a result the Zundel ion suits well for the
+purpose of demonstrating the capability of our approach to the calculation of
+PT rate in HB. For the Zundel ion the distance between the oxygen atoms ROO
+is not a fixed and predetermined value but can be varied in a wide range. In the
+present article the case ROO = 3.0 A is chosen because it provides sufficiently
+high barrier to exclude the contribution of the over-barrier transition into PT
+rate constant even at high temperature. This choice is in accord with the aim
+of the article to study the effect of an external vibration on the tunneling
+contribution into the rate constant.
+The paper is organized as follows. In the preliminary Sec.2 we remind some
+results of the Weiner’s theory in the form suitable for our analysis. In Sec.3
+we briefly summarize the results of [25] which are necessary for the calculation
+of PT rate in HB. In Sec.4 we derive the expression for the PT rate constant.
+In Sec.5 the results are discussed and the conclusions are summarized. In
+Appendix some technical information is presented.
+2
+Weiner’s theory
+In the Weiner’s theory [18,19] the proton position is described by the sta-
+tionary one-dimensional SE (which we write in the dimensionless form) with
+symmetric DWP U(x) which has the solutions for the energy levels ǫq and the
+corresponding wave functions ψq(x)
+ψ′′
+q (x) + [ǫq − U(x)] ψq(x) = 0
+(1)
+The rate constant consists of the contribution from the tunneling process and
+that from the over-barrier transition. Concerning the former the Weiner’s the-
+ory deals with two important values. The first one is the probability flux to the
+right of particles in the left well when the particle is in the q-th state Jq. The
+second one is the quantum transmission coefficient, i.e., the fraction of those
+right-moving particles which are transmitted to the right well | Tq |2. Accord-
+ing to [18,19] the reaction rate constant is a result of Boltzmann averaging of
+4
+
+the product Jq | Tq |2 calculated over the doublets
+k =
+
+
+∞
+�
+q=0
+e−βǫq
+
+
+−1 
+
+
+N
+�
+n=0
+e−βǫ2nJ2n | T2n |2 +
+∞
+�
+m=2N+2
+e−βǫm
+
+
+
+(2)
+where n = 0, 1, 2, ..., N , ǫ2n is the energy for the level 2n described by the wave
+function ψ2n(x). In the Weiner’s theory the quantum transmission coefficient
+is calculated for the doublets which are counted by the even energy levels. For
+this reason n is fixed to be even in the first sum in the curly brackets (see the
+text below the formula (3.1) in Sec.III of [19] the formulas from which are used
+in the present article). The first sum in the curly brackets corresponds to the
+contribution due to the tunneling process in the reaction rate. It is over the
+energy levels below the barrier top for which the notions of J2n and | T2n |2
+have sense. In (2) it is suggested by Weiner that the quantum transmission
+coefficient of the lower level in the doublet is determined by the splitting of
+the energy levels in it. Thus N +1 is the number of doublets below the barrier
+top and ǫ2N is the lower energy level in the last doublet in this region. As a
+result only the sum over doublets (i.e., even levels q = 2n) is left. The second
+sum in the curly brackets corresponds to the over-barrier transition and ǫ2N+2
+is the the first energy level above the barrier top. The Weiner’s theory is based
+on the quasi-classical approximation of the solution of SE [19]
+ψ2n(x) =
+B2n
+�
+P2n(x)
+cos
+
+
+x
+�
+0
+dξ P2n(ξ) + S2n
+
+
+(3)
+for x ≥ 0. Taking into account that for even energy levels the wave function
+is symmetric (ψ′
+2n(0)) one obtains that
+tan S2n = − P ′
+2n(0)
+2P 2
+2n(0)
+(4)
+The function Pq(x) satisfies the so-called Milne equation
+P 2
+q + U(x) + 1
+2
+
+P ′′
+q
+Pq
+− 3
+2
+�
+P ′
+q
+�2
+P 2
+q
+
+ = ǫq
+(5)
+The expression for | T2n |2 follows from (3.5) of [19]
+| T2n |2= ψ2
+2n(0)P2n(0)
+B2
+2n
+(6)
+5
+
+The expression for J2n is given by (2.14) of [19]
+J2n = B2
+2n
+2
+(7)
+In the particular case P ′
+2n(0) = 0 (which will be pertinent in our further
+consideration) it follows from (4) that
+S2n = 0
+(8)
+Substitution of the results into (2) yields
+k =
+
+
+∞
+�
+q=0
+e−βǫq
+
+
+−1 
+
+
+1
+2
+N
+�
+n=0
+e−βǫ2nB2
+2n +
+∞
+�
+m=2N+2
+e−βǫm
+
+
+
+(9)
+It is worthy to note that in the original Weiner’s approach both ǫq and ψq(x)
+are unknown and all efforts are directed to obtain formulas that do not con-
+tain values like ψq(0) or ψ′
+q(0). In contrast for TDWP the exact solution of
+SE ψq(x) as well as the corresponding energy levels ǫq are available and we
+make use of the them. Also it should be stressed that the Weiner’s theory is
+originally written for the infinite range of the space variable −∞ < x < ∞
+with the requirement | ψq(x) |→ 0 at x → ±∞. In our case of TDWP we
+have the requirement | ψq(x) |→ 0 at x → ±π/2. For this reason we apply the
+corresponding formulas to the case −π/2 ≤ x ≤ π/2. We consider SE (1) in
+this range with the dimensionless form of the symmetric TDWP [39]
+U(x) =
+�
+m2 − 1
+4
+�
+tan2 x − p2 sin2 x
+(10)
+Here m is an integer number and p is a real number. The two parameters of
+TDWP m and p are related to two main characteristics of the potential energy
+surface, i.e., the barrier height and the barrier width (see Appendix). The
+example of TDWP for intermolecular HB in the Zundel ion with ROO = 3.0 A
+distance between oxygen atoms is presented in Fig.1. For TDWP the exact
+solution of SE is available [39]
+ψq(x) = cos1/2 x ¯Sm(q+m) (p; sin x)
+(11)
+q = 0, 1, 2, ... and ¯Sm(q+m) (p; s) is the normalized angular prolate spheroidal
+function [42]. It is implemented in Mathematica as SpheroidalPS[(q+m), m, ip, s]
+(note that the latter is a non-normalized one). The energy levels are
+ǫq = λm(q+m) (p) + 1
+2 − m2 − p2
+(12)
+6
+
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+�7
+0=57.9;ωmax≈0.0064
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ6
+0=22.85
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ4
+0=-57.705
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ5
+0=-58.386
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ2
+0=-191.406
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ3
+0=-191.782
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ0
+0=-346.192919
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ1
+0=-346.197925
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ7
+0=77.2;ωc≈0.0112
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ6
+0=32.94
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ5
+0=-39.296
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ4
+0=-45.915
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ3
+0=-163.7194636
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ2
+0=-163.8071785
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ1
+0=-306.41765635
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Λ0
+0=-306.41765771
+-0.5
+0.5
+x
+-400
+-300
+-200
+-100
+U(x)
+Zundel ion ROO=3.0 A
+Fig. 1. The trigonometric double-well potential (10) at the values of the parameters
+m = 57; p = 76. The parameters are chosen to describe the hydrogen bond in the
+Zundel ion H5O+
+2 (oxonium hydrate) for the case ROO = 3.0 ˚A (they are extracted
+from the data of quantum chemistry [46]). Several energy levels are indicated for the
+coupling constant α = 0.3. They are given by (19) and calculated at two frequencies
+of the external oscillator (for that of the main peak ωmax ≈ 0.0064 depicted by long
+dashes and for the critical frequency ωc ≈ 0.0112 depicted by short dashes).
+Here λm(q+m) (p) is the spectrum of eigenvalues for ¯Sm(q+m) (p; s). It is imple-
+mented in Mathematica as λm(q+m) (p) ≡ SpheroidalEigenvalue[(q+m), m, ip].
+For TDWP the position of the right minimum is defined by the requirement
+cos xmin =
+�m2 − 1/4
+p2
+�1/4
+(13)
+3
+Solution of two-dimensional Schr¨odinger equation with trigono-
+metric double-well potential
+For the two-dimensional SE the wave function is the function of the proton
+coordinate x and that of the oscillator z. The interaction Hamiltonian for
+various types of the mode coupling can be schematically depicted by the form
+αf(x)g(z) where α is the coupling constant. For the symmetric mode coupling
+it is αzx2, for anti-symmetric and squeezed mode couplings it is αzx and αz2x2
+respectively. In the present article we consider the case of the symmetric mode
+coupling (see a comment on other types of interaction in Sec.5). For TDWP
+it is more natural for the mathematical convenience to make for x in the
+interaction term the transformation x → sin x so that the coupling term is
+αz sin2 x. In fact the external vibration always interacts with some function
+f(x) of the proton coordinate x (e.g., the with the dipole moment if the
+vibration is produced by an electro-magnetic field). The term αzx2 for the
+symmetric mode coupling means that only the linear approximation for the
+function f(x) ≈ x is taken into account. However the linear approximation can
+be valid within the interval of a sufficiently small x only. In our opinion it is
+7
+
+reasonable to go beyond the linear approximation, i.e., to make the replacing
+x −→ sin x at −π/2 ≤ x ≤ π/2. In the case of the dipole moment it deflects to
+slower growth than the linear one (see, e.g., Fig. 10.54 in [47]. The necessity to
+go beyond the framework of the linear approximation for HB in the Zundel ion
+was stressed in [22]. To achieve this goal we model such deflection by replacing
+the linear term by the trigonometric one x −→ sin x at −π/2 ≤ x ≤ π/2. Then
+the dimensionless form of the two-dimensional SE with the symmetric mode
+coupling and TDWP is [25]
+�
+δ ∂2
+∂z2 + ∂2
+∂x2 + Λ −
+�
+m2 − 1
+4
+�
+tan2 x + p2 sin2 x−
+ω2z2
+2
+− αz sin2 x
+�
+Φ(x, z) = 0
+(14)
+Here ω is the frequency of the oscillator coupled to the proton coordinate. The
+dimensionless variables and parameters are discussed in Appendix for the case
+when the oscillator is produced by the heavy atoms stretching mode in HB.
+The solution of (14) in the adiabatic approximation corresponding to the q-th
+state of the particle in TDWP and the j-th state of the oscillator is [25]
+Φ(x, q, z, j) ≈ ϕq
+j(z)ψq(x)
+(15)
+Here the quantum number q quantizes the states of the particle in TDWP and
+ψq(x) is given by (11). The quantum number j in (15) quantizes the excitation
+states of the oscillator
+ϕq
+j(z) ≈ A exp
+�
+−
+ω
+2
+√
+2δ
+�
+z + cq
+ω2
+�2� j!(−2)j
+(2j)! H2j
+
+
+�2ω2
+δ
+�1/4 �
+z + cq
+ω2
+�
+(16)
+j = 0, 1, 2, ... , Hn(x) is the Hermit polynomial and A is a normalization
+constant. Making use of N2.20.16.6 from [48] we obtain
+A−2 =
+�j!(−2)j
+(2j)!
+�2
+24j
+�√
+2δ
+ω Γ
+�
+2j + 1
+2
+�
+2F1
+�
+−2j, −2j, 1
+2 − 2j; −1
+2
+�
+(17)
+where
+2F1 (a, b, c; x) is the hypergeometric function. The coefficient cq is
+cq = α
+1
+�
+−1
+dη η2 � ¯Sm(q+m) (p; η)
+�2
+(18)
+8
+
+The energy levels corresponding to (15) are [25]
+Λj
+q ≈ λm(q+m) (p) + 1
+2 − m2 − p2 − (cq)2
+2ω2 + (4j + 1)ω
+�
+δ
+2
+(19)
+4
+Proton transfer rate constant
+We introduce the dimensionless inverse temperature β (for its expression
+via dimensional parameters of the model see Appendix). Further we restrict
+ourselves to the relatively high temperature range 200 K ≤ T ≤ 400 K
+(0.0345 ≤ β ≤ 0.069) in which the Boltzmann statistics is valid. Then the
+partition function is calculated with the help of the energy levels Λk
+q given by
+the formula (19)
+Z(β, ω) =
+�
+q
+∞
+�
+j=0
+exp
+�
+−βΛj
+q(ω)
+�
+=
+∞
+�
+j=0
+e
+−β
+�
+(4j+1)ω√
+δ
+2+ 1
+2−m2−p2�
+�
+q
+e
+−β
+�
+λm(q+m)(p)− (cq)2
+2ω2
+�
+(20)
+With the help of (16) we calculate the average value of z
+< z >q=
+∞
+�
+−∞
+dz z
+�
+ϕq
+j(z)
+�2 = − cq
+ω2
+(21)
+We define the auxiliary parameter
+˜pq =
+�
+p2 − α < z >q
+(22)
+and the auxiliary TDWP
+˜U(x) =
+�
+m2 − 1
+4
+�
+tan2 x −
+�
+p2 − α < z >q
+�
+sin2 x
+(23)
+We introduce the auxiliary wave function ˜ψq(x) which satisfies SE
+˜ψ′′
+q (x) +
+�
+λm(q+m) (˜pq) + 1
+2 − m2 − ˜p2
+q − ˜U(x)
+�
+˜ψq(x) = 0
+(24)
+9
+
+Its solution is (11) with taking into account the replacement p →
+�
+p2 − α < z >q
+˜ψq(x) = cos1/2 x ¯Sm(q+m)
+��
+p2 − α < z >q; sin x
+�
+(25)
+We seek the solution of (16) in the form
+Φ(x, q, z, j) ≈ ϕq
+j(z)
+Bq
+�
+Pq(x)
+cos
+
+
+x
+�
+0
+dξ Pq(ξ) + Sq
+
+
+(26)
+We take into account the equation for ϕq
+j(z) (see argumentation in [25])
+�
+δ d2
+dz2 − ǫq + Λj
+q − ω2z2
+2
+− cqz
+�
+ϕq
+j(z) = 0
+(27)
+Substituting (26) into (14) and replacing z by its average value given by (21)
+(z →< z >q) we obtain the Milne equation for Pq(x)
+P 2
+q + 1
+2
+
+P ′′
+q
+Pq
+− 3
+2
+�
+P ′
+q
+�2
+P 2
+q
+
+ = ǫq + cq < z >q − ˜U(x)
+(28)
+We seek its approximate solution in the form
+Pq(x) ≈
+Dq
+˜ψ2q(x)
+(29)
+where Dq is a constant to be determined later. It is noteworthy that Pq(x)
+from (29) yields P ′
+2n(0) = 0 because for even energy levels the wave function
+is symmetric ( ˜ψ′
+2n(0) = 0). Hence (4) yields that S2n = 0 in (26). Substitution
+of (29) in (28) results in the relationship
+D2
+q
+˜ψ4q(x) = λm(q+m) (p) − λm(q+m) (˜pq) + < z >q (cq − α)
+(30)
+We require that the approximate solution of SE (3) (i.e., the corresponding
+term in (26)) coincides with our exact solution (11) for TDWP in the crucial
+points x = 0 and x = xmin. The exact wave function ψq(x) given by (11) is
+a normalized function in the range −π/2 ≤ x ≤ π/2 and its known values
+ψq(0) and ψq(xmin) further replace the corresponding unknown values for the
+approximation (3). The requirement for (30) to be satisfied at x = 0 yields
+Dq = ˜ψ2
+q(0)
+�
+λm(q+m) (p) − λm(q+m) (˜pq) + < z >q (cq − α)
+(31)
+10
+
+With the help of thus defined P2n(x) we calculate from (3) (with taking into
+account that P ′
+2n(0) = 0 yielding (8)) the wave function at xmin
+ψ2n(xmin) =
+B2n
+�
+P2n(xmin)
+cos
+
+
+xmin
+�
+0
+dξ P2n(ξ)
+
+
+(32)
+As a result we obtain
+B2
+2n = ψ2
+2n(xmin)D2n
+˜ψ2
+2n(xmin)
+cos−2
+
+D2n
+xmin
+�
+0
+dξ
+˜ψ2
+2n(ξ)
+
+
+(33)
+Then the expression for the two-dimensional generalization of (9) takes the
+form
+k(β, ω) ≈
+1
+Z(β, ω)
+�1
+2
+�
+n
+∞
+�
+j=0
+exp
+�
+−βΛj
+2n(ω)
+� ψ2
+2n(xmin)D2n
+˜ψ2
+2n(xmin)
+×
+cos−2
+
+D2n
+xmin
+�
+0
+dξ
+˜ψ2
+2n(ξ)
+
+ +
+∞
+�
+l=2N+2
+∞
+�
+j=0
+exp
+�
+−βΛj
+l (ω)
+��
+(34)
+It should be stressed that the summation over j yields the same factor as
+that in the partition function Z(β, ω) and as a result they are canceled out.
+Substituting (25) and (31) in (34) we finally obtain
+k(β, ω) ≈
+
+
+
+∞
+�
+q=0
+e
+−β
+�
+λm(q+m)(p)− (cq)2
+2ω2
+�
+
+
+−1 �1
+2
+N
+�
+n=0
+e
+−β
+�
+λm(2n+m)(p)− (c2n)2
+2ω2
+�
+×
+cos−2
+��
+λm(2n+m) (p) − λm(2n+m)
+��
+p2 − α < z >2n
+�
++ < z >2n (c2n − α) ×
+¯S2
+m(2n+m)
+��
+p2 − α < z >2n; 0
+� xmin
+�
+0
+dξ cos−1 ξ
+¯S2
+m(2n+m)
+�√p2 − α < z >2n; sin ξ
+�
+�
+×
+¯S2
+m(2n+m)
+�√p2 − α < z >2n; 0
+� ¯S2
+m(2n+m) (p; sin xmin)
+¯S2
+m(2n+m)
+�√p2 − α < z >2n; sin xmin
+�
+×
+�
+λm(2n+m) (p) − λm(2n+m)
+��
+p2 − α < z >2n
+�
++ < z >2n (c2n − α)+
+∞
+�
+l=2N+2
+e
+−β
+�
+λm(l+m)(p)− (cl)2
+2ω2
+��
+(35)
+11
+
+where < z >2n is given by (21). The sum over n is that over the doublets
+below the barrier top (see the discussion below (2)).
+5
+Results and discussion
+Notwithstanding to be cumbersome the formula (35) is easily programmed in
+Mathematica because the crucial elements λm(q+m) (p) and ¯Sm(q+m) (p; s) are
+implemented in this software package. We take the rate constant k(β, ωref)
+for the internal stretching mode of the heavy atoms in the Zundel ion (with
+a fixed frequency ωref) as a natural reference point. For this object there are
+potential energy surfaces for several ROO as a result of quantum chemical
+ab initio calculations [45], [46]. For the cases of HB in the Zundel ion with
+ROO = 2.5 A, ROO = 2.6 A and ROO = 2.7 A the authors of [20] provide the
+estimates of the dimensional coupling constant λ (a21 in their Table.1) as 0.1
+a.u., 0.1 a.u. and 0.05 a.u. respectively. For the dimensional frequency Ω/2
+(a02 in their Table.1) they present the value 0.039 a.u. for all three distances.
+From here we obtain the reference value of α = 0.6 at ROO = 2.5 A and
+α = 0.3 at ROO = 2.7 A. Also from the above results of [20] we obtain that
+the reference value for the dimensionless frequency of O-O stretching mode is
+ωref = 1.4. In the present article we are interested in the effect of the external
+vibration on the tunneling process. To make the contribution into PT rate
+constant from the over-barrier transition to be negligible compared with the
+tunneling one even at T = 400 K (β = 0.0345) we restrict ourselves by the
+case of PT for HB in the Zundel ion with very high barrier. For this reason
+we consider ROO = 3.0 A from data of [46]. TDWP for this case is depicted
+Fig.1. We carry out the parametric analysis of PT rate constant for HB in
+the Zundel ion with large ROO = 3.0 A taking the above mentioned reference
+value α = 0.3 and varying ω in the range 0.005 ≤ ω ≤ ωref. In this case there
+are three doublets below the barrier top (see Fig.1) that means N = 2 in (35).
+In the calculations of the rate constant we also take into account two levels
+above the barrier top (i.e., replace ∞ in (35) by Nmax = 7) to make sure that
+the contribution of the over-barrier transition can be discarded.
+Fig.2 shows that at decreasing the frequency from the reference value ωref
+we obtain the monotonic increase of PT rate constant in agreement with the
+conclusion of [28]. However there is a critical value of the frequency ωc (for
+α = 0.3 this value is ωc ≈ 0.01125) below which a drastic change of the
+behavior takes place. In Fig.3, Fig.4 and Fig.5 the dependence of PT rate
+constant on the frequency of the oscillator at ω < ωc is depicted. For a given
+value of the coupling constant α at the corresponding ωc there is Λ0
+1 = Λ0
+0 and
+a reversal in the total energy levels takes place (the energy levels of TDWP
+do not depend on ω and retain their natural order ǫq+1 > ǫq). At ω > ωc we
+have the normal sequence Λ0
+2n+1 > Λ0
+2n where n = 0, 1, 2, ... while at ω < ωc
+12
+
+ 
+
+
+
+
+ 
+-1.5
+-1.0
+-0.5
+0.0
+log10 ω
+-9
+-8
+-7
+-6
+-5
+log10 k(β, ω)
+ β=0.0345 (T=400 K)
+▲ ▲
+▲
+▲
+▲
+▲ ▲
+-1.5
+-1.0
+-0.5
+0.0
+log10 ω
+-9
+-8
+-7
+-6
+-5
+log10 k(β, ω)
+▲ β=0.069 (T=200 K)
+Fig. 2. The dependence of proton transfer rate constant on the frequency of the
+external oscillator above the critical value ω > ωc in the Zundel ion H5O+
+2 (oxonium
+hydrate) with ROO = 3.0 A. The critical frequency is ωc ≈ 0.01125 for the value
+of the coupling constant α = 0.3 between the proton coordinate and that of the
+oscillator.
+
+
+
+
+   
+
+-2.2
+-2.1
+-2.0
+-1.9
+log10 ω
+-5
+0
+5
+10
+15
+log10 k(β, ω)
+ β=0.0345 (T=400 K)
+Fig. 3. The dependence of proton transfer rate constant on the frequency of the
+external oscillator below the critical value ω < ωc in the Zundel ion H5O+
+2 (oxonium
+hydrate) with ROO = 3.0 A at high temperature (T = 400 K (β = 0.0345). The
+critical frequency is ωc ≈ 0.01125 for the value of the coupling constant α = 0.3
+between the proton coordinate and that of the oscillator. Low resolution picture of
+the whole interval 0.005 < ω < ωc.
+an anomalous picture Λ0
+2n+1 < Λ0
+2n occurs at first for the ground state doublet
+(n = 0) and at further decrease of the frequency for higher ones below the
+barrier top (see, e.g., the case for ωmax in Fig.1). This transformation leads to
+an extraordinary alteration in the behavior of PT rate constant. Fig.3, Fig.4
+and Fig.5 vividly exhibit that in this case there are very rich manifestations
+of resonant activation, i.e., the bell-shaped peaks of PT rate enhancement
+by the external vibration at its symmetric coupling to the proton coordinate.
+The height of the main peak at ωmax = 0.006429109800232905 is temperature
+dependent (e.g., k(0.046, ωmax)/k(0.046, ωref) = 5.31 · 1022 at T = 300K and
+k(0.0345, ωmax)/k(0.0345, ωref) = 3.13 · 1023 at T = 400K). Fig.3 shows that
+13
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+  
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+-2.14
+-2.12
+-2.10
+-2.08
+-2.06
+log10 ω
+-4.0
+-3.5
+-3.0
+-2.5
+-2.0
+log10 k(β, ω)
+ β=0.0345 (T=400 K)
+Fig. 4. The dependence of proton transfer rate constant on the frequency of the
+external oscillator below the critical value ω < ωc in the Zundel ion H5O+
+2 (oxonium
+hydrate) with ROO = 3.0 A at high temperature (T = 400 K (β = 0.0345). The
+critical frequency is ωc ≈ 0.01125 for the value of the coupling constant α = 0.3
+between the proton coordinate and that of the oscillator. High resolution picture of
+the interval 0.00693 < ω < 0.009.
+▲▲ ▲▲ ▲▲▲▲
+▲▲▲
+▲▲▲▲▲▲
+▲
+▲▲▲▲▲▲▲▲
+▲▲▲▲▲▲▲▲ ▲
+▲ ▲
+▲
+▲
+-2.2
+-2.1
+-2.0
+-1.9
+-1.8
+-1.7
+log10 ω
+-5
+0
+5
+10
+15
+log10 k(β, ω)
+▲ β=0.069 (T=200 K)
+Fig. 5. The dependence of proton transfer rate constant on the frequency of the
+external oscillator below the critical value ω < ωc in the Zundel ion H5O+
+2 (oxonium
+hydrate) with ROO = 3.0 A at low temperature (T = 200 K (β = 0.069). The critical
+frequency is ωc ≈ 0.01125 for the value of the coupling constant α = 0.3 between
+the proton coordinate and that of the oscillator.
+at high temperature β = 0.0345 the approach to the main peak from the side
+of higher frequencies is not smooth. There is a sequence of comb-like regions of
+increasing intensity with the decrease of the frequency ω (see Fig.4 for higher
+resolution picture). The intensity of these combs decreases with the decrease of
+temperature and at β = 0.069 they are not discernable (see Fig.5). At α = 0.3
+and ω < ωc the ground state doublet approaches the bottom of TDWP (see,
+e.g., Fig.1) and at ω < 0.005 the former becomes below the latter.
+By attaining the resonance condition ωmax one can obtain a very efficient
+mechanism of PT rate enhancement. For α = 0.3 we have the acceleration
+up to 23 orders of magnitude compared with the reference value ωref = 1.4.
+14
+
+The mathematical reason for the phenomenon of such PT resonance acti-
+vation is in the fact that the function ¯Sm(2n+m)
+��
+p2 − α
+�
+−c2n
+ω2
+�
+; sin xmin
+�
+taking place in the denominators of (35) becomes extremely small for the
+second doublet n = 1 at ω = ωmax. In our opinion the descriptive physical
+origin of the phenomenon can be revealed from the following empirical obser-
+vation. Let us consider the wave functions in the left and the right wells for
+the j-th doublet (j = 1, 2, 3) defined as usual ψ(j)
+R = 1/
+√
+2
+�
+ψ(j)
++ − ψ(j)
+−
+�
+and
+ψ(j)
+L
+= 1/
+√
+2
+�
+ψ(j)
++ + ψ(j)
+−
+�
+respectively where ψ(j)
+±
+are given by (11). Here +
+means the upper energy level in the doublet while − means the lower one.
+Then we recall the notion of the Rabi frequency in energetic units (multiplied
+by the Planck constant) as the module of the interaction energy, i.e., that of
+the product of the electromagnetic field strength and the matrix element of
+the dipole moment for the transition between the corresponding energy lev-
+els. The dimensional resonance condition Ef − Ei = ℏΩRabi
+if
+=| Hint | in the
+dimensionless form is ǫf − ǫi =
+√
+2δ ωRabi
+if
+=| hint |. Analogously we equate
+the difference between the energy levels ǫ(j)
++ − ǫ(j)
+− in the j-th doublet and the
+module of the matrix element of the interaction energy term | αz sin2 x | from
+(14) with the functions ψ(j)
+R and ψ(j)
+L . For sin2 x we take the value of its matrix
+element
+< ψ(j)
+R | sin2 x | ψ(j)
+L >=
+π/2
+�
+−π/2
+dx ψ(j)
+R sin2 x ψ(j)
+L
+(36)
+For z we take the average < z >− given by (21), i.e., < z >−= −c(j)
+− /ω2. As
+a result we have an empirical relationship
+ǫ(j)
++ − ǫ(j)
+− = α | −c(j)
+− < ψ(j)
+R | sin2 x | ψ(j)
+L >|
+ω2
+(37)
+From (37) we obtain the resonance frequency ω(j)
+±
+ω(j)
+± =
+�
+�
+�
+�
+�α | −c(j)
+− < ψ(j)
+R | sin2 x | ψ(j)
+L >|
+ǫ(j)
++ − ǫ(j)
+−
+(38)
+At α = 0.3 and δ = 1/8 we have for j = 2, i.e., for the second doublet
+ω(2)
+±
+= 0.00673 which is rather close to ωmax ≈ 0.00643 for the main peak.
+In our opinion such quantitative agreement can not be fortuitous. It suggests
+the physical interpretation of PT resonant activation as an analog of the Rabi
+transition between the left and the right wells under the influence of the vibra-
+tion with a suitable frequency applied to the proton in DWP. The case j = 1
+15
+
+yields the resonance frequency ω(1)
+± = 0.00795 which is within the range of the
+right comb-like region in Fig.4. Constructing various matrix elements between
+wave functions of different doublets for both wells ψin
+R = 1/
+√
+2 (ψi − ψn) and
+ψlm
+L = 1/
+√
+2 (ψl + ψm) yields
+ǫ(j)
++ − ǫ(j)
+− = α | −ck < ψin
+R | sin2 x | ψlm
+L >|
+ω2
+(39)
+where j = 1, 2, 3 and {k, i, n, l, m = 0, 1, 2, 3, 4, 5}. Then we obtain for the
+third doublet j = 3 the resonance frequencies: 0.00722 at
+{l = 2, m = 5, i = 1, k = n = 0}; 0.00725 at {k = l = 0, m = 3, i = 1, n = 4};
+0.00753 at
+{l = 1, k = m = 4, i = 2, n = 5} which are within the range of the left comb-
+like region in Fig.4. Constructing various matrix elements between wave func-
+tions of different doublets for the left well yields
+ǫ(j)
++ − ǫ(j)
+− = α | −ck < ψlm
+L | sin2 x | ψl′m′
+L
+>|
+ω2
+(40)
+Then we obtain for the third doublet j = 3 the resonance frequencies: 0.00787
+at
+{l = 1, k = m = 4, l′ = 5, m′ = 4} which is within the range of the right comb-
+like region in Fig.4 and 0.00723 at {l = 0, m = 3, l′ = 4, k = m′ = 5} which is
+within the range of the left comb-like region in Fig.4. In our opinion these
+numerous resonance frequencies provide qualitative explanation of severe os-
+cillations in Fig.4.
+Also in this connection it is worthy to note that for the symmetric mode cou-
+pling (Hint = λZX2) the effect of resonant activation results from the term
+− (cq)2 /2ω2 in the total energy levels Λk
+q (19). The energy levels of TDWP ǫq
+are re-normalized due to the coupling of the proton coordinate to the oscilla-
+tor. For anti-symmetric (Hint = λZX) and squeezed (Hint = λZ2X2) mode
+couplings this effect is absent. In the former case the coupling strength is zero
+c{as}
+q
+= 0 due to the symmetry of the wave functions [25] that leads to the
+actual lack of the crucial term −
+�
+c{as}
+q
+�2 / (2ω2) in the formula (19) for Λk
+q. In
+the latter case the expression for Λk
+q [25]
+Λk
+q|{sq} ≈ λm(q+m) (p) + 1
+2 − m2 − p2 + (4k + 1)
+�
+�
+�
+�δ
+�
+ω2 + 2c{sq}
+q
+�
+2
+(41)
+does not contain the required term at all. For instance the interaction of the
+proton in HB with an IR laser field in the dipole approximation (the dipole
+moment d ∝ x) belongs to the anti-symmetric type and does not fit our
+16
+
+requirement for PT resonant activation by a low-frequency vibration (high-
+frequency Rabi transitions between different doublets stimulated by an IR
+laser field certainly can considerably interfere PT process). Only taking into
+account that a realistic dipole moment contains the appropriate higher or-
+der contributions (d ∝ x + const x2 + ...) may provide the required type of
+interaction in this case.
+We conclude that the suggested approach enables one to obtain an analytically
+tractable expression for proton transfer rate constant in a hydrogen bond. It
+is based on the Schr¨odinger equation with the model Hamiltonian taking into
+account only the proton coordinate and an external oscillator coupled to it
+(the heavy atoms stretching mode, a low-frequency vibration of the protein
+scaffold in an enzyme, etc). The literature data from quantum chemical ab
+initio calculations of the potential energy surface are transformed into the
+parameters of the model trigonometric double-well potential. For the two-
+dimensional Schr¨odinger equation with this potential the analytic solution
+within the framework of the standard adiabatic approximation is available.
+The parameters of the model for the Zundel ion in the case of the heavy atoms
+stretching mode are extracted from the literature data on IR spectroscopy and
+serve as a reference point. The approach yields the pronounced resonant effect
+of proton transfer acceleration in some frequency range of the oscillator (below
+the corresponding critical value of the frequency) at its symmetric coupling
+to the proton coordinate. The phenomenon is absent for anti-symmetric and
+squeezed mode couplings.
+6
+Appendix
+In dimensional units the one-dimensional SE for a quantum particle with the
+reduced mass M (proton in our case of usual HB or deuterium in the case of
+a deuterated HB) has the form
+d2ψ(X)
+dX2
++ 2M
+ℏ2 [E − V (X)] ψ(X) = 0
+(42)
+where −L ≤ X ≤ L and V (X) is a DWP. The latter is assumed to be infinite
+at the boundaries of the finite interval for the spatial variable X = ±L. The
+dimensionless values for the distance x, the potential U(x) and the energy ǫ
+are introduced as follows
+x = πX
+2L ;
+U(x) = 8ML2
+ℏ2π2 V (X);
+ǫ = 8ML2E
+ℏ2π2
+(43)
+where −π/2 ≤ x ≤ π/2. As a result we obtain the dimensionless SE (1). In
+17
+
+the case of the trigonometric DWP (10) the transformation formulas for the
+parameters {m, p} into {B, D} (B is the barrier height and D is the barrier
+width) are [24]
+p =
+√
+B
+1 − [cos (D/2)]2;
+m2 − 1
+4 =
+B [cos (D/2)]4
+�
+1 − [cos (D/2)]2�2
+(44)
+The Hamiltonian of the two-dimensional SE includes the spatial variable Z
+(e.g., that for the reduced mass of the heavy atoms in HB which in the case of
+the Zundel ion is the O-O stretching mode) of the harmonic potential ΩZ2/2
+with the frequency Ω. We introduce the dimensionless distance x = πX/ (2L)
+where −π/2 ≤ x ≤ π/2 and dimensionless coordinate z = πZ/ (2L) where
+−∞ < z < ∞. The dimensionless coupling constant α in (14) for the symmet-
+ric mode coupling (this case was proved in [20] to be pertinent for the Zundel
+ion), the dimensionless inverse temperature β in (20), (34) and (35) and the
+dimensionless frequency ω in (14) are
+α = 26λML5
+ℏ2π5
+;
+β =
+ℏ2π2
+8ML2kBT ;
+ω = 4√2MµL2Ω
+ℏπ2
+;
+µ =
+M1M2
+M1 + M2
+(45)
+Here λ is a dimensional coupling constant for the case of the symmetric mode
+coupling term (λZX2) and µ is the reduced mass of the heavy atoms in HB
+A1 − H · · · A2. In the case of the Zundel ion it is µ = MO/2. As a result δ in
+(14) is δ = M/µ = 2M/MO. Taking the proton mass M = 1 a.u and that of
+the oxygen atom MO = 16 a.u. we have δ = 1/8.
+Acknowledgements. The author is grateful to Prof. Yu.F. Zuev for helpful dis-
+cussions. The work was supported from the government assignment for FRC
+Kazan Scientific Center of RAS.
+18
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+[44] C.M. Porto, G.A. Barros, L.C. Santana, A.C. Moralles, N.H. Morgon,
+J.Mol.Model. 28 (2022) 293-301.
+[45] Q. Yu, J.M. Bowman, J.Phys.Chem.Lett. 7 (2016) 5259-5265.
+[46] Z.-H. Xu, Atomistic simulations of proton transport in the gas and condensed
+phases: spectroscopy, reaction kinetics and Grotthuss mechanism, PhD thesis,
+Basel, 2018.
+20
+
+[47] P. Atkins, J. de Paula, R.Friedman, Quanta, Matter, and Change. A molecular
+approach to physical chemistry, Freeman, 2009.
+[48] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and series. Special
+functions., 2-d ed., FIZMATLIT, Moscow, 2003.
+21
+
diff --git a/9dE4T4oBgHgl3EQfDQsU/content/tmp_files/load_file.txt b/9dE4T4oBgHgl3EQfDQsU/content/tmp_files/load_file.txt
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@@ -0,0 +1,1113 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf,len=1112
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='04867v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='chem-ph]  12 Jan 2023  Model for vibrationally enhanced tunneling of  proton transfer in hydrogen bond  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Sitnitsky,  Kazan Institute of Biochemistry and Biophysics, FRC Kazan Scientific Center of  RAS, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' 30, 420111, Russian Federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' e-mail: sitnitsky@kibb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='knc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='ru  Abstract  Theoretical analysis of the effect of an external vibration on proton transfer (PT) in  a hydrogen bond (HB) is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' It is based on the two-dimensional Schr¨odinger  equation with trigonometric double-well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Its solution obtained within the  framework of the standard adiabatic approximation is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' An analytic for-  mula is derived that provides the calculation of PT rate with the help of elements  implemented in Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' We exemplify the general theory by calculating PT  rate constant for the intermolecular HB in the Zundel ion H5O+  2 (oxonium hydrate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  This object enables one to explore a wide range of the HB lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Below some crit-  ical value of the frequency of the external vibration the calculated PT rate yields  extremely rich resonant behavior (multiple manifestations of bell-shaped peaks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' It  takes place at symmetric coupling of the external vibration to the proton coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  This phenomenon is absent for anti-symmetric and squeezed mode couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  Key words: Schr¨odinger equation, double-well potential, quantum tunneling,  spheroidal function, Zundel ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  1  Introduction  Proton transfer (PT) in hydrogen bonds (HB) is one of the main processes  in the reaction rate theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' It takes place in the most important biological  molecules such as proteins (participating in some enzymatic reactions) and  DNA (arguably participating in the occurrence of mutations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In particular  the phenomenon of vibrationally enhanced (or assisted or promoted) tunnel-  ing at PT (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', resonant acceleration of the process by a coupled oscillation  in some frequency range) [1], [2] is of great interest especially in regard of its  Email address: sitnitsky@kibb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='knc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='ru ( A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Sitnitsky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  Preprint submitted to Chemical Physics Letters  13 January 2023  possible role in a mechanism for enzymatic hydrogen transfer [3-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Within  the context of enzyme catalysis it is a specific case of the more general trend  named ”rate-promoting vibration” [6-11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' There are several cases in which a  vibration can be coupled to the proton coordinate in HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' First of all there  is the heavy atoms stretching mode which is an intrinsic degree of freedom  in HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' It is thoroughly studied theoretically since the pioneer articles [12],  [13], [1] dealing with vibrationally promoted PT in solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Unfortunately it  is an internal vibration and its fixed frequency is not an experimentally con-  trollable parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Then there are external vibrations exerted on HB and  provided either by protein scaffold for HB in enzymes or by some means from  the researcher’s toolkit for HB in model compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' One of the most effi-  cient ways for such purpose is the usage of the IR electromagnetic field of an  optical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The phenomenon of resonant activation (or, in contrast, sup-  pression) of reaction rates is widely discussed for modifying chemical kinetics  by optical cavities (for recent articles in this field which is sometimes called  vibrational polariton chemistry see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', [14-16] and refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The reso-  nance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', maximal cavity induced enhancement of the reaction rate under  the vibrational resonance condition is produced in this case by mixing the  electromagnetic field with quantum states of molecular systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The cavity is  equivalent to a harmonic oscillator of a given frequency coupled to the molec-  ular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The Hamiltonian of the molecule degree of freedom coupled to  the field oscillator in the electric dipole approximation of light-matter interac-  tion has the same structure as those used for PT coupled to the heavy atoms  stretching mode in HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In this regard constructing reliable theoretical models  of PT which take into account the possibility of varying the frequency of the  external vibration exerted on HB is a long-standing problem for the reaction  rate theory and seems to be of interest for perspectives of various application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  A proton in HB is known to be sufficiently light to exhibit full-fledged quan-  tum behavior leading to tunneling effect, energy levels splitting, etc (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=',  [12,13,17-28] and refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Physical models of PT based on simplified  Hamiltonians take their peculiar place in the enormous amount of literature on  HB including also ab initio calculations by the methods of quantum chemistry,  DFT, their combination with molecular dynamics simulations (considering nu-  clei as classical Newtonian particles), QM/MM, chemical physics approaches  within the framework of modern trends in TST, quantum-classical Liouville  dynamics, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In physical models of PT the reaction coordinate is singled out  and studied separately from the environment for which various approxima-  tions are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The problem of PT rate estimate is inevitably reduced to  a one-dimensional path on the potential energy surface (PES) and as a result  to a one-dimensional cross-section of PES for HB which usually has the form  of a double-well potential (DWP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Quantum mechanical models of PT are  motivated by the necessity to take into account other (than the reaction co-  ordinate) internal dynamic modes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', the heavy atoms stretching mode and  to account for vibrationally and/or thermally assisted tunneling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The modern  2  physical approach to taking into account dissipative effects at tunneling is  based on the Lindblad master equation (describing the dynamics of Marko-  vian open quantum systems) for the time evolution of the density matrix and  Caldeira-Leggett model of the thermal bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For PT such scheme was initiated  in [12,13] and by now it has been thoroughly studied within the framework of  the general context for the reaction rate theory (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', [17] and refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  The problem of the rate-promoting vibration for PT is considered with the  help of this theory in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The authors obtained the desired increase of the PT  rate at adding the vibrational mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' However they came to a conclusion that  the lower its frequency the stronger the enhancement of PT rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Our aim is to  find out the conceptual possibility of resonant activation (bell-shaped peaks)  with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In the present article we avoid the complications of the above  mentioned theory (ensuing from the necessity to deal with numerous evolution  equations for the density matrix elements) which seem to be unimportant for  our aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For calculating PT rate we make use of the Weiner’s theory [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  Our model of HB is maximally simple and constructed in an ad hoc manner  for studying the effect of PT resonant activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' It corresponds to HB in a  gas phase and does not touch upon the effects of environment taking place  in solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' It deals only with two salient degrees of freedom, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', the proton  coordinate and that of an oscillator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', the heavy atoms stretching mode or  an external vibration) with symmetric coupling between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' We treat both  degrees of freedom quantum-mechanically by solving the corresponding two-  dimensional Schr¨odinger equation (SE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' We make use of literature data for  the one-dimensional cross-section of PES from quantum-chemical calculations  and model it by a suitable phenomenological DWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For the case of the heavy  atoms stretching mode we use literature data of IR-spectroscopy for HB to  determine its frequency and the strength of proton coordinate coupling to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  The calculation of PT rate in HB requires the knowledge of the energy levels  which are the eigenvalues of the corresponding SE with DWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' PT is a typical  example of a quantum particle in DWP which is an omnipresent problem in  physics and chemistry [23,26,29-41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Most DWPs used for the analysis of HB  and composed of polynomials, exponentials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', the double Morse potential)  or their combinations are amenable only to numerical solutions (even in one-  dimensional case let alone its two-dimensional generalization) or approximate  analytic approaches like the quasi-classical (WKB) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' This restriction  was inevitable until 2010-s because of the lack of a convenient DWP for which  SE would have an exact analytic solution (see [29] and refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Since then  a number of exactly solvable DWPs suitable for chemical problems (taking  infinite values at the boundaries of the spatial variable interval) appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For  them analytic solutions of SE are feasible via the confluent Heun’s function  [23], [31-38] or the spheroidal function [39,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The latter is a well-studied  special function of mathematical physics [42] implemented in Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  The case which is amenable to the treatment by both functions [23,31,39]  makes use of the so-called trigonometric DWP (TDWP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In the previous years  3  TDWP was applied to numerous objects [23,24,25,31,39,40,43,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The aim  of the present article is to show that TDWP enables one to construct an  analytically tractable model for PT resonant activation in HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' We exemplify  the general theory by the analysis of PT rate for intermolecular HB in the  Zundel ion H5O+  2 (oxonium hydrate H2O · · · H · · · OH2 in which the proton is  equally shared between two water molecules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For the Zundel ion the detailed  data of IR spectroscopy [20-22] along with the quantum chemical ab initio  calculations [45,46] are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' As a result the Zundel ion suits well for the  purpose of demonstrating the capability of our approach to the calculation of  PT rate in HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For the Zundel ion the distance between the oxygen atoms ROO  is not a fixed and predetermined value but can be varied in a wide range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In the  present article the case ROO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0 A is chosen because it provides sufficiently  high barrier to exclude the contribution of the over-barrier transition into PT  rate constant even at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' This choice is in accord with the aim  of the article to study the effect of an external vibration on the tunneling  contribution into the rate constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In the preliminary Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2 we remind some  results of the Weiner’s theory in the form suitable for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3  we briefly summarize the results of [25] which are necessary for the calculation  of PT rate in HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='4 we derive the expression for the PT rate constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5 the results are discussed and the conclusions are summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In  Appendix some technical information is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  2  Weiner’s theory  In the Weiner’s theory [18,19] the proton position is described by the sta-  tionary one-dimensional SE (which we write in the dimensionless form) with  symmetric DWP U(x) which has the solutions for the energy levels ǫq and the  corresponding wave functions ψq(x)  ψ′′  q (x) + [ǫq − U(x)] ψq(x) = 0  (1)  The rate constant consists of the contribution from the tunneling process and  that from the over-barrier transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Concerning the former the Weiner’s the-  ory deals with two important values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The first one is the probability flux to the  right of particles in the left well when the particle is in the q-th state Jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The  second one is the quantum transmission coefficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', the fraction of those  right-moving particles which are transmitted to the right well | Tq |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Accord-  ing to [18,19] the reaction rate constant is a result of Boltzmann averaging of  4  the product Jq | Tq |2 calculated over the doublets  k =  \uf8ee  \uf8f0  ∞  �  q=0  e−βǫq  \uf8f9  \uf8fb  −1 \uf8f1  \uf8f2  \uf8f3  N  �  n=0  e−βǫ2nJ2n | T2n |2 +  ∞  �  m=2N+2  e−βǫm  \uf8fc  \uf8fd  \uf8fe  (2)  where n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', N , ǫ2n is the energy for the level 2n described by the wave  function ψ2n(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In the Weiner’s theory the quantum transmission coefficient  is calculated for the doublets which are counted by the even energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For  this reason n is fixed to be even in the first sum in the curly brackets (see the  text below the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='III of [19] the formulas from which are used  in the present article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The first sum in the curly brackets corresponds to the  contribution due to the tunneling process in the reaction rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' It is over the  energy levels below the barrier top for which the notions of J2n and | T2n |2  have sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In (2) it is suggested by Weiner that the quantum transmission  coefficient of the lower level in the doublet is determined by the splitting of  the energy levels in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Thus N +1 is the number of doublets below the barrier  top and ǫ2N is the lower energy level in the last doublet in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' As a  result only the sum over doublets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', even levels q = 2n) is left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The second  sum in the curly brackets corresponds to the over-barrier transition and ǫ2N+2  is the the first energy level above the barrier top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The Weiner’s theory is based  on the quasi-classical approximation of the solution of SE [19]  ψ2n(x) =  B2n  �  P2n(x)  cos  \uf8eb  \uf8ed  x  �  0  dξ P2n(ξ) + S2n  \uf8f6  \uf8f8  (3)  for x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Taking into account that for even energy levels the wave function  is symmetric (ψ′  2n(0)) one obtains that  tan S2n = − P ′  2n(0)  2P 2  2n(0)  (4)  The function Pq(x) satisfies the so-called Milne equation  P 2  q + U(x) + 1  2  \uf8ee  \uf8ef\uf8f0P ′′  q  Pq  − 3  2  �  P ′  q  �2  P 2  q  \uf8f9  \uf8fa\uf8fb = ǫq  (5)  The expression for | T2n |2 follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5) of [19]  | T2n |2= ψ2  2n(0)P2n(0)  B2  2n  (6)  5  The expression for J2n is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='14) of [19]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='J2n = B2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='In the particular case P ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2n(0) = 0 (which will be pertinent in our further  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='consideration) it follows from (4) that  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='S2n = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='Substitution of the results into (2) yields  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='k =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8ee  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8f0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e−βǫq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8f9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8fb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='−1 \uf8f1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8f2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8f3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='n=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e−βǫ2nB2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2n +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='m=2N+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e−βǫm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8fc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8fd  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8fe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(9)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='It is worthy to note that in the original Weiner’s approach both ǫq and ψq(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='are unknown and all efforts are directed to obtain formulas that do not con-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='tain values like ψq(0) or ψ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In contrast for TDWP the exact solution of  SE ψq(x) as well as the corresponding energy levels ǫq are available and we  make use of the them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Also it should be stressed that the Weiner’s theory is  originally written for the infinite range of the space variable −∞ < x < ∞  with the requirement | ψq(x) |→ 0 at x → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In our case of TDWP we  have the requirement | ψq(x) |→ 0 at x → ±π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For this reason we apply the  corresponding formulas to the case −π/2 ≤ x ≤ π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' We consider SE (1) in  this range with the dimensionless form of the symmetric TDWP [39]  U(x) =  �  m2 − 1  4  �  tan2 x − p2 sin2 x  (10)  Here m is an integer number and p is a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The two parameters of  TDWP m and p are related to two main characteristics of the potential energy  surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', the barrier height and the barrier width (see Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The  example of TDWP for intermolecular HB in the Zundel ion with ROO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0 A  distance between oxygen atoms is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For TDWP the exact  solution of SE is available [39]  ψq(x) = cos1/2 x ¯Sm(q+m) (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' sin x)  (11)  q = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' and ¯Sm(q+m) (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' s) is the normalized angular prolate spheroidal  function [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' It is implemented in Mathematica as SpheroidalPS[(q+m), m, ip, s]  (note that the latter is a non-normalized one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The energy levels are  ǫq = λm(q+m) (p) + 1  2 − m2 − p2  (12)  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  �7  0=57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='ωmax≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0064  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ6  0=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ4  0=-57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='705  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ5  0=-58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='386  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ2  0=-191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='406  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ3  0=-191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='782  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ0  0=-346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='192919  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ1  0=-346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='197925  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ7  0=77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='ωc≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0112  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ6  0=32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ5  0=-39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='296  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ4  0=-45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='915  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ3  0=-163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='7194636  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ2  0=-163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='8071785  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ1  0=-306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='41765635  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Λ0  0=-306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='41765771  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  x  400  300  200  100  U(x)  Zundel ion ROO=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0 A  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The trigonometric double-well potential (10) at the values of the parameters  m = 57;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' p = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The parameters are chosen to describe the hydrogen bond in the  Zundel ion H5O+  2 (oxonium hydrate) for the case ROO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0 ˚A (they are extracted  from the data of quantum chemistry [46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Several energy levels are indicated for the  coupling constant α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' They are given by (19) and calculated at two frequencies  of the external oscillator (for that of the main peak ωmax ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0064 depicted by long  dashes and for the critical frequency ωc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0112 depicted by short dashes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  Here λm(q+m) (p) is the spectrum of eigenvalues for ¯Sm(q+m) (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' It is imple-  mented in Mathematica as λm(q+m) (p) ≡ SpheroidalEigenvalue[(q+m), m, ip].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  For TDWP the position of the right minimum is defined by the requirement  cos xmin =  �m2 − 1/4  p2  �1/4  (13)  3  Solution of two-dimensional Schr¨odinger equation with trigono-  metric double-well potential  For the two-dimensional SE the wave function is the function of the proton  coordinate x and that of the oscillator z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The interaction Hamiltonian for  various types of the mode coupling can be schematically depicted by the form  αf(x)g(z) where α is the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For the symmetric mode coupling  it is αzx2, for anti-symmetric and squeezed mode couplings it is αzx and αz2x2  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In the present article we consider the case of the symmetric mode  coupling (see a comment on other types of interaction in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For TDWP  it is more natural for the mathematical convenience to make for x in the  interaction term the transformation x → sin x so that the coupling term is  αz sin2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In fact the external vibration always interacts with some function  f(x) of the proton coordinate x (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', the with the dipole moment if the  vibration is produced by an electro-magnetic field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The term αzx2 for the  symmetric mode coupling means that only the linear approximation for the  function f(x) ≈ x is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' However the linear approximation can  be valid within the interval of a sufficiently small x only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In our opinion it is  7  reasonable to go beyond the linear approximation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', to make the replacing  x −→ sin x at −π/2 ≤ x ≤ π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In the case of the dipole moment it deflects to  slower growth than the linear one (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='54 in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The necessity to  go beyond the framework of the linear approximation for HB in the Zundel ion  was stressed in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' To achieve this goal we model such deflection by replacing  the linear term by the trigonometric one x −→ sin x at −π/2 ≤ x ≤ π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Then  the dimensionless form of the two-dimensional SE with the symmetric mode  coupling and TDWP is [25]  �  δ ∂2  ∂z2 + ∂2  ∂x2 + Λ −  �  m2 − 1  4  �  tan2 x + p2 sin2 x−  ω2z2  2  − αz sin2 x  �  Φ(x, z) = 0  (14)  Here ω is the frequency of the oscillator coupled to the proton coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The  dimensionless variables and parameters are discussed in Appendix for the case  when the oscillator is produced by the heavy atoms stretching mode in HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  The solution of (14) in the adiabatic approximation corresponding to the q-th  state of the particle in TDWP and the j-th state of the oscillator is [25]  Φ(x, q, z, j) ≈ ϕq  j(z)ψq(x)  (15)  Here the quantum number q quantizes the states of the particle in TDWP and  ψq(x) is given by (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The quantum number j in (15) quantizes the excitation  states of the oscillator  ϕq  j(z) ≈ A exp  �  −  ω  2  √  2δ  �  z + cq  ω2  �2� j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' (−2)j  (2j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' H2j  \uf8eb  \uf8ed  �2ω2  δ  �1/4 �  z + cq  ω2  �\uf8f6  \uf8f8(16)  j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' , Hn(x) is the Hermit polynomial and A is a normalization  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Making use of N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='6 from [48] we obtain  A−2 =  �j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' (−2)j  (2j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  �2  24j  �√  2δ  ω Γ  �  2j + 1  2  �  2F1  �  −2j, −2j, 1  2 − 2j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' −1  2  �  (17)  where  2F1 (a, b, c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' x) is the hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The coefficient cq is  cq = α  1  �  −1  dη η2 � ¯Sm(q+m) (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' η)  �2  (18)  8  The energy levels corresponding to (15) are [25]  Λj  q ≈ λm(q+m) (p) + 1  2 − m2 − p2 − (cq)2  2ω2 + (4j + 1)ω  �  δ  2  (19)  4  Proton transfer rate constant  We introduce the dimensionless inverse temperature β (for its expression  via dimensional parameters of the model see Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Further we restrict  ourselves to the relatively high temperature range 200 K ≤ T ≤ 400 K  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0345 ≤ β ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='069) in which the Boltzmann statistics is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Then the  partition function is calculated with the help of the energy levels Λk  q given by  the formula (19)  Z(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' ω) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='j=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='exp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='−βΛj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q(ω)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='j=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='−β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(4j+1)ω√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='δ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2−m2−p2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='−β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='λm(q+m)(p)− (cq)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2ω2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(20)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='With the help of (16) we calculate the average value of z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='< z >q=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='−∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='dz z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='ϕq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='j(z)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�2 = − cq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='ω2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(21)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='We define the auxiliary parameter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='˜pq =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='p2 − α < z >q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(22)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='and the auxiliary TDWP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='˜U(x) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='m2 − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='tan2 x −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='p2 − α < z >q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='sin2 x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(23)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='We introduce the auxiliary wave function ˜ψq(x) which satisfies SE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='˜ψ′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q (x) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='λm(q+m) (˜pq) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2 − m2 − ˜p2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q − ˜U(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='˜ψq(x) = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(24)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='Its solution is (11) with taking into account the replacement p →  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='p2 − α < z >q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='˜ψq(x) = cos1/2 x ¯Sm(q+m)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='p2 − α < z >q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' sin x  �  (25)  We seek the solution of (16) in the form  Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' j) ≈ ϕq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='j(z)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='Bq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='Pq(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='cos  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='dξ Pq(ξ) + Sq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(26)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='We take into account the equation for ϕq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='j(z) (see argumentation in [25])  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='δ d2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='dz2 − ǫq + Λj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q − ω2z2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='− cqz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='ϕq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='j(z) = 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(27)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='Substituting (26) into (14) and replacing z by its average value given by (21)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(z →< z >q) we obtain the Milne equation for Pq(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='P 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8ee  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8ef\uf8f0P ′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='Pq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='− 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='P ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='P 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8f9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8fa\uf8fb = ǫq + cq < z >q − ˜U(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(28)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='We seek its approximate solution in the form  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='Pq(x) ≈  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='Dq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='˜ψ2q(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(29)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='where Dq is a constant to be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' It is noteworthy that Pq(x)  from (29) yields P ′  2n(0) = 0 because for even energy levels the wave function  is symmetric ( ˜ψ′  2n(0) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Hence (4) yields that S2n = 0 in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Substitution  of (29) in (28) results in the relationship  D2  q  ˜ψ4q(x) = λm(q+m) (p) − λm(q+m) (˜pq) + < z >q (cq − α)  (30)  We require that the approximate solution of SE (3) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', the corresponding  term in (26)) coincides with our exact solution (11) for TDWP in the crucial  points x = 0 and x = xmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The exact wave function ψq(x) given by (11) is  a normalized function in the range −π/2 ≤ x ≤ π/2 and its known values  ψq(0) and ψq(xmin) further replace the corresponding unknown values for the  approximation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The requirement for (30) to be satisfied at x = 0 yields  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='Dq = ˜ψ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='q(0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='λm(q+m) (p) − λm(q+m) (˜pq) + < z >q (cq − α)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(31)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='With the help of thus defined P2n(x) we calculate from (3) (with taking into  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='account that P ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2n(0) = 0 yielding (8)) the wave function at xmin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='ψ2n(xmin) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='B2n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='P2n(xmin)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='cos  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='xmin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='dξ P2n(ξ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(32)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='As a result we obtain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='B2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2n = ψ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2n(xmin)D2n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='˜ψ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2n(xmin)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='cos−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8edD2n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='xmin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='dξ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='˜ψ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2n(ξ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='(33)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='Then the expression for the two-dimensional generalization of (9) takes the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='form  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='k(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' ω) ≈  1  Z(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' ω)  �1  2  �  n  ∞  �  j=0  exp  �  −βΛj  2n(ω)  � ψ2  2n(xmin)D2n  ˜ψ2  2n(xmin)  ×  cos−2  \uf8eb  \uf8edD2n  xmin  �  0  dξ  ˜ψ2  2n(ξ)  \uf8f6  \uf8f8 +  ∞  �  l=2N+2  ∞  �  j=0  exp  �  −βΛj  l (ω)  ��  (34)  It should be stressed that the summation over j yields the same factor as  that in the partition function Z(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' ω) and as a result they are canceled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  Substituting (25) and (31) in (34) we finally obtain  k(β, ω) ≈  \uf8f1  \uf8f2  \uf8f3  ∞  �  q=0  e  −β  �  λm(q+m)(p)− (cq)2  2ω2  �\uf8fc  \uf8fd  \uf8fe  −1 �1  2  N  �  n=0  e  −β  �  λm(2n+m)(p)− (c2n)2  2ω2  �  ×  cos−2  ��  λm(2n+m) (p) − λm(2n+m)  ��  p2 − α < z >2n  �  + < z >2n (c2n − α) ×  ¯S2  m(2n+m)  ��  p2 − α < z >2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' 0  � xmin  �  0  dξ cos−1 ξ  ¯S2  m(2n+m)  �√p2 − α < z >2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' sin ξ  �  �  ×  ¯S2  m(2n+m)  �√p2 − α < z >2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' 0  � ¯S2  m(2n+m) (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' sin xmin)  ¯S2  m(2n+m)  �√p2 − α < z >2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' sin xmin  �  ×  �  λm(2n+m) (p) − λm(2n+m)  ��  p2 − α < z >2n  �  + < z >2n (c2n − α)+  ∞  �  l=2N+2  e  −β  �  λm(l+m)(p)− (cl)2  2ω2  ��  (35)  11  where < z >2n is given by (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The sum over n is that over the doublets  below the barrier top (see the discussion below (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  5  Results and discussion  Notwithstanding to be cumbersome the formula (35) is easily programmed in  Mathematica because the crucial elements λm(q+m) (p) and ¯Sm(q+m) (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' s) are  implemented in this software package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' We take the rate constant k(β, ωref)  for the internal stretching mode of the heavy atoms in the Zundel ion (with  a fixed frequency ωref) as a natural reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For this object there are  potential energy surfaces for several ROO as a result of quantum chemical  ab initio calculations [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For the cases of HB in the Zundel ion with  ROO = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5 A, ROO = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='6 A and ROO = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='7 A the authors of [20] provide the  estimates of the dimensional coupling constant λ (a21 in their Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1) as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='05 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For the dimensional frequency Ω/2  (a02 in their Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1) they present the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='039 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' for all three distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  From here we obtain the reference value of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='6 at ROO = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5 A and  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3 at ROO = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Also from the above results of [20] we obtain that  the reference value for the dimensionless frequency of O-O stretching mode is  ωref = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In the present article we are interested in the effect of the external  vibration on the tunneling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' To make the contribution into PT rate  constant from the over-barrier transition to be negligible compared with the  tunneling one even at T = 400 K (β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0345) we restrict ourselves by the  case of PT for HB in the Zundel ion with very high barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For this reason  we consider ROO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0 A from data of [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' TDWP for this case is depicted  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' We carry out the parametric analysis of PT rate constant for HB in  the Zundel ion with large ROO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0 A taking the above mentioned reference  value α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3 and varying ω in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='005 ≤ ω ≤ ωref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In this case there  are three doublets below the barrier top (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1) that means N = 2 in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  In the calculations of the rate constant we also take into account two levels  above the barrier top (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', replace ∞ in (35) by Nmax = 7) to make sure that  the contribution of the over-barrier transition can be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2 shows that at decreasing the frequency from the reference value ωref  we obtain the monotonic increase of PT rate constant in agreement with the  conclusion of [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' However there is a critical value of the frequency ωc (for  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3 this value is ωc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='01125) below which a drastic change of the  behavior takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5 the dependence of PT rate  constant on the frequency of the oscillator at ω < ωc is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For a given  value of the coupling constant α at the corresponding ωc there is Λ0  1 = Λ0  0 and  a reversal in the total energy levels takes place (the energy levels of TDWP  do not depend on ω and retain their natural order ǫq+1 > ǫq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' At ω > ωc we  have the normal sequence Λ0  2n+1 > Λ0  2n where n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' while at ω < ωc  12  \uf750 \uf750  \uf750  \uf750\uf750\uf750  \uf750  \uf750  \uf750 \uf750  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0  log10 ω  9  8  7  6  5  log10 k(β, ω)  \uf750 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0345 (T=400 K)  ▲ ▲  ▲  ▲  ▲  ▲ ▲  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0  log10 ω  9  8  7  6  5  log10 k(β, ω)  ▲ β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='069 (T=200 K)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The dependence of proton transfer rate constant on the frequency of the  external oscillator above the critical value ω > ωc in the Zundel ion H5O+  2 (oxonium  hydrate) with ROO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The critical frequency is ωc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='01125 for the value  of the coupling constant α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3 between the proton coordinate and that of the  oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  \uf750  \uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750  \uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750  \uf750  \uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750 \uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750 \uf750 \uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750\uf750  \uf750  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='9  log10 ω  5  0  5  10  15  log10 k(β, ω)  \uf750 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0345 (T=400 K)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The dependence of proton transfer rate constant on the frequency of the  external oscillator below the critical value ω < ωc in the Zundel ion H5O+  2 (oxonium  hydrate) with ROO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0 A at high temperature (T = 400 K (β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0345).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The  critical frequency is ωc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='01125 for the value of the coupling constant α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3  between the proton coordinate and that of the oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Low resolution picture of  the whole interval 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='005 < ω < ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  an anomalous picture Λ0  2n+1 < Λ0  2n occurs at first for the ground state doublet  (n = 0) and at further decrease of the frequency for higher ones below the  barrier top (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', the case for ωmax in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' This transformation leads to  an extraordinary alteration in the behavior of PT rate constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='4  and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5 vividly exhibit that in this case there are very rich manifestations  of resonant activation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', the bell-shaped peaks of PT rate enhancement  by the external vibration at its symmetric coupling to the proton coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  The height of the main peak at ωmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='006429109800232905 is temperature  dependent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', k(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='046, ωmax)/k(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='046, ωref) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='31 · 1022 at T = 300K and  k(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0345, ωmax)/k(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0345, ωref) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='13 · 1023 at T = 400K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3 shows that  13  \uf750  \uf750\uf750\uf750\uf750\uf750\uf750\uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750\uf750  \uf750  \uf750  \uf750  \uf750\uf750\uf750  \uf750  \uf750\uf750  \uf750  \uf750 \uf750 \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  \uf750  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='08  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='06  log10 ω  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0  log10 k(β, ω)  \uf750 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0345 (T=400 K)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The dependence of proton transfer rate constant on the frequency of the  external oscillator below the critical value ω < ωc in the Zundel ion H5O+  2 (oxonium  hydrate) with ROO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0 A at high temperature (T = 400 K (β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0345).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The  critical frequency is ωc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='01125 for the value of the coupling constant α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3  between the proton coordinate and that of the oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' High resolution picture of  the interval 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='00693 < ω < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  ▲▲ ▲▲ ▲▲▲▲  ▲▲▲  ▲▲▲▲▲▲  ▲  ▲▲▲▲▲▲▲▲  ▲▲▲▲▲▲▲▲ ▲  ▲ ▲  ▲  ▲  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='7  log10 ω  5  0  5  10  15  log10 k(β, ω)  ▲ β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='069 (T=200 K)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The dependence of proton transfer rate constant on the frequency of the  external oscillator below the critical value ω < ωc in the Zundel ion H5O+  2 (oxonium  hydrate) with ROO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0 A at low temperature (T = 200 K (β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='069).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The critical  frequency is ωc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='01125 for the value of the coupling constant α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3 between  the proton coordinate and that of the oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  at high temperature β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='0345 the approach to the main peak from the side  of higher frequencies is not smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' There is a sequence of comb-like regions of  increasing intensity with the decrease of the frequency ω (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='4 for higher  resolution picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The intensity of these combs decreases with the decrease of  temperature and at β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='069 they are not discernable (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' At α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3  and ω < ωc the ground state doublet approaches the bottom of TDWP (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='1) and at ω < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='005 the former becomes below the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  By attaining the resonance condition ωmax one can obtain a very efficient  mechanism of PT rate enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3 we have the acceleration  up to 23 orders of magnitude compared with the reference value ωref = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  14  The mathematical reason for the phenomenon of such PT resonance acti-  vation is in the fact that the function ¯Sm(2n+m)  ��  p2 − α  �  −c2n  ω2  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' sin xmin  �  taking place in the denominators of (35) becomes extremely small for the  second doublet n = 1 at ω = ωmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In our opinion the descriptive physical  origin of the phenomenon can be revealed from the following empirical obser-  vation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Let us consider the wave functions in the left and the right wells for  the j-th doublet (j = 1, 2, 3) defined as usual ψ(j)  R = 1/  √  2  �  ψ(j)  + − ψ(j)  −  �  and  ψ(j)  L  = 1/  √  2  �  ψ(j)  + + ψ(j)  −  �  respectively where ψ(j)  ±  are given by (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Here +  means the upper energy level in the doublet while − means the lower one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  Then we recall the notion of the Rabi frequency in energetic units (multiplied  by the Planck constant) as the module of the interaction energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', that of  the product of the electromagnetic field strength and the matrix element of  the dipole moment for the transition between the corresponding energy lev-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The dimensional resonance condition Ef − Ei = ℏΩRabi  if  =| Hint | in the  dimensionless form is ǫf − ǫi =  √  2δ ωRabi  if  =| hint |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Analogously we equate  the difference between the energy levels ǫ(j)  + − ǫ(j)  − in the j-th doublet and the  module of the matrix element of the interaction energy term | αz sin2 x | from  (14) with the functions ψ(j)  R and ψ(j)  L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For sin2 x we take the value of its matrix  element  < ψ(j)  R | sin2 x | ψ(j)  L >=  π/2  �  −π/2  dx ψ(j)  R sin2 x ψ(j)  L  (36)  For z we take the average < z >− given by (21), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', < z >−= −c(j)  − /ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' As  a result we have an empirical relationship  ǫ(j)  + − ǫ(j)  − = α | −c(j)  − < ψ(j)  R | sin2 x | ψ(j)  L >|  ω2  (37)  From (37) we obtain the resonance frequency ω(j)  ±  ω(j)  ± =  �  �  �  �  �α | −c(j)  − < ψ(j)  R | sin2 x | ψ(j)  L >|  ǫ(j)  + − ǫ(j)  −  (38)  At α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='3 and δ = 1/8 we have for j = 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', for the second doublet  ω(2)  ±  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='00673 which is rather close to ωmax ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='00643 for the main peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  In our opinion such quantitative agreement can not be fortuitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' It suggests  the physical interpretation of PT resonant activation as an analog of the Rabi  transition between the left and the right wells under the influence of the vibra-  tion with a suitable frequency applied to the proton in DWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The case j = 1  15  yields the resonance frequency ω(1)  ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='00795 which is within the range of the  right comb-like region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Constructing various matrix elements between  wave functions of different doublets for both wells ψin  R = 1/  √  2 (ψi − ψn) and  ψlm  L = 1/  √  2 (ψl + ψm) yields  ǫ(j)  + − ǫ(j)  − = α | −ck < ψin  R | sin2 x | ψlm  L >|  ω2  (39)  where j = 1, 2, 3 and {k, i, n, l, m = 0, 1, 2, 3, 4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Then we obtain for the  third doublet j = 3 the resonance frequencies: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='00722 at  {l = 2, m = 5, i = 1, k = n = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='00725 at {k = l = 0, m = 3, i = 1, n = 4};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='00753 at  {l = 1, k = m = 4, i = 2, n = 5} which are within the range of the left comb-  like region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Constructing various matrix elements between wave func-  tions of different doublets for the left well yields  ǫ(j)  + − ǫ(j)  − = α | −ck < ψlm  L | sin2 x | ψl′m′  L  >|  ω2  (40)  Then we obtain for the third doublet j = 3 the resonance frequencies: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='00787  at  {l = 1, k = m = 4, l′ = 5, m′ = 4} which is within the range of the right comb-  like region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='00723 at {l = 0, m = 3, l′ = 4, k = m′ = 5} which is  within the range of the left comb-like region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In our opinion these  numerous resonance frequencies provide qualitative explanation of severe os-  cillations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  Also in this connection it is worthy to note that for the symmetric mode cou-  pling (Hint = λZX2) the effect of resonant activation results from the term  − (cq)2 /2ω2 in the total energy levels Λk  q (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The energy levels of TDWP ǫq  are re-normalized due to the coupling of the proton coordinate to the oscilla-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For anti-symmetric (Hint = λZX) and squeezed (Hint = λZ2X2) mode  couplings this effect is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In the former case the coupling strength is zero  c{as}  q  = 0 due to the symmetry of the wave functions [25] that leads to the  actual lack of the crucial term −  �  c{as}  q  �2 / (2ω2) in the formula (19) for Λk  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In  the latter case the expression for Λk  q [25]  Λk  q|{sq} ≈ λm(q+m) (p) + 1  2 − m2 − p2 + (4k + 1)  �  �  �  �δ  �  ω2 + 2c{sq}  q  �  2  (41)  does not contain the required term at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For instance the interaction of the  proton in HB with an IR laser field in the dipole approximation (the dipole  moment d ∝ x) belongs to the anti-symmetric type and does not fit our  16  requirement for PT resonant activation by a low-frequency vibration (high-  frequency Rabi transitions between different doublets stimulated by an IR  laser field certainly can considerably interfere PT process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Only taking into  account that a realistic dipole moment contains the appropriate higher or-  der contributions (d ∝ x + const x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=') may provide the required type of  interaction in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  We conclude that the suggested approach enables one to obtain an analytically  tractable expression for proton transfer rate constant in a hydrogen bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' It  is based on the Schr¨odinger equation with the model Hamiltonian taking into  account only the proton coordinate and an external oscillator coupled to it  (the heavy atoms stretching mode, a low-frequency vibration of the protein  scaffold in an enzyme, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The literature data from quantum chemical ab  initio calculations of the potential energy surface are transformed into the  parameters of the model trigonometric double-well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' For the two-  dimensional Schr¨odinger equation with this potential the analytic solution  within the framework of the standard adiabatic approximation is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  The parameters of the model for the Zundel ion in the case of the heavy atoms  stretching mode are extracted from the literature data on IR spectroscopy and  serve as a reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The approach yields the pronounced resonant effect  of proton transfer acceleration in some frequency range of the oscillator (below  the corresponding critical value of the frequency) at its symmetric coupling  to the proton coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The phenomenon is absent for anti-symmetric and  squeezed mode couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  6  Appendix  In dimensional units the one-dimensional SE for a quantum particle with the  reduced mass M (proton in our case of usual HB or deuterium in the case of  a deuterated HB) has the form  d2ψ(X)  dX2  + 2M  ℏ2 [E − V (X)] ψ(X) = 0  (42)  where −L ≤ X ≤ L and V (X) is a DWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The latter is assumed to be infinite  at the boundaries of the finite interval for the spatial variable X = ±L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The  dimensionless values for the distance x, the potential U(x) and the energy ǫ  are introduced as follows  x = πX  2L ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  U(x) = 8ML2  ℏ2π2 V (X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  ǫ = 8ML2E  ℏ2π2  (43)  where −π/2 ≤ x ≤ π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' As a result we obtain the dimensionless SE (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In  17  the case of the trigonometric DWP (10) the transformation formulas for the  parameters {m, p} into {B, D} (B is the barrier height and D is the barrier  width) are [24]  p =  √  B  1 − [cos (D/2)]2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  m2 − 1  4 =  B [cos (D/2)]4  �  1 − [cos (D/2)]2�2  (44)  The Hamiltonian of the two-dimensional SE includes the spatial variable Z  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=', that for the reduced mass of the heavy atoms in HB which in the case of  the Zundel ion is the O-O stretching mode) of the harmonic potential ΩZ2/2  with the frequency Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' We introduce the dimensionless distance x = πX/ (2L)  where −π/2 ≤ x ≤ π/2 and dimensionless coordinate z = πZ/ (2L) where  −∞ < z < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The dimensionless coupling constant α in (14) for the symmet-  ric mode coupling (this case was proved in [20] to be pertinent for the Zundel  ion), the dimensionless inverse temperature β in (20), (34) and (35) and the  dimensionless frequency ω in (14) are  α = 26λML5  ℏ2π5  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  β =  ℏ2π2  8ML2kBT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  ω = 4√2MµL2Ω  ℏπ2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  µ =  M1M2  M1 + M2  (45)  Here λ is a dimensional coupling constant for the case of the symmetric mode  coupling term (λZX2) and µ is the reduced mass of the heavy atoms in HB  A1 − H · · · A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' In the case of the Zundel ion it is µ = MO/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' As a result δ in  (14) is δ = M/µ = 2M/MO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Taking the proton mass M = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='u and that of  the oxygen atom MO = 16 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' we have δ = 1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The author is grateful to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Zuev for helpful dis-  cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' The work was supported from the government assignment for FRC  Kazan Scientific Center of RAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
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+page_content=' Zhu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
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+page_content=' Jelic, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
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+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
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+page_content=' Ibrahim, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
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+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
+page_content=' 86 (2018) 180-185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
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+page_content=' Dong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
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+page_content=' Sun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQfDQsU/content/2301.04867v1.pdf'}
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+ CIRED workshop on E-mobility and power distribution systems 
+Porto, 2-3 June 2022 
+ 
+Paper n°  1347 
+ 
+ 
+CIRED 2022 Workshop  
+1/4 
+ESTIMATING REQUIRED FLEXIBILITY FOR SECURE DISTRIBUTION GRID 
+OPERATION CONSIDERING THE UNCERTAINTIES OF EV AND PV  
+ 
+ 
+ 
+Manijeh ALIPOUR  
+Omid ALIZADEH-MOUSAVI  
+ 
+Depsys, Puidoux - Switzerland 
+Depsys, Puidoux - Switzerland 
+ 
+manijeh.alipour@depsys.com 
+omid.mousavi@depsys.com  
+ 
+ABSTRACT 
+Renewable energy productions and electrification of 
+mobility are promising solutions to reduce greenhouse gas 
+emissions. Their effective integration in a power grid 
+encounters several challenges. The uncertain nature of 
+renewable energy productions as well as stochastic 
+consumptions of electric vehicles introduce remarkable 
+intermittency to a distribution grid and results in bi-
+uncertain characteristics of both  supply and demand sides. 
+One way to verify the secure grid operation within 
+acceptable voltage and loading levels is to assess its 
+required flexibility considering possible boundaries of 
+uncertain variables. In this paper, first a comprehensive 
+linear model of distribution grid considering all pertaining 
+constraints is presented. Then, a flexibility estimation 
+technique is proposed based on the feasibility study of the 
+uncertain space of photovoltaic power productions and 
+load 
+containing 
+electric 
+vehicles. 
+The 
+proposed 
+methodology uses grid monitoring data to determine grid 
+state and to model uncertain parameters. The model is 
+applied on a real low voltage (LV) system equipped with 
+grid monitoring devices.  
+INTRODUCTION 
+The deployment of renewable energy productions has 
+many advantages, comprising economic convenience, 
+reducing the reliance on fossil fuel markets (especially, gas 
+and oil) and environmental friendliness. In addition, the 
+wide penetration of renewable energy sources accelerates 
+occupation in the EU, by job creation in different ‘green’ 
+technologies. Aim behind the European Green Deal 
+(COM(2019) 640 final) is to be the world’s first climate-
+neutral continent by 2050. In spite of renewable 
+generation’s significant advantages, it has the defects of 
+uncertainty and fluctuation. Besides to renewable 
+generations, electric vehicles as a new variable load 
+increase the intermittency of distribution grid and leads to 
+bi-uncertain characteristics of both demand and supply 
+sides. With the increased penetration of these uncertainty 
+sources, the modern power system is confronted with a 
+challenge to preserve the reliability, security, and quality 
+of supply. Thus, a more flexible distribution grid is 
+required. 
+The distribution grid flexibility can be defined from 
+different perspectives [1]. In [2], the flexibility is specified 
+as the degree to which a system can change its electricity 
+consumption and production in response to anticipated or 
+unanticipated variabilities. The flexibility envelope 
+method is presented in [3] to assess the flexibility potential 
+of individual power system assets and their aggregation at 
+the system level. A flexibility measure is developed in [4] 
+which indicates the largest uncertainty deviation that a 
+system can bear. In [5] an algorithm is proposed which 
+allocates aggregate-level control decisions amongst 
+individual 
+systems 
+economically. 
+Concerning 
+the 
+flexibility estimation of distribution grids, an index is 
+proposed in [6] that is related to specific viewpoints such 
+as power regulation ability and energy balance ability of 
+distribution grids. In [7] the net load uncertainty is 
+considered in the flexibility assessment of distribution 
+grids.  
+The secure grid operation can be verified by assessing the 
+required flexibility considering boundaries of uncertain 
+variables. A hyper-rectangle is implemented for heat 
+exchanger networks [8] to define the multi-dimensional 
+region of uncertain variables. In this paper, the flexibility 
+and the lack of flexibility are quantified by determining 
+acceptable boundaries of uncertain parameters in 
+distribution grids. 
+A large portion of uncertainty resources are not monitored 
+by the Distribution System Operator (DSO) and as a 
+consequence, the level of available flexibility of the grid 
+as well as the lack of flexibility caused by the uncertain 
+resources are not identified. In such a situation, the power 
+flows from unmonitored and uncontrolled resources may 
+cause voltage violations and congestions. Therefore, the 
+monitoring of grid and the control of resources connected 
+to the grid are vital for the secure operation of distribution 
+grid.  
+The DSOs around the world are going through the roll-out 
+of smart metering and grid monitoring devices [9, 10]. 
+However, equipping all the nodes of distribution grid with 
+monitoring devices is practically impossible due to costs 
+of required underlying infrastructure. In addition, although 
+all the grid topology information is usually available in the 
+DSO’s Geographic Information System (GIS), a trustable 
+and up-to-date data of LV grid topology and parameters is 
+not available to take decisions for the secure grid operation 
+[11]. To face these challenges, the grid sensitivity 
+coefficients of a subset of grid nodes can be used as an 
+approximation of the power flow model. The sensitivity 
+coefficients can be calculated between a plurality of 
+measurement nodes without using the information of grid 
+parameters [13], hereafter called model-less approach. 
+This approach significantly reduces the required number 
+of monitored nodes, thus reducing the cost of required grid 
+monitoring infrastructure, and does not rely on the 
+availability of an accurate and up-to-date grid parameters. 
+This paper proposes a flexibility estimation model for 
+distribution grids. The main contributions are as follows: 
+1) The proposed flexibility estimation method is based on 
+the feasibility study of the uncertain space of load 
+containing electric vehicles and photovoltaic powers. It 
+allows evaluating the flexibility and lack of flexibility 
+based on the coverage of the feasible space to the 
+
+CIRED CIRED workshop on E-mobility and power distribution systems 
+Porto, 2-3 June 2022 
+ 
+Paper n°  1347 
+ 
+ 
+CIRED 2022 Workshop  
+2/4 
+uncertain region. 
+2) The projection of each direction in the uncertain space 
+to the feasible space is illustrated and investigated. The 
+illustrated projection to the feasible space can assist in 
+determining the reason of flexibility inadequacy. 
+3) The linear power flow model based on the sensitivity 
+coefficients is used to model the grid operation 
+constraints. It properly accounts for the secure grid 
+operation within acceptable voltage levels and without 
+congestions.  
+4) The proposed model for the calculation of the 
+flexibility index is a linear and convex mathematical 
+optimization problem that can be solved efficiently 
+with non-commercial solvers and the optimality of the 
+solution is guaranteed.  
+The remaining parts of the paper is structured as following. 
+Next section provides the definition of distribution grid 
+flexibility. Then, the proposed mathematical formulation 
+is presented. The results of applying the method on a real 
+LV grid are evaluated. Finally, the conclusions and 
+outcomes are discussed. 
+THE FLEXIBILITY OF DISTRIBUTION GRID 
+In this paper, the flexibility of distribution grid is described 
+as the capability of distribution grid to effectively cope 
+with multiple uncertainties of grid operation. Fig. 1 
+illustrates the structure of the proposed flexibility 
+estimation algorithm in a LV distribution grid. The power 
+grid contains Electric Vehicles (EV), EV charging 
+stations, photovoltaic arrays and commercial load. In the 
+proposed method, grid monitoring data are used to 
+calculate the grid state and the sensitivity coefficients as 
+well as to model uncertain parameters. Fig. 1 shows the 
+way that flexibility quantification is realized using the grid 
+monitoring data. 
+ 
+Flexibility assessment and quantification
+MV/LV
+DSO
+ Real-time data acquisition
+Monitoring device
+ 
+Fig. 1. Proposed monitoring based flexibility 
+quantification structure 
+ 
+In fact, the proposed model is accomplished by detailed 
+and accurate information of the resources from the grid 
+monitoring facilities. Fig. 2 shows the concept of proposed 
+flexibility quantification method. The uncertain space is 
+calculated using historic data and prediction methods. The 
+margin of feasible space characterizes the maximum 
+feasible deviation from the expected operation point ‘O’. 
+Any operation point in the uncertain space can be 
+characterized by two components: the direction vector and 
+the normalized variation value. The direction matrix 
+expresses the direction vector, in which each diagonal 
+element indicates an uncertain variable, and the range of 
+element’s value is [−1, 1] [12]. In order to guarantee that 
+direction matrix elements represent a unique direction, at 
+least one of the direction matrix elements must be set to -
+1 or 1. This will refrain the duplication of direction 
+matrices with the same direction like diag(1, 0.5) and 
+diag(0.8, 0.4).  
+Any point outside the feasible region indicates an 
+infeasible grid operation point where the technical 
+constraints of the grid are not satisfied. The minimum 
+feasible deviation direction stands for the critical direction. 
+The boundary point, shown by ‘C’ in Fig.2, corresponding 
+to the critical direction is the flexibility index which 
+reveals the adequacy or insufficiency of flexibility 
+resources in the system. 
+O
+C
+Feasible Region
+Max
+pv
+P
+min
+pv
+P
+min
+lP
+Max
+lP
+Uncertain Region
+pv
+P
+lP
+pv
+P
+lP
+ 
+Fig. 2. The concept of the proposed flexibility 
+quantification method 
+MATHEMATICAL FORMULATION 
+A mathematical optimization problem is formulated to 
+determine the feasible space. The objective function is the 
+minimum value of maximum feasible variation in the 
+direction D (𝛽𝐷) of feasible operation region. In addition, 
+the flexibility (F) is defined as the minimum value of 𝛽𝐷. 
+ 
+𝐹 = 𝑚𝑖𝑛𝛽𝐷 
+(1) 
+𝛽𝐷 = 𝑚𝑎𝑥(𝛽) 
+(2) 
+𝑋 = 𝑋̃ + 𝛽𝐷∆𝑋 
+(3) 
+ 
+where 𝑋 and 𝑋̃ represent the uncertain parameter and its 
+expected value which are distribution grid’s load and PV 
+generation. ∆𝑋 indicates the expected value and 
+maximum/minimum value of uncertain parameter’s 
+difference. If the flexibility index is equal to one (F=1), the 
+flexibility resources are enough for the secure operation of 
+the grid. In addition, it means that feasible region covers 
+the uncertain region. However, if the flexibility index is 
+less than one (F <1), there is not sufficient margin for the 
+technically secure grid operation and additional flexibility 
+resources are required. The direction matrix D is a 
+diagonal matrix that can be defined as: 
+𝐷 =
+[
+ 
+ 
+ 𝑑1
+𝑑2
+⋱
+𝑑𝑛]
+ 
+ 
+ 
+ 
+(4) 
+
+CIRED CIRED workshop on E-mobility and power distribution systems 
+Porto, 2-3 June 2022 
+ 
+Paper n°  1347 
+ 
+ 
+CIRED 2022 Workshop  
+3/4 
+where diagonal element di is the direction on the ith 
+uncertain parameter and n is the number of uncertain 
+parameters.  
+For the flexibility estimation a linear power flow model is 
+used based on the sensitivity coefficients calculated using 
+the model-less approach. The formulation is described 
+below. 
+Δ𝑉𝑚 = ∑ [𝐾𝑚,𝑛
+𝑉𝑃 Δ𝑃𝑛 + 𝐾𝑚,𝑛
+𝑉𝑄 Δ𝑄𝑛]
+𝑛
+ 
+(5) 
+Δ𝐼𝑙 = ∑ [𝐾𝑙,𝑛
+𝐼𝑃Δ𝑃𝑛 + 𝐾𝑙,𝑛
+𝐼𝑄Δ𝑄𝑛 ]
+𝑛
+ 
+(6) 
+𝑉𝑚𝑖𝑛 ≤ 𝑉𝑚
+0 + ∆𝑉𝑚 ≤ 𝑉𝑚𝑎𝑥 
+(7) 
+−𝐼𝑚𝑎𝑥 ≤ 𝐼𝑙
+0 + ∆𝐼𝑙
+≤ 𝐼𝑚𝑎𝑥 
+(8) 
+∆𝑉𝑚 = 𝑉𝑚 − 𝑉𝑚
+0 
+(9) 
+∆𝐼𝑙
+= 𝐼𝑙 − 𝐼𝑙
+0 
+(10) 
+0 ≤ 𝑆𝑔𝑟𝑖𝑑 ≤ 𝑠𝑇𝑟𝑎𝑓𝑜
+𝑚𝑎𝑥  
+(11) 
+The voltage and current sensitivity coefficients are used in 
+equations (5) and (6), respectively, to model the impacts 
+of nodal active and reactive power changes (Δ𝑃𝑛 and 
+Δ𝑄𝑛 ) on the nodal voltage variations (Δ𝑉𝑚) and branch 
+current variations (Δ𝐼𝑙). The sensitivity coefficients, i.e., 
+𝐾𝑉𝑃, 𝐾𝑉𝑄, 𝐾𝐼𝑃𝑎𝑛𝑑 𝐾𝐼𝑄, are computed around the grid 
+operation point corresponding to voltage 𝑉𝑚
+0 and current 
+𝐼𝑙,𝑡
+0 . The voltage level at each node and the branch current 
+should be within the allowed limits as given in (7) and (8), 
+respectively. The voltage and current deviations can be 
+calculated as given by equations (9) and (10). The 
+constraint (11) limits the apparent power flow in the 
+MV/LV transformer (𝑆𝑔𝑟𝑖𝑑) by the capacity of transformer 
+(𝑠𝑚𝑎𝑥
+𝑇𝑟𝑎𝑓𝑜). 
+RESULTS AND DISCUSSIONS 
+In this section, the performance of proposed flexibility 
+calculation method is tested and validated in a modified 
+LV grid of Switzerland. The LV grid includes a PV unit 
+with known capacity to the DSO, and connected to bus 101 
+with the capacity of 72 kVA. Further, EVSEs are 
+connected to the buses 102 and 106. The grid with several 
+GridEye monitoring devices is illustrated in Fig. 3. The 
+real-time information of the aggregated loads and 
+productions are provided by the GridEye monitoring 
+devices. Figs. 4 and 5 illustrate the 10-minute granularity 
+of PV generation and load profiles used in the simulations. 
+In this work, the DSO’s objective is to estimate the 
+distribution grid’s flexibility value, the value of flexibility 
+insufficiency and the periods with insufficient flexibility. 
+ 
+ 
+Fig. 3. The LV grid topology with grid monitoring devices 
+ 
+Fig. 4. Load profile of LV grid  
+ 
+ 
+Fig. 5. PV generation at node 101  
+ 
+Table 1 provides the simulation results for time slots that 
+there is insufficient flexibility in the grid, considering the 
+uncertainties of load and PVs. For the remaining time slots, 
+the flexibility value is equal to one indicating that there is 
+adequate flexibility in the grid. At the time periods from 
+09:20 to 10:10 and from 17:40 to 18:20 on Nov 28, 2021 
+due to the high level of load and the thermal capacity of 
+lines, the flexibility is less than one. The uncertain and 
+feasible regions for two sample time slots are depicted in 
+Figs. 6 and 7. The value of flexibility at time slot 18:00 
+(Fig. 7) has the lowest value 0.49. In this time slot, the load 
+level is higher than the average consumption and the PV 
+generation has the lowest value. Moreover, at time slot 
+09:40 (Fig. 6), the load is at its highest value and the value 
+of flexibility is 0.74. At this time slot, although the load 
+has the highest value, the flexibility index is higher than 
+the one at time slot 18:00. The higher level of flexibility is 
+due to the high level of PV generation at this time which is 
+20.4 kW.  
+  
+
+CIRED100
+103
+104
+101
+Low voltage grid
+105
+102
+EVSE
+Grid monitoring device
+EVSE240-
+220
+(M)peo
+200
+180
+160-
+00:00
+06:00
+12:00
+18:00
+Nov28,2021
+Time20-
+15-
+PV (kW)
+10-
+5-
+00:00
+06:00
+12:00
+18:00
+Nov28,2021
+Time CIRED workshop on E-mobility and power distribution systems 
+Porto, 2-3 June 2022 
+ 
+Paper n°  1347 
+ 
+ 
+CIRED 2022 Workshop  
+4/4 
+Table 1. Flexibility results for insecure time slots 
+Time slot 
+Flexibility 
+Nov 28, 2021, 09:20 
+0.93 
+Nov 28, 2021, 09:30 
+0.84 
+Nov 28, 2021, 09:40 
+0.74 
+Nov 28, 2021, 09:50 
+0.84 
+Nov 28, 2021, 10:00 
+0.80 
+Nov 28, 2021, 10:10 
+0.80 
+Nov 28, 2021, 17:40 
+0.67 
+Nov 28, 2021, 17:50 
+0.57 
+Nov 28, 2021, 18:00 
+0.49 
+Nov 28, 2021, 18:10 
+0.72 
+Nov 28, 2021, 18:20 
+0.90 
+ 
+ 
+Fig. 6. Feasible and uncertain regions on Nov 28, 2021 at 
+09:40. 
+ 
+ 
+Fig. 7. Feasible and uncertain regions on Nov 28, 2021 at 
+18:00. 
+CONCLUSIONS 
+In this paper, a flexibility estimation methodology is 
+proposed taking into account the feasible grid operation 
+region and the uncertain region of load containing electric 
+vehicle and photovoltaic powers. The method is applicable 
+by using the information of grid monitoring devices. The 
+simulation results on the LV grid illustrated the inadequate 
+flexibility in time slots with the higher level of load and 
+the lower level of PV generation than the average level.  
+However, the model can detect other factors like voltage 
+limit violation regarding the grid constraints that limit the 
+grid’s flexibility. The outcome of proposed model informs 
+the grid operator regarding the time slots with sufficient 
+and insufficient flexibility along with the value of 
+insufficiency considering the uncertainty space. 
+ 
+Acknowledgments 
+This project is supported by the European Union’s Horizon 
+2020 programme under the Marie Sklodowska-Curie grant 
+agreement no. 101026259. 
+ 
+REFERENCES 
+ 
+[1] North American Electric Reliability Corporation, 
+"Accommodating high levels of variable generation" 
+(North American Electric Reliability Corp, 2009) 
+[2] International Energy Agency, "Harnessing variable 
+renewables: A guide to the balancing challenge" 
+(Organisation for Economic Co-operation and 
+Development, 2011) 
+[3] H. Nosair, F. Bouffard, 2015. "Flexibility envelopes for 
+power 
+system 
+operational 
+planning", 
+IEEE 
+Transactions on Sustainable Energy, 6(3), pp.800-
+809.  
+[4] J. Zhao, T. Zheng, E. Litvinov, 2015. "A unified 
+framework for defining and measuring flexibility in 
+power system". IEEE Transactions on power systems, 
+31(1), 339-347  
+[5] F.L. Müller, J. Szabó, O. Sundström, and J. Lygeros, 
+2017. "Aggregation and disaggregation of energetic 
+flexibility from distributed energy resources", IEEE 
+Transactions on Smart Grid, 10(2), 1205-1214..  
+[6] F. Chen, C. Huang, L. Wang, C. Zhu, C. Wang, and 
+Xie, N., 2017, November. "Flexibility evaluation of 
+distribution network with high penetration of variable 
+generations." In 2017 IEEE Conference on Energy 
+Internet and Energy System Integration (EI2),  1-6  
+[7] A. Majzoobi, A. Khodaei, 2016. "Application of 
+microgrids in supporting distribution grid flexibility." 
+IEEE Transactions on Power Systems, 32(5), 
+pp.3660-3669. 
+[8] Li, J. , Du, J. , Zhao, Z. , et al.: "An Efficient Method 
+for Flexibility Analysis of Large -scale Nonconvex 
+Heat Exchanger Networks", Industrial & Engineering 
+Chemistry Research , 2015, 54, (43),  10757 -10767 
+[9] European Commission. "Smart Metering deployment 
+in the European Union", http://bit.ly/2MbDfeL, 
+accessed Feb 2021.  
+[10] Swiss Federal Office of Energy. "Smart Grids", 
+http://bit.ly/3pB8vS5, accessed Feb 2021  
+[11] L. Richaud, R. Pellerej, C. Benoit, and E .Ramos, 
+CIRED 2019, "Analysis of voltage patterns for 
+topology identification and GIS correction". 
+[12] L. Jilong, J. Du, Z. Zhao, and P. Yao, 2015, "Efficient 
+method for flexibility analysis of large-scale 
+nonconvex heat exchanger networks. " Industrial & 
+Engineering Chemistry Research 54, no. 43, 10757-
+10767. 
+[13] Method of determining mutual voltage sensitivity 
+coefficients between a plurality of measuring nodes of 
+an electric power network, US patent, 0003811, Jan. 
+2, 2020.
+ 
+
+CIRED300-
+Load (kW)
+Uncertain region
+250
+Feasible region
+200-
+16
+18
+20
+22
+24
+26
+PV(kW)300-
+Load (kW)
+Uncertain region
+250
+Feasible region
+200
+16
+18
+20
+22
+24
+26
+PV(kW)
\ No newline at end of file
diff --git a/AtE2T4oBgHgl3EQfRQeF/content/tmp_files/load_file.txt b/AtE2T4oBgHgl3EQfRQeF/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f3c7334ca03189a8fa39e06212b357ed654d8afd
--- /dev/null
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@@ -0,0 +1,207 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf,len=206
+page_content='CIRED workshop on E-mobility and power distribution systems  Porto, 2-3 June 2022  Paper n°  1347  CIRED 2022 Workshop  1/4  ESTIMATING REQUIRED FLEXIBILITY FOR SECURE DISTRIBUTION GRID  OPERATION CONSIDERING THE UNCERTAINTIES OF EV AND PV  Manijeh ALIPOUR  Omid ALIZADEH-MOUSAVI  Depsys, Puidoux - Switzerland  Depsys, Puidoux - Switzerland  manijeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='alipour@depsys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='com  omid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='mousavi@depsys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='com  ABSTRACT  Renewable energy productions and electrification of  mobility are promising solutions to reduce greenhouse gas  emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Their effective integration in a power grid  encounters several challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The uncertain nature of  renewable energy productions as well as stochastic  consumptions of electric vehicles introduce remarkable  intermittency to a distribution grid and results in bi-  uncertain characteristics of both  supply and demand sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  One way to verify the secure grid operation within  acceptable voltage and loading levels is to assess its  required flexibility considering possible boundaries of  uncertain variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In this paper, first a comprehensive  linear model of distribution grid considering all pertaining  constraints is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Then, a flexibility estimation  technique is proposed based on the feasibility study of the  uncertain space of photovoltaic power productions and  load  containing  electric  vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  The  proposed  methodology uses grid monitoring data to determine grid  state and to model uncertain parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The model is  applied on a real low voltage (LV) system equipped with  grid monitoring devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  INTRODUCTION  The deployment of renewable energy productions has  many advantages, comprising economic convenience,  reducing the reliance on fossil fuel markets (especially, gas  and oil) and environmental friendliness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In addition, the  wide penetration of renewable energy sources accelerates  occupation in the EU, by job creation in different ‘green’  technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Aim behind the European Green Deal  (COM(2019) 640 final) is to be the world’s first climate-  neutral continent by 2050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In spite of renewable  generation’s significant advantages, it has the defects of  uncertainty and fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Besides to renewable  generations, electric vehicles as a new variable load  increase the intermittency of distribution grid and leads to  bi-uncertain characteristics of both demand and supply  sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' With the increased penetration of these uncertainty  sources, the modern power system is confronted with a  challenge to preserve the reliability, security, and quality  of supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Thus, a more flexible distribution grid is  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  The distribution grid flexibility can be defined from  different perspectives [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In [2], the flexibility is specified  as the degree to which a system can change its electricity  consumption and production in response to anticipated or  unanticipated variabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The flexibility envelope  method is presented in [3] to assess the flexibility potential  of individual power system assets and their aggregation at  the system level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' A flexibility measure is developed in [4]  which indicates the largest uncertainty deviation that a  system can bear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In [5] an algorithm is proposed which  allocates aggregate-level control decisions amongst  individual  systems  economically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  Concerning  the  flexibility estimation of distribution grids, an index is  proposed in [6] that is related to specific viewpoints such  as power regulation ability and energy balance ability of  distribution grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In [7] the net load uncertainty is  considered in the flexibility assessment of distribution  grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  The secure grid operation can be verified by assessing the  required flexibility considering boundaries of uncertain  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' A hyper-rectangle is implemented for heat  exchanger networks [8] to define the multi-dimensional  region of uncertain variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In this paper, the flexibility  and the lack of flexibility are quantified by determining  acceptable boundaries of uncertain parameters in  distribution grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  A large portion of uncertainty resources are not monitored  by the Distribution System Operator (DSO) and as a  consequence, the level of available flexibility of the grid  as well as the lack of flexibility caused by the uncertain  resources are not identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In such a situation, the power  flows from unmonitored and uncontrolled resources may  cause voltage violations and congestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Therefore, the  monitoring of grid and the control of resources connected  to the grid are vital for the secure operation of distribution  grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  The DSOs around the world are going through the roll-out  of smart metering and grid monitoring devices [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  However, equipping all the nodes of distribution grid with  monitoring devices is practically impossible due to costs  of required underlying infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In addition, although  all the grid topology information is usually available in the  DSO’s Geographic Information System (GIS), a trustable  and up-to-date data of LV grid topology and parameters is  not available to take decisions for the secure grid operation  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' To face these challenges, the grid sensitivity  coefficients of a subset of grid nodes can be used as an  approximation of the power flow model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The sensitivity  coefficients can be calculated between a plurality of  measurement nodes without using the information of grid  parameters [13], hereafter called model-less approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  This approach significantly reduces the required number  of monitored nodes, thus reducing the cost of required grid  monitoring infrastructure, and does not rely on the  availability of an accurate and up-to-date grid parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  This paper proposes a flexibility estimation model for  distribution grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The main contributions are as follows:  1) The proposed flexibility estimation method is based on  the feasibility study of the uncertain space of load  containing electric vehicles and photovoltaic powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' It  allows evaluating the flexibility and lack of flexibility  based on the coverage of the feasible space to the  CIRED CIRED workshop on E-mobility and power distribution systems  Porto, 2-3 June 2022  Paper n°  1347  CIRED 2022 Workshop  2/4  uncertain region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  2) The projection of each direction in the uncertain space  to the feasible space is illustrated and investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The  illustrated projection to the feasible space can assist in  determining the reason of flexibility inadequacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  3) The linear power flow model based on the sensitivity  coefficients is used to model the grid operation  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' It properly accounts for the secure grid  operation within acceptable voltage levels and without  congestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  4) The proposed model for the calculation of the  flexibility index is a linear and convex mathematical  optimization problem that can be solved efficiently  with non-commercial solvers and the optimality of the  solution is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  The remaining parts of the paper is structured as following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  Next section provides the definition of distribution grid  flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Then, the proposed mathematical formulation  is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The results of applying the method on a real  LV grid are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Finally, the conclusions and  outcomes are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  THE FLEXIBILITY OF DISTRIBUTION GRID  In this paper, the flexibility of distribution grid is described  as the capability of distribution grid to effectively cope  with multiple uncertainties of grid operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 1  illustrates the structure of the proposed flexibility  estimation algorithm in a LV distribution grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The power  grid contains Electric Vehicles (EV), EV charging  stations, photovoltaic arrays and commercial load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In the  proposed method, grid monitoring data are used to  calculate the grid state and the sensitivity coefficients as  well as to model uncertain parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 1 shows the  way that flexibility quantification is realized using the grid  monitoring data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  Flexibility assessment and quantification  MV/LV  DSO  Real-time data acquisition  Monitoring device  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Proposed monitoring based flexibility  quantification structure  In fact, the proposed model is accomplished by detailed  and accurate information of the resources from the grid  monitoring facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 2 shows the concept of proposed  flexibility quantification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The uncertain space is  calculated using historic data and prediction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The  margin of feasible space characterizes the maximum  feasible deviation from the expected operation point ‘O’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  Any operation point in the uncertain space can be  characterized by two components: the direction vector and  the normalized variation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The direction matrix  expresses the direction vector, in which each diagonal  element indicates an uncertain variable, and the range of  element’s value is [−1, 1] [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In order to guarantee that  direction matrix elements represent a unique direction, at  least one of the direction matrix elements must be set to -  1 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' This will refrain the duplication of direction  matrices with the same direction like diag(1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='5) and  diag(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  Any point outside the feasible region indicates an  infeasible grid operation point where the technical  constraints of the grid are not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The minimum  feasible deviation direction stands for the critical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  The boundary point, shown by ‘C’ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='2, corresponding  to the critical direction is the flexibility index which  reveals the adequacy or insufficiency of flexibility  resources in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  O  C  Feasible Region  Max  pv  P  min  pv  P  min  lP  Max  lP  Uncertain Region  pv  P  lP  pv  P  lP  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The concept of the proposed flexibility  quantification method  MATHEMATICAL FORMULATION  A mathematical optimization problem is formulated to  determine the feasible space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The objective function is the  minimum value of maximum feasible variation in the  direction D (𝛽𝐷) of feasible operation region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In addition,  the flexibility (F) is defined as the minimum value of 𝛽𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  𝐹 = 𝑚𝑖𝑛𝛽𝐷  (1)  𝛽𝐷 = 𝑚𝑎𝑥(𝛽)  (2)  𝑋 = 𝑋̃ + 𝛽𝐷∆𝑋  (3)  where 𝑋 and 𝑋̃ represent the uncertain parameter and its  expected value which are distribution grid’s load and PV  generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' ∆𝑋 indicates the expected value and  maximum/minimum value of uncertain parameter’s  difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' If the flexibility index is equal to one (F=1), the  flexibility resources are enough for the secure operation of  the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In addition, it means that feasible region covers  the uncertain region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' However, if the flexibility index is  less than one (F <1), there is not sufficient margin for the  technically secure grid operation and additional flexibility  resources are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The direction matrix D is a  diagonal matrix that can be defined as:  𝐷 =  [  𝑑1  𝑑2  ⋱  𝑑𝑛]  (4)  CIRED CIRED workshop on E-mobility and power distribution systems  Porto, 2-3 June 2022  Paper n°  1347  CIRED 2022 Workshop  3/4  where diagonal element di is the direction on the ith  uncertain parameter and n is the number of uncertain  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  For the flexibility estimation a linear power flow model is  used based on the sensitivity coefficients calculated using  the model-less approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The formulation is described  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  Δ𝑉𝑚 = ∑ [𝐾𝑚,𝑛  𝑉𝑃 Δ𝑃𝑛 + 𝐾𝑚,𝑛  𝑉𝑄 Δ𝑄𝑛]  𝑛  (5)  Δ𝐼𝑙 = ∑ [𝐾𝑙,𝑛  𝐼𝑃Δ𝑃𝑛 + 𝐾𝑙,𝑛  𝐼𝑄Δ𝑄𝑛 ]  𝑛  (6)  𝑉𝑚𝑖𝑛 ≤ 𝑉𝑚  0 + ∆𝑉𝑚 ≤ 𝑉𝑚𝑎𝑥  (7)  −𝐼𝑚𝑎𝑥 ≤ 𝐼𝑙  0 + ∆𝐼𝑙  ≤ 𝐼𝑚𝑎𝑥  (8)  ∆𝑉𝑚 = 𝑉𝑚 − 𝑉𝑚  0  (9)  ∆𝐼𝑙  = 𝐼𝑙 − 𝐼𝑙  0  (10)  0 ≤ 𝑆𝑔𝑟𝑖𝑑 ≤ 𝑠𝑇𝑟𝑎𝑓𝑜  𝑚𝑎𝑥  (11)  The voltage and current sensitivity coefficients are used in  equations (5) and (6), respectively, to model the impacts  of nodal active and reactive power changes (Δ𝑃𝑛 and  Δ𝑄𝑛 ) on the nodal voltage variations (Δ𝑉𝑚) and branch  current variations (Δ𝐼𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The sensitivity coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=',  𝐾𝑉𝑃, 𝐾𝑉𝑄, 𝐾𝐼𝑃𝑎𝑛𝑑 𝐾𝐼𝑄, are computed around the grid  operation point corresponding to voltage 𝑉𝑚  0 and current  𝐼𝑙,𝑡  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The voltage level at each node and the branch current  should be within the allowed limits as given in (7) and (8),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The voltage and current deviations can be  calculated as given by equations (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The  constraint (11) limits the apparent power flow in the  MV/LV transformer (𝑆𝑔𝑟𝑖𝑑) by the capacity of transformer  (𝑠𝑚𝑎𝑥  𝑇𝑟𝑎𝑓𝑜).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  RESULTS AND DISCUSSIONS  In this section, the performance of proposed flexibility  calculation method is tested and validated in a modified  LV grid of Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The LV grid includes a PV unit  with known capacity to the DSO, and connected to bus 101  with the capacity of 72 kVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Further, EVSEs are  connected to the buses 102 and 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The grid with several  GridEye monitoring devices is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The  real-time information of the aggregated loads and  productions are provided by the GridEye monitoring  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 4 and 5 illustrate the 10-minute granularity  of PV generation and load profiles used in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  In this work, the DSO’s objective is to estimate the  distribution grid’s flexibility value, the value of flexibility  insufficiency and the periods with insufficient flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The LV grid topology with grid monitoring devices  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Load profile of LV grid  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' PV generation at node 101  Table 1 provides the simulation results for time slots that  there is insufficient flexibility in the grid, considering the  uncertainties of load and PVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' For the remaining time slots,  the flexibility value is equal to one indicating that there is  adequate flexibility in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' At the time periods from  09:20 to 10:10 and from 17:40 to 18:20 on Nov 28, 2021  due to the high level of load and the thermal capacity of  lines, the flexibility is less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The uncertain and  feasible regions for two sample time slots are depicted in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The value of flexibility at time slot 18:00  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 7) has the lowest value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' In this time slot, the load  level is higher than the average consumption and the PV  generation has the lowest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Moreover, at time slot  09:40 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 6), the load is at its highest value and the value  of flexibility is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' At this time slot, although the load  has the highest value, the flexibility index is higher than  the one at time slot 18:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The higher level of flexibility is  due to the high level of PV generation at this time which is  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='4 kW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  CIRED100  103  104  101  Low voltage grid  105  102  EVSE  Grid monitoring device  EVSE240-  220  (M)peo  200  180  160-  00:00  06:00  12:00  18:00  Nov28,2021  Time20-  15-  PV (kW)  10-  5-  00:00  06:00  12:00  18:00  Nov28,2021  Time CIRED workshop on E-mobility and power distribution systems  Porto, 2-3 June 2022  Paper n°  1347  CIRED 2022 Workshop  4/4  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Flexibility results for insecure time slots  Time slot  Flexibility  Nov 28, 2021, 09:20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='93  Nov 28, 2021, 09:30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='84  Nov 28, 2021, 09:40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='74  Nov 28, 2021, 09:50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='84  Nov 28, 2021, 10:00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='80  Nov 28, 2021, 10:10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='80  Nov 28, 2021, 17:40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='67  Nov 28, 2021, 17:50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='57  Nov 28, 2021, 18:00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='49  Nov 28, 2021, 18:10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='72  Nov 28, 2021, 18:20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='90  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Feasible and uncertain regions on Nov 28, 2021 at  09:40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' Feasible and uncertain regions on Nov 28, 2021 at  18:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  CONCLUSIONS  In this paper, a flexibility estimation methodology is  proposed taking into account the feasible grid operation  region and the uncertain region of load containing electric  vehicle and photovoltaic powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The method is applicable  by using the information of grid monitoring devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The  simulation results on the LV grid illustrated the inadequate  flexibility in time slots with the higher level of load and  the lower level of PV generation than the average level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  However, the model can detect other factors like voltage  limit violation regarding the grid constraints that limit the  grid’s flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' The outcome of proposed model informs  the grid operator regarding the time slots with sufficient  and insufficient flexibility along with the value of  insufficiency considering the uncertainty space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content='  Acknowledgments  This project is supported by the European Union’s Horizon  2020 programme under the Marie Sklodowska-Curie grant  agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
+page_content=' 101026259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfRQeF/content/2301.03779v1.pdf'}
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+Learning Personalized Brain Functional Connectivity of MDD
+Patients from Multiple Sites via Federated Bayesian Networks
+Shuai Liu ∗
+Xiao Guo †
+Shun Qi ‡
+Huaning Wang §
+Xiangyu Chang ¶
+January 9, 2023
+Abstract
+Identifying functional connectivity biomarkers of major depressive disorder (MDD) patients is
+essential to advance understanding of the disorder mechanisms and early intervention. However,
+due to the small sample size and the high dimension of available neuroimaging data, the
+performance of existing methods is often limited. Multi-site data could enhance the statistical
+power and sample size, while they are often subject to inter-site heterogeneity and data-sharing
+policies. In this paper, we propose a federated joint estimator, NOTEARS-PFL, for simultaneous
+learning of multiple Bayesian networks (BNs) with continuous optimization, to identify disease-
+induced alterations in MDD patients. We incorporate information shared between sites and
+site-specific information into the proposed federated learning framework to learn personalized
+BN structures by introducing the group fused lasso penalty. We develop the alternating direction
+method of multipliers, where in the local update step, the neuroimaging data is processed at
+each local site. Then the learned network structures are transmitted to the center for the global
+update. In particular, we derive a closed-form expression for the local update step and use the
+iterative proximal projection method to deal with the group fused lasso penalty in the global
+update step. We evaluate the performance of the proposed method on both synthetic and
+real-world multi-site rs-fMRI datasets. The results suggest that the proposed NOTEARS-PFL
+yields superior effectiveness and accuracy than the comparable methods.
+1
+Introduction
+Major depressive disorder (MDD) is one of the most prevalent, costly, and disabling mental disorders
+worldwide [Cassano and Fava, 2002, Jia et al., 2010]. It is often characterized by persistent sadness,
+worthlessness, hopelessness, anxiety, and cognitive impairments [Fu et al., 2008]. The negative
+psychology of MDD can lead to severe consequences, including suicide [Jia et al., 2015]. The
+∗Department of Information Systems and Intelligent Business, School of Management, Xi’an Jiaotong University;
+email: hljliushuai@stu.xjtu.edu.cn.
+†Corresponding author: Center for Modern Statistics, School of Mathematics, Northwest University; email:
+xiaoguo@nwu.edu.cn.
+‡Key Laboratory of Biomedical Information Engineering of Ministry of Education, Institute of Health and
+Rehabilitation Science, School of Life Science and Technology, Xi’an Jiaotong University; email: qishun@xjtu.edu.cn.
+§Department of Psychiatry, Xijing Hospital, Air Force Medical University; email: xskzhu@fmmu.edu.cn.
+¶Center for Intelligent Decision-Making and Machine Learning, School of Management, Xi’an Jiaotong University;
+email: xiangyuchang@xjtu.edu.cn.
+1
+arXiv:2301.02423v1  [cs.LG]  6 Jan 2023
+
+productivity loss and disability due to MDD also pose a severe burden to society and affected
+families [Vos et al., 2017]. Therefore, understanding the pathogenesis of MDD is crucial for effective
+intervention, diagnosis, and treatment [Belmaker and Agam, 2008]. Diagnostic neuroimaging has
+been shown to be an effective modality to understand the underlying pathological mechanisms
+and cognitive information of brain diseases [Liu et al., 2016, 2017], and has been used for early
+identification of MDD [Smith et al., 2013]. Resting-state functional magnetic resonance imaging (rs-
+fMRI) is one of the most widely used imaging modes for MDD identification [Hamilton et al., 2011, Liu
+et al., 2013, Lin et al., 2016]. It can detect abnormal brain functional connectivity in MDD patients,
+allowing non-invasive investigation of the disease [Ye et al., 2015]. Functional connectivity (FC),
+an essential biomarker in neurological disorders, is traditionally defined as the Pearson correlation
+between different regions of interest (ROIs) [Ryali et al., 2012]. However, this association still does
+not truly determine the direction of interactions between ROIs. An accurate understanding of this
+directional information is critical to understanding the brain functional integration of MDD [Price
+et al., 2017]. To further investigate FC with rs-fMRI data, a number of approaches beyond the
+Pearson correlation are proposed. For example, Kandilarova et al. [2018] used a dynamic causal
+modeling (DCM) method to reveal differences of FC in anterior insula between healthy control (HC)
+participants and MDD patients. Hamilton et al. [2011] used Granger causality (GC) analysis method
+to investigate the aberrant patterns of FC in MDD, and identify the limbic inhibition of dorsal cortex
+as a key biomarker for MDD diagnosis . Nevertheless, these methods are not robust for analyzing
+specific individuals or large-scale ROIs [Molenaar, 2004, Friston et al., 2011].
+Bayesian network (BN), an approach for studying the conditional dependencies between variables
+in a system via a directed acyclic graph (DAG), has been used for detecting abnormal brain FC
+in MDD patients [Koller and Friedman, 2009, Ide et al., 2014, Liu et al., 2022]. Compared with
+DCM and GC methods, BN benefits from utilizing more samples to improve the robustness and
+stability of the resulting FC. Note that the amount of rs-fMRI data collected at a single imaging site
+is limited due to the difficult and expensive data acquisition in practice. Therefore, it is difficult
+to provide enough data from a single site to make a good BN estimate. Fortunately, multiple
+rs-fMRI datasets from different sites provide the possibility for enhancing the quality of learned BNs.
+However, multi-site data are often subject to inter-site heterogeneity and data-sharing policies [Li
+et al., 2020, Tong et al., 2022]. How to learn more helpful heterogeneous information from multi-site
+data with considering data-sharing policies is still one of the research hotspots of BN learning.
+In this paper, we consider the problem of learning BN structures using multiple datasets. Our
+initial interest arises from discovering noninvasive biomarkers inherent in multi-site rs-fMRI data.
+Generally, data collected at multiple imaging sites have a common disease of interest (such as MDD),
+which results in that they may share similar disease-induced connectivity abnormalities. Therefore,
+we consider leveraging data from multi-site to learn multiple BNs. We are committed to addressing
+the three challenges: (1) computational inefficiency of the structural learning algorithms of
+BNs, which partially comes from the DAG constraint and the resulting combinatorial optimization
+problem; (2) large heterogeneity of multi-site rs-fMRI data, which is caused by inter-site variation
+in scanners, image acquisition protocols, and patient populations across sites [Wang et al., 2022];
+(3) data sharing barriers, that is, patients worry that sharing their medical data will lead to
+personal information disclosure [Li et al., 2020]. The government is likely to promulgate some privacy
+protection regulations and inflict severe penalties for medical data breach events [Jin et al., 2019].
+To address the above issues, we propose a federated joint estimator for the simultaneous learning
+of multiple BNs with continuous optimization. In pursuit of computational efficiency, we convert the
+2
+
+traditional combinatorial BN structure learning problem into a continuous numerical optimization
+problem inspired by the NOTEARS method [Zheng et al., 2018]. To capture the heterogeneity, we
+learn multiple BNs from multi-site data but with similar structures by introducing the group fused
+lasso penalty instead of learning a single BN. To lighten the data-sharing barriers, we study the
+structure learning of multiple BNs in the federated learning (FL) setting, which is a natural method to
+tackle the data-sharing problem [Pfitzner et al., 2021]. We refer to the proposed NOTEARS-based
+Personalized Federated Learning framework of learning multiple BNs’ structures as NOTEARS-PFL,
+whose whole protocol is shown in Figure 1.
+Figure 1: Illustration of the learning framework of NOTEARS-PFL. Each site stores and processes
+rs-fMRI data locally. In the t-th iteration, each site estimates the local weighted adjacency matrix
+W k
+(t) through the local update step. Then, each site transmits the estimated local matrix to the
+center. Center updates the global weighted adjacency matrices ˜Z(t) through the global update step
+based on the received ˜
+W(t) = {W 1
+(t), . . . , W K
+(t)}. Finally, the center transmits the estimated global
+matrix to each site for the (t + 1)-th iteration update.
+3
+
+Center
+Global model: Z(t) ← argmin [L, (W(t),z,O(t-1))
+①(t) ←[W(t),Z(t), O(t-1))
+Wh
+()
+Z2
+02
+-1),0h
+W(t)
+-1)
+Local model: W()  arg win [L, (wK, z(-1), (-1))
++2
+data
+data
+data
+rs-fMRI
+preprocessing
+rs-fMRI
+preprocessing
+preprocessing
+rs-fMRI
+scan
+ scan
+scan
+Site K
+Site 1
+Site 2The merits of the proposed NOTEARS-PFL are multi-fold.
+• NOTEARS-PFL is capable of learning site-specific BNs from multi-site rs-fMRI data, which
+makes full use of the multi-site data while still being flexible enough to capture the site-specific
+BN structures.
+• NOTEARS-PFL is subtly fitted in an FL framework without sharing and exchanging the
+original data.
+• NOTEARS-PFL overcomes the computational intractability of traditional BNs by transforming
+the discrete combinatorial optimization to continuous numerical optimization.
+• NOTEARS-PFL can effectively detect the common abnormal FC in MDD patients among
+multiple sites, especially the connectivity between the insula, thalamus cortex, and middle
+temporal gyrus. Meanwhile, NOTEARS-PFL can also identify the site-specific abnormal FC
+due to the differences in scanners, protocols, and patient populations.
+The paper is organized as follows. In Section 2, we briefly review directed FC estimation methods
+for rs-fMRI data and BN structure learning with continuous optimization. We present the formulation
+and optimization approach of NOTEARS-PFL in Section 3. We conduct simulation experiments to
+validate NOTEARS-PFL in Section 4, and analyze the real-world multi-site rs-fMRI data in Section
+5. Finally, we conclude this paper in Section 6.
+2
+Related Work
+2.1
+Directed FC Estimation Methods for rs-fMRI Data
+The directed FC, also known as effective FC, helps depict the abnormalities or dysfunction of the
+brain connectivity network [Smith, 2012] and shows how information flows in the brain connectivity
+network [Henry and Gates, 2017]. Based on rs-fMRI data, many directed FC estimation methods
+have been developed, including the following four types.
+(1) Granger causality (GC). GC establishes a model in the framework of vector autoregression
+(VAR) and determines the edges to be added after considering the autoregressive effect of each region
+[Granger, 1969, Liao et al., 2009]. GC is applied to detect the effects of psychostimulants on youths
+with attention deficit hyperactivity disorder (ADHD) [Peterson et al., 2009]. However, when GC is
+applied to analyze specific individuals, heterogeneity across individuals may lead to spurious results
+[Molenaar, 2004].
+(2) Dynamic Causal Modeling (DCM). DCM is a method to capture nonlinear relationships
+between brain regions that may be affected by different stimuli during scanning [Friston et al., 2003],
+and has been used to detect the propagation of epileptic seizures [Murta et al., 2012]. The limitation
+of DCM is that it cannot handle large amounts of ROIs or analysis group rs-fMRI data. [Friston
+et al., 2011].
+(3) Constraint-based methods. Constraint-based methods estimate directed edges by conditional
+independence tests. Common constraint-based methods include Peter Spirtes, and Clark Glymour
+(PC) algorithm [Spirtes and Glymour, 1991], conservative PC (CPC) [Ramsey et al., 2006], fast
+causal inference (FCI) [Spirtes, 2001], and cyclic causal discovery (CCD) [Richardson, 1996]. The
+PC algorithm can obtain reliable networks on data aggregated across individuals [Smith et al., 2011].
+4
+
+However, according to results from Smith et al. [2011], PC can detect the existence of edges but can
+not accurately determine the directions at the individual level.
+(4) Score-based methods. Score-based methods define a score function that measures how well the
+BN structure fits the observed data and searches for the best network structure. Standard score-based
+methods include greedy equivalent search (GES) [Chickering, 2002], independent multiple-sample
+greedy equivalent search (IMaGES) [Ramsey et al., 2010], and group iterative multiple model
+Estimation (GIMME) [Gates and Molenaar, 2012]. Score-based methods typically require large
+samples to identify edges and directions, which limits the utility of these methods when using rs-fMRI
+data for individual-level analysis [Mumford and Ramsey, 2014]. Meanwhile, the performance of
+score-based methods is poor on data with a large number of ROIs, which makes it difficult to analyze
+the complete parcellation of the brain. Due to the combinatorial acyclicity constraint of DAG, it is
+often hard to find BN with the optimal score. And its scalability is also limited [Liu et al., 2022].
+2.2
+BN Structure Learning with Continuous Optimization
+In this work, we focus on BN learning.
+BN is a probabilistic graphical model that captures
+dependencies among a collection of random variables. Each model is associated with a directed
+network G = (V, E), where the vertex set V corresponds to the random variables and the edge set E
+corresponds to the set of conditional independence [Drton and Maathuis, 2017].
+BN models the conditional independence among random variables via Markov properties [Cowell,
+1998, Marrelec et al., 2004]. Denote X = (X1, . . . , Xd)⊤ that satisfies the local Markov property
+with respect to a DAG G if
+Xv⊥XndG(v)\paG(v) | XpaG(v),
+where paG(v) = {w ∈ V : {w, v} ∈ E} denotes the parents of node v in G, ndG(v) = V \deG(v)
+denotes the non-descendants of v in G with deG(v) = {w ∈ V : w = v or v → . . . → w in G}
+representing the descendants of v in G.
+Alternatively, if X is a continuous random vector and it has a density P(X) with respect to a
+product measure, then the local Markov property is equivalent to the following factorization property
+[Verma and Pearl, 1990],
+P(X) =
+d
+�
+v=1
+P
+�
+Xv | XpaG(v)
+�
+.
+(1)
+The structure learning of BN is to estimate the DAG G using the observations from the density
+P(X). Score-based methods seek the optimal G by minimizing the score function S(W) with respect
+to the corresponding weighted adjacency matrix W ∈ Rd×d, subject to that the induced graph g(W )
+is a DAG; see the LHS of equation (2). The DAG constraint initiates a combinatorial optimization
+problem, which is computationally costly when d is large.
+Recently, Zheng et al. [2018] proposed a NOTEARS method which enforces the acyclicity via
+setting a smooth function h(W ) exactly at zero; see the RHS of equation (2). By this novel
+characterization of acyclicity, the resulting optimization problem can be solved efficiently using
+standard algorithms.
+minW ∈Rd×d S(W ),
+minW ∈Rd×d S(W ),
+s.t. g(W ) ∈ DAGs,
+s.t. h(W ) = 0.
+(2)
+5
+
+Currently, NOTEARS has been extended to handle problems in many situations. For example,
+DAG Graph Neural Network (DAG-GNN) extends NOTEARS to solve nonlinear cases by incor-
+porating neural networks [Yu et al., 2019]. Graph AutoEncoder (GAE) extends NOTEARS and
+DAG-GNN into a graph autoencoder model, facilitating vector-valued variables [Ng et al., 2019].
+Pamfil et al. [2020] proposed DYNOTEARS method to learn dynamic BNs from time-series data.
+Huang et al. [2020] used reinforcement learning to find the optimal DAGs from incomplete data
+based on NOTEARS.
+To the best of our knowledge, few studies have focused on using NOTEARS to learn personalized
+BNs from fMRI data. Zhang et al. [2022] proposed a joint BN estimation model to detect abnormal
+directed FC in schizophrenia (SZ) patients. However, the joint estimation model proposed by Zhang
+et al. [2022] is not applicable for the FL setting. Ng and Zhang [2022] proposed NOTEARS-ADMM
+model to estimate the structure of BN in FL setting. However, they did not consider the data
+heterogeneity, which can lead to model bias [Zhao et al., 2018, Jiang et al., 2022]. Compared with
+Zhang et al. [2022] and Ng and Zhang [2022], the proposed NOTEARS-PFL is powerful because it
+captures the data heterogeneity, overcomes the data sharing barriers, and attains computational
+efficiency, simultaneously.
+3
+NOTEARS-based Personalized Federated Learning Framework
+3.1
+Preliminary
+The BN associated with a DAG G can also be thought of as a structural equation model [Bollen,
+1989]
+Xi = fi(W T
+i X) + Zi, i = 1, 2, . . . , d,
+(3)
+where W = (W1, . . . , Wd) ∈ Rd×d denotes the weighted adjacency matrix whose nonzero coefficients
+correspond to the directed edges in G, fi’s are functions related to the conditional distributions of
+Xi’s, and Zi’s are jointly independent random noises. Equation (3) is equivalent to the local Markov
+property and factorization property introduced in Section 2 [Drton and Maathuis, 2017].
+We consider the linear structural equation model
+Xi = W T
+i X + Zi, i = 1, 2, . . . , d.
+(4)
+Actually, the multivariate Gaussian distribution can be modeled as equation (4) with Zi’s being
+independent Gaussian noises.
+Let the data matrix x ∈ Rn×d. Zheng et al. [2018] formulates the structure learning of BNs as
+min
+W ∈Rd×d
+1
+2n ∥x − xW ∥2
+F + λ∥W ∥1,
+s.t.
+h(W ) = 0,
+(5)
+where
+h(W ) := tr
+�
+eW ⊙W �
+− d = 0,
+(6)
+⊙ denotes the Hadamard product, eA = �∞
+k=0
+1
+k!Ak denotes the the matrix exponential operator
+[Leonard, 1996], ∥A∥F denotes the Frobenius norm, ∥A∥1 refers to the entry-wise ℓ1 norm and the
+tuning parameter λ > 0 controls the amount of sparsity.
+6
+
+3.2
+Problem Formulation
+We now proceed to formulate our NOTEARS-PFL method, which can efficiently learn multiple
+heterogeneous BNs without sharing data.
+We consider the setting in which there is a fixed set of K sites in total, and each site has its
+own local dataset. The k-th site holds nk i.i.d. samples, denoted by xk =
+�
+xk
+i
+�nk
+i=1 ∈ Rnk×d. Let
+D = {x1, . . . , xK} be the K overall observational datasets. The goal is to infer the K weighted
+adjacency matrices ˜
+W0 = {W 1
+0 , . . . , W K
+0 } (correspond to the BN structures) using D.
+To incorporate the heterogeneity among K sites, we consider the following optimization problem:
+min
+˜
+W
+K
+�
+k=1
+1
+2nk
+���xk − xkW k���
+2
+F + R( ˜
+W ),
+s.t.
+h(W k) = 0, k = 1, . . . , K,
+(7)
+where h(W k) is defined in the same way as equation (6) and R( ˜
+W ) is a penalty function for inducing
+the similarity and sparsity within ˜
+W .
+With the data sharing constraint, we formulate equation (7) into the framework of ADMM,
+which is a natural solution to distributed learning and FL [Ng and Zhang, 2022]. Define a set of
+consensus variables ˜Z = {Z1, . . . , ZK}, which can be thought of as the global weighted adjacency
+matrices, in contrast to the local weighted adjacency matrices ˜
+W . Then we rewrite equation (7) as
+min
+˜
+W , ˜
+Z
+K
+�
+k=1
+L(W k, xk) + R( ˜Z),
+s.t.
+h(Zk) = 0,
+W k = Zk,
+k = 1, . . . , K,
+(8)
+where
+L(W k, xk) :=
+1
+2nk
+���xk − xkW k���
+2
+F ,
+(9)
+h(Zk) is defined in the same way as equation (6), and R( ˜Z) is defined by
+R( ˜Z) := λ1
+K
+�
+k=1
+∥Zk∥1 + λ2
+K−1
+�
+k=1
+∥Zk+1 − Zk∥F ,
+(10)
+where λ1 > 0 controls the sparsity of Zk and λ2 > 0 controls the similarity within ˜Z.
+The general optimization procedure can be summarized as follows. First, each site estimates the
+local weighted adjacency matrix W k by using local dataset xk. Then, instead of transmitting the
+original data, each site only transmits its local weighted adjacency matrix to the center. After that,
+the center updates the global weighted adjacency matrices ˜Z0 = {Z1
+0, . . . , ZK
+0 }, based on received
+˜
+W0, Finally, the center transmits the updated ˜Z to each site, which performs the next update locally
+based W k = Zk.
+In the next subsection, we provide the computational details of NOTEARS-PFL.
+7
+
+3.3
+Optimization Algorithm
+By applying the augmented Lagrangian method [Bertsekas, 2014], the equation (8) can be written as
+L
+�
+˜
+W , ˜Z, Θ
+�
+=
+K
+�
+k=1
+L(W k, xk) + R( ˜Z) +
+K
+�
+k=1
+αkh(Zk) + ρ1
+2
+K
+�
+k=1
+|h(Zk)|2 +
+K
+�
+k=1
+tr(βk(W k − Zk)⊤)
++ ρ2
+2
+K
+�
+k=1
+���W k − Zk���
+2
+F ,
+(11)
+where ˜Θ = {ρ1, ρ2, α1, . . . , αK, β1, . . . , βK}, ρ1, ρ2 are the penalty coefficients, α1, . . . , αK and
+β1, . . . , βK are the Lagrange multiplies.
+At the t-th iteration, we can solve equation (11) as follows:
+(1) Local update:
+˜
+W(t) ← arg min
+˜
+W
+�
+L
+�
+˜
+W , ˜Z(t−1), ˜Θ(t−1)
+��
+.
+(12)
+(2) Global update:
+˜Z(t) ← arg min
+˜
+Z
+�
+L
+�
+˜
+W(t), ˜Z, ˜Θ(t−1)
+��
+.
+(13)
+(3) Global update:
+˜Θ(t) ←
+�
+˜
+W(t), ˜Z(t), ˜Θ(t−1)
+�
+.
+(14)
+3.3.1
+Federated Local Update of ˜
+W
+Taking the derivative of equation (11) with respect to ˜
+W , we can update each W k
+(t) of ˜
+W(t) as
+W k
+(t) := arg min
+W k
+(L(W k, xk) + ρ2,(t−1)
+2
+���W k − Zk
+(t−1)
+���
+2
+F + tr(βk,(t−1)(W k − Zk
+(t−1))⊤)).
+(15)
+Let U k =
+1
+nk (xk)⊤xk and assume U k + ρ2I is invertible. Equation (15) has a closed form expression
+W k
+(t) = (U k + ρ2,(t−1)I)−1(ρ2,(t−1)Zk
+(t−1) − βk,(t−1) + U k),
+(16)
+where the details of the derivation are given in Appendix A.
+3.3.2
+Federated Global Update of ˜Z
+Similar to equation (15), we can update each Zk
+(t) of ˜Z(t) as
+Zk
+(t) := arg min
+Zk
+{L( ˜Z) + R( ˜Z)},
+(17)
+8
+
+Algorithm 1 Dykstra-like iterative proximal algorithm (DIPA)
+Input: K: Number of site; ˜Z: global adjacency matrices; ˜
+W : local adjacency matrices; N: The
+number of iterations; ϵ: Tolerance.
+Output: ˜Z: The result of equation (19).
+1: ∇Lk
+�
+Zk�
+= equation (20).
+2: for k = 1, . . . , K do
+3:
+U k = Zk − 1
+C ∇L
+�
+Zk�
+.
+4: end for
+5: ¯U ← Transform (U k).
+6: Set A0 = ¯U, Mn = 0, Qn = 0.
+7: for n = 0, . . . , N do
+8:
+V(n) ← proxR2
+�
+A(n) + M(n)
+�
+, see equation (28).
+9:
+M(n+1) ← A(n) + M(n) − V(n).
+10:
+A(n+1) ← proxR1
+�
+V(n) + Q(n)
+�
+, see equation (25).
+11:
+Q(n+1) ← V(n) + Q(n) − A(n+1).
+12:
+Break: if
+��A(n+1) − A(n)
+��
+F < ϵ.
+13: end for
+14: ¯U ← A(N+1).
+15: ˜Z ← Inverse-Transform ( ¯U).
+16: return ˜Z.
+where
+L( ˜Z) :=
+K
+�
+k=1
+αk,(t−1)h(Zk) + ρ1,(t−1)
+2
+K
+�
+k=1
+|h(Zk)|2 + ρ2,(t−1)
+2
+K
+�
+k=1
+���W k
+(t) − Zk���
+2
+F )
++
+K
+�
+k=1
+tr(βk,(t−1)(W k
+(t) − Zk)⊤).
+Due to the acyclicity term h(Zk), we are not able to derive a closed-form solution for equation
+(17). To tackle this issue, we utilize a proximal gradient method which constructs a quadratic
+approximation of L( ˜Z) at ˜Z(t−1), and updates ˜Z(t) by
+˜Z(t) = arg min
+˜
+Z
+{L( ˜Z(t−1)) +
+K
+�
+k=1
+�
+Zk − Zk
+(t−1), ∇Lk(Zk
+(t−1))
+�
++ C
+2
+K
+�
+k=1
+���Zk − Zk
+(t−1)
+���
+2
+F + R( ˜Z)},
+(18)
+where ∇Lk(·) is the gradient of Lk(·), C > 0 is the Lipschitz constant of ∇Lk(·). After some simple
+9
+
+manipulations of equation (18), e.g., ignoring constant terms of Zk
+(t−1), we have
+˜Z(t) = arg min
+˜
+Z
+{C
+2
+K
+�
+k=1
+����Zk −
+�
+Zk
+(t−1) − 1
+L∇Lk
+�
+Zk
+t−1
+������
+2
+F
++ R( ˜Z)},
+= proxCR
+� K
+�
+k=1
+�
+Zk
+(t−1) − 1
+L∇Lk
+�
+Zk
+(t−1)
+���
+,
+(19)
+where
+∇Lk
+�
+Zk
+(t−1)
+�
+= αk,(t−1)∇h(Zk
+(t−1))+ρ1,(t−1)h(Zk
+(t−1))∇h(Zk
+(t−1))−βk,(t−1)+ρ2,(t−1)(Zk
+(t−1)−W k),
+(20)
+and proxg(v) = arg min
+x
+( 1
+2∥x − v∥2
+F + g(x)) denotes the proximal operator.
+Unlike the calculation of W k, we cannot separate the equation (19) by k because of the grouped
+constraint R( ˜Z). Instead, we must estimate the whole set of matrices ˜Z(t) jointly. Inspired by
+Gibberd and Nelson [2017], we use the following iterative proximal projection step to solve.
+Let U k = Zk
+(t−1) − 1
+C ∇Lk
+�
+Zk
+(t−1)
+�
+, and perform transform procedure (see AppendixB) for
+U k → ¯U. Then, equation (19) can be re-written as
+min
+¯
+Z
+1
+2
+�� ¯Z − ¯U
+��2
+F
+�
+��
+�
+L( ¯
+Z)
++ λ1∥ ¯Z∥1
+� �� �
+R1( ¯
+Z)
++ λ2∥D ¯Z∥2,1
+�
+��
+�
+R2( ¯
+Z)
+,
+(21)
+where D ∈ R(K−1)×K is a backwards difference matrix of the form Di,i = −1, Di,i+1 = 1 for i =
+1, . . . , K − 1, and ∥A∥2,1 := �
+k ∥Ak,·∥2 denotes the group ℓ2,1 norm. Actually, equation (21) is
+equivalent to the Group-Fused Lasso Signal Approximator (GFLSA) defined by Gibberd and Nelson
+[2017]. We denote the objective in equation (21) by f
+� ¯U; λ1, λ2
+�
+= proxR1+R2
+� ¯U
+�
+. Then, we adopt
+the Dykstra-like iterative proximal algorithm (DIPA) [Combettes and Pesquet, 2011] to handle
+proxR1+R2
+� ¯U
+�
+and find a feasible solution for both the group fused lasso penalty and the lasso
+penalty. We summarize the details of iteration in Algorithm 1.
+3.3.3
+Federated Global Update of ˜Θ
+The update of ˜Θ follows the following iterative steps
+βk,t := βk,t−1 + ρ2,t−1
+�
+W k
+(t) − Zk
+(t)
+�
+,
+αk,t := αk,t−1 + ρ1,t−1h
+�
+Zk
+(t)
+�
+,
+ρ1,t := γ1ρ1,t−1,
+ρ2,t := γ2ρ2,t−1,
+(22)
+where γ1, γ1 ∈ R are hyperparameters that respectively control the increasing speeds of the coefficients
+ρ1, ρ2. The overall ADMM algorithm of NOTEARS-PFL is thereby given in Algorithm 2.
+10
+
+Algorithm 2 ADMM of NOTEARS-PFL
+Input: K: Number of site; xk: Original data; ˜Z(0): Initial global adjacency matrices; ρ1,0, ρ2,0,
+α1,0, . . . , αK,0, β1,0, . . . , βK,0: Initial parameters; T: The maximum of iterations; ϵ: Tolerance.
+Output:
+ˆ
+Zk: Estimated adjacency matrix.
+for iterations t = 1 to T do
+Local update step:
+for site k = 1, . . . , K do
+Update W k
+(t) according to equation (16).
+end for
+Global update step:
+for variables ˜Z(t−1), Θ(t−1) in the center do
+Zk
+(t) ← DIPA (W k
+(t), Zk
+(t−1)).
+Update ˜Θ(t−1) according to equation (22).
+end for
+Break: if
+K
+�
+k=1
+���Zk
+(t) − Zk
+(t−1)
+���
+F
+���Zk
+(t)
+���
+2
+< ϵ.
+end for
+return ˜Z = (Z1
+(T), ..., ZK
+(T)).
+11
+
+4
+Simulations
+4.1
+Simulation Design of Synthetic Data
+We simulate synthetic data according to equation (4). First, we generate a random graph Gtruth
+based on Erdös–Rényi (ER) model [Erdös et al., 1960]. To obtain a set of related graphs, we apply
+perturbations to Gtruth to create K similar but different graphs ˜G = {G1, . . . , GK}. The details of
+the definition of perturbation are shown in Appendix C, and we denote the perturbation level as pl.
+Given ˜G, we assign uniformly random edge weights to obtain K weights matrices
+˜
+W =
+{W 1, . . . , W K}, where the non-zero entries are sampled uniformly at random form [−2, −0.5]∪[0.5, 2].
+Given ˜
+W , we generate the datasets D = {x1, . . . , xK} based on equation (4) with standard Gaussian
+noise.
+To evaluate and compare the performance of the proposed method, we apply the NOTEARS
+[Zheng et al., 2018] method to estimate each network independently from each site’s local data
+(NOTEARS-SIG for short), and apply the NOTEARS method to learn a common network structure
+for all contexts (NOTEARS-AVG for short), i.e., an “average” network that treats all site’s data as
+samples in one dataset. We also compare the performance between NOTEARS-ADMM [Ng and
+Zhang, 2022] and the proposed method.
+We use the following four metrics to evaluate the learning performance of different methods (see
+Appendix C for a detailed definition of metrics).
+• Edge arrowhead error:
+Error =
+FP + FN
+TP + TN + FP + FN .
+(23)
+• Edge arrowhead precision:
+Precision =
+TP
+TP + FP .
+• Edge arrowhead F-score:
+F-score = 2 · Precision · Recall
+Precision + Recall ,
+where Recall = TP/(TP + FN).
+• Edge arrowhead structural Hamming distance (SHD): Given two structures, this
+measure is the minimum number of edges needed to convert the learned graph into the true
+one.
+We compare the performance of the proposed NOTEARS-PFL, NOTEARS-SIG, NOTEARS-
+AVG, and NOTEARS-ADMM on the following three aspects.
+Performance vs. the number of variable: We fix the number of sites K = 10, perturbation
+level pl = 10%, set the total sample size nt = �K
+k=1 nk equals to three times the number of variable d
+(nt = 3d, nk = 0.3d), and compare the model performance of network structure learning for different
+variable number d = 10, 20, 30, 40, 50, 60, 70, 80.
+12
+
+Performance vs. the number of site: We fix the number of variable d = 50, perturbation
+level pl = 10%. We set the total sample size nt = 256, and distributed evenly across the number of
+site K = 2, 4, 8, 16, 32, 64.
+Performance vs. perturbation level: We fix the number of site K = 10, the number of
+variable d = 30, sample size nt = 3d, and vary the perturbation level pl = 5%, 10%, 15%, 20%, 30%.
+The evaluation metrics for one simulation run are averaged over the K sites’ dataset, and we
+repeat each experiment 10 times. The final evaluation metrics are thereby averaged over the ten
+simulation runs.
+(a)
+(b)
+(d)
+(c)
+Figure 2: Simulation results for synthetic data under different variable numbers. (a) Average edge
+arrowhead error (lower is better); (b) Average edge arrowhead SHD (lower is better); (c) Average
+edge arrowhead precision (higher is better); (d) Average edge arrowhead F-score (higher is better).
+In the synthetic data experiment, we first evaluate the effect of the number of variables d on
+the performance of each compared method. The results are shown in Figure 2. It is clear that
+NOTEARS-PFL performs the best among all the methods. The average arrowhead adjacency error of
+NOTEARS-PFL has a relatively low level ranging from 0.02 to 0.13. In contrast, for NOTEARS-SIG,
+the average arrowhead adjacency error is far larger than NOTEARS-PFL, ranging from 0.04 to
+0.43. This may be because the sample size of a single dataset is too small for NOTEARS-SIG to
+give a satisfactory estimate. The same result is observed in the average arrowhead SHD. For the
+other two metrics, NOTEARS-PFL consistently results in higher average arrowhead precision and
+F-score than the other three methods. Compared with NOTEARS-SIG and NOTEARS-AVG, the
+average arrowhead precision of NOTEARS-PFL is also more stable with respect to different variable
+numbers, indicating that it can successfully identify most of the edges in high-dimensional settings.
+We also test the effect of the number of sites K on the performance of each method. The
+results are shown in Figure 3. Clearly, NOTEARS-PFL performs better than NOTEARS-SIG,
+NOTEARS-AVG, and NOTEARS-ADMM for all the tested cases. As K increases, the performance
+13
+
+0.4
+100
+0.3 -
+80 -
+SHD
+60
+0.2
+40 -
+0.1 -
+20 -
+0.0 -
+0
+20
+50
+10
+30
+40
+60
+70
+80
+10
+20
+30
+40
+50
+60
+70
+80
+Number of variables
+Number of variables
+0.9
+1.0-
+0.8 -
+0.7
+0.8
+0.6
+ion
+0.5
+Preci
+0.4 -
+0.4
+0.3
+0.2
+0.2
+0.1
+10
+20
+30
+40
+50
+80
+60
+70
+10
+20
+30
+40
+50
+60
+70
+80
+Number of variables
+Number of variables
+NOTEARS-PFL
+NOTEARS-SIG
+NOTEARS-AVG
+NOTEARS-ADMM(a)
+(b)
+(d)
+(c)
+Figure 3: Simulation results for synthetic data under different site numbers. (a) Average edge
+arrowhead error (lower is better); (b) Average edge arrowhead SHD (lower is better); (c) Average
+edge arrowhead precision (higher is better); (d) Average edge arrowhead F-score (higher is better).
+(a)
+(b)
+(d)
+(c)
+Figure 4: Simulation results for synthetic data under different perturbation levels (a) Average edge
+arrowhead error (lower is better); (b) Average edge arrowhead SHD (lower is better); (c) Average
+edge arrowhead precision (higher is better); (d) Average edge arrowhead F-score (higher is better).
+14
+
+0.10
+120
+100:
+0.08
+80 -
+0.06
+SHD
+60
+0.04
+40 -
+0.02
+20 -
+-0
+0.00
+10
+20
+50
+60
+0
+10
+20
+30
+40
+50
+60
+0
+30
+40
+Number of sites
+Number of sites
+1.0
+0.8
+0.8
+0.6
+Precisic
+0.4
+0.4
+0.2 -
+0.2
+0.0 -
+0
+10
+20
+30
+40
+50
+60
+0
+10
+20
+30
+50
+60
+40
+Number of sites
+Number of sites
+NOTEARS-PFL
+NOTEARS-SIG
+NOTEARS-AVG
+NOTEARS-ADMM0.08:
+35 -
+0.07
+30
+0.06 -
+25 
+0.05
+D
+0.04
+15-
+0.03
+10 -
+0.02
+5 -
+0.01
+5%
+10%
+15%
+20%
+30%
+5%
+10%
+15%
+20%
+30%
+Perturbation levels
+Perturbation levels
+1.00
+0.9
+0.95
+0.8
+0.90
+0.7
+0.85
+ision
+core
+0.80
+Precis
+0.6
+S
+F
+0.75
+0.5
+0.70 -
+0.4
+0.65
+0.60
+0.3
+5%
+10%
+15%
+20%
+30%
+5%
+10%
+15%
+20%
+30%
+Perturbation levels
+Perturbation levels
+NOTEARS-PFL
+NOTEARS-SIG
+NOTEARS-AVG
+NOTEARS-ADMMof NOTEARS-SIG, NOTEARS-AVG decline rapidly. Although the performance of NOTEARS-PFL
+also decreases with the increase of K, it still provides a relatively stable and satisfying performance.
+As expected, NOTEARS-SIG does not perform better at larger K. A possible reason is that
+NOTEARS-SIG only utilizes the information of a single dataset whose sample size decreases with
+the increase of K. On the contrary, NOTEARS-PFL works well in the setting with a large site
+number because information can be exchanged during optimization. These results are also consistent
+with the existing literature [Ng and Zhang, 2022].
+Finally, we examine the effect of the perturbation level pl on the performance of each method.
+The results are shown in Figure 4. It is clear that NOTEARS-PFL significantly outperforms the other
+three methods. It can be seen that with the increase of perturbation level, average arrowhead error
+and SHD increase significantly, while average arrowhead precision and F-score decrease. When the
+perturbation level is relatively low, the difference between the four methods is relatively small, but
+when the perturbation level is relatively large, the advantage of NOTEARS-PFL is more significant.
+5
+Application to Real-world rs-fMRI Data
+5.1
+Data and Preprocessing
+(1) Data acquisition: The multi-site rs-fMRI datasets in this study were obtained from the
+DecNef Project Brain Data Repository1, collected as part of the Japanese Strategic Research Program
+for the Promotion of Brain Science (SRPBS) database project [Tanaka et al., 2021]. This dataset
+consists of 255 MDD patients (135 men versus 120 women) and 791 HC participants (426 men versus
+365 women) from 8 sites. Some samples were discarded as some sites only have rs-fMRI data on
+HC participants. As a result, 238 MDD patients and 475 HC participants from 5 sites were used
+in this study (see Table 3). The demographic characteristics of all subjects in both datasets are
+summarized in Table 4.
+(2) Data preprocessing : Each subject of the data underwent a single rs-fMRI session and a
+structural MRI session. During the fMRI scan, participants were asked to relax, stay awake and
+think about anything in particular. Detailed imaging parameters of rs-fMRI and T1-weighted (T1w)
+structural MRI at each site are shown in Table 5. We applied the following preprocessing steps
+to the data by using DPARSF [Yan and Zang, 2010] software2. For each participant, we removed
+the first 10 MRI volumes. Slice-time correction and head motion were corrected on the remaining
+images. The T1-weighted images were unified segment segmentation and diffeomorphic anatomical
+registration. The rs-fMRI data were smoothed with a 6mm full-width at half-maximum Gaussian
+kernel and the Friston 24-parameter model was used to regress head motion effects further. Then,
+all the time series were filtered by linear detrending and a temporal band-pass filter (0.01 - 0.08 Hz).
+(3) Feature extraction: After preprocessing, we estimated the time series of 116 ROIs based
+on the automated anatomical labeling (AAL) template [Tzourio-Mazoyer et al., 2002]. Details of
+cortical and subcortical regions of interest (ROIs) defined in the AAL template are presented in
+Table 6 and Table 7. We selected 56 ROIs from 116 ROIs because these regions are considered to
+be related to MDD in the literature. The names of the selected ROIs are shown in bold in Table
+6 and Table 7. The voxel-wise fMRI time courses were averaged into one regional average time
+course within each ROI. The data for each participant was then in the form of an X× 56 matrix (X
+1https://bicr-resource.atr.jp/srpbsopen/
+2http://www.rfmri.org/DPARSF
+15
+
+sampling points in each regional average fMRI time course and 56 selected ROIs).
+5.2
+Experimental Setup
+We randomly select 70% of MDD patients and HC participants from each site and aggregate their
+rs-fMRI data into the MDD discovery dataset and HC discovery dataset for that site, respectively.
+Similarly, we aggregate the remaining 30% into the MDD validation dataset and HC validation
+dataset for that site, respectively. We analyze the 5 sites’ discovery datasets of MDD patients, which
+are regarded as 5 related tasks. Then we can obtain a brain connection network of MDD for each
+site. To study the differences in brain FC, the brain connection networks of HC for five sites, which
+are viewed as another set of related tasks, are also obtained using the proposed NOTEARS-PFL
+method. According to the estimated BN structures, we analyze the abnormal FC in MDD by the
+following aspects.
+5.2.1
+Important Nodes Identification
+By identifying important nodes, we reveal the aberrant nodal characteristics of the brain functional
+connectivity networks of MDD from multi-site datasets.
+5.2.2
+Overlapping Connections Identification
+By identifying overlapping connections, we obtain the common functional connections for MDD
+patients from multiple sites. By comparing with the overlapping connections of HC participants, we
+further investigate the altered functional connections in MDD patients, which provide the potential
+biomarkers for clinic treatment of MDD.
+5.2.3
+Site-specific Connections Identification
+By identifying the site-specific connections in different sites, we investigate the specific FC alterations
+in MDD at each site due to the differences in image acquisition protocols and patient populations at
+different sites.
+5.2.4
+Comparison with Other Methods
+We randomly select 5 MDD patients and 5 HC participants from each site’s MDD validation dataset
+and HC validation dataset and repeat such selection 10 times. We then have 10 datasets containing
+MDD data and HC data from the 5 sites. Based on these datasets, we compare NOTEARS-PFL with
+NOTEARS-SIG and existing standard analysis methods of rs-fMRI data: GES [Chickering, 2002],
+PC [Spirtes and Glymour, 1991], and GC [Granger, 1969]. To further test whether NOTEARS-PFL
+is able to learn the differences between MDD patients and HC participants.
+5.3
+Experimental Results
+In this section, we use NOTEARS-PFL method to analyze the multi-site rs-fMRI data from 5 sites.
+We can obtain the brain functional connection network of MDD patients for each site. To study the
+differences in brain FC, we also obtain the brain functional connection network of HC participants
+for each site.
+16
+
+5.3.1
+Results of Important Nodes
+To better understand the differences in brain FC between MDD patients and HC participants through
+the estimated directed brain networks from multi-site rs-fMRI data, we further use the connection
+degree to identify some important nodes. The connection degree (including in and out degree) is a
+commonly used measure of functional connectivity [Liu et al., 2022]. More specifically, it represents
+the amount of directed functional connections. The greater the connection degree, the higher the
+effective FC. Identifying important nodes further helps to discover the nodes associated with the
+disease. Table 1 shows the differences in total connection degrees between MDD patients and HC
+participants at 5 sites. The total out connection degrees in Table 1 are 657 for MDD and 478 for
+HC, indicating that MDD has 37.4% effective FC than HC. Similarly, MDD also has 23.4% more
+effective FC than HC, with a total in connection degrees of 516 and 418, respectively. The rise of FC
+is an apparent robust biomarker of MDD disease [Zamoscik et al., 2014], which can also be found in
+our results.
+Table 1: The top 10 out and in connection degrees of MDD patients and HC participants
+Index
+ROI
+Abbreviation
+Degree
+Index
+ROI
+Abbreviation
+Degree
+MDD (Out)
+77
+Thalamus
+THA.L
+75
+HC (Out)
+55
+Fusiform gyrus
+FFG.L
+58
+29
+Insula
+INS.L
+74
+37
+Hippocampus
+HIP.L
+57
+35
+Cingulate gyurs, posterior part
+PCG.L
+72
+31
+Cingulate gyrus, anterior part
+ACG.L
+55
+33
+Cingulate gyrus, mid part
+DCG.L
+68
+38
+Hippocampus
+HIP.R
+49
+34
+Cingulate gyrus, mid part
+DCG.R
+67
+9
+Middle frontal gyrus, orbital
+HIP.R
+47
+31
+Cingulate gyrus, anterior part
+ACG.L
+63
+29
+Insula
+INS.L
+46
+72
+Caudate
+CAU.R
+62
+77
+Thalamus
+THA.L
+45
+71
+Caudate
+CAU.L
+61
+33
+Cingulate gyrus, mid part
+DCG.L
+43
+55
+Fusiform gyrus
+FFG.L
+58
+56
+Fusiform gyrus
+FFG.R
+40
+79
+Heschl gyrus
+HES.L
+57
+79
+Heschl gyrus
+HES.L
+38
+Total
+657
+Total
+478
+MDD (In)
+24
+Superior frontal gyrus, medial
+SFGmed.R
+79
+HC (In)
+3
+Superior frontal gyrus, dorsolateral
+SFGdor.L
+55
+90
+Inferior temporal gyrus
+ITG.R
+62
+9
+Middle frontal gyrus, orbital
+ORBmid.L
+48
+26
+Superior frontal gyrus, medial orbital ORBsupmed.R
+59
+26
+Superior frontal gyrus, medial orbital ORBsupmed.R
+45
+89
+Inferior temporal gyrus
+ITG.L
+54
+90
+Inferior temporal gyrus
+ITG.R
+44
+3
+Superior frontal gyrus, dorsolateral
+SFGdor.L
+48
+10
+Middle frontal gyrus, orbital
+ORBmid.R
+41
+46
+Cuneus
+CUN.R
+46
+14
+Inferior frontal gyrus, triangular
+IFGtriang.R
+40
+11
+Inferior frontal gyrus, opercular
+IFGoperc.L
+45
+24
+Superior frontal gyrus, medial
+SFGmed.R
+39
+25
+Superior frontal gyrus, medial orbital ORBsupmed.L
+42
+1
+Precentral gyrus
+PreCG.L
+38
+41
+Amygdala
+AMYG.L
+41
+89
+Inferior temporal gyrus
+ITG.L
+36
+32
+Cingulate gyrus, anterior part
+ACG.R
+40
+6
+Superior frontal gyrus, orbital
+ORBsup.R
+32
+Total
+516
+Total
+418
+From Table 1, we can find some ROI nodes with high connection degrees, such as THA.L
+(thalamus), INS.L (insula), SFGmed.R (superior frontal gyrus, medial) in MDD, and FFG.L (fusiform
+gyrus) and HIP.L (hippocampus) in HC. Specifically, THA.L and INS.L have more outgoing
+connections in MDD than in HC. It indicates that THA.L and INS.L of MDD have a higher essential
+impact on other ROI nodes than HC. The result from NOTEARS-PFL is in line with literature
+[Kang et al., 2018, Avery et al., 2014, Porta-Casteràs et al., 2021]. According to Kang et al. [2018],
+emerging evidence indicates that the enhanced thalamus FC is an important feature of the underlying
+pathophysiology of MDD. This abnormal FC is related to the core clinical symptoms of MDD, such as
+anxiety, memory, and attention. Our result further implies that the impaired FC of the hippocampus
+is potentially an important biomarker of MDD. Compared to HC, the bilateral hippocampus in
+MDD lack FC. As the core region of the limbic system, the hippocampus plays an important role in
+regulating motivation and emotion. The mode of emotion regulation in MDD may be related to the
+decreased FC of the hippocampus [Cao et al., 2012]. SFGmed.R (superior frontal gyrus, medial) and
+ITG.R (inferior temporal gyrus) have more incoming connections than outgoing connections, which
+indicates that SFGmed.R and ITG.R are mainly affected by other ROI nodes. These findings are
+also consistent with the physiological discoveries of MDD disease [Porta-Casteràs et al., 2021].
+17
+
+5.3.2
+Results of Overlapping Connections
+Figure 5 depicts the overlapping connections between the directed networks of 5 sites of MDD
+patients and HC participants. The overlapping connection refers to the connection shared by five
+sites. We can use it to study the universal connection between MDD and compare the differences
+between MDD and HC. From Figure 5, we can find some abnormal functional connections in MDD:
+INS.R (insula) → ORBmid.R (middle frontal gyrus, orbital), INS.R (insula) → INS.L (insula), INS.R
+(insula) → ACG.R (cingulate gyrus, anterior part), PCG.L (cingulate gyurs, posterior part) →
+PCG.R (cingulate gyurs, posterior part), THA.L (thalamus) → CAU.R (caudate), STG.R (superior
+temporal gyrus) → MTG.R (middle temporal gyrus). Three common functional connections are
+associated with the insula, which further demonstrates the important role of the insula in identifying
+abnormal functional connections in MDD. The insula has been shown to be a potential primary
+biomarker for the diagnosis of MDD [McGrath et al., 2013]. In MDD patients, the insula show
+increased FC with other limbic or paralimbic structures, particularly with the ventromedial prefrontal
+cortex (vmPFC) and orbitofrontal cortex (OFC) [Avery et al., 2014, Drevets et al., 2008], which is
+also illustrated in our results.
+Figure 5: The overlapping connections between the directed networks of 5 sites of MDD patients
+and HC participants.
+In addition to the abnormal insula structure in MDD patients, we also observe the abnormalities
+of FC in posterior cingulate gyurs, thalamus, inferior temporal gyrus, and middle temporal gyrus,
+which are also consistent with findings in the works of literature [Khundakar and Thomas, 2009,
+Zhao et al., 2014]. These cortices belong to the default mode network (DMN) and are considered
+to contribute greatly to MDD [Gong and He, 2015]. The abnormalities of FC in DMN have been
+reported in many MDD studies [Vasudev et al., 2018], especially the frontal lobe and temporal
+lobe. These lobes are suggested to be closely associated to the cognitive and information-processing
+abilities of MDD patients [Brzezicka, 2013].
+18
+
+R
+L
+R
+L
+SFGdo
+MFG.R
+CG.1
+R
+14 1
+13 11 9
+14
+INS.
+ACGL
+ACG.R
+DCG.L
+3
+DCG.R
+3
+PCG.L
+、PCG.R
+3
+%
+HIP.R
+ Frontal
+41 39 37
+PHG.L
+40
+PHG.R
+ Insula and Cingulate Gyri
+42
+Temporal
+41
+ CUN.R
+CUN.L
+4
+Occipital
+Pariental
+ Subcortial region
+5
+FFG.R
+18
+5
+LL 6L T8
+TVHL
+58 Z8 68 06 88 98
+STG.R
+MTG.L
+MTG.R
+ITG.L
+MTG.R
+ITG.L
+(a) MDD patients
+(b) HC participantsFigure 6: The site-specific connections between the directed networks of 5 sites of MDD patients.
+Site 1: Center of Innovation in Hiroshima (COI), Site 2: Kyoto University (KUT), Site 3: University
+of Tokyo (UTO), Site 4: Hiroshima Kajikawa Hospital (HKH), Site 5: Hiroshima University Hospital
+(HUH).
+5.3.3
+Results of Site-specific Connections
+In this study, we are also interested in the connections specific to a site. A site-specific connection is
+a connection that only exists on one site. By identifying the site-specific connections in different
+sites, we can better understand the specificity of FC in MDD patients due to the differences in image
+acquisition protocols and patient populations at different sites. The site-specific connections of 5
+sites of MDD patients are visualized in Figure 6. In the results of site 1, we can find the increased
+functional connection from frontal to pariental and from frontal to temporal, especially the increased
+afferent connections to precuneus. The precuneus is considered one of the hubs of DMN, which is
+generally associated with the rumination of negative and sad thoughts in MDD [Cheng et al., 2018].
+It has been shown to play a key role in identifying MDD [Peng et al., 2015]. A study of effective
+FC in MDD has shown a significant increase in forward connectivity from the middle and inferior
+temporal cortical areas to the precuneus [Öngür et al., 2003], which is consistent with our study (the
+connections IFGoperc.L → PCUN.R, IFGtriang.L → PCUN.L).
+Similarly, in site 2, we find the enhanced FC within the frontal cortex, particularly the connections
+associated with the orbitofrontal cortex (the connections ORBsup.L → ORBsupmed.L, MFG.L →
+ORBmid.R, AMYG.L → ORBmid.L). The increased FC in the orbitofrontal cortex in MDD patients
+is associated with emotionally negative self-perception [Cheng et al., 2016]. The orbitofrontal cortex
+is also a part of the brain’s affective network (AN), and the increased functional connections of the
+orbitofrontal cortex in MDD have been consistently reported [Townsend et al., 2010]. The different
+effective FC patterns of the orbitofrontal cortex can also be used to reveal different pathophysiological
+mechanisms of different depression types or groups.
+19
+
+Sitel
+Site4
+Site2
+Site5
+Site3Table 2: Two-sample proportion tests for published overlapped connections.
+Connection
+NOTEARS-PFL NOTEARS-SIG
+GES
+PC
+GC
+INS.R→INS.L
+0.0002
+0.0016
+0.0022 0.0512 0.0057
+INS.R→ORBmid.R
+0.0012
+0.0029
+0.0055 0.0234 0.0133
+THA.L →CAU.R
+0.0016
+0.0040
+0.0126 0.1636 0.0264
+PCG.L→PCG.R
+0.0040
+0.0091
+0.0283 0.0984 0.0549
+INS.R →ACG.R
+0.0049
+0.0234
+0.0133 0.2625 0.0289
+STG.R→MTG.R
+0.0055
+0.0264
+0.0549 0.0721 0.0567
+MTG.L→ORBmid.R
+0.0126
+0.0283
+0.0264 0.1556 0.1018
+5.3.4
+Comparison Results with Other Methods
+To further test whether NOTEARS-PFL is able to learn the differences between MDD patients and
+HC participants, we use different methods to learn the BN structures from the test validation dataset
+established in Section 5. Based on the learned BN structures, we perform a two-sample proportion
+test for each published overlapping connection of MDD in Figure 5, and report the p-values in Table
+2. The smaller p value indicates that NOTEARS-PFL recognizes the connection more easily in MDD
+group than in HC group. We also included test results for NOTEARS-SIG, GES, PC, and GS for
+comparison. In order to ensure the consistency of measurement standards, we control these methods
+to learn the same sparsity network structure.
+6
+Conclusion
+In this work, we focus on learning the heterogeneous brain functional connectivity in MDD patients
+using multi-site data efficiently and with the data sharing constraint. To this end, we developed a
+toolbox called NOTEARS-PFL. To achieve the model heterogeneity and computational efficiency,
+NOTEARS-PFL is formulated as minimizing a group fused lasso penalized score function with a
+continuous constraint for DAG. Considering the data-sharing barriers, the objective is solved in
+a federated learning framework using the ADMM, where original data is not exchanged during
+the optimization process. The algorithm is fast because the local update step has a closed-form
+expression, and the global update step can be solved using an efficient iterative proximal projection
+method. Extensive experiments on synthetic data and real-world multi-site rs-fMRI datasets with
+MDD demonstrate the excellent performance of the proposed method. We highlight the usefulness
+of NOTERAS-PFL in analyzing multi-site rs-fMRI data and facilitating the study of MDD.
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+26
+
+Supplementary Materials
+A
+Derivation of Federated Local Update Step
+To solve the subproblem as shown in equation (15), we first drop the subscript t to lighten the
+notation, which leads to
+min
+W k f(W k) := L(W k, xk) + tr(βk(W k − Zk)⊤) + ρ2
+2
+���W k − Zk���
+2
+F .
+(24)
+Let U k =
+1
+nk (xk)⊤xk. The first term of the equation (24) can be written as
+L(W k, xk) =
+1
+2nk
+���xk − xkW k���
+2
+F ,
+=
+1
+2nk
+tr((xk − xkW k)⊤(xk − xkW k)),
+=
+1
+2nk
+tr((xk)⊤xk − (W k)⊤(xk)⊤xk − (xk)⊤xkW k + (W k)⊤(xk)⊤xkW k),
+= 1
+2 tr
+�
+U k − (W k)⊤U k − U kW k + (W k)⊤U kW k�
+,
+= 1
+2 tr
+�
+U k − 2(W k)⊤U k + (W k)⊤U kW k�
+.
+Similarly, the third term of equation (24) can be written as
+ρ2
+2 ∥W k − Zk∥2
+F = ρ2
+2 tr
+��
+W k − Zk�⊤ �
+W k − Zk��
+,
+= ρ2
+2 tr
+�
+(W k)⊤W k − 2(W k)⊤Zk + (Zk)⊤Zk�
+.
+Therefore, we have
+f(W k) = 1
+2 tr(U k − 2(W k)⊤U k + (W k)⊤U kW k) + tr(βk(W k)⊤ − βk(Zk)⊤) + ρ2
+2 tr((W k)⊤W k
+− 2(W k)⊤Zk + (Zk)⊤Zk),
+= 1
+2 tr(−2(W k)⊤U k + (W k)⊤U kW k) + tr(βk(W k)⊤)ρ2
+2 tr((W k)⊤W k − 2(W k)⊤Zk + c,
+and its derivative is given by
+∇W kf
+�
+W k�
+= 1
+2(−2U k + ((U k)⊤ + U k)W k) + βk + ρ2
+2 (2W k − 2Zk),
+= −U k + U kW k + βk + ρ2W k − ρ2Zk,
+= (U k + ρ2I)W k − U k + βk − ρ2Zk.
+Assuming U k + ρ2I is invertible, solving ∇W kf
+�
+W k�
+= 0 yields the solution
+W k = (U k + ρ2I)−1(ρ2Zk − βk + U k).
+27
+
+B
+Details on the Federated Global Update Step
+We provide details of the DIPA algorithm in the federated global update step. For convenience, we
+introduce the following procedures and discuss how to transform K matrices into one matrix, so
+that we can re-write equation (19) into vector form.
+Transform: Given K matrices
+˜
+M = {M1, . . . , MK}. For each k = 1, . . . , K, Mk ∈ Rd×d.
+Transform Mk into vector form, denote as mk =
+�
+Mk
+i,j | for i, j = 1, . . . , d
+�⊤
+. Then, construct a
+matrix
+¯
+M =
+�
+m1, . . . , mK�⊤ ∈ RK×d2.
+Inverse-Transform: The inverse-transform reverses the steps of the transform, which to
+transform the matrix
+¯
+M ∈ RK×d2 to K matrices
+˜
+M = {M1, . . . , MK}.
+The core of DIPA algorithm is to compute the separate proximal operators for R1 and R2. And
+then the iterative projection can be used to find a feasible point. The proximal operator for the ℓ1
+term proxR1
+� ¯U
+�
+is given by the soft-thresholding operator [Tibshirani, 1996].
+proxR1
+� ¯U; λ1
+�
+= arg min
+¯
+Z
+1
+2
+�� ¯Z − ¯U
+��2
+F + λ1∥ ¯Z∥1,
+= sign( ¯U) ⊙ max
+�
+| ¯U| − λ1, 0
+�
+.
+(25)
+where the max and sign functions act in an element-wise manner and ⊙ denotes element-wise
+multiplication.
+The calculation of group-fused lasso proximal operator proxR2
+� ¯U
+�
+is more complex, and there
+is no obvious closed-form solution. We tackle this through a block-coordinate descent approach
+[Bleakley and Vert, 2011]. Then, the proximal operator for the group smoothing aspect of the
+regularizer can be written as:
+proxR2
+� ¯U; λ2
+�
+= arg min
+¯
+Z
+1
+2
+�� ¯Z − ¯U
+��2
+F + λ2∥D ¯Z∥2,1.
+(26)
+Let A = D ¯Z and construct ¯Z as a sum of differences via Zk,• = a+�K−1
+i=1 Ai,•, ( where a = Z1,•).
+Then we can interpret the proximal operator as a group lasso probem [Bleakley and Vert, 2011].
+A := arg min
+A
+1
+2∥ ¯Ucen − ¯RcenA∥2
+F + λ2∥A∥2,1,
+(27)
+where Xcen denotes a column centered matrix and R ∈ RK×(K−1) is a matrix with entries Ri,j = 1
+for i > j and 0 otherwise. The problem above can be solved through a block-coordinate descent
+strategy, sequentially updating the solution for each block Ak,• for k = 1, . . . , K − 1. We can then
+construct a solution for ¯Z by summing the differences. And the optimal value for a is given by
+a = 11,T (U − RA). Correspondingly, the proximal operator for R2 is
+proxR2
+� ¯U; λ2
+�
+=
+�
+�a⊤, (a + A1,•)⊤ , . . . ,
+�
+a +
+K−1
+�
+i=1
+Ai,•
+�⊤�
+�
+⊤
+.
+(28)
+C
+Details of the Synthetic Data Generating
+We followed the similar synthetic data generate procedure as in [Liu et al., 2022]. The key is how to
+create similar but different graphs by applying perturbations.
+28
+
+Perturbation level: Denote the adjacency matrix of Gtruth as At = (at
+ij) ∈ Rd×d, where at
+ij = 1
+if there is an edge from node i to node j, and at
+ij = 0 otherwise, i, j = 1, . . . , d. Then the generated
+BN, denoted by Ggen, is constructed in the following way. Let Ag be the adjacency matrix of Ggen,
+and define the perturbation level as the ratio of the number of changed edges in Ag to the number
+of all possible edges. For example, if the perturbation level is 10%, we first set Ag = At, and then
+randomly select 10% of the off-diagonal elements at
+ij(i ̸= j) in At. For each selected element at
+ij,
+if at
+ij = 0, then we set ag
+ij = 1 (adding edge); if at
+ij = 1, then with probability 1/2 we set ag
+ij = 0
+(deleting edge), otherwise let ag
+ji = 1 (reversing edge). We generate K graphs ˜G = {G1, . . . , GK} by
+such a perturbation scheme.
+Evaluation metric: Let {G1
+output, . . . , GK
+output} represent the K BN structures obtained from
+NOTEARS-PFL, NOTEARS-SIG, NOTEARS-AVG, NOTEARS-ADMM. The four basic statistics
+in the evaluation metrics are defined as follows: true positives (TP), the number of edges that are
+both in G1
+output and G1; false positives (FP), the number of edges that are present in G1
+output but
+not in G1; false negatives (FN), the number of edges that are present in G1 but not in G1
+output; and
+true negatives (TN), the number of vertex pairs that are neither edges in G1 nor in G1
+output.
+29
+
+D
+Supplementary Tables
+Table 3: Data summary of the datasets used in this study.
+Center of Innovation in Hiroshima
+(COI)
+Kyoto University
+(KUT)
+University of Tokyo
+(UTO)
+Hiroshima Kajikawa Hospital
+(HKH)
+Hiroshima University Hospital
+(HUH)
+Total Subject
+194
+175
+158
+62
+124
+MDD Subject
+70
+16
+62
+33
+57
+HC Subject
+124
+159
+96
+29
+67
+MDD Percentage
+36%
+9%
+39%
+53%
+46%
+rs-fMRI Frames
+240
+240
+240
+107
+143
+Table 4: Demographic characteristics of participants in multi-site datasets.
+Site
+Number Male/Female
+Age
+BDI
+MDD COI
+70
+31/39
+45.0 ± 12.5
+26.2 ± 9.9
+KUT
+16
+10 6
+33.79±10.82
+27.7 ± 10.1
+UTO
+62
+36/26
+38.74 ±11.62 21.5±11.3
+HKH
+33
+20/13
+44.8±11.5
+28.5±8.7
+HUH
+57
+32/25
+43.3±12.2
+30.9±9.0
+HC
+COI
+124
+46/78
+51.9 ± 13.4
+8.2 ± 6.3
+KUT
+159
+93/66
+36.51± 13.59 6.0 ± 5.4
+UTO
+96
+33/63
+46.65±15.54
+6.6 ± 6.5
+HKH
+29
+12 17
+45.4±9.5
+5.1±4.6
+HUH
+67
+30/37
+45.6±9.4
+5.6±4.3
+Table 5: Imaging protocols for rs-fMRI and structural MRI in multi-site datasets.
+COI
+KUT
+UTO
+HKH
+HUH
+rs-fMRI
+MRI scanner
+Siemens
+Verio
+Siemens
+TimTrio
+GE
+MR750w
+Siemens
+Spectra
+GE
+Sigma HDxt
+Magnetic field strength
+3.0T
+3.0T
+3.0T
+3.0T
+3.0T
+Number of channels per coil
+12
+32
+24
+12
+8
+FoV (mm)
+212
+212
+212
+192
+256
+Matrix
+64 x 64
+64 x 64
+64 x 64
+64 x 64
+64 x 64
+Number of slices
+40
+40
+40
+38
+32
+Number of volumes
+240
+240
+240
+107
+143
+In-plane resolution (mm)
+3.3 x 3.3
+3.3125 X 3.3125
+3.3 x 3.3
+3.0 x 3.0
+4.0 x 4.0
+Slice thickness (mm)
+3.2
+3.2
+3.2
+3
+4
+Slice gap (mm)
+0.8
+0.8
+0.8
+0
+0
+TR (ms)
+2500
+2500
+2500
+2700
+2000
+TE (ms)
+30
+30
+30
+31
+27
+Flip angel (deg)
+80
+80
+80
+90
+90
+Slice acquisition order
+Ascending
+Ascending
+Ascending
+Ascending
+Ascending
+Total scan
+time
+10 min + 10s
+(dummy)
+10 min + 10s
+(dummy)
+10 min + 10s
+(dummy)
+4 min. 46s. + 14s
+(dummy)
+4 min. 46s. + 14s
+(dummy)
+Eye closed/fixate
+Fixate
+Fixate
+Fixate
+Fixate
+Fixate
+Structural MRI FoV (mm)
+256
+225x 240
+240
+256
+256
+Matrix
+256 x 256
+240 x 256
+256 x 256
+256 x 256
+256 x 256
+Voxel size mm3
+1 x 1x 1
+0.9375 x 0.9375 x 1
+1 x 1x 1.2
+1 x 1x 1
+1 x 1x 1
+TR (ms)
+2300
+2000
+7.7
+1900
+6812
+TE (ms)
+2.98
+3.4
+3.1
+2.38
+1986
+TI (ms)
+900
+990
+400
+900
+450
+Flip angel (deg)
+9
+8
+11
+10
+20
+30
+
+Table 6: Description of ROIs in AAL template and their Montreal Neurological Institute (MNI)
+coordinates.
+Index
+Region name
+Abbreviation
+Lobe
+x
+y
+z
+1
+Precentral gyrus
+PreCG.L
+Frontal
+-38.65 -5.68
+50.94
+2
+Precentral gyrus
+PreCG.R
+Frontal
+41.37
+-8.21
+52.09
+3
+Superior frontal gyrus, dorsolateral
+SFGdor.L
+Frontal
+-18.45 34.81
+42.20
+4
+Superior frontal gyrus, dorsolateral
+SFGdor.R
+Frontal
+21.90
+31.12
+43.82
+5
+Superior frontal gyrus, orbital
+ORBsup.L
+Frontal
+-16.56 47.32 -13.31
+6
+Superior frontal gyrus, orbital
+ORBsup.R
+Frontal
+18.49
+48.10 -14.02
+7
+Middle frontal gyrus
+MFG.L
+Frontal
+-33.43 32.73
+35.46
+8
+Middle frontal gyrus
+MFG.R
+Frontal
+37.59
+33.06
+34.04
+9
+Middle frontal gyrus, orbital
+ORBmid.L
+Frontal
+-30.65 50.43
+-9.62
+10
+Middle frontal gyrus, orbital
+ORBmid.R
+Frontal
+33.18
+52.59 -10.73
+11
+Inferior frontal gyrus, opercular
+IFGoperc.L
+Frontal
+-48.43 12.73
+19.02
+12
+Inferior frontal gyrus, opercular
+IFGoperc.R
+Frontal
+50.20
+14.98
+21.41
+13
+Inferior frontal gyrus, triangular
+IFGtriang.L
+Frontal
+-45.58 29.91
+13.99
+14
+Inferior frontal gyrus, triangular
+IFGtriang.R
+Frontal
+50.33
+30.16
+14.17
+15
+Inferior frontal gyrus, orbital
+ORBinf.L
+Frontal
+-35.98 30.71 -12.11
+16
+Inferior frontal gyrus, orbital
+ORBinf.R
+Frontal
+41.22
+32.23 -11.91
+17
+Rolandic operculum
+ROL.L
+Frontal
+-47.16 -8.48
+13.95
+18
+Rolandic operculum
+ROL.R
+Frontal
+52.65
+-6.25
+14.63
+19
+Supplementary motor area
+SMA.L
+Frontal
+-5.32
+4.85
+61.38
+20
+Supplementary motor area
+SMA.R
+Frontal
+8.62
+0.17
+61.85
+21
+Olfactory cortex
+OLF.L
+Frontal
+-8.06
+15.05 -11.46
+22
+Olfactory cortex
+OLF.R
+Frontal
+10.43
+15.91 -11.26
+23
+Superior frontal gyrus, medial
+SFGmed.L
+Frontal
+-4.80
+49.17
+30.89
+24
+Superior frontal gyrus, medial
+SFGmed.R
+Frontal
+9.10
+50.84
+30.22
+25
+Superior frontal gyrus, medial orbital ORBsupmed.L
+Frontal
+-5.17
+54.06
+-7.40
+26
+Superior frontal gyrus, medial orbital ORBsupmed.R
+Frontal
+8.16
+51.67
+-7.13
+27
+Gyrus rectus
+REC.L
+Frontal
+-5.08
+37.07 -18.14
+28
+Gyrus rectus
+REC.R
+Frontal
+8.35
+35.64 -18.04
+29
+Insula
+INS.L
+Insula and Cingulate Gyri -35.13
+6.65
+3.44
+30
+Insula
+INS.R
+Insula and Cingulate Gyri 39.02
+6.25
+2.08
+31
+Cingulate gyrus, anterior part
+ACG.L
+Insula and Cingulate Gyri -4.04
+35.40
+13.95
+32
+Cingulate gyrus, anterior part
+ACG.R
+Insula and Cingulate Gyri
+8.46
+37.01
+15.84
+33
+Cingulate gyrus, mid part
+DCG.L
+Insula and Cingulate Gyri -5.48 -14.92 41.57
+34
+Cingulate gyrus, mid part
+DCG.R
+Insula and Cingulate Gyri
+8.02
+-8.83
+39.79
+35
+Cingulate gyurs, posterior part
+PCG.L
+Insula and Cingulate Gyri -4.85 -42.92 24.67
+36
+Cingulate gyurs, posterior part
+PCG.R
+Insula and Cingulate Gyri
+7.44
+-41.81 21.87
+37
+Hippocampus
+HIP.L
+Temporal
+-25.03 -20.74 -10.13
+38
+Hippocampus
+HIP.R
+Temporal
+29.23 -19.78 -10.33
+39
+Parahippocampus
+PHG.L
+Temporal
+-21.17 -15.95 -20.70
+40
+Parahippocampus
+PHG.R
+Temporal
+25.38 -15.15 -20.47
+41
+Amygdala
+AMYG.L
+Temporal
+-23.27 -0.67 -17.14
+42
+Amygdala
+AMYG.R
+Temporal
+27.32
+0.64
+-17.50
+43
+Calcarine fissure and surrounding cortex
+CAL.L
+Occipital
+-7.14 -78.67
+6.44
+44
+Calcarine fissure and surrounding cortex
+CAL.R
+Occipital
+15.99 -73.15
+9.40
+45
+Cuneus
+CUN.L
+Occipital
+-5.93 -80.13 27.22
+46
+Cuneus
+CUN.R
+Occipital
+13.51 -79.36 28.23
+47
+Lingual gyrus
+LING.L
+Occipital
+-14.62 -67.56 -4.63
+48
+Lingual gyrus
+LING.R
+Occipital
+16.29 -66.93 -3.87
+49
+Superior occipital lobe
+SOG.L
+Occipital
+-16.54 -84.26 28.17
+50
+Superior occipital lobe
+SOG.R
+Occipital
+24.29 -80.85 30.59
+51
+Middle occipital lobe
+MOG.L
+Occipital
+-32.39 -80.73 16.11
+52
+Middle occipital lobe
+MOG.R
+Occipital
+37.39 -79.70 19.42
+53
+Inferior occipital lobe
+IOG.L
+Occipital
+-36.36 -78.29 -7.84
+54
+Inferior occipital lobe
+IOG.R
+Occipital
+38.16 -81.99 -7.61
+55
+Fusiform gyrus
+FFG.L
+Temporal
+-31.16 -40.30 -20.23
+56
+Fusiform gyrus
+FFG.R
+Temporal
+33.97 -39.10 -20.18
+57
+Postcentral gyrus
+PoCG.L
+Pariental
+-42.46 -22.63 48.92
+58
+Postcentral gyrus
+PoCG.R
+Pariental
+41.43 -25.49 52.55
+59
+Superior pariental gyrus
+SPG.L
+Pariental
+-23.45 -59.56 58.96
+60
+Superior pariental gyrus
+SPG.R
+Pariental
+26.11 -59.18 62.06
+61
+Inferior pariental gyrus
+IPL.L
+Pariental
+-42.80 -45.82 46.74
+62
+Inferior pariental gyrus
+IPL.R
+Pariental
+46.46 -46.29 49.54
+63
+Supramarginal gyrus
+SMG.L
+Pariental
+-55.79 -33.64 30.45
+31
+
+Table 7: Continue table: Description of ROIs in AAL template and their Montreal Neurological
+Institute (MNI) coordinates.
+Index
+Region name
+Abbreviation
+Lobe
+x
+y
+z
+64
+Supramarginal gyrus
+SMG.R
+Pariental
+57.61 -31.50 34.48
+65
+Angular gyrus
+ANG.L
+Pariental
+-44.14 -60.82 35.59
+66
+Angular gyrus
+ANG.R
+Pariental
+45.51 -59.98 38.63
+67
+Precuneus
+PCUN.L
+Pariental
+-7.24 -56.07 48.01
+68
+Precuneus
+PCUN.R
+Pariental
+9.98
+-56.05 43.77
+69
+Paracentral lobule
+PCL.L
+Frontal
+-7.63 -25.36 70.07
+70
+Paracentral lobule
+PCL.R
+Frontal
+7.48
+-31.59 68.09
+71
+Caudate
+CAU.L
+Subcortial region -11.46 11.00
+9.24
+72
+Caudate
+CAU.R
+Subcortial region 14.84
+12.07
+9.42
+73
+Putamen
+PUT.L
+Subcortial region -23.91
+3.86
+2.40
+74
+Putamen
+PUT.R
+Subcortial region 27.78
+4.91
+2.46
+75
+Pallidum
+PAL.L
+Subcortial region -17.75 -0.03
+0.21
+76
+Pallidum
+PAL.R
+Subcortial region 21.20
+0.18
+0.23
+77
+Thalamus
+THA.L
+Subcortial region -10.85 -17.56
+7.98
+78
+Thalamus
+THA.R
+Temporal
+13.00 -17.55
+8.09
+79
+Heschl gyrus
+HES.L
+Temporal
+-41.99 -18.88
+9.98
+80
+Heschl gyrus
+HES.R
+Temporal
+45.86 -17.15 10.41
+81
+Superior temporal gyrus
+STG.L
+Temporal
+-53.16 -20.68
+7.13
+82
+Superior temporal gyrus
+STG.R
+Temporal
+58.15 -21.78
+6.80
+83
+Temporal pole: superior temporal gyrus
+TPOsup.L
+Temporal
+-39.88 15.14 -20.18
+84
+Temporal pole: superior temporal gyrus
+TPOsup.R
+Temporal
+48.25
+14.75 -16.86
+85
+Middle temporal gyrus
+MTG.L
+Temporal
+-55.52 -33.80 -2.20
+86
+Middle temporal gyrus
+MTG.R
+Temporal
+57.47 -37.23 -1.47
+87
+Temporal pole: middle temporal gyrus
+TPOmid.L
+Temporal
+-36.32 14.59 -34.08
+88
+Temporal pole: middle temporal gyrus
+TPOmid.R
+Temporal
+44.22
+14.55 -32.23
+89
+Inferior temporal gyrus
+ITG.L
+Temporal
+-49.77 -28.05 -23.17
+90
+Inferior temporal gyrus
+ITG.R
+Temporal
+53.69 -31.07 -22.32
+91
+Cerebellum crus 1
+CRBLCrus1.L
+Posterior Fossa
+-36.07 -66.72 -28.93
+92
+Cerebellum crus 1
+CRBLCrus1.R
+Posterior Fossa
+37.46 -67.14 -29.55
+93
+Cerebellum crus 2
+CRBLCrus2.L
+Posterior Fossa
+-28.64 -73.26 -38.2
+94
+Cerebellum crus 2
+CRBLCrus2.R
+Posterior Fossa
+32.06 -69.02 -39.95
+95
+Cerebellum 3
+CRBL3.L
+Posterior Fossa
+-8.8
+-37.22 -18.58
+96
+Cerebellum 3
+CRBL3.R
+Posterior Fossa
+12.32 -34.47 -19.39
+97
+Cerebellum 4
+CRBL45.L
+Posterior Fossa
+-15
+-43.49 -16.93
+98
+Cerebellum 4
+CRBL45.R
+Posterior Fossa
+17.2
+-42.86 -18.15
+99
+Cerebellum 6
+CRBL6.L
+Posterior Fossa
+-23.24 -59.1 -22.13
+100
+Cerebellum 6
+CRBL6.R
+Posterior Fossa
+24.69 -58.32 -23.65
+101
+Cerebellum 7
+CRBL7b.L
+Posterior Fossa
+-32.36 -59.82 -45.45
+102
+Cerebellum 7
+CRBL7b.R
+Posterior Fossa
+33.14 -63.18 -48.46
+103
+Cerebellum 8
+CRBL8.L
+Posterior Fossa
+-25.75 -54.52 -47.68
+104
+Cerebellum 8
+CRBL8.R
+Posterior Fossa
+25.06 -56.34 -49.47
+105
+Cerebellum 9
+CRBL9.L
+Posterior Fossa
+-10.95 -48.95 -45.9
+106
+Cerebellum 9
+CRBL9.R
+Posterior Fossa
+9.46
+-49.5 -46.33
+107
+Cerebellum 10
+CRBL10.L
+Posterior Fossa
+-22.61 -33.8 -41.76
+108
+Cerebellum 10
+CRBL10.R
+Posterior Fossa
+25.99 -33.84 -41.35
+109
+Vermis12
+Vermis12
+Posterior Fossa
+0.76
+-38.79 -20.05
+110
+Vermis3
+Vermis3
+Posterior Fossa
+1.38
+-39.93 -11.4
+111
+Vermis45
+Vermis45
+Posterior Fossa
+1.22
+-52.36 -6.11
+112
+Vermis6
+Vermis6
+Posterior Fossa
+1.14
+-67.06 -15.12
+113
+Vermis7
+Vermis7
+Posterior Fossa
+1.15
+-71.93 -25.14
+114
+Vermis8
+Vermis8
+Posterior Fossa
+1.15
+-64.43 -34.08
+115
+Vermis9
+Vermis9
+Posterior Fossa
+0.86
+-54.87 -34.9
+116
+Vermis10
+Vermis10
+Posterior Fossa
+0.36
+-45.8 -31.68
+32
+
+Table 8: The overlapping connections between the directed networks of 5 sites of MDD patients and
+HC participants.
+From
+To
+Index
+ROI
+Lobe
+Index
+ROI
+Lobe
+MDD
+10
+ORBmid.R
+Frontal
+6
+ORBsup.R
+Frontal
+24
+SFGmed.R
+Frontal
+33
+DCG.L
+Insula and Cingulate Gyri
+30
+INS.R
+Insula and Cingulate Gyri
+10
+ORBmid.R
+Frontal
+30
+INS.R
+Insula and Cingulate Gyri
+29
+INS.L
+Insula and Cingulate Gyri
+30
+INS.R
+Insula and Cingulate Gyri
+32
+ACG.R
+Insula and Cingulate Gyri
+33
+DCG.L
+Insula and Cingulate Gyri
+3
+SFGdor.L
+Frontal
+35
+PCG.L
+Insula and Cingulate Gyri
+36
+PCG.R
+Insula and Cingulate Gyri
+46
+CUN.R
+Occipital
+45
+CUN.L
+Occipital
+56
+FFG.R
+Temporal
+11
+IFGoperc.L
+Frontal
+68
+PCUN.R
+Pariental
+46
+CUN.R
+Occipital
+68
+PCUN.R
+Pariental
+26
+ORBsupmed.R
+Frontal
+71
+CAU.L
+Subcortial region
+6
+ORBsup.R
+Frontal
+77
+THA.L
+Subcortial region
+72
+CAU.R
+Subcortial region
+82
+STG.R
+Temporal
+86
+MTG.R
+Temporal
+85
+MTG.L
+Temporal
+10
+ORBmid.R
+Frontal
+89
+ITG.L
+Temporal
+1
+PreCG.L
+Frontal
+HC
+1
+PreCG.L
+Frontal
+26
+ORBsupmed.R
+Frontal
+8
+MFG.R
+Frontal
+9
+ORBmid.L
+Frontal
+23
+SFGmed.L
+Frontal
+24
+SFGmed.R
+Frontal
+31
+ACG.L
+Insula and Cingulate Gyri
+23
+SFGmed.L
+Frontal
+34
+DCG.R
+Insula and Cingulate Gyri
+67
+PCUN.L
+Pariental
+38
+HIP.R
+Temporal
+39
+PHG.L
+Temporal
+40
+PHG.R
+Temporal
+3
+SFGdor.L
+Frontal
+82
+STG.R
+Temporal
+8
+MFG.R
+Frontal
+86
+MTG.R
+Temporal
+89
+ITG.L
+Temporal
+33
+
+Table 9: The site-specific connections between the directed networks of 5 sites of MDD patients.
+From
+To
+Index
+ROI
+Lobe
+Index
+ROI
+Lobe
+Site 1
+1
+PreCG.L
+Frontal
+10
+ORBmid.R
+Frontal
+1
+PreCG.L
+Frontal
+67
+PCUN.L
+Pariental
+2
+PreCG.R
+Frontal
+67
+PCUN.L
+Pariental
+5
+ORBsup.L
+Frontal
+39
+PHG.L
+Temporal
+6
+ORBsup.R
+Frontal
+78
+THA.R
+Temporal
+7
+MFG.L
+Frontal
+71
+CAU.L
+Subcortial region
+8
+MFG.R
+Frontal
+82
+STG.R
+Temporal
+11
+IFGoperc.L
+Frontal
+68
+PCUN.R
+Pariental
+13
+IFGtriang.L
+Frontal
+67
+PCUN.L
+Pariental
+16
+ORBinf.R
+Frontal
+56
+FFG.R
+Temporal
+24
+SFGmed.R
+Frontal
+72
+CAU.R
+Subcortial region
+26
+ORBsupmed.R
+Frontal
+42
+AMYG.R
+Temporal
+29
+INS.L
+Insula and Cingulate Gyri
+8
+MFG.R
+Frontal
+42
+AMYG.R
+Temporal
+87
+TPOmid.L
+Temporal
+55
+FFG.L
+Temporal
+68
+PCUN.R
+Pariental
+77
+THA.L
+Subcortial region
+31
+ACG.L
+Insula and Cingulate Gyri
+90
+ITG.R
+Temporal
+56
+FFG.R
+Temporal
+Site2
+3
+SFGdor.L
+Frontal
+7
+MFG.L
+Frontal
+4
+SFGdor.R
+Frontal
+1
+PreCG.L
+Frontal
+5
+ORBsup.L
+Frontal
+25
+ORBsupmed.L
+Frontal
+7
+MFG.L
+Frontal
+10
+ORBmid.R
+Frontal
+15
+ORBinf.L
+Frontal
+11
+IFGoperc.L
+Frontal
+24
+SFGmed.R
+Frontal
+3
+SFGdor.L
+Frontal
+29
+INS.L
+Insula and Cingulate Gyri
+9
+ORBmid.L
+Frontal
+29
+INS.L
+Insula and Cingulate Gyri
+16
+ORBinf.R
+Frontal
+31
+ACG.L
+Insula and Cingulate Gyri
+5
+ORBsup.L
+Frontal
+77
+THA.L
+Subcortial region
+13
+IFGtriang.L
+Frontal
+81
+STG.L
+Temporal
+24
+SFGmed.R
+Frontal
+85
+MTG.L
+Temporal
+14
+IFGtriang.R
+Frontal
+33
+DCG.L
+Insula and Cingulate Gyri
+35
+PCG.L
+Insula and Cingulate Gyri
+37
+HIP.L
+Temporal
+12
+IFGoperc.R
+Frontal
+89
+ITG.L
+Temporal
+42
+AMYG.R
+Temporal
+56
+FFG.R
+Temporal
+35
+PCG.L
+Insula and Cingulate Gyri
+86
+MTG.R
+Temporal
+5
+ORBsup.L
+Frontal
+Site3
+1
+PreCG.L
+Frontal
+14
+IFGtriang.R
+Frontal
+2
+PreCG.R
+Frontal
+57
+PoCG.L
+Pariental
+5
+ORBsup.L
+Frontal
+26
+ORBsupmed.R
+Frontal
+7
+MFG.L
+Frontal
+35
+PCG.L
+Insula and Cingulate Gyri
+11
+IFGoperc.L
+Frontal
+23
+SFGmed.L
+Frontal
+13
+IFGtriang.L
+Frontal
+46
+CUN.R
+Occipital
+23
+SFGmed.L
+Frontal
+9
+ORBmid.L
+Frontal
+29
+INS.L
+Insula and Cingulate Gyri
+3
+SFGdor.L
+Frontal
+33
+DCG.L
+Insula and Cingulate Gyri
+42
+AMYG.R
+Temporal
+33
+DCG.L
+Insula and Cingulate Gyri
+45
+CUN.L
+Occipital
+35
+PCG.L
+Insula and Cingulate Gyri
+85
+MTG.L
+Temporal
+40
+PHG.R
+Temporal
+11
+IFGoperc.L
+Frontal
+39
+PHG.L
+Temporal
+6
+ORBsup.R
+Frontal
+72
+CAU.R
+Subcortial region
+56
+FFG.R
+Temporal
+78
+THA.R
+Temporal
+2
+PreCG.R
+Frontal
+80
+HES.R
+Temporal
+82
+STG.R
+Temporal
+Site4
+2
+PreCG.R
+Frontal
+24
+SFGmed.R
+Frontal
+5
+ORBsup.L
+Frontal
+4
+SFGdor.R
+Frontal
+7
+MFG.L
+Frontal
+29
+INS.L
+Insula and Cingulate Gyri
+9
+ORBmid.L
+Frontal
+57
+PoCG.L
+Pariental
+12
+IFGoperc.R
+Frontal
+14
+IFGtriang.R
+Frontal
+15
+ORBinf.L
+Frontal
+7
+MFG.L
+Frontal
+24
+SFGmed.R
+Frontal
+12
+IFGoperc.R
+Frontal
+30
+INS.R
+Insula and Cingulate Gyri
+23
+SFGmed.L
+Frontal
+31
+ACG.L
+Insula and Cingulate Gyri
+16
+ORBinf.R
+Frontal
+32
+ACG.R
+Insula and Cingulate Gyri
+68
+PCUN.R
+Pariental
+34
+DCG.R
+Insula and Cingulate Gyri
+5
+ORBsup.L
+Frontal
+37
+HIP.L
+Temporal
+29
+INS.L
+Insula and Cingulate Gyri
+40
+PHG.R
+Temporal
+16
+ORBinf.R
+Frontal
+41
+AMYG.L
+Temporal
+3
+SFGdor.L
+Frontal
+45
+CUN.L
+Occipital
+72
+CAU.R
+Subcortial region
+56
+FFG.R
+Temporal
+7
+MFG.L
+Frontal
+68
+PCUN.R
+Pariental
+15
+ORBinf.L
+Frontal
+87
+TPOmid.L
+Temporal
+33
+DCG.L
+Insula and Cingulate Gyri
+90
+ITG.R
+Temporal
+86
+MTG.R
+Temporal
+34
+
+Table 10: Continue table: The site-specific connections between the directed networks of 5 sites of
+MDD patients.
+From
+To
+Index
+ROI
+Lobe
+Index
+ROI
+Lobe
+Site 5
+1
+PreCG.L
+Frontal
+24
+SFGmed.R
+Frontal
+2
+PreCG.R
+Frontal
+46
+CUN.R
+Occipital
+7
+MFG.L
+Frontal
+2
+PreCG.R
+Frontal
+10
+ORBmid.R
+Frontal
+25
+ORBsupmed.L
+Frontal
+13
+IFGtriang.L
+Frontal
+15
+ORBinf.L
+Frontal
+15
+ORBinf.L
+Frontal
+4
+SFGdor.R
+Frontal
+24
+SFGmed.R
+Frontal
+10
+ORBmid.R
+Frontal
+26
+ORBsupmed.R
+Frontal
+77
+THA.L
+Subcortial region
+33
+DCG.L
+Insula and Cingulate Gyri
+31
+ACG.L
+Insula and Cingulate Gyri
+34
+DCG.R
+Insula and Cingulate Gyri
+38
+HIP.R
+Temporal
+36
+PCG.R
+Insula and Cingulate Gyri
+39
+PHG.L
+Temporal
+39
+PHG.L
+Temporal
+4
+SFGdor.R
+Frontal
+46
+CUN.R
+Occipital
+77
+THA.L
+Subcortial region
+56
+FFG.R
+Temporal
+32
+ACG.R
+Insula and Cingulate Gyri
+67
+PCUN.L
+Pariental
+4
+SFGdor.R
+Frontal
+78
+THA.R
+Temporal
+39
+PHG.L
+Temporal
+86
+MTG.R
+Temporal
+16
+ORBinf.R
+Frontal
+89
+ITG.L
+Temporal
+12
+IFGoperc.R
+Frontal
+35
+
diff --git a/EtE0T4oBgHgl3EQfgwHQ/content/tmp_files/load_file.txt b/EtE0T4oBgHgl3EQfgwHQ/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9b336cd08a6f270f8b19eb98408e390cfe8a9abf
--- /dev/null
+++ b/EtE0T4oBgHgl3EQfgwHQ/content/tmp_files/load_file.txt
@@ -0,0 +1,1876 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf,len=1875
+page_content='Learning Personalized Brain Functional Connectivity of MDD  Patients from Multiple Sites via Federated Bayesian Networks  Shuai Liu ∗  Xiao Guo †  Shun Qi ‡  Huaning Wang §  Xiangyu Chang ¶  January 9, 2023  Abstract  Identifying functional connectivity biomarkers of major depressive disorder (MDD) patients is  essential to advance understanding of the disorder mechanisms and early intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' However,  due to the small sample size and the high dimension of available neuroimaging data, the  performance of existing methods is often limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Multi-site data could enhance the statistical  power and sample size, while they are often subject to inter-site heterogeneity and data-sharing  policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' In this paper, we propose a federated joint estimator, NOTEARS-PFL, for simultaneous  learning of multiple Bayesian networks (BNs) with continuous optimization, to identify disease-  induced alterations in MDD patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We incorporate information shared between sites and  site-specific information into the proposed federated learning framework to learn personalized  BN structures by introducing the group fused lasso penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We develop the alternating direction  method of multipliers, where in the local update step, the neuroimaging data is processed at  each local site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Then the learned network structures are transmitted to the center for the global  update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' In particular, we derive a closed-form expression for the local update step and use the  iterative proximal projection method to deal with the group fused lasso penalty in the global  update step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We evaluate the performance of the proposed method on both synthetic and  real-world multi-site rs-fMRI datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The results suggest that the proposed NOTEARS-PFL  yields superior effectiveness and accuracy than the comparable methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  1  Introduction  Major depressive disorder (MDD) is one of the most prevalent, costly, and disabling mental disorders  worldwide [Cassano and Fava, 2002, Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' It is often characterized by persistent sadness,  worthlessness, hopelessness, anxiety, and cognitive impairments [Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The negative  psychology of MDD can lead to severe consequences, including suicide [Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The  ∗Department of Information Systems and Intelligent Business, School of Management, Xi’an Jiaotong University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  email: hljliushuai@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  †Corresponding author: Center for Modern Statistics, School of Mathematics, Northwest University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' email:  xiaoguo@nwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  ‡Key Laboratory of Biomedical Information Engineering of Ministry of Education, Institute of Health and  Rehabilitation Science, School of Life Science and Technology, Xi’an Jiaotong University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' email: qishun@xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  §Department of Psychiatry, Xijing Hospital, Air Force Medical University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' email: xskzhu@fmmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  ¶Center for Intelligent Decision-Making and Machine Learning, School of Management, Xi’an Jiaotong University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  email: xiangyuchang@xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='02423v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='LG]  6 Jan 2023  productivity loss and disability due to MDD also pose a severe burden to society and affected  families [Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Therefore, understanding the pathogenesis of MDD is crucial for effective  intervention, diagnosis, and treatment [Belmaker and Agam, 2008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Diagnostic neuroimaging has  been shown to be an effective modality to understand the underlying pathological mechanisms  and cognitive information of brain diseases [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2016, 2017], and has been used for early  identification of MDD [Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Resting-state functional magnetic resonance imaging (rs-  fMRI) is one of the most widely used imaging modes for MDD identification [Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2011, Liu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2013, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' It can detect abnormal brain functional connectivity in MDD patients,  allowing non-invasive investigation of the disease [Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Functional connectivity (FC),  an essential biomarker in neurological disorders, is traditionally defined as the Pearson correlation  between different regions of interest (ROIs) [Ryali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' However, this association still does  not truly determine the direction of interactions between ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' An accurate understanding of this  directional information is critical to understanding the brain functional integration of MDD [Price  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' To further investigate FC with rs-fMRI data, a number of approaches beyond the  Pearson correlation are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' For example, Kandilarova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' [2018] used a dynamic causal  modeling (DCM) method to reveal differences of FC in anterior insula between healthy control (HC)  participants and MDD patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' [2011] used Granger causality (GC) analysis method  to investigate the aberrant patterns of FC in MDD, and identify the limbic inhibition of dorsal cortex  as a key biomarker for MDD diagnosis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Nevertheless, these methods are not robust for analyzing  specific individuals or large-scale ROIs [Molenaar, 2004, Friston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Bayesian network (BN), an approach for studying the conditional dependencies between variables  in a system via a directed acyclic graph (DAG), has been used for detecting abnormal brain FC  in MDD patients [Koller and Friedman, 2009, Ide et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2014, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Compared with  DCM and GC methods, BN benefits from utilizing more samples to improve the robustness and  stability of the resulting FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Note that the amount of rs-fMRI data collected at a single imaging site  is limited due to the difficult and expensive data acquisition in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Therefore, it is difficult  to provide enough data from a single site to make a good BN estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Fortunately, multiple  rs-fMRI datasets from different sites provide the possibility for enhancing the quality of learned BNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  However, multi-site data are often subject to inter-site heterogeneity and data-sharing policies [Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2020, Tong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' How to learn more helpful heterogeneous information from multi-site  data with considering data-sharing policies is still one of the research hotspots of BN learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  In this paper, we consider the problem of learning BN structures using multiple datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Our  initial interest arises from discovering noninvasive biomarkers inherent in multi-site rs-fMRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Generally, data collected at multiple imaging sites have a common disease of interest (such as MDD),  which results in that they may share similar disease-induced connectivity abnormalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Therefore,  we consider leveraging data from multi-site to learn multiple BNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We are committed to addressing  the three challenges: (1) computational inefficiency of the structural learning algorithms of  BNs, which partially comes from the DAG constraint and the resulting combinatorial optimization  problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' (2) large heterogeneity of multi-site rs-fMRI data, which is caused by inter-site variation  in scanners, image acquisition protocols, and patient populations across sites [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2022];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (3) data sharing barriers, that is, patients worry that sharing their medical data will lead to  personal information disclosure [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The government is likely to promulgate some privacy  protection regulations and inflict severe penalties for medical data breach events [Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  To address the above issues, we propose a federated joint estimator for the simultaneous learning  of multiple BNs with continuous optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' In pursuit of computational efficiency, we convert the  2  traditional combinatorial BN structure learning problem into a continuous numerical optimization  problem inspired by the NOTEARS method [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' To capture the heterogeneity, we  learn multiple BNs from multi-site data but with similar structures by introducing the group fused  lasso penalty instead of learning a single BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' To lighten the data-sharing barriers, we study the  structure learning of multiple BNs in the federated learning (FL) setting, which is a natural method to  tackle the data-sharing problem [Pfitzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We refer to the proposed NOTEARS-based  Personalized Federated Learning framework of learning multiple BNs’ structures as NOTEARS-PFL,  whose whole protocol is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Figure 1: Illustration of the learning framework of NOTEARS-PFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Each site stores and processes  rs-fMRI data locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' In the t-th iteration, each site estimates the local weighted adjacency matrix  W k  (t) through the local update step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Then, each site transmits the estimated local matrix to the  center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Center updates the global weighted adjacency matrices ˜Z(t) through the global update step  based on the received ˜  W(t) = {W 1  (t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , W K  (t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Finally, the center transmits the estimated global  matrix to each site for the (t + 1)-th iteration update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  3  Center  Global model: Z(t) ← argmin [L, (W(t),z,O(t-1))  ①(t) ←[W(t),Z(t), O(t-1))  Wh  ()  Z2  02  1),0h  W(t)  1)  Local model: W()  arg win [L, (wK, z(-1), (-1))  +2  data  data  data  rs-fMRI  preprocessing  rs-fMRI  preprocessing  preprocessing  rs-fMRI  scan  scan  scan  Site K  Site 1  Site 2The merits of the proposed NOTEARS-PFL are multi-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  NOTEARS-PFL is capable of learning site-specific BNs from multi-site rs-fMRI data, which  makes full use of the multi-site data while still being flexible enough to capture the site-specific  BN structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  NOTEARS-PFL is subtly fitted in an FL framework without sharing and exchanging the  original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  NOTEARS-PFL overcomes the computational intractability of traditional BNs by transforming  the discrete combinatorial optimization to continuous numerical optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  NOTEARS-PFL can effectively detect the common abnormal FC in MDD patients among  multiple sites, especially the connectivity between the insula, thalamus cortex, and middle  temporal gyrus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Meanwhile, NOTEARS-PFL can also identify the site-specific abnormal FC  due to the differences in scanners, protocols, and patient populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' In Section 2, we briefly review directed FC estimation methods  for rs-fMRI data and BN structure learning with continuous optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We present the formulation  and optimization approach of NOTEARS-PFL in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We conduct simulation experiments to  validate NOTEARS-PFL in Section 4, and analyze the real-world multi-site rs-fMRI data in Section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Finally, we conclude this paper in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  2  Related Work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1  Directed FC Estimation Methods for rs-fMRI Data  The directed FC, also known as effective FC, helps depict the abnormalities or dysfunction of the  brain connectivity network [Smith, 2012] and shows how information flows in the brain connectivity  network [Henry and Gates, 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Based on rs-fMRI data, many directed FC estimation methods  have been developed, including the following four types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (1) Granger causality (GC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' GC establishes a model in the framework of vector autoregression  (VAR) and determines the edges to be added after considering the autoregressive effect of each region  [Granger, 1969, Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' GC is applied to detect the effects of psychostimulants on youths  with attention deficit hyperactivity disorder (ADHD) [Peterson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' However, when GC is  applied to analyze specific individuals, heterogeneity across individuals may lead to spurious results  [Molenaar, 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (2) Dynamic Causal Modeling (DCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' DCM is a method to capture nonlinear relationships  between brain regions that may be affected by different stimuli during scanning [Friston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2003],  and has been used to detect the propagation of epileptic seizures [Murta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The limitation  of DCM is that it cannot handle large amounts of ROIs or analysis group rs-fMRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' [Friston  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (3) Constraint-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Constraint-based methods estimate directed edges by conditional  independence tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Common constraint-based methods include Peter Spirtes, and Clark Glymour  (PC) algorithm [Spirtes and Glymour, 1991], conservative PC (CPC) [Ramsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2006], fast  causal inference (FCI) [Spirtes, 2001], and cyclic causal discovery (CCD) [Richardson, 1996].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The  PC algorithm can obtain reliable networks on data aggregated across individuals [Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  4  However, according to results from Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' [2011], PC can detect the existence of edges but can  not accurately determine the directions at the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (4) Score-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Score-based methods define a score function that measures how well the  BN structure fits the observed data and searches for the best network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Standard score-based  methods include greedy equivalent search (GES) [Chickering, 2002], independent multiple-sample  greedy equivalent search (IMaGES) [Ramsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2010], and group iterative multiple model  Estimation (GIMME) [Gates and Molenaar, 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Score-based methods typically require large  samples to identify edges and directions, which limits the utility of these methods when using rs-fMRI  data for individual-level analysis [Mumford and Ramsey, 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Meanwhile, the performance of  score-based methods is poor on data with a large number of ROIs, which makes it difficult to analyze  the complete parcellation of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Due to the combinatorial acyclicity constraint of DAG, it is  often hard to find BN with the optimal score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' And its scalability is also limited [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  BN Structure Learning with Continuous Optimization  In this work, we focus on BN learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  BN is a probabilistic graphical model that captures  dependencies among a collection of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Each model is associated with a directed  network G = (V, E), where the vertex set V corresponds to the random variables and the edge set E  corresponds to the set of conditional independence [Drton and Maathuis, 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  BN models the conditional independence among random variables via Markov properties [Cowell,  1998, Marrelec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Denote X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , Xd)⊤ that satisfies the local Markov property  with respect to a DAG G if  Xv⊥XndG(v)\\paG(v) | XpaG(v),  where paG(v) = {w ∈ V : {w, v} ∈ E} denotes the parents of node v in G, ndG(v) = V \\deG(v)  denotes the non-descendants of v in G with deG(v) = {w ∈ V : w = v or v → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' → w in G}  representing the descendants of v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Alternatively, if X is a continuous random vector and it has a density P(X) with respect to a  product measure, then the local Markov property is equivalent to the following factorization property  [Verma and Pearl, 1990],  P(X) =  d  �  v=1  P  �  Xv | XpaG(v)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (1)  The structure learning of BN is to estimate the DAG G using the observations from the density  P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Score-based methods seek the optimal G by minimizing the score function S(W) with respect  to the corresponding weighted adjacency matrix W ∈ Rd×d, subject to that the induced graph g(W )  is a DAG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' see the LHS of equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The DAG constraint initiates a combinatorial optimization  problem, which is computationally costly when d is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Recently, Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' [2018] proposed a NOTEARS method which enforces the acyclicity via  setting a smooth function h(W ) exactly at zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' see the RHS of equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' By this novel  characterization of acyclicity, the resulting optimization problem can be solved efficiently using  standard algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  minW ∈Rd×d S(W ),  minW ∈Rd×d S(W ),  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' g(W ) ∈ DAGs,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' h(W ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (2)  5  Currently, NOTEARS has been extended to handle problems in many situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' For example,  DAG Graph Neural Network (DAG-GNN) extends NOTEARS to solve nonlinear cases by incor-  porating neural networks [Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Graph AutoEncoder (GAE) extends NOTEARS and  DAG-GNN into a graph autoencoder model, facilitating vector-valued variables [Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Pamfil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' [2020] proposed DYNOTEARS method to learn dynamic BNs from time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' [2020] used reinforcement learning to find the optimal DAGs from incomplete data  based on NOTEARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  To the best of our knowledge, few studies have focused on using NOTEARS to learn personalized  BNs from fMRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' [2022] proposed a joint BN estimation model to detect abnormal  directed FC in schizophrenia (SZ) patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' However, the joint estimation model proposed by Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' [2022] is not applicable for the FL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Ng and Zhang [2022] proposed NOTEARS-ADMM  model to estimate the structure of BN in FL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' However, they did not consider the data  heterogeneity, which can lead to model bias [Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2018, Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Compared with  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' [2022] and Ng and Zhang [2022], the proposed NOTEARS-PFL is powerful because it  captures the data heterogeneity, overcomes the data sharing barriers, and attains computational  efficiency, simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  3  NOTEARS-based Personalized Federated Learning Framework  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1  Preliminary  The BN associated with a DAG G can also be thought of as a structural equation model [Bollen,  1989]  Xi = fi(W T  i X) + Zi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , d,  (3)  where W = (W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , Wd) ∈ Rd×d denotes the weighted adjacency matrix whose nonzero coefficients  correspond to the directed edges in G, fi’s are functions related to the conditional distributions of  Xi’s, and Zi’s are jointly independent random noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Equation (3) is equivalent to the local Markov  property and factorization property introduced in Section 2 [Drton and Maathuis, 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  We consider the linear structural equation model  Xi = W T  i X + Zi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (4)  Actually, the multivariate Gaussian distribution can be modeled as equation (4) with Zi’s being  independent Gaussian noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Let the data matrix x ∈ Rn×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' [2018] formulates the structure learning of BNs as  min  W ∈Rd×d  1  2n ∥x − xW ∥2  F + λ∥W ∥1,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  h(W ) = 0,  (5)  where  h(W ) := tr  �  eW ⊙W �  − d = 0,  (6)  ⊙ denotes the Hadamard product, eA = �∞  k=0  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='Ak denotes the the matrix exponential operator  [Leonard, 1996], ∥A∥F denotes the Frobenius norm, ∥A∥1 refers to the entry-wise ℓ1 norm and the  tuning parameter λ > 0 controls the amount of sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  Problem Formulation  We now proceed to formulate our NOTEARS-PFL method, which can efficiently learn multiple  heterogeneous BNs without sharing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  We consider the setting in which there is a fixed set of K sites in total, and each site has its  own local dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The k-th site holds nk i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' samples, denoted by xk =  �  xk  i  �nk  i=1 ∈ Rnk×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Let  D = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , xK} be the K overall observational datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The goal is to infer the K weighted  adjacency matrices ˜  W0 = {W 1  0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , W K  0 } (correspond to the BN structures) using D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  To incorporate the heterogeneity among K sites, we consider the following optimization problem:  min  ˜  W  K  �  k=1  1  2nk  ���xk − xkW k���  2  F + R( ˜  W ),  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  h(W k) = 0, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , K,  (7)  where h(W k) is defined in the same way as equation (6) and R( ˜  W ) is a penalty function for inducing  the similarity and sparsity within ˜  W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  With the data sharing constraint, we formulate equation (7) into the framework of ADMM,  which is a natural solution to distributed learning and FL [Ng and Zhang, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Define a set of  consensus variables ˜Z = {Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , ZK}, which can be thought of as the global weighted adjacency  matrices, in contrast to the local weighted adjacency matrices ˜  W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Then we rewrite equation (7) as  min  ˜  W , ˜  Z  K  �  k=1  L(W k, xk) + R( ˜Z),  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  h(Zk) = 0,  W k = Zk,  k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , K,  (8)  where  L(W k, xk) :=  1  2nk  ���xk − xkW k���  2  F ,  (9)  h(Zk) is defined in the same way as equation (6), and R( ˜Z) is defined by  R( ˜Z) := λ1  K  �  k=1  ∥Zk∥1 + λ2  K−1  �  k=1  ∥Zk+1 − Zk∥F ,  (10)  where λ1 > 0 controls the sparsity of Zk and λ2 > 0 controls the similarity within ˜Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  The general optimization procedure can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' First, each site estimates the  local weighted adjacency matrix W k by using local dataset xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Then, instead of transmitting the  original data, each site only transmits its local weighted adjacency matrix to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' After that,  the center updates the global weighted adjacency matrices ˜Z0 = {Z1  0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , ZK  0 }, based on received  ˜  W0, Finally, the center transmits the updated ˜Z to each site, which performs the next update locally  based W k = Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  In the next subsection, we provide the computational details of NOTEARS-PFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3  Optimization Algorithm  By applying the augmented Lagrangian method [Bertsekas, 2014], the equation (8) can be written as  L  �  ˜  W , ˜Z, Θ  �  =  K  �  k=1  L(W k, xk) + R( ˜Z) +  K  �  k=1  αkh(Zk) + ρ1  2  K  �  k=1  |h(Zk)|2 +  K  �  k=1  tr(βk(W k − Zk)⊤)  + ρ2  2  K  �  k=1  ���W k − Zk���  2  F ,  (11)  where ˜Θ = {ρ1, ρ2, α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , αK, β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , βK}, ρ1, ρ2 are the penalty coefficients, α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , αK and  β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , βK are the Lagrange multiplies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  At the t-th iteration, we can solve equation (11) as follows:  (1) Local update:  ˜  W(t) ← arg min  ˜  W  �  L  �  ˜  W , ˜Z(t−1), ˜Θ(t−1)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (12)  (2) Global update:  ˜Z(t) ← arg min  ˜  Z  �  L  �  ˜  W(t), ˜Z, ˜Θ(t−1)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (13)  (3) Global update:  ˜Θ(t) ←  �  ˜  W(t), ˜Z(t), ˜Θ(t−1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (14)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1  Federated Local Update of ˜  W  Taking the derivative of equation (11) with respect to ˜  W , we can update each W k  (t) of ˜  W(t) as  W k  (t) := arg min  W k  (L(W k, xk) + ρ2,(t−1)  2  ���W k − Zk  (t−1)  ���  2  F + tr(βk,(t−1)(W k − Zk  (t−1))⊤)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (15)  Let U k =  1  nk (xk)⊤xk and assume U k + ρ2I is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Equation (15) has a closed form expression  W k  (t) = (U k + ρ2,(t−1)I)−1(ρ2,(t−1)Zk  (t−1) − βk,(t−1) + U k),  (16)  where the details of the derivation are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  Federated Global Update of ˜Z  Similar to equation (15), we can update each Zk  (t) of ˜Z(t) as  Zk  (t) := arg min  Zk  {L( ˜Z) + R( ˜Z)},  (17)  8  Algorithm 1 Dykstra-like iterative proximal algorithm (DIPA)  Input: K: Number of site;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' ˜Z: global adjacency matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' ˜  W : local adjacency matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' N: The  number of iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' ϵ: Tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Output: ˜Z: The result of equation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  1: ∇Lk  �  Zk�  = equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  2: for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , K do  3:  U k = Zk − 1  C ∇L  �  Zk�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  4: end for  5: ¯U ← Transform (U k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  6: Set A0 = ¯U, Mn = 0, Qn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  7: for n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , N do  8:  V(n) ← proxR2  �  A(n) + M(n)  �  , see equation (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  9:  M(n+1) ← A(n) + M(n) − V(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  10:  A(n+1) ← proxR1  �  V(n) + Q(n)  �  , see equation (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  11:  Q(n+1) ← V(n) + Q(n) − A(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  12:  Break: if  ��A(n+1) − A(n)  ��  F < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  13: end for  14: ¯U ← A(N+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  15: ˜Z ← Inverse-Transform ( ¯U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  16: return ˜Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  where  L( ˜Z) :=  K  �  k=1  αk,(t−1)h(Zk) + ρ1,(t−1)  2  K  �  k=1  |h(Zk)|2 + ρ2,(t−1)  2  K  �  k=1  ���W k  (t) − Zk���  2  F )  +  K  �  k=1  tr(βk,(t−1)(W k  (t) − Zk)⊤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Due to the acyclicity term h(Zk), we are not able to derive a closed-form solution for equation  (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' To tackle this issue, we utilize a proximal gradient method which constructs a quadratic  approximation of L( ˜Z) at ˜Z(t−1), and updates ˜Z(t) by  ˜Z(t) = arg min  ˜  Z  {L( ˜Z(t−1)) +  K  �  k=1  �  Zk − Zk  (t−1), ∇Lk(Zk  (t−1))  �  + C  2  K  �  k=1  ���Zk − Zk  (t−1)  ���  2  F + R( ˜Z)},  (18)  where ∇Lk(·) is the gradient of Lk(·), C > 0 is the Lipschitz constant of ∇Lk(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' After some simple  9  manipulations of equation (18), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', ignoring constant terms of Zk  (t−1), we have  ˜Z(t) = arg min  ˜  Z  {C  2  K  �  k=1  ����Zk −  �  Zk  (t−1) − 1  L∇Lk  �  Zk  t−1  ������  2  F  + R( ˜Z)},  = proxCR  � K  �  k=1  �  Zk  (t−1) − 1  L∇Lk  �  Zk  (t−1)  ���  ,  (19)  where  ∇Lk  �  Zk  (t−1)  �  = αk,(t−1)∇h(Zk  (t−1))+ρ1,(t−1)h(Zk  (t−1))∇h(Zk  (t−1))−βk,(t−1)+ρ2,(t−1)(Zk  (t−1)−W k),  (20)  and proxg(v) = arg min  x  ( 1  2∥x − v∥2  F + g(x)) denotes the proximal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Unlike the calculation of W k, we cannot separate the equation (19) by k because of the grouped  constraint R( ˜Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Instead, we must estimate the whole set of matrices ˜Z(t) jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Inspired by  Gibberd and Nelson [2017], we use the following iterative proximal projection step to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Let U k = Zk  (t−1) − 1  C ∇Lk  �  Zk  (t−1)  �  , and perform transform procedure (see AppendixB) for  U k → ¯U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Then, equation (19) can be re-written as  min  ¯  Z  1  2  �� ¯Z − ¯U  ��2  F  �  ��  �  L( ¯  Z)  + λ1∥ ¯Z∥1  � �� �  R1( ¯  Z)  + λ2∥D ¯Z∥2,1  �  ��  �  R2( ¯  Z)  ,  (21)  where D ∈ R(K−1)×K is a backwards difference matrix of the form Di,i = −1, Di,i+1 = 1 for i =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , K − 1, and ∥A∥2,1 := �  k ∥Ak,·∥2 denotes the group ℓ2,1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Actually, equation (21) is  equivalent to the Group-Fused Lasso Signal Approximator (GFLSA) defined by Gibberd and Nelson  [2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We denote the objective in equation (21) by f  � ¯U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' λ1, λ2  �  = proxR1+R2  � ¯U  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Then, we adopt  the Dykstra-like iterative proximal algorithm (DIPA) [Combettes and Pesquet, 2011] to handle  proxR1+R2  � ¯U  �  and find a feasible solution for both the group fused lasso penalty and the lasso  penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We summarize the details of iteration in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3  Federated Global Update of ˜Θ  The update of ˜Θ follows the following iterative steps  βk,t := βk,t−1 + ρ2,t−1  �  W k  (t) − Zk  (t)  �  ,  αk,t := αk,t−1 + ρ1,t−1h  �  Zk  (t)  �  ,  ρ1,t := γ1ρ1,t−1,  ρ2,t := γ2ρ2,t−1,  (22)  where γ1, γ1 ∈ R are hyperparameters that respectively control the increasing speeds of the coefficients  ρ1, ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The overall ADMM algorithm of NOTEARS-PFL is thereby given in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  10  Algorithm 2 ADMM of NOTEARS-PFL  Input: K: Number of site;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' xk: Original data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' ˜Z(0): Initial global adjacency matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' ρ1,0, ρ2,0,  α1,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , αK,0, β1,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , βK,0: Initial parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' T: The maximum of iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' ϵ: Tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Output:  ˆ  Zk: Estimated adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  for iterations t = 1 to T do  Local update step:  for site k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , K do  Update W k  (t) according to equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  end for  Global update step:  for variables ˜Z(t−1), Θ(t−1) in the center do  Zk  (t) ← DIPA (W k  (t), Zk  (t−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Update ˜Θ(t−1) according to equation (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  end for  Break: if  K  �  k=1  ���Zk  (t) − Zk  (t−1)  ���  F  ���Zk  (t)  ���  2  < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  end for  return ˜Z = (Z1  (T), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', ZK  (T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  11  4  Simulations  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1  Simulation Design of Synthetic Data  We simulate synthetic data according to equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' First, we generate a random graph Gtruth  based on Erdös–Rényi (ER) model [Erdös et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 1960].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' To obtain a set of related graphs, we apply  perturbations to Gtruth to create K similar but different graphs ˜G = {G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , GK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The details of  the definition of perturbation are shown in Appendix C, and we denote the perturbation level as pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Given ˜G, we assign uniformly random edge weights to obtain K weights matrices  ˜  W =  {W 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , W K}, where the non-zero entries are sampled uniformly at random form [−2, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='5]∪[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='5, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Given ˜  W , we generate the datasets D = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , xK} based on equation (4) with standard Gaussian  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  To evaluate and compare the performance of the proposed method, we apply the NOTEARS  [Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2018] method to estimate each network independently from each site’s local data  (NOTEARS-SIG for short), and apply the NOTEARS method to learn a common network structure  for all contexts (NOTEARS-AVG for short), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', an “average” network that treats all site’s data as  samples in one dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We also compare the performance between NOTEARS-ADMM [Ng and  Zhang, 2022] and the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  We use the following four metrics to evaluate the learning performance of different methods (see  Appendix C for a detailed definition of metrics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Edge arrowhead error:  Error =  FP + FN  TP + TN + FP + FN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (23)  Edge arrowhead precision:  Precision =  TP  TP + FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Edge arrowhead F-score:  F-score = 2 · Precision · Recall  Precision + Recall ,  where Recall = TP/(TP + FN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Edge arrowhead structural Hamming distance (SHD): Given two structures, this  measure is the minimum number of edges needed to convert the learned graph into the true  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  We compare the performance of the proposed NOTEARS-PFL, NOTEARS-SIG, NOTEARS-  AVG, and NOTEARS-ADMM on the following three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Performance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' the number of variable: We fix the number of sites K = 10, perturbation  level pl = 10%, set the total sample size nt = �K  k=1 nk equals to three times the number of variable d  (nt = 3d, nk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3d), and compare the model performance of network structure learning for different  variable number d = 10, 20, 30, 40, 50, 60, 70, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  12  Performance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' the number of site: We fix the number of variable d = 50, perturbation  level pl = 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We set the total sample size nt = 256, and distributed evenly across the number of  site K = 2, 4, 8, 16, 32, 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Performance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' perturbation level: We fix the number of site K = 10, the number of  variable d = 30, sample size nt = 3d, and vary the perturbation level pl = 5%, 10%, 15%, 20%, 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  The evaluation metrics for one simulation run are averaged over the K sites’ dataset, and we  repeat each experiment 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The final evaluation metrics are thereby averaged over the ten  simulation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (a)  (b)  (d)  (c)  Figure 2: Simulation results for synthetic data under different variable numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' (a) Average edge  arrowhead error (lower is better);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' (b) Average edge arrowhead SHD (lower is better);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' (c) Average  edge arrowhead precision (higher is better);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' (d) Average edge arrowhead F-score (higher is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  In the synthetic data experiment, we first evaluate the effect of the number of variables d on  the performance of each compared method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The results are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' It is clear that  NOTEARS-PFL performs the best among all the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The average arrowhead adjacency error of  NOTEARS-PFL has a relatively low level ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='02 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' In contrast, for NOTEARS-SIG,  the average arrowhead adjacency error is far larger than NOTEARS-PFL, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='04 to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' This may be because the sample size of a single dataset is too small for NOTEARS-SIG to  give a satisfactory estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The same result is observed in the average arrowhead SHD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' For the  other two metrics, NOTEARS-PFL consistently results in higher average arrowhead precision and  F-score than the other three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Compared with NOTEARS-SIG and NOTEARS-AVG, the  average arrowhead precision of NOTEARS-PFL is also more stable with respect to different variable  numbers, indicating that it can successfully identify most of the edges in high-dimensional settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  We also test the effect of the number of sites K on the performance of each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The  results are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Clearly, NOTEARS-PFL performs better than NOTEARS-SIG,  NOTEARS-AVG, and NOTEARS-ADMM for all the tested cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' As K increases, the performance  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3 -  80 -  SHD  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  40 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1 -  20 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0 -  0  20  50  10  30  40  60  70  80  10  20  30  40  50  60  70  80  Number of variables  Number of variables  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='8 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='6  ion  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='5  Preci  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1  10  20  30  40  50  80  60  70  10  20  30  40  50  60  70  80  Number of variables  Number of variables  NOTEARS-PFL  NOTEARS-SIG  NOTEARS-AVG  NOTEARS-ADMM(a)  (b)  (d)  (c)  Figure 3: Simulation results for synthetic data under different site numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' (a) Average edge  arrowhead error (lower is better);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' (b) Average edge arrowhead SHD (lower is better);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' (c) Average  edge arrowhead precision (higher is better);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' (d) Average edge arrowhead F-score (higher is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (a)  (b)  (d)  (c)  Figure 4: Simulation results for synthetic data under different perturbation levels (a) Average edge  arrowhead error (lower is better);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' (b) Average edge arrowhead SHD (lower is better);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' (c) Average  edge arrowhead precision (higher is better);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' (d) Average edge arrowhead F-score (higher is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='10  120  100:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='08  80 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='06  SHD  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='04  40 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='02  20 -  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='00  10  20  50  60  0  10  20  30  40  50  60  0  30  40  Number of sites  Number of sites  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='6  Precisic  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0 -  0  10  20  30  40  50  60  0  10  20  30  50  60  40  Number of sites  Number of sites  NOTEARS-PFL  NOTEARS-SIG  NOTEARS-AVG  NOTEARS-ADMM0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='08:  35 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='07  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='06 -  25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='05  D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='04  15-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='03  10 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='02  5 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='01  5%  10%  15%  20%  30%  5%  10%  15%  20%  30%  Perturbation levels  Perturbation levels  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='85  ision  core  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='80  Precis  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='6  S  F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='70 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3  5%  10%  15%  20%  30%  5%  10%  15%  20%  30%  Perturbation levels  Perturbation levels  NOTEARS-PFL  NOTEARS-SIG  NOTEARS-AVG  NOTEARS-ADMMof NOTEARS-SIG, NOTEARS-AVG decline rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Although the performance of NOTEARS-PFL  also decreases with the increase of K, it still provides a relatively stable and satisfying performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  As expected, NOTEARS-SIG does not perform better at larger K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' A possible reason is that  NOTEARS-SIG only utilizes the information of a single dataset whose sample size decreases with  the increase of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' On the contrary, NOTEARS-PFL works well in the setting with a large site  number because information can be exchanged during optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' These results are also consistent  with the existing literature [Ng and Zhang, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Finally, we examine the effect of the perturbation level pl on the performance of each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  The results are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' It is clear that NOTEARS-PFL significantly outperforms the other  three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' It can be seen that with the increase of perturbation level, average arrowhead error  and SHD increase significantly, while average arrowhead precision and F-score decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' When the  perturbation level is relatively low, the difference between the four methods is relatively small, but  when the perturbation level is relatively large, the advantage of NOTEARS-PFL is more significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  5  Application to Real-world rs-fMRI Data  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1  Data and Preprocessing  (1) Data acquisition: The multi-site rs-fMRI datasets in this study were obtained from the  DecNef Project Brain Data Repository1, collected as part of the Japanese Strategic Research Program  for the Promotion of Brain Science (SRPBS) database project [Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' This dataset  consists of 255 MDD patients (135 men versus 120 women) and 791 HC participants (426 men versus  365 women) from 8 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Some samples were discarded as some sites only have rs-fMRI data on  HC participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' As a result, 238 MDD patients and 475 HC participants from 5 sites were used  in this study (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The demographic characteristics of all subjects in both datasets are  summarized in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (2) Data preprocessing : Each subject of the data underwent a single rs-fMRI session and a  structural MRI session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' During the fMRI scan, participants were asked to relax, stay awake and  think about anything in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Detailed imaging parameters of rs-fMRI and T1-weighted (T1w)  structural MRI at each site are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We applied the following preprocessing steps  to the data by using DPARSF [Yan and Zang, 2010] software2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' For each participant, we removed  the first 10 MRI volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Slice-time correction and head motion were corrected on the remaining  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The T1-weighted images were unified segment segmentation and diffeomorphic anatomical  registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The rs-fMRI data were smoothed with a 6mm full-width at half-maximum Gaussian  kernel and the Friston 24-parameter model was used to regress head motion effects further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Then,  all the time series were filtered by linear detrending and a temporal band-pass filter (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='01 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='08 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (3) Feature extraction: After preprocessing, we estimated the time series of 116 ROIs based  on the automated anatomical labeling (AAL) template [Tzourio-Mazoyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Details of  cortical and subcortical regions of interest (ROIs) defined in the AAL template are presented in  Table 6 and Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We selected 56 ROIs from 116 ROIs because these regions are considered to  be related to MDD in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The names of the selected ROIs are shown in bold in Table  6 and Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The voxel-wise fMRI time courses were averaged into one regional average time  course within each ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The data for each participant was then in the form of an X× 56 matrix (X  1https://bicr-resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='atr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='jp/srpbsopen/  2http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='rfmri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='org/DPARSF  15  sampling points in each regional average fMRI time course and 56 selected ROIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  Experimental Setup  We randomly select 70% of MDD patients and HC participants from each site and aggregate their  rs-fMRI data into the MDD discovery dataset and HC discovery dataset for that site, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Similarly, we aggregate the remaining 30% into the MDD validation dataset and HC validation  dataset for that site, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We analyze the 5 sites’ discovery datasets of MDD patients, which  are regarded as 5 related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Then we can obtain a brain connection network of MDD for each  site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' To study the differences in brain FC, the brain connection networks of HC for five sites, which  are viewed as another set of related tasks, are also obtained using the proposed NOTEARS-PFL  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' According to the estimated BN structures, we analyze the abnormal FC in MDD by the  following aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1  Important Nodes Identification  By identifying important nodes, we reveal the aberrant nodal characteristics of the brain functional  connectivity networks of MDD from multi-site datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  Overlapping Connections Identification  By identifying overlapping connections, we obtain the common functional connections for MDD  patients from multiple sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' By comparing with the overlapping connections of HC participants, we  further investigate the altered functional connections in MDD patients, which provide the potential  biomarkers for clinic treatment of MDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3  Site-specific Connections Identification  By identifying the site-specific connections in different sites, we investigate the specific FC alterations  in MDD at each site due to the differences in image acquisition protocols and patient populations at  different sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4  Comparison with Other Methods  We randomly select 5 MDD patients and 5 HC participants from each site’s MDD validation dataset  and HC validation dataset and repeat such selection 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We then have 10 datasets containing  MDD data and HC data from the 5 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Based on these datasets, we compare NOTEARS-PFL with  NOTEARS-SIG and existing standard analysis methods of rs-fMRI data: GES [Chickering, 2002],  PC [Spirtes and Glymour, 1991], and GC [Granger, 1969].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' To further test whether NOTEARS-PFL  is able to learn the differences between MDD patients and HC participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3  Experimental Results  In this section, we use NOTEARS-PFL method to analyze the multi-site rs-fMRI data from 5 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  We can obtain the brain functional connection network of MDD patients for each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' To study the  differences in brain FC, we also obtain the brain functional connection network of HC participants  for each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  16  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1  Results of Important Nodes  To better understand the differences in brain FC between MDD patients and HC participants through  the estimated directed brain networks from multi-site rs-fMRI data, we further use the connection  degree to identify some important nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The connection degree (including in and out degree) is a  commonly used measure of functional connectivity [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' More specifically, it represents  the amount of directed functional connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The greater the connection degree, the higher the  effective FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Identifying important nodes further helps to discover the nodes associated with the  disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Table 1 shows the differences in total connection degrees between MDD patients and HC  participants at 5 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The total out connection degrees in Table 1 are 657 for MDD and 478 for  HC, indicating that MDD has 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4% effective FC than HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Similarly, MDD also has 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4% more  effective FC than HC, with a total in connection degrees of 516 and 418, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The rise of FC  is an apparent robust biomarker of MDD disease [Zamoscik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2014], which can also be found in  our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Table 1: The top 10 out and in connection degrees of MDD patients and HC participants  Index  ROI  Abbreviation  Degree  Index  ROI  Abbreviation  Degree  MDD (Out)  77  Thalamus  THA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  75  HC (Out)  55  Fusiform gyrus  FFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  58  29  Insula  INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  74  37  Hippocampus  HIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  57  35  Cingulate gyurs, posterior part  PCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  72  31  Cingulate gyrus, anterior part  ACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  55  33  Cingulate gyrus, mid part  DCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  68  38  Hippocampus  HIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  49  34  Cingulate gyrus, mid part  DCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  67  9  Middle frontal gyrus, orbital  HIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  47  31  Cingulate gyrus, anterior part  ACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  63  29  Insula  INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  46  72  Caudate  CAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  62  77  Thalamus  THA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  45  71  Caudate  CAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  61  33  Cingulate gyrus, mid part  DCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  43  55  Fusiform gyrus  FFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  58  56  Fusiform gyrus  FFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  40  79  Heschl gyrus  HES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  57  79  Heschl gyrus  HES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  38  Total  657  Total  478  MDD (In)  24  Superior frontal gyrus, medial  SFGmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  79  HC (In)  3  Superior frontal gyrus, dorsolateral  SFGdor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  55  90  Inferior temporal gyrus  ITG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  62  9  Middle frontal gyrus, orbital  ORBmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  48  26  Superior frontal gyrus, medial orbital ORBsupmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  59  26  Superior frontal gyrus, medial orbital ORBsupmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  45  89  Inferior temporal gyrus  ITG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  54  90  Inferior temporal gyrus  ITG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  44  3  Superior frontal gyrus, dorsolateral  SFGdor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  48  10  Middle frontal gyrus, orbital  ORBmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  41  46  Cuneus  CUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  46  14  Inferior frontal gyrus, triangular  IFGtriang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  40  11  Inferior frontal gyrus, opercular  IFGoperc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  45  24  Superior frontal gyrus, medial  SFGmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  39  25  Superior frontal gyrus, medial orbital ORBsupmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  42  1  Precentral gyrus  PreCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  38  41  Amygdala  AMYG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  41  89  Inferior temporal gyrus  ITG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  36  32  Cingulate gyrus, anterior part  ACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  40  6  Superior frontal gyrus, orbital  ORBsup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  32  Total  516  Total  418  From Table 1, we can find some ROI nodes with high connection degrees, such as THA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  (thalamus), INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L (insula), SFGmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R (superior frontal gyrus, medial) in MDD, and FFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L (fusiform  gyrus) and HIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L (hippocampus) in HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Specifically, THA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L and INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L have more outgoing  connections in MDD than in HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' It indicates that THA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L and INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L of MDD have a higher essential  impact on other ROI nodes than HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The result from NOTEARS-PFL is in line with literature  [Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2018, Avery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2014, Porta-Casteràs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' According to Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' [2018],  emerging evidence indicates that the enhanced thalamus FC is an important feature of the underlying  pathophysiology of MDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' This abnormal FC is related to the core clinical symptoms of MDD, such as  anxiety, memory, and attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Our result further implies that the impaired FC of the hippocampus  is potentially an important biomarker of MDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Compared to HC, the bilateral hippocampus in  MDD lack FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' As the core region of the limbic system, the hippocampus plays an important role in  regulating motivation and emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The mode of emotion regulation in MDD may be related to the  decreased FC of the hippocampus [Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' SFGmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R (superior frontal gyrus, medial) and  ITG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R (inferior temporal gyrus) have more incoming connections than outgoing connections, which  indicates that SFGmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R and ITG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R are mainly affected by other ROI nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' These findings are  also consistent with the physiological discoveries of MDD disease [Porta-Casteràs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  17  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  Results of Overlapping Connections  Figure 5 depicts the overlapping connections between the directed networks of 5 sites of MDD  patients and HC participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The overlapping connection refers to the connection shared by five  sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We can use it to study the universal connection between MDD and compare the differences  between MDD and HC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' From Figure 5, we can find some abnormal functional connections in MDD:  INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R (insula) → ORBmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R (middle frontal gyrus, orbital), INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R (insula) → INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L (insula), INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  (insula) → ACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R (cingulate gyrus, anterior part), PCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L (cingulate gyurs, posterior part) →  PCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R (cingulate gyurs, posterior part), THA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L (thalamus) → CAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R (caudate), STG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R (superior  temporal gyrus) → MTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R (middle temporal gyrus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Three common functional connections are  associated with the insula, which further demonstrates the important role of the insula in identifying  abnormal functional connections in MDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The insula has been shown to be a potential primary  biomarker for the diagnosis of MDD [McGrath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' In MDD patients, the insula show  increased FC with other limbic or paralimbic structures, particularly with the ventromedial prefrontal  cortex (vmPFC) and orbitofrontal cortex (OFC) [Avery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2014, Drevets et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2008], which is  also illustrated in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Figure 5: The overlapping connections between the directed networks of 5 sites of MDD patients  and HC participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  In addition to the abnormal insula structure in MDD patients, we also observe the abnormalities  of FC in posterior cingulate gyurs, thalamus, inferior temporal gyrus, and middle temporal gyrus,  which are also consistent with findings in the works of literature [Khundakar and Thomas, 2009,  Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' These cortices belong to the default mode network (DMN) and are considered  to contribute greatly to MDD [Gong and He, 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The abnormalities of FC in DMN have been  reported in many MDD studies [Vasudev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2018], especially the frontal lobe and temporal  lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' These lobes are suggested to be closely associated to the cognitive and information-processing  abilities of MDD patients [Brzezicka, 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  18  R  L  R  L  SFGdo  MFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  CG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1  R  14 1  13 11 9  14  INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  ACGL  ACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  DCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  3  DCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  3  PCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  、PCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  3  %  HIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  41 39 37  PHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  40  PHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Insula and Cingulate Gyri  42  Temporal  41  CUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  CUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  4  Occipital  Pariental  Subcortial region  5  FFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  18  5  LL 6L T8  TVHL  58 Z8 68 06 88 98  STG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  MTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  MTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  ITG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  MTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  ITG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  (a) MDD patients  (b) HC participantsFigure 6: The site-specific connections between the directed networks of 5 sites of MDD patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Site 1: Center of Innovation in Hiroshima (COI), Site 2: Kyoto University (KUT), Site 3: University  of Tokyo (UTO), Site 4: Hiroshima Kajikawa Hospital (HKH), Site 5: Hiroshima University Hospital  (HUH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3  Results of Site-specific Connections  In this study, we are also interested in the connections specific to a site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' A site-specific connection is  a connection that only exists on one site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' By identifying the site-specific connections in different  sites, we can better understand the specificity of FC in MDD patients due to the differences in image  acquisition protocols and patient populations at different sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The site-specific connections of 5  sites of MDD patients are visualized in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' In the results of site 1, we can find the increased  functional connection from frontal to pariental and from frontal to temporal, especially the increased  afferent connections to precuneus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The precuneus is considered one of the hubs of DMN, which is  generally associated with the rumination of negative and sad thoughts in MDD [Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  It has been shown to play a key role in identifying MDD [Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' A study of effective  FC in MDD has shown a significant increase in forward connectivity from the middle and inferior  temporal cortical areas to the precuneus [Öngür et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2003], which is consistent with our study (the  connections IFGoperc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L → PCUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R, IFGtriang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L → PCUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Similarly, in site 2, we find the enhanced FC within the frontal cortex, particularly the connections  associated with the orbitofrontal cortex (the connections ORBsup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L → ORBsupmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L, MFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L →  ORBmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R, AMYG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L → ORBmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The increased FC in the orbitofrontal cortex in MDD patients  is associated with emotionally negative self-perception [Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The orbitofrontal cortex  is also a part of the brain’s affective network (AN), and the increased functional connections of the  orbitofrontal cortex in MDD have been consistently reported [Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The different  effective FC patterns of the orbitofrontal cortex can also be used to reveal different pathophysiological  mechanisms of different depression types or groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  19  Sitel  Site4  Site2  Site5  Site3Table 2: Two-sample proportion tests for published overlapped connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Connection  NOTEARS-PFL NOTEARS-SIG  GES  PC  GC  INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R→INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0057  INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R→ORBmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0029  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0133  THA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L →CAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0040  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0264  PCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L→PCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0040  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0091  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0549  INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R →ACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0049  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0234  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0289  STG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R→MTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0055  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0264  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
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+page_content='0721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0567  MTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L→ORBmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0126  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0283  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1018  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4  Comparison Results with Other Methods  To further test whether NOTEARS-PFL is able to learn the differences between MDD patients and  HC participants, we use different methods to learn the BN structures from the test validation dataset  established in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Based on the learned BN structures, we perform a two-sample proportion  test for each published overlapping connection of MDD in Figure 5, and report the p-values in Table  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The smaller p value indicates that NOTEARS-PFL recognizes the connection more easily in MDD  group than in HC group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We also included test results for NOTEARS-SIG, GES, PC, and GS for  comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' In order to ensure the consistency of measurement standards, we control these methods  to learn the same sparsity network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  6  Conclusion  In this work, we focus on learning the heterogeneous brain functional connectivity in MDD patients  using multi-site data efficiently and with the data sharing constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' To this end, we developed a  toolbox called NOTEARS-PFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' To achieve the model heterogeneity and computational efficiency,  NOTEARS-PFL is formulated as minimizing a group fused lasso penalized score function with a  continuous constraint for DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Considering the data-sharing barriers, the objective is solved in  a federated learning framework using the ADMM, where original data is not exchanged during  the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The algorithm is fast because the local update step has a closed-form  expression, and the global update step can be solved using an efficient iterative proximal projection  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Extensive experiments on synthetic data and real-world multi-site rs-fMRI datasets with  MDD demonstrate the excellent performance of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We highlight the usefulness  of NOTERAS-PFL in analyzing multi-site rs-fMRI data and facilitating the study of MDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  References  Jason A Avery, Wayne C Drevets, Scott E Moseman, Jerzy Bodurka, Joel C Barcalow, and W Kyle  Simmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Major depressive disorder is associated with abnormal interoceptive activity and  functional connectivity in the insula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Biological Psychiatry, 76(3):258–266, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Robert H Belmaker and Galila Agam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Major depressive disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' New England Journal of Medicine,  358(1):55–68, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  20  Dimitri P Bertsekas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Constrained optimization and Lagrange multiplier methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Academic Press,  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Kevin Bleakley and Jean-Philippe Vert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The group fused lasso for multiple change-point detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  arXiv preprint arXiv:1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
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+page_content='  Kenneth A Bollen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Structural equations with latent variables, volume 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' John Wiley & Sons, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Aneta Brzezicka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Integrative deficits in depression and in negative mood states as a result of  fronto-parietal network dysfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
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+page_content='  Xiaohua Cao, Zhifen Liu, Cheng Xu, Jianying Li, Qiang Gao, Ning Sun, Yong Xu, Yan Ren, Chunxia  Yang, and Kerang Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Disrupted resting-state functional connectivity of the hippocampus  in medication-naive patients with major depressive disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Journal of Affective Disorders, 141  (2-3):194–203, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Paolo Cassano and Maurizio Fava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
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+page_content=' Causal networks: Semantics and expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
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+page_content=' Changes of functional  brain networks in major depressive disorder: a graph theoretical analysis of resting-state fMRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
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+page_content=' Dag-gnn: Dag structure learning with graph neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
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+page_content='  Vera Zamoscik, Silke Huffziger, Ulrich Ebner-Priemer, Christine Kuehner, and Peter Kirsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Increased  involvement of the parahippocampal gyri in a sad mood predicts future depressive symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Social Cognitive and Affective Neuroscience, 9(12):2034–2040, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
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+page_content=' Brain grey matter abnormalities in medication-free patients with major depressive disorder:  a meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
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+page_content=' Federated  learning with non-iid data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' arXiv preprint arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='00582, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Xun Zheng, Bryon Aragam, Pradeep K Ravikumar, and Eric P Xing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Dags with no tears: Continuous  optimization for structure learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Advances in Neural Information Processing Systems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  26  Supplementary Materials  A  Derivation of Federated Local Update Step  To solve the subproblem as shown in equation (15), we first drop the subscript t to lighten the  notation, which leads to  min  W k f(W k) := L(W k, xk) + tr(βk(W k − Zk)⊤) + ρ2  2  ���W k − Zk���  2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (24)  Let U k =  1  nk (xk)⊤xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The first term of the equation (24) can be written as  L(W k, xk) =  1  2nk  ���xk − xkW k���  2  F ,  =  1  2nk  tr((xk − xkW k)⊤(xk − xkW k)),  =  1  2nk  tr((xk)⊤xk − (W k)⊤(xk)⊤xk − (xk)⊤xkW k + (W k)⊤(xk)⊤xkW k),  = 1  2 tr  �  U k − (W k)⊤U k − U kW k + (W k)⊤U kW k�  ,  = 1  2 tr  �  U k − 2(W k)⊤U k + (W k)⊤U kW k�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Similarly, the third term of equation (24) can be written as  ρ2  2 ∥W k − Zk∥2  F = ρ2  2 tr  ��  W k − Zk�⊤ �  W k − Zk��  ,  = ρ2  2 tr  �  (W k)⊤W k − 2(W k)⊤Zk + (Zk)⊤Zk�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Therefore, we have  f(W k) = 1  2 tr(U k − 2(W k)⊤U k + (W k)⊤U kW k) + tr(βk(W k)⊤ − βk(Zk)⊤) + ρ2  2 tr((W k)⊤W k  − 2(W k)⊤Zk + (Zk)⊤Zk),  = 1  2 tr(−2(W k)⊤U k + (W k)⊤U kW k) + tr(βk(W k)⊤)ρ2  2 tr((W k)⊤W k − 2(W k)⊤Zk + c,  and its derivative is given by  ∇W kf  �  W k�  = 1  2(−2U k + ((U k)⊤ + U k)W k) + βk + ρ2  2 (2W k − 2Zk),  = −U k + U kW k + βk + ρ2W k − ρ2Zk,  = (U k + ρ2I)W k − U k + βk − ρ2Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Assuming U k + ρ2I is invertible, solving ∇W kf  �  W k�  = 0 yields the solution  W k = (U k + ρ2I)−1(ρ2Zk − βk + U k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  27  B  Details on the Federated Global Update Step  We provide details of the DIPA algorithm in the federated global update step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' For convenience, we  introduce the following procedures and discuss how to transform K matrices into one matrix, so  that we can re-write equation (19) into vector form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Transform: Given K matrices  ˜  M = {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , MK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' For each k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , K, Mk ∈ Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Transform Mk into vector form, denote as mk =  �  Mk  i,j | for i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , d  �⊤  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Then, construct a  matrix  ¯  M =  �  m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , mK�⊤ ∈ RK×d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Inverse-Transform: The inverse-transform reverses the steps of the transform, which to  transform the matrix  ¯  M ∈ RK×d2 to K matrices  ˜  M = {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , MK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  The core of DIPA algorithm is to compute the separate proximal operators for R1 and R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' And  then the iterative projection can be used to find a feasible point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The proximal operator for the ℓ1  term proxR1  � ¯U  �  is given by the soft-thresholding operator [Tibshirani, 1996].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  proxR1  � ¯U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' λ1  �  = arg min  ¯  Z  1  2  �� ¯Z − ¯U  ��2  F + λ1∥ ¯Z∥1,  = sign( ¯U) ⊙ max  �  | ¯U| − λ1, 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (25)  where the max and sign functions act in an element-wise manner and ⊙ denotes element-wise  multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  The calculation of group-fused lasso proximal operator proxR2  � ¯U  �  is more complex, and there  is no obvious closed-form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We tackle this through a block-coordinate descent approach  [Bleakley and Vert, 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Then, the proximal operator for the group smoothing aspect of the  regularizer can be written as:  proxR2  � ¯U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' λ2  �  = arg min  ¯  Z  1  2  �� ¯Z − ¯U  ��2  F + λ2∥D ¯Z∥2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (26)  Let A = D ¯Z and construct ¯Z as a sum of differences via Zk,• = a+�K−1  i=1 Ai,•, ( where a = Z1,•).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Then we can interpret the proximal operator as a group lasso probem [Bleakley and Vert, 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  A := arg min  A  1  2∥ ¯Ucen − ¯RcenA∥2  F + λ2∥A∥2,1,  (27)  where Xcen denotes a column centered matrix and R ∈ RK×(K−1) is a matrix with entries Ri,j = 1  for i > j and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The problem above can be solved through a block-coordinate descent  strategy, sequentially updating the solution for each block Ak,• for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We can then  construct a solution for ¯Z by summing the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' And the optimal value for a is given by  a = 11,T (U − RA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Correspondingly, the proximal operator for R2 is  proxR2  � ¯U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' λ2  �  =  �  �a⊤, (a + A1,•)⊤ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' ,  �  a +  K−1  �  i=1  Ai,•  �⊤�  �  ⊤  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  (28)  C  Details of the Synthetic Data Generating  We followed the similar synthetic data generate procedure as in [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The key is how to  create similar but different graphs by applying perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  28  Perturbation level: Denote the adjacency matrix of Gtruth as At = (at  ij) ∈ Rd×d, where at  ij = 1  if there is an edge from node i to node j, and at  ij = 0 otherwise, i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Then the generated  BN, denoted by Ggen, is constructed in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' Let Ag be the adjacency matrix of Ggen,  and define the perturbation level as the ratio of the number of changed edges in Ag to the number  of all possible edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' For example, if the perturbation level is 10%, we first set Ag = At, and then  randomly select 10% of the off-diagonal elements at  ij(i ̸= j) in At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' For each selected element at  ij,  if at  ij = 0, then we set ag  ij = 1 (adding edge);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' if at  ij = 1, then with probability 1/2 we set ag  ij = 0  (deleting edge), otherwise let ag  ji = 1 (reversing edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' We generate K graphs ˜G = {G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , GK} by  such a perturbation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Evaluation metric: Let {G1  output, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' , GK  output} represent the K BN structures obtained from  NOTEARS-PFL, NOTEARS-SIG, NOTEARS-AVG, NOTEARS-ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' The four basic statistics  in the evaluation metrics are defined as follows: true positives (TP), the number of edges that are  both in G1  output and G1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' false positives (FP), the number of edges that are present in G1  output but  not in G1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' false negatives (FN), the number of edges that are present in G1 but not in G1  output;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' and  true negatives (TN), the number of vertex pairs that are neither edges in G1 nor in G1  output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  29  D  Supplementary Tables  Table 3: Data summary of the datasets used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Center of Innovation in Hiroshima  (COI)  Kyoto University  (KUT)  University of Tokyo  (UTO)  Hiroshima Kajikawa Hospital  (HKH)  Hiroshima University Hospital  (HUH)  Total Subject  194  175  158  62  124  MDD Subject  70  16  62  33  57  HC Subject  124  159  96  29  67  MDD Percentage  36%  9%  39%  53%  46%  rs-fMRI Frames  240  240  240  107  143  Table 4: Demographic characteristics of participants in multi-site datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Site  Number Male/Female  Age  BDI  MDD COI  70  31/39  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='5  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='9  KUT  16  10 6  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='79±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='82  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='7 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1  UTO  62  36/26  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='74 ±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='62 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='5±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3  HKH  33  20/13  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='8±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='5  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='5±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='7  HUH  57  32/25  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='9±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0  HC  COI  124  46/78  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='9 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3  KUT  159  93/66  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='51± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='59 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4  UTO  96  33/63  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='65±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='54  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='6 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='5  HKH  29  12 17  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='6  HUH  67  30/37  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='6±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='6±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3  Table 5: Imaging protocols for rs-fMRI and structural MRI in multi-site datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  COI  KUT  UTO  HKH  HUH  rs-fMRI  MRI scanner  Siemens  Verio  Siemens  TimTrio  GE  MR750w  Siemens  Spectra  GE  Sigma HDxt  Magnetic field strength  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0T  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0T  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0T  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0T  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0T  Number of channels per coil  12  32  24  12  8  FoV (mm)  212  212  212  192  256  Matrix  64 x 64  64 x 64  64 x 64  64 x 64  64 x 64  Number of slices  40  40  40  38  32  Number of volumes  240  240  240  107  143  In-plane resolution (mm)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3 x 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3125 X 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3125  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3 x 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0 x 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0 x 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='0  Slice thickness (mm)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  3  4  Slice gap (mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='8  0  0  TR (ms)  2500  2500  2500  2700  2000  TE (ms)  30  30  30  31  27  Flip angel (deg)  80  80  80  90  90  Slice acquisition order  Ascending  Ascending  Ascending  Ascending  Ascending  Total scan  time  10 min + 10s  (dummy)  10 min + 10s  (dummy)  10 min + 10s  (dummy)  4 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' 46s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' + 14s  (dummy)  4 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' 46s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content=' + 14s  (dummy)  Eye closed/fixate  Fixate  Fixate  Fixate  Fixate  Fixate  Structural MRI FoV (mm)  256  225x 240  240  256  256  Matrix  256 x 256  240 x 256  256 x 256  256 x 256  256 x 256  Voxel size mm3  1 x 1x 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='9375 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='9375 x 1  1 x 1x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='2  1 x 1x 1  1 x 1x 1  TR (ms)  2300  2000  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='7  1900  6812  TE (ms)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='98  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='38  1986  TI (ms)  900  990  400  900  450  Flip angel (deg)  9  8  11  10  20  30  Table 6: Description of ROIs in AAL template and their Montreal Neurological Institute (MNI)  coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Index  Region name  Abbreviation  Lobe  x  y  z  1  Precentral gyrus  PreCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='65 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='68  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='94  2  Precentral gyrus  PreCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='37  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='21  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='09  3  Superior frontal gyrus, dorsolateral  SFGdor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='45 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='81  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='20  4  Superior frontal gyrus, dorsolateral  SFGdor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='90  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='12  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='82  5  Superior frontal gyrus, orbital  ORBsup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='56 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='32 -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='31  6  Superior frontal gyrus, orbital  ORBsup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='49  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='10 -14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='02  7  Middle frontal gyrus  MFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='43 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='73  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='46  8  Middle frontal gyrus  MFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='59  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='06  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='04  9  Middle frontal gyrus, orbital  ORBmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='65 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='43  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='62  10  Middle frontal gyrus, orbital  ORBmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='18  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='59 -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='73  11  Inferior frontal gyrus, opercular  IFGoperc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='43 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='73  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='02  12  Inferior frontal gyrus, opercular  IFGoperc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='20  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='98  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='41  13  Inferior frontal gyrus, triangular  IFGtriang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='58 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='91  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='99  14  Inferior frontal gyrus, triangular  IFGtriang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='33  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='16  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='17  15  Inferior frontal gyrus, orbital  ORBinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='98 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='71 -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='11  16  Inferior frontal gyrus, orbital  ORBinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='22  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='23 -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='91  17  Rolandic operculum  ROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='16 -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='48  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='95  18  Rolandic operculum  ROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='65  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='25  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='63  19  Supplementary motor area  SMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='32  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='85  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='38  20  Supplementary motor area  SMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='17  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='85  21  Olfactory cortex  OLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='06  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='05 -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='46  22  Olfactory cortex  OLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='43  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='91 -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='26  23  Superior frontal gyrus, medial  SFGmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='80  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='17  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='89  24  Superior frontal gyrus, medial  SFGmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='10  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='84  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='22  25  Superior frontal gyrus, medial orbital ORBsupmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='17  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='06  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='40  26  Superior frontal gyrus, medial orbital ORBsupmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='16  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='67  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='13  27  Gyrus rectus  REC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='08  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='07 -18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='14  28  Gyrus rectus  REC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='35  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='64 -18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='04  29  Insula  INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Insula and Cingulate Gyri -35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='13  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='65  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='44  30  Insula  INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Insula and Cingulate Gyri 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='02  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='08  31  Cingulate gyrus, anterior part  ACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Insula and Cingulate Gyri -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='04  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='40  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='95  32  Cingulate gyrus, anterior part  ACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Insula and Cingulate Gyri  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='46  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='01  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='84  33  Cingulate gyrus, mid part  DCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Insula and Cingulate Gyri -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='48 -14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='92 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='57  34  Cingulate gyrus, mid part  DCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Insula and Cingulate Gyri  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='02  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='83  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='79  35  Cingulate gyurs, posterior part  PCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Insula and Cingulate Gyri -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='85 -42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='92 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='67  36  Cingulate gyurs, posterior part  PCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Insula and Cingulate Gyri  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='44  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='81 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='87  37  Hippocampus  HIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Temporal  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='03 -20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='74 -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='13  38  Hippocampus  HIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Temporal  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='23 -19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='78 -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='33  39  Parahippocampus  PHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Temporal  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='17 -15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='95 -20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='70  40  Parahippocampus  PHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Temporal  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='38 -15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='15 -20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='47  41  Amygdala  AMYG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Temporal  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='27 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='67 -17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='14  42  Amygdala  AMYG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Temporal  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='64  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='50  43  Calcarine fissure and surrounding cortex  CAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Occipital  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='14 -78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='67  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='44  44  Calcarine fissure and surrounding cortex  CAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Occipital  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='99 -73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='15  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='40  45  Cuneus  CUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Occipital  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='93 -80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='13 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='22  46  Cuneus  CUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Occipital  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='51 -79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='36 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='23  47  Lingual gyrus  LING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Occipital  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='62 -67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='56 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='63  48  Lingual gyrus  LING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Occipital  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='29 -66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='93 -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='87  49  Superior occipital lobe  SOG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Occipital  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='54 -84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='26 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='17  50  Superior occipital lobe  SOG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Occipital  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='29 -80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='85 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='59  51  Middle occipital lobe  MOG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Occipital  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='39 -80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='73 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='11  52  Middle occipital lobe  MOG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Occipital  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='39 -79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='70 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='42  53  Inferior occipital lobe  IOG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Occipital  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='36 -78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='29 -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='84  54  Inferior occipital lobe  IOG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Occipital  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='16 -81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='99 -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='61  55  Fusiform gyrus  FFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Temporal  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='16 -40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='30 -20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='23  56  Fusiform gyrus  FFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Temporal  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='97 -39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='10 -20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='18  57  Postcentral gyrus  PoCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Pariental  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='46 -22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='63 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='92  58  Postcentral gyrus  PoCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Pariental  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='43 -25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='49 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='55  59  Superior pariental gyrus  SPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Pariental  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='45 -59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='56 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='96  60  Superior pariental gyrus  SPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Pariental  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='11 -59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='18 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='06  61  Inferior pariental gyrus  IPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Pariental  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='80 -45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='82 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='74  62  Inferior pariental gyrus  IPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Pariental  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='46 -46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='29 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='54  63  Supramarginal gyrus  SMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Pariental  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='79 -33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='64 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='45  31  Table 7: Continue table: Description of ROIs in AAL template and their Montreal Neurological  Institute (MNI) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='  Index  Region name  Abbreviation  Lobe  x  y  z  64  Supramarginal gyrus  SMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Pariental  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='61 -31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='50 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='48  65  Angular gyrus  ANG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Pariental  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='14 -60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='82 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='59  66  Angular gyrus  ANG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Pariental  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='51 -59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='98 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='63  67  Precuneus  PCUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Pariental  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='24 -56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='07 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='01  68  Precuneus  PCUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Pariental  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='98  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='05 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='77  69  Paracentral lobule  PCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Frontal  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='63 -25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='36 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='07  70  Paracentral lobule  PCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Frontal  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='48  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='59 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='09  71  Caudate  CAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Subcortial region -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='46 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='00  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='24  72  Caudate  CAU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Subcortial region 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='84  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='07  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='42  73  Putamen  PUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Subcortial region -23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='91  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='86  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='40  74  Putamen  PUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Subcortial region 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='78  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='91  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='46  75  Pallidum  PAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Subcortial region -17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='75 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='21  76  Pallidum  PAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Subcortial region 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='23  77  Thalamus  THA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Subcortial region -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='85 -17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='56  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='98  78  Thalamus  THA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Temporal  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='00 -17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='55  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='09  79  Heschl gyrus  HES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Temporal  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='99 -18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='88  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='98  80  Heschl gyrus  HES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Temporal  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='86 -17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='15 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='41  81  Superior temporal gyrus  STG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Temporal  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='16 -20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='68  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='13  82  Superior temporal gyrus  STG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Temporal  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='15 -21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='78  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='80  83  Temporal pole: superior temporal gyrus  TPOsup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Temporal  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='88 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='14 -20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='18  84  Temporal pole: superior temporal gyrus  TPOsup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Temporal  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='25  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='75 -16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='86  85  Middle temporal gyrus  MTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Temporal  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='52 -33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='80 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='20  86  Middle temporal gyrus  MTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Temporal  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='47 -37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='23 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='47  87  Temporal pole: middle temporal gyrus  TPOmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Temporal  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='32 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='59 -34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='08  88  Temporal pole: middle temporal gyrus  TPOmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Temporal  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='22  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='55 -32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='23  89  Inferior temporal gyrus  ITG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Temporal  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='77 -28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='05 -23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='17  90  Inferior temporal gyrus  ITG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Temporal  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='69 -31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='07 -22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='32  91  Cerebellum crus 1  CRBLCrus1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Posterior Fossa  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='07 -66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='72 -28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='93  92  Cerebellum crus 1  CRBLCrus1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='R  Posterior Fossa  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='46 -67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='14 -29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='55  93  Cerebellum crus 2  CRBLCrus2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='L  Posterior Fossa  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='64 -73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
+page_content='26 -38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE0T4oBgHgl3EQfgwHQ/content/2301.02423v1.pdf'}
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+arXiv:2301.03128v1  [cs.IT]  9 Jan 2023
+1
+Compress-and-Forward via Multilevel Coding
+and Trellis Coded Quantization
+Heping Wan, Anders Høst-Madsen, Fellow, IEEE, and Aria Nosratinia, Fellow,
+IEEE
+Abstract
+Compress-forward (CF) relays can improve communication rates even when the relay cannot decode
+the source signal. Efficient implementation of CF is a topic of contemporary interest, in part because
+of its potential impact on wireless technologies such as cloud-RAN. There exists a gap between
+the performance of CF implementations in the high spectral efficiency regime and the corresponding
+information theoretic achievable rates. We begin by re-framing a dilemma causing this gap, and propose
+an approach for its mitigation. We utilize trellis coded quantization (TCQ) at the relay together with
+multi-level coding at the source and relay, in a manner that facilitates the calculation of bit LLRs at
+the destination for joint decoding. The contributions of this work include designing TCQ for end-to-
+end relay performance, since a distortion-minimizing TCQ is suboptimum. The reported improvements
+include a 1dB gain over prior results for PSK modulation.
+Index Terms
+Compress-forward relay, coded modulation, multilevel coding, trellis coded quantization
+I. INTRODUCTION
+This paper addresses compress-forward (CF) relaying under high-spectral efficiency (coded
+modulation), where implementations of CF show a performance gap to the best known (informa-
+tion theoretic) achievable rates. The nature of the difficulties is succinctly explained as follows.
+This work of H. Wan and A. Nosratinia was supported by the grant 1711689 from the National Science Foundation. The
+work of A. Host-Madsen was supported by the grant 1923751 from the National Science Foundation.
+H. Wan and A. Nosratinia are with the Department of Electrical Engineering, The University of Texas at Dallas, Richardson,
+TX, USA,Email: Heping.Wan@utdallas.edu, aria@utdallas.edu A. Host-Madsen is with the Department of Electrical Engineering,
+University of Hawaii, Manoa, Honolulu, HI, USA Email: ahm@hawaii.edu
+DRAFT
+
+2
+The best theoretical achievable rates employ either Wyner-Ziv compression at the relay, or joint
+decoding at the destination [1], [2]. Since practical Wyner-Ziv coding implementations have had
+some difficulty in approaching the theoretical limits, joint decoding at the destination has been
+pursued [3], [4] via iterative decoding. Iterative decoding requires bit-level LLRs, which are
+easy to obtain under scalar quantization but difficult with Vector Quantization (VQ). Thus, joint
+decoding approaches so far have utilized scalar quantization and given up on the shaping gain
+that is only provided by vector quantization. The current paper addresses this issue and, via a
+new approach, improves the performance of CF implementations.
+We begin with a brief survey of literature. In [5] the relay quantizes soft information of
+received values followed by a multi-level coding with convolutional codes for each level, which
+serves as both Slepian-Wolf compression as well as error protection. The works [6]–[8], feature
+half-duplex relay, Wyner-Ziv coding, and successive decoding at destination. [6], [7] employ
+LDPC codes at the source and irregular-repeat accumulation (IRA) codes at the relay. [8] uses
+a nested construction of polar codes at the relay. Liu et al. [9] implements Wyner-Ziv coding
+via a superposition structure.
+An alternative strategy maps the quantized sequence to a relay codeword without Wyner-
+Ziv encoding [1], [2], [10], but with joint decoding at destination. This strategy has the same
+achievable rate as Wyner-Ziv encoding with successive cancellation decoding. The following
+works utilized scalar quantization and joint decoding. Chakrabarti et al. [11] designed quantizers
+by maximizing the mutual information between the source and the quantizer output instead of
+minimizing the distortion. Nagpal et al. [3] used LDPC codes, and Nagpal et al. [4] used Low
+Density Generator Matrix (LDGM) codes to map the scalar quantized symbols to codewords.
+These binary modulations were extended to coded modulation via bit-interleaved coded mod-
+ulation (BICM). The present authors employed this approach in [12] with scalar quantization,
+multilevel coding, and LDPC codes, and joint decoding.
+As mentioned earlier, scalar quantization is easier to manage in soft decoding, but sacrifices
+the shaping gain. A general vector quantizer (VQ) captures the shaping gain in principle, but
+makes the calculation of bit LLRs intractable. To resolve this tension, we utilize trellis coded
+quantization (TCQ) [13]. In this technique, trellis coding captures shaping gain while likelihood
+exchange is enabled by the BCJR algorithm [14].
+We utilize this technique for the design of coded modulation for full-duplex compress-forward
+relays in the AWGN channel. We employ multilevel coding (MLC) [15] as a convenient and
+DRAFT
+
+3
+flexible method for the implementation of coded modulation in this multi-node network. We
+analyze the proper assignment of rates to the component level-wise binary codes for near-
+optimal error protection; this is in contrast with [3], [4] which were limited to the same rate
+at all levels. Part of the contribution of this work is a practical design of the TCQ that targets
+end-to-end relay performance, since a distortion-minimizing TCQ proved insufficient. Quantizer
+optimization, and the related computation of input-output distributions for the soft decoding
+of TCQ, are elucidated. We use LDPC codes for error protection at the source, and map the
+quantized sequence at the relay to a codeword whose parity bits are transmitted. A joint iterative
+decoding graphical model and corresponding information exchange algorithm are described for
+the joint decoding at the destination.
+In summary, contributions of this letter include: TCQ for the CF relay to achieve shaping
+gain while allowing LLR exchange at the iterative decoder, MLC per-level rate analysis for CF
+relaying, iterative joint decoding, and demonstrating the gains of the proposed approach.
+II. SETUP, NOTATION, AND PRELIMINARIES1
+In the point-to-point channel, MLC is implemented by splitting the data stream into m bit-
+streams for a 2m-ary constellation. Each sub-stream i ∈ {1, 2, ..., m} is encoded independently.
+At each time instance, the outputs of the (binary) encoders are combined to construct vector
+[A1, A2, ...Am] which is then mapped to a constellation point X and transmitted. At the desti-
+nation, Y is observed. The channel is described by conditional distribution p(y|x). The mutual
+information between the input and output is given by
+I(X; Y ) = I(A1, A2, ..., Am; Y ) =
+m
+�
+i=1
+I(Ai; Y |Ai−1),
+(1)
+where Ai−1 ≜ [A1, A2, ..., Ai−1] with A0 representing a constant, and the chain rule for the
+mutual information and a one-to-one relationship between X and [A1, A2, ..., Am] are used.
+Equation (1) suggests a multistage decoding, and the original channel is decomposed into m
+levels where the information bits of level i is recovered using channel observations and the
+output of decoders of preceding levels. Therefore, the rate assigned to level i should be less than
+or equal to I(Ai; Y |Ai−1).
+1This section appears here for completeness and follows [12] in its entirety.
+DRAFT
+
+4
+Fig. 1. Full-duplex Gaussian relay channel
+The three-node AWGN full-duplex relay channel is shown in Fig. 1. The source transmits
+X1 ∈ A satisfying power constraint Ps to the destination and relay. The relay transmits X2 ∈ B
+satisfying power constraint Pr to the destination. A and B denote the constellation alphabets
+of X1 and X2. n2 and n3 are AWGN with zero mean and variance N2 and N3. The received
+signals at the relay and destination are
+Y2 = H12X1 + n2,
+(2)
+Y3 = H13X1 + H23X2 + n3,
+(3)
+where H13, H12 and H23 are channel coefficients as illustrated in Fig. 1.
+In block t, the source maps one of 2nR messages to a length-n codeword X(t)
+1 , and emits X(t)
+1
+to the destination and relay. Due to causality, the relay quantizes the sequence received in block
+t − 1. ˜Y(t−1)
+2
+of length n denotes the quantized binning of Y(t−1)
+2
+. Then ˜Y(t−1)
+2
+is mapped to
+a length-n codeword X(t−1)
+2
+and transmitted to the destination. The destination uses the joint
+decoding to recover the source message sent at block t − 1. The achievable rate is [16, Section
+16.7]
+R < max I(X1; ˜Y2, Y3|X2),
+(4)
+subject to
+I(Y2; ˜Y2|X2, Y3) ≤ I(X2; Y3), where the maximum is over joint distributions
+p(x1)p(x2)p(˜y2|x2, y2).
+When MLC is implemented in the CF relaying under 2m-QAM/PSK modulation for both
+source and relay, the data stream at the source and the quantized binning at the relay are split
+into m bit-streams. Each of m binary bit-streams at the source/relay is encoded independently.
+Due to one-to-one relationships between X1 ↔ [A1, A2, ..., Am], ˜Y2 ↔ [B1, B2, ..., Bm], and
+DRAFT
+
+5
+X2 ↔ [C1, C2, ..., Cm], multistage decoding will induce a decomposition of the original channel,
+and the achievable rate in (4) can be expressed as
+R ≤ max
+m
+�
+i=1
+I(Ai; Bm, Y3|Cm, Ai−1).
+(5)
+subject to the constraint:
+m
+�
+i=1
+I(Y2; Bi|Cm, Y3, Bi−1) ≤
+m
+�
+i=1
+I(Ci; Y3|Ci−1) .
+(6)
+The rate assigned to each level is denoted Ri, requiring
+Ri ≤ I(Ai; Bm, Y3|Cm, Ai−1)
+(7)
+III. MULTILEVEL REALIZATION OF CF RELAYING
+The message to be transmitted is partitioned into m bit-streams. Each bit-stream is indepen-
+dently encoded with binary LDPC code of rate Ri, i = 1, . . . , m, where Ri should satisfy (7).
+At each time instance, binary vector [A1, A2, ...Am] is mapped to a symbol of the transmitted
+sequence X1 by a 2m-QAM/PSK modulator. At the relay, the quantized binning ˜Y2 is split into
+m bit-streams. Each bit-stream is encoded through a systematic rate-1/2 binary LDPC code.
+The parity bits of the codeword are combined via a 2m-QAM/PSK modulator, to construct X2
+which is transmitted to the destination at the next block.
+A. Quantization at the Relay
+The key impediment to CF performance has been the implementation of compression at the
+relay. Wyner-Ziv implementations have not approached fundamental limits, and joint decoding
+has concentrated on scalar quantization to facilitate LLR calculation, thus losing shaping gain
+of the quantizer at the relay. We provide a new solution via TCQ that reconciles this issue by
+providing shaping gain as well as feasible LLR calculation for joint decoding.
+Trellis source coding aims to compress a sequence y = [y1, y2, . . . , yn] sampled from a
+memoryless source with marginal distribution p(y) to ˜y = [˜y1, ˜y2, . . . , ˜yn] at a rate of ˜R. The
+reconstruction values are ˆy ∈ ˆY. Trellis source coding is to find a path determined by ˜y which
+generates a reconstruction sequence ˆy with minimum distortion.
+TCQ [13] is a practical implementation of trellis source coding that was inspired by trellis
+coded modulation (TCM) [17]. In TCM, to send m information bits in each signaling interval,
+the traditional 2m- point signal constellation is expanded to 2m+1 points and partitioned into 2m
+DRAFT
+
+6
+subsets according to Ungerboeck’s set partitioning idea. m input bits are passed through a rate
+m/(m + 1) trellis encoder among which m − 1 bits are used to select the set partition, and the
+remaining bit determines the symbol from the set partition.
+B. Obtaining p(ˆy2|y2) and Quantizer Optimization
+For iterative decoding at the destination, one needs a soft version of TCQ, for which a
+logical candidate is the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm. To implement the BCJR,
+the (marginal) conditional distribution p(ˆy2|y2) is needed [14], where ˆy2 ∈ ˆY2 represents re-
+construction value and | ˆY2| = 2m+1. Unfortunately, this distribution is not readily available and
+its calculation is not straight forward. In this subsection, we highlight the different approaches
+for calculating this value, and present the technique that we utilized, which to our experience
+produces the best results.
+Rate-distortion theory [18] suggests an optimal source encoder coming from the solution of
+the following minimization
+R(D) =
+min
+p(ˆy|y):E{d(y,ˆy)}≤DI(Y, ˆY ),
+where the minimization is over all conditional distributions p(ˆy|y) for which the joint distribution
+p(ˆy, y) = p(y)p(ˆy|y) satisfies the expected distortion constraint D, and d(y, ˆy) is the distortion
+measure function. p(ˆy|y) can be interpreted as the probability that a given source value y is
+represented by a reconstruction value ˆy.
+With the resulting p(ˆy|y) obtained from the above optimization, the encoding procedure is
+equivalent to finding a ˆy(n) that maximizes p(ˆy(n), y). For trellis source coding, the BCJR algo-
+rithm [14] can be used to find the reconstruction sequence ˆy and the corresponding compressed
+bitstream ˜y.
+However, minimizing distortion is not necessarily a good strategy for CF relaying, because the
+ultimate goal is for relaying to give maximal help in decoding at destination. Counterintuitively,
+this is not always the same as low distortion, which only preserves the fidelity of a reconstruction
+of y2. Our simulations indicate that Lloyd-Max reconstruction values and p(ˆy2|y2) obtained
+from rate-distortion optimization together produce mediocre performance.2 This seems to suggest
+jointly designing p(ˆy2|y2) and the trellis structure, but this is too unwieldy for practical quantizer
+2This issue has also been pointed out by others, e.g., Chakrabarti et al. [11] produced a comparison in which a quantizer with
+worse mean-squared distortion resulted in better relay performance.
+DRAFT
+
+7
+design. Our results are obtained by choosing a good trellis from the literature and determining
+quantizer boundaries by maximizing the CF (end-to-end) achievable rate expression, which
+is related to the quantization boundaries through the conditional distribution p(ˆy2|y2). The
+boundaries were determined via a simple discrete search that maintained the symmetry of the
+quantizer and utilized the trellis and the reconstruction values in Fig. 4.
+C. Decoding
+As mentioned earlier, the destination decodes the source signal as well as the quantized signal
+at the relay; information theory arguments show that this joint decoding permits the relay to avoid
+dirty-paper type encoding without a rate penalty. The joint decoder is implemented iteratively via
+a graphical model for the exchange of information within, and between, the component LDPC
+tanner graphs.
+The desination begins by seeking a codeword x1 according to maximum a posteriori proba-
+bility p(x1|y3); this search is performed via a bit-wise MAP decoder with decoding rule:
+ˆx1,j = arg max
+x1,j∈A
+�
+∼x1,j
+p(x1|y3),
+for all j = 1, ..., n,. Recall that ˆy2 are the quantized values while ˜y2 is the compressed bit
+sequence. These two representations are equivalent given the quantization strategy, however,
+˜y2 is a more convenient representation for the relay’s channel encoder as well as for iterative
+decoding at destination. p(x1|y3) can be decomposed as follows:
+p(x1|y3) =
+1
+p(y3)
+�
+˜y2,x2
+p(x1, x2, ˜y2, y3),
+(8)
+where
+p(x1, x2, ˜y2, y3) = p(y3|x1, x2, ˜y2)p(x1, x2, ˜y2),
+(9)
+(a)
+= p(y3|x1, x2)p(x1, x2, ˜y2),
+(10)
+= p(y3|x1, x2)p(x2|x1, ˜y2)p(˜y2|x1)p(x1),
+(11)
+(b)= p(y3|x1, x2)p(x2|˜y2)p(˜y2|x1)p(x1),
+(12)
+(c)
+∝ p(y3|x1, x2)1{x2 = CR(˜y2)}p(˜y2|x1)1{x1 ∈ CS},
+(13)
+where (a) follows because y3 is only dependent on x1, x2, (b) is due to Markov x1 ↔ ˜y2 ↔ x2,
+and (c) is due to p(x1) =
+1{x1∈CS}
+2nR
+being a uniform distribution over the elements of the source
+DRAFT
+
+8
+codebook CS, and p(x2|˜y2) =
+1{x2 = CR(˜y2)}, i.e., the output of error-control coding at the
+relay CR being a deterministic function of its input.
+Fig. 2. The simplified destination part of the channel model
+During block t, the destination receives a version of X(t−1)
+2
+that is contaminated with noise
+and interference, which carries information about X(t−1)
+1
+, which is independent of X(t)
+1 . This
+independence allows successive interference cancellation, simplifying the channel model at des-
+tination [3]:
+Y23 = H23X2 + n′
+3,
+(14)
+Y13 = H13X1 + n3,
+(15)
+which is shown in Fig. 2. At block t, the destination jointly decodes X(t−1)
+1
+and X(t−1)
+2
+on the
+basis of Y(t)
+23 and Y(t−1)
+13
+where the processing of Y(t)
+23 treats X(t)
+1
+as zero-mean Gaussian noise
+with power H2
+12Ps, and Y(t−1)
+13
+is processed by subtracting X(t−2)
+2
+decoded during block t−1 from
+Y(t−1)
+3
+. This operation can decompose Y3 to two links Y13 and Y23 with independent additive
+white Gaussian noise n′
+3 of variance N3 + H2
+12Ps and n3 of variance N3. Using p(y3|x1.x2) =
+p(y31|x1)p(y32|x2), for LLR calculation at destination, (13) can be rewritten as
+n
+�
+j=1
+p(y13,j|x1,j)
+n
+�
+k=1
+p(y23,k|x2,k)
+·
+1{x1 ∈ CS}p(˜y2|x1)1{x2 = CR(˜y2)},
+(16)
+Enabled by this decomposition, joint decoding can be performed by a repeated application
+of two LDPC decoders whose cost function is based on p(y13|x1) and p(y23|x2), as well as
+enforcing the relationship between x1 and x2 through the term p(˜y2|x1) together with the relay
+LPDC coding x2 = CR(˜y2). Using the destination observations y13 and y23 (after successive in-
+terference cancellation) a three-way iterative decoding is performed whose information exchange
+is described by Fig. 3, where each LDPC decoder block represents a Tanner graph.
+DRAFT
+
+9
+Fig. 3. The joint iterative decoding graphical model
+The two LDPC decoder components follow well-known principles, therefore their internal
+structure is omitted in Fig. 3 for simplicity. The iterative decoding also needs propagating soft in-
+formation through p(˜y2|x1), which is not straight forward, therefore we break it into two compo-
+nents. Considering the Markov chain x1 ↔ ˆy2 ↔ ˜y2, we have p(˜y2|x1) = �
+ˆy2
+p(˜y2|ˆy2)p(ˆy2|x1).
+The information exchange across the trellis, shown by p(˜y2|ˆy2), is calculated using the BCJR
+algorithm. The information exchange between x1 and ˆy2 is through p(ˆy2|x1). This marginal
+distribution of quantized values is described as a weighted sum since TCQ employs several
+scalar quantizers, whose number we denote with K. The quantizer choice at each sample has a
+probability P(Qi|x1). The overall marginal distribution is given by:
+p(ˆy2|x1) =
+K
+�
+i=1
+P(Qi|x1)
+�
+ˆy2=Qi(y2)
+p(y2|x1) dy2
+(17)
+The probability P(Qi|x1) at each sample is driven by the contamination of x1 with noise prior to
+quantization, as well as the action of the trellis. The overall effect is difficult to model, therefore
+we obtain an empirical estimate for P(Qi|x1) by a monte carlo simulation of the channel x1 → y2,
+applying the TCQ, and collecting the empirical frequency of occurrence of each quantizer Qi
+conditioned on x1.
+The destination begins by initializing p(y13) and p(y23) according to channel observations.
+Then, we update the soft estimate of x1 at the destination, via executing the sum-product
+algorithm for CS. From this, an updated soft estimate for ˆy2 is obtained via
+(17). Then, the
+DRAFT
+
+10
+BCJR algorithm, induced by the compression trellis, is used to update the soft estimate of ˜y2.
+Then, another sum-product is executed utilizing the observation y23 and the structure of the
+relay-destination LDPC code, to update ˜y2. We then repeat backward, updating ˆy2 and x1, and
+then another sum-product iteration on x1.
+To summarize, the process consists of two sum-product algorithms operating on the source
+LDPC code and the relay LDPC code, and information being exchanged between them via
+the method mentioned above. Our algorithm performs one bit-to-check and one check-to-bit
+operation in each sum-product before propagating the information to the other LDPC decoder.
+At the end of the process, we perform a multistage hard decoding of the MLC.
+IV. SIMULATIONS
+This section presents simulation results supporting the findings of this letter. The following
+values underlie the simulations: N3 = 1, N2 = 8, H13 = 1, H12 = 2, and H23 = 11. Furthermore,
+in all experiments the source and the relay transmitters emit the same power. The reported SNRs
+for simulations are generated by fixing the noise power and varying the power of the source/relay.
+Error control coding is achieved via DVB-S2 LDPC codes with block length 64,800. At the
+relay, the first 32,400 quantized bits are encoded by a rate-1/2 DVB-S2 code, and the remaining
+quantized bits are encoded with a second, similar, DVB-S2 code. The parity bits of these two
+encoders are then concatenated for transmission.
+Our experiments involve 16-QAM at the normalized rate of 3.4 bits/s/Hz and 16-PSK at the
+rate of 3.27 bits/s/Hz. Each modulation is attached to a 4-level multi-level code, which in the
+case of 16-QAM have rates 0.9, 0.8, 0.9, and 0.8, and in the case of 16-PSK have rates 0.9, 0.9,
+0.8, and 2/3.
+The relay utilizes a TCQ with an 8-state trellis whose generator matrix is:
+
+
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+D
+1
+0
+0
+0
+1
+D2
+D
+
+
+.
+The quantizer reconstruction values, shown in Fig. 4, have a cardinality that is twice the size
+of the 16-QAM/PSK constellation used for source/relay modulation. For the 16-QAM example,
+extending to a standard 32AMPM would present inconvenient cell boundaries, therefore we
+experimented with the shown 32-point extension as well as a symmetric extension with 64
+DRAFT
+
+11
+Fig. 4. The reconstruction values for CF relaying using 16-QAM (a), and 16-PSK (b), where the circles represent 16-QAM/PSK
+constellation.
+points, finding the latter produces negligible gains over the former. The design of quantizer bin
+boundaries is according to the optimization mentioned in Section III-B.
+We compare our results with that of bit-interleaved coded modulation (BICM) [3], [4], where
+the component LDPC codes in the sub-channels have the same rate by definition. Since the
+granularity of available DVB-2 codes are limited, we simulated BICM at the closest rate possible
+with available codes. A rate-5/6 DVB-S2 code is used for 16-QAM BICM, resulting in an overall
+rate of 3.33 bits/s/Hz, and a rate-0.8 DVB-S2 code was used for 16-PSK resulting in overall rate
+of 3.2 bits/s/Hz. In addition to the BICM results, we also display the results of a preliminary
+version of this work from [12] which uses only scalar quantization. In addition, rates are carefully
+chosen to avoid any unfair advantage to the method of this letter in reporting error rates.
+Simulation results are displayed in Fig. 5. The observations and insights arising from these
+simulations are as follows:
+• The proposed method outperforms the error rate of BICM under 16-QAM and 16-PSK,
+even though the simulated BICM rates were slightly advantageous.
+• Superior performance of MLC over BICM was known in point-to-point channels, but is
+affirmed in this letter for the end-to-end performance of a CF relaying.
+• It is also established that TCQ can outperform scalar quantization in CF relaying. Although
+at first sight this may seem a natural outcome, past work [11] implies that utilizing vector
+quantization in the context of CF has been remarkably difficult. In that sense, the result
+reported herein has significant practical novelty.
+DRAFT
+
+12
+9
+10
+11
+12
+13
+14
+15
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+BER
+16-QAM, TCQ
+16-QAM, scalar
+16-QAM, BICM
+16-PSK, TCQ
+16-PSK, scalar
+16-PSK, BICM
+Fig. 5. Simulation results using 16-QAM/PSK with N3 = 1, N2 = 8, H13 = 1, H12 = 2, H23 = 11, and Pr = Ps.
+• Approximately 1dB improvement is observed for 16-PSK. The improvement for 16-QAM
+is more modest.
+V. DISCUSSION AND CONCLUSION
+This paper significantly improves known implementations of compress-forward via a combi-
+nation of trellis coded quantization at the relay, multi-level coding with LDPC component codes,
+and iterative joint decoding of the source and relay signals at the destination. The utilization
+of TCQ to facilitate the generation of bit LLRs at the destination is a key contribution of this
+work. Among others, this work shows a 1dB improvement for compress-forward with PSK
+modulations.
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+DRAFT
+
diff --git a/HdE1T4oBgHgl3EQfXgSr/content/tmp_files/load_file.txt b/HdE1T4oBgHgl3EQfXgSr/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..6f599553659f854bf019b5bda63267a895511127
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf,len=457
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='03128v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='IT]  9 Jan 2023  1  Compress-and-Forward via Multilevel Coding  and Trellis Coded Quantization  Heping Wan, Anders Høst-Madsen, Fellow, IEEE, and Aria Nosratinia, Fellow,  IEEE  Abstract  Compress-forward (CF) relays can improve communication rates even when the relay cannot decode  the source signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Efficient implementation of CF is a topic of contemporary interest, in part because  of its potential impact on wireless technologies such as cloud-RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' There exists a gap between  the performance of CF implementations in the high spectral efficiency regime and the corresponding  information theoretic achievable rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' We begin by re-framing a dilemma causing this gap, and propose  an approach for its mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' We utilize trellis coded quantization (TCQ) at the relay together with  multi-level coding at the source and relay, in a manner that facilitates the calculation of bit LLRs at  the destination for joint decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The contributions of this work include designing TCQ for end-to-  end relay performance, since a distortion-minimizing TCQ is suboptimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The reported improvements  include a 1dB gain over prior results for PSK modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Index Terms  Compress-forward relay, coded modulation, multilevel coding, trellis coded quantization  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' INTRODUCTION  This paper addresses compress-forward (CF) relaying under high-spectral efficiency (coded  modulation), where implementations of CF show a performance gap to the best known (informa-  tion theoretic) achievable rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The nature of the difficulties is succinctly explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  This work of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Wan and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Nosratinia was supported by the grant 1711689 from the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The  work of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Host-Madsen was supported by the grant 1923751 from the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Wan and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Nosratinia are with the Department of Electrical Engineering, The University of Texas at Dallas, Richardson,  TX, USA,Email: Heping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='Wan@utdallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='edu, aria@utdallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='edu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Host-Madsen is with the Department of Electrical Engineering,  University of Hawaii, Manoa, Honolulu, HI, USA Email: ahm@hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='edu  DRAFT  2  The best theoretical achievable rates employ either Wyner-Ziv compression at the relay, or joint  decoding at the destination [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Since practical Wyner-Ziv coding implementations have had  some difficulty in approaching the theoretical limits, joint decoding at the destination has been  pursued [3], [4] via iterative decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Iterative decoding requires bit-level LLRs, which are  easy to obtain under scalar quantization but difficult with Vector Quantization (VQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Thus, joint  decoding approaches so far have utilized scalar quantization and given up on the shaping gain  that is only provided by vector quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The current paper addresses this issue and, via a  new approach, improves the performance of CF implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  We begin with a brief survey of literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' In [5] the relay quantizes soft information of  received values followed by a multi-level coding with convolutional codes for each level, which  serves as both Slepian-Wolf compression as well as error protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The works [6]–[8], feature  half-duplex relay, Wyner-Ziv coding, and successive decoding at destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' [6], [7] employ  LDPC codes at the source and irregular-repeat accumulation (IRA) codes at the relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' [8] uses  a nested construction of polar codes at the relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' [9] implements Wyner-Ziv coding  via a superposition structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  An alternative strategy maps the quantized sequence to a relay codeword without Wyner-  Ziv encoding [1], [2], [10], but with joint decoding at destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' This strategy has the same  achievable rate as Wyner-Ziv encoding with successive cancellation decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The following  works utilized scalar quantization and joint decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Chakrabarti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' [11] designed quantizers  by maximizing the mutual information between the source and the quantizer output instead of  minimizing the distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Nagpal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' [3] used LDPC codes, and Nagpal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' [4] used Low  Density Generator Matrix (LDGM) codes to map the scalar quantized symbols to codewords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  These binary modulations were extended to coded modulation via bit-interleaved coded mod-  ulation (BICM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The present authors employed this approach in [12] with scalar quantization,  multilevel coding, and LDPC codes, and joint decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  As mentioned earlier, scalar quantization is easier to manage in soft decoding, but sacrifices  the shaping gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' A general vector quantizer (VQ) captures the shaping gain in principle, but  makes the calculation of bit LLRs intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' To resolve this tension, we utilize trellis coded  quantization (TCQ) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' In this technique, trellis coding captures shaping gain while likelihood  exchange is enabled by the BCJR algorithm [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  We utilize this technique for the design of coded modulation for full-duplex compress-forward  relays in the AWGN channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' We employ multilevel coding (MLC) [15] as a convenient and  DRAFT  3  flexible method for the implementation of coded modulation in this multi-node network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' We  analyze the proper assignment of rates to the component level-wise binary codes for near-  optimal error protection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' this is in contrast with [3], [4] which were limited to the same rate  at all levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Part of the contribution of this work is a practical design of the TCQ that targets  end-to-end relay performance, since a distortion-minimizing TCQ proved insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Quantizer  optimization, and the related computation of input-output distributions for the soft decoding  of TCQ, are elucidated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' We use LDPC codes for error protection at the source, and map the  quantized sequence at the relay to a codeword whose parity bits are transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' A joint iterative  decoding graphical model and corresponding information exchange algorithm are described for  the joint decoding at the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  In summary, contributions of this letter include: TCQ for the CF relay to achieve shaping  gain while allowing LLR exchange at the iterative decoder, MLC per-level rate analysis for CF  relaying, iterative joint decoding, and demonstrating the gains of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' SETUP, NOTATION, AND PRELIMINARIES1  In the point-to-point channel, MLC is implemented by splitting the data stream into m bit-  streams for a 2m-ary constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Each sub-stream i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=', m} is encoded independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  At each time instance, the outputs of the (binary) encoders are combined to construct vector  [A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='Am] which is then mapped to a constellation point X and transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' At the desti-  nation, Y is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The channel is described by conditional distribution p(y|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The mutual  information between the input and output is given by  I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Y ) = I(A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=', Am;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Y ) =  m  �  i=1  I(Ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Y |Ai−1),  (1)  where Ai−1 ≜ [A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=', Ai−1] with A0 representing a constant, and the chain rule for the  mutual information and a one-to-one relationship between X and [A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=', Am] are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Equation (1) suggests a multistage decoding, and the original channel is decomposed into m  levels where the information bits of level i is recovered using channel observations and the  output of decoders of preceding levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Therefore, the rate assigned to level i should be less than  or equal to I(Ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Y |Ai−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  1This section appears here for completeness and follows [12] in its entirety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  DRAFT  4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Full-duplex Gaussian relay channel  The three-node AWGN full-duplex relay channel is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The source transmits  X1 ∈ A satisfying power constraint Ps to the destination and relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The relay transmits X2 ∈ B  satisfying power constraint Pr to the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' A and B denote the constellation alphabets  of X1 and X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' n2 and n3 are AWGN with zero mean and variance N2 and N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The received  signals at the relay and destination are  Y2 = H12X1 + n2,  (2)  Y3 = H13X1 + H23X2 + n3,  (3)  where H13, H12 and H23 are channel coefficients as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  In block t, the source maps one of 2nR messages to a length-n codeword X(t)  1 , and emits X(t)  1  to the destination and relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Due to causality, the relay quantizes the sequence received in block  t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' ˜Y(t−1)  2  of length n denotes the quantized binning of Y(t−1)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Then ˜Y(t−1)  2  is mapped to  a length-n codeword X(t−1)  2  and transmitted to the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The destination uses the joint  decoding to recover the source message sent at block t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The achievable rate is [16, Section  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='7]  R < max I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' ˜Y2, Y3|X2),  (4)  subject to  I(Y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' ˜Y2|X2, Y3) ≤ I(X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Y3), where the maximum is over joint distributions  p(x1)p(x2)p(˜y2|x2, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  When MLC is implemented in the CF relaying under 2m-QAM/PSK modulation for both  source and relay, the data stream at the source and the quantized binning at the relay are split  into m bit-streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Each of m binary bit-streams at the source/relay is encoded independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Due to one-to-one relationships between X1 ↔ [A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=', Am], ˜Y2 ↔ [B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=', Bm], and  DRAFT  5  X2 ↔ [C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=', Cm], multistage decoding will induce a decomposition of the original channel,  and the achievable rate in (4) can be expressed as  R ≤ max  m  �  i=1  I(Ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Bm, Y3|Cm, Ai−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  (5)  subject to the constraint:  m  �  i=1  I(Y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Bi|Cm, Y3, Bi−1) ≤  m  �  i=1  I(Ci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Y3|Ci−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  (6)  The rate assigned to each level is denoted Ri, requiring  Ri ≤ I(Ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Bm, Y3|Cm, Ai−1)  (7)  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' MULTILEVEL REALIZATION OF CF RELAYING  The message to be transmitted is partitioned into m bit-streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Each bit-stream is indepen-  dently encoded with binary LDPC code of rate Ri, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' , m, where Ri should satisfy (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  At each time instance, binary vector [A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='Am] is mapped to a symbol of the transmitted  sequence X1 by a 2m-QAM/PSK modulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' At the relay, the quantized binning ˜Y2 is split into  m bit-streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Each bit-stream is encoded through a systematic rate-1/2 binary LDPC code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  The parity bits of the codeword are combined via a 2m-QAM/PSK modulator, to construct X2  which is transmitted to the destination at the next block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Quantization at the Relay  The key impediment to CF performance has been the implementation of compression at the  relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Wyner-Ziv implementations have not approached fundamental limits, and joint decoding  has concentrated on scalar quantization to facilitate LLR calculation, thus losing shaping gain  of the quantizer at the relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' We provide a new solution via TCQ that reconciles this issue by  providing shaping gain as well as feasible LLR calculation for joint decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Trellis source coding aims to compress a sequence y = [y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' , yn] sampled from a  memoryless source with marginal distribution p(y) to ˜y = [˜y1, ˜y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' , ˜yn] at a rate of ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The  reconstruction values are ˆy ∈ ˆY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Trellis source coding is to find a path determined by ˜y which  generates a reconstruction sequence ˆy with minimum distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  TCQ [13] is a practical implementation of trellis source coding that was inspired by trellis  coded modulation (TCM) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' In TCM, to send m information bits in each signaling interval,  the traditional 2m- point signal constellation is expanded to 2m+1 points and partitioned into 2m  DRAFT  6  subsets according to Ungerboeck’s set partitioning idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' m input bits are passed through a rate  m/(m + 1) trellis encoder among which m − 1 bits are used to select the set partition, and the  remaining bit determines the symbol from the set partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Obtaining p(ˆy2|y2) and Quantizer Optimization  For iterative decoding at the destination, one needs a soft version of TCQ, for which a  logical candidate is the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' To implement the BCJR,  the (marginal) conditional distribution p(ˆy2|y2) is needed [14], where ˆy2 ∈ ˆY2 represents re-  construction value and | ˆY2| = 2m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Unfortunately, this distribution is not readily available and  its calculation is not straight forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' In this subsection, we highlight the different approaches  for calculating this value, and present the technique that we utilized, which to our experience  produces the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Rate-distortion theory [18] suggests an optimal source encoder coming from the solution of  the following minimization  R(D) =  min  p(ˆy|y):E{d(y,ˆy)}≤DI(Y, ˆY ),  where the minimization is over all conditional distributions p(ˆy|y) for which the joint distribution  p(ˆy, y) = p(y)p(ˆy|y) satisfies the expected distortion constraint D, and d(y, ˆy) is the distortion  measure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' p(ˆy|y) can be interpreted as the probability that a given source value y is  represented by a reconstruction value ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  With the resulting p(ˆy|y) obtained from the above optimization, the encoding procedure is  equivalent to finding a ˆy(n) that maximizes p(ˆy(n), y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' For trellis source coding, the BCJR algo-  rithm [14] can be used to find the reconstruction sequence ˆy and the corresponding compressed  bitstream ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  However, minimizing distortion is not necessarily a good strategy for CF relaying, because the  ultimate goal is for relaying to give maximal help in decoding at destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Counterintuitively,  this is not always the same as low distortion, which only preserves the fidelity of a reconstruction  of y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Our simulations indicate that Lloyd-Max reconstruction values and p(ˆy2|y2) obtained  from rate-distortion optimization together produce mediocre performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='2 This seems to suggest  jointly designing p(ˆy2|y2) and the trellis structure, but this is too unwieldy for practical quantizer  2This issue has also been pointed out by others, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=', Chakrabarti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' [11] produced a comparison in which a quantizer with  worse mean-squared distortion resulted in better relay performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  DRAFT  7  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Our results are obtained by choosing a good trellis from the literature and determining  quantizer boundaries by maximizing the CF (end-to-end) achievable rate expression, which  is related to the quantization boundaries through the conditional distribution p(ˆy2|y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The  boundaries were determined via a simple discrete search that maintained the symmetry of the  quantizer and utilized the trellis and the reconstruction values in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Decoding  As mentioned earlier, the destination decodes the source signal as well as the quantized signal  at the relay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' information theory arguments show that this joint decoding permits the relay to avoid  dirty-paper type encoding without a rate penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The joint decoder is implemented iteratively via  a graphical model for the exchange of information within, and between, the component LDPC  tanner graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  The desination begins by seeking a codeword x1 according to maximum a posteriori proba-  bility p(x1|y3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' this search is performed via a bit-wise MAP decoder with decoding rule:  ˆx1,j = arg max  x1,j∈A  �  ∼x1,j  p(x1|y3),  for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=', n,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Recall that ˆy2 are the quantized values while ˜y2 is the compressed bit  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' These two representations are equivalent given the quantization strategy, however,  ˜y2 is a more convenient representation for the relay’s channel encoder as well as for iterative  decoding at destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' p(x1|y3) can be decomposed as follows:  p(x1|y3) =  1  p(y3)  �  ˜y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='x2  p(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' ˜y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' y3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  (8)  where  p(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' ˜y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' y3) = p(y3|x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' ˜y2)p(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' ˜y2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  (9)  (a)  = p(y3|x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' x2)p(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' ˜y2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  (10)  = p(y3|x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' x2)p(x2|x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' ˜y2)p(˜y2|x1)p(x1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  (11)  (b)= p(y3|x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' x2)p(x2|˜y2)p(˜y2|x1)p(x1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  (12)  (c)  ∝ p(y3|x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' x2)1{x2 = CR(˜y2)}p(˜y2|x1)1{x1 ∈ CS},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  (13)  where (a) follows because y3 is only dependent on x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' (b) is due to Markov x1 ↔ ˜y2 ↔ x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  and (c) is due to p(x1) =  1{x1∈CS}  2nR  being a uniform distribution over the elements of the source  DRAFT  8  codebook CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' and p(x2|˜y2) =  1{x2 = CR(˜y2)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=', the output of error-control coding at the  relay CR being a deterministic function of its input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The simplified destination part of the channel model  During block t, the destination receives a version of X(t−1)  2  that is contaminated with noise  and interference, which carries information about X(t−1)  1  , which is independent of X(t)  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' This  independence allows successive interference cancellation, simplifying the channel model at des-  tination [3]:  Y23 = H23X2 + n′  3,  (14)  Y13 = H13X1 + n3,  (15)  which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' At block t, the destination jointly decodes X(t−1)  1  and X(t−1)  2  on the  basis of Y(t)  23 and Y(t−1)  13  where the processing of Y(t)  23 treats X(t)  1  as zero-mean Gaussian noise  with power H2  12Ps, and Y(t−1)  13  is processed by subtracting X(t−2)  2  decoded during block t−1 from  Y(t−1)  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' This operation can decompose Y3 to two links Y13 and Y23 with independent additive  white Gaussian noise n′  3 of variance N3 + H2  12Ps and n3 of variance N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Using p(y3|x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='x2) =  p(y31|x1)p(y32|x2), for LLR calculation at destination, (13) can be rewritten as  n  �  j=1  p(y13,j|x1,j)  n  �  k=1  p(y23,k|x2,k) 1{x1 ∈ CS}p(˜y2|x1)1{x2 = CR(˜y2)},  (16)  Enabled by this decomposition, joint decoding can be performed by a repeated application  of two LDPC decoders whose cost function is based on p(y13|x1) and p(y23|x2), as well as  enforcing the relationship between x1 and x2 through the term p(˜y2|x1) together with the relay  LPDC coding x2 = CR(˜y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Using the destination observations y13 and y23 (after successive in-  terference cancellation) a three-way iterative decoding is performed whose information exchange  is described by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 3, where each LDPC decoder block represents a Tanner graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  DRAFT  9  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The joint iterative decoding graphical model  The two LDPC decoder components follow well-known principles, therefore their internal  structure is omitted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 3 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The iterative decoding also needs propagating soft in-  formation through p(˜y2|x1), which is not straight forward, therefore we break it into two compo-  nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Considering the Markov chain x1 ↔ ˆy2 ↔ ˜y2, we have p(˜y2|x1) = �  ˆy2  p(˜y2|ˆy2)p(ˆy2|x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  The information exchange across the trellis, shown by p(˜y2|ˆy2), is calculated using the BCJR  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The information exchange between x1 and ˆy2 is through p(ˆy2|x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' This marginal  distribution of quantized values is described as a weighted sum since TCQ employs several  scalar quantizers, whose number we denote with K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The quantizer choice at each sample has a  probability P(Qi|x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The overall marginal distribution is given by:  p(ˆy2|x1) =  K  �  i=1  P(Qi|x1)  �  ˆy2=Qi(y2)  p(y2|x1) dy2  (17)  The probability P(Qi|x1) at each sample is driven by the contamination of x1 with noise prior to  quantization, as well as the action of the trellis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The overall effect is difficult to model, therefore  we obtain an empirical estimate for P(Qi|x1) by a monte carlo simulation of the channel x1 → y2,  applying the TCQ, and collecting the empirical frequency of occurrence of each quantizer Qi  conditioned on x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  The destination begins by initializing p(y13) and p(y23) according to channel observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Then, we update the soft estimate of x1 at the destination, via executing the sum-product  algorithm for CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' From this, an updated soft estimate for ˆy2 is obtained via  (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Then, the  DRAFT  10  BCJR algorithm, induced by the compression trellis, is used to update the soft estimate of ˜y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Then, another sum-product is executed utilizing the observation y23 and the structure of the  relay-destination LDPC code, to update ˜y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' We then repeat backward, updating ˆy2 and x1, and  then another sum-product iteration on x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  To summarize, the process consists of two sum-product algorithms operating on the source  LDPC code and the relay LDPC code, and information being exchanged between them via  the method mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Our algorithm performs one bit-to-check and one check-to-bit  operation in each sum-product before propagating the information to the other LDPC decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  At the end of the process, we perform a multistage hard decoding of the MLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' SIMULATIONS  This section presents simulation results supporting the findings of this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The following  values underlie the simulations: N3 = 1, N2 = 8, H13 = 1, H12 = 2, and H23 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Furthermore,  in all experiments the source and the relay transmitters emit the same power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The reported SNRs  for simulations are generated by fixing the noise power and varying the power of the source/relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Error control coding is achieved via DVB-S2 LDPC codes with block length 64,800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' At the  relay, the first 32,400 quantized bits are encoded by a rate-1/2 DVB-S2 code, and the remaining  quantized bits are encoded with a second, similar, DVB-S2 code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The parity bits of these two  encoders are then concatenated for transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Our experiments involve 16-QAM at the normalized rate of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='4 bits/s/Hz and 16-PSK at the  rate of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='27 bits/s/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Each modulation is attached to a 4-level multi-level code, which in the  case of 16-QAM have rates 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='9, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='8, and in the case of 16-PSK have rates 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='9,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='8, and 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  The relay utilizes a TCQ with an 8-state trellis whose generator matrix is:  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  1  0  0  0  0  0  1  0  0  0  0  0  D  1  0  0  0  1  D2  D  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  The quantizer reconstruction values, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 4, have a cardinality that is twice the size  of the 16-QAM/PSK constellation used for source/relay modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' For the 16-QAM example,  extending to a standard 32AMPM would present inconvenient cell boundaries, therefore we  experimented with the shown 32-point extension as well as a symmetric extension with 64  DRAFT  11  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The reconstruction values for CF relaying using 16-QAM (a), and 16-PSK (b), where the circles represent 16-QAM/PSK  constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  points, finding the latter produces negligible gains over the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The design of quantizer bin  boundaries is according to the optimization mentioned in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  We compare our results with that of bit-interleaved coded modulation (BICM) [3], [4], where  the component LDPC codes in the sub-channels have the same rate by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Since the  granularity of available DVB-2 codes are limited, we simulated BICM at the closest rate possible  with available codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' A rate-5/6 DVB-S2 code is used for 16-QAM BICM, resulting in an overall  rate of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='33 bits/s/Hz, and a rate-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='8 DVB-S2 code was used for 16-PSK resulting in overall rate  of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='2 bits/s/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' In addition to the BICM results, we also display the results of a preliminary  version of this work from [12] which uses only scalar quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' In addition, rates are carefully  chosen to avoid any unfair advantage to the method of this letter in reporting error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Simulation results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The observations and insights arising from these  simulations are as follows:  The proposed method outperforms the error rate of BICM under 16-QAM and 16-PSK,  even though the simulated BICM rates were slightly advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Superior performance of MLC over BICM was known in point-to-point channels, but is  affirmed in this letter for the end-to-end performance of a CF relaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  It is also established that TCQ can outperform scalar quantization in CF relaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Although  at first sight this may seem a natural outcome, past work [11] implies that utilizing vector  quantization in the context of CF has been remarkably difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' In that sense, the result  reported herein has significant practical novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  DRAFT  12  9  10  11  12  13  14  15  10-8  10-7  10-6  10-5  10-4  10-3  10-2  10-1  BER  16-QAM, TCQ  16-QAM, scalar  16-QAM, BICM  16-PSK, TCQ  16-PSK, scalar  16-PSK, BICM  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Simulation results using 16-QAM/PSK with N3 = 1, N2 = 8, H13 = 1, H12 = 2, H23 = 11, and Pr = Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  Approximately 1dB improvement is observed for 16-PSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The improvement for 16-QAM  is more modest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' DISCUSSION AND CONCLUSION  This paper significantly improves known implementations of compress-forward via a combi-  nation of trellis coded quantization at the relay, multi-level coding with LDPC component codes,  and iterative joint decoding of the source and relay signals at the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' The utilization  of TCQ to facilitate the generation of bit LLRs at the destination is a key contribution of this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
+page_content=' Among others, this work shows a 1dB improvement for compress-forward with PSK  modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdE1T4oBgHgl3EQfXgSr/content/2301.03128v1.pdf'}
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+arXiv:2301.08680v1  [cs.DS]  20 Jan 2023
+Online Dependent Rounding Schemes
+Joseph (Seffi) Naor∗1, Aravind Srinivasan†2, and David Wajc‡3
+1Technion – Israel Institute of Technology
+2University of Maryland, College Park
+3Google Research
+January 23, 2023
+Abstract
+We study the abstract problem of rounding fractional bipartite b-matchings online. The
+input to the problem is an unknown fractional bipartite b-matching, which is exposed node-by-
+node on one side. The objective is to maximize the rounding ratio of the output matching M,
+which is the minimum over all fractional b-matching x, and edges e, of the ratio Pr[e ∈ M]/xe.
+In offline settings, dependent rounding schemes achieving a ratio of one and strong negative
+correlation properties are known (e.g., Gandhi et al., FOCS’02, J.ACM’06 and Chekuri et al.,
+FOCS’10), and have found numerous applications. Motivated by online applications, we present
+online dependent-rounding schemes (ODRSes) for b-matching.
+For the basic case of a single offline node, i.e., uniform matroids (a.k.a. level sets), we achieve
+an online rounding ratio of one. Moreover, we show that our ODRS yields the same distribution
+as (offline) pivotal sampling (Srinivasan, FOCS’01, J.ACM’06), and so inherits the latter’s known
+strong correlation properties. In arbitrary bipartite graphs, for which online rounding ratios of
+one are impossible, we show that a combination of our level-set ODRS with repeated invocations
+of offline contention resolution schemes (CRSes) yields a rounding ratio of 1 − 1
+e ≈ 0.632. Our
+main technical contribution is an ODRS breaking this pervasive bound, yielding rounding ratios
+of 0.646 and 0.652 for b-matchings and simple matchings, respectively. We obtain these results
+by grouping nodes and using CRSes for negatively-correlated distributions, together with a new
+method we call group discount and individual markup, analyzed using the theory of negative
+association.
+Extending and applying our ODRSes, we obtain a number of new results, including (directly)
+the first online bipartite edge-weighted matching algorithms with ML predictions, and:
+1. The first results for online edge coloring multigraphs under adversarial arrivals, adding to
+the recent exciting work on online edge coloring, e.g., Kulkarni et al. (STOC’22), Bhat-
+tacharya et al. (SODA’21) and Cohen et al. (FOCS’19).
+2. Improved polytime online approximation of the optimum (computationally-unbounded)
+online algorithm for online stochastic weighted bipartite matching, improving on recent
+works of Papadimitriou et al. (EC’21) and Braverman et al. (EC’22).
+3. A generalization of a result of Srinivasan (SODA’07) that improved a previous result of
+Swamy and Shmoys (FOCS’05, SICOMP’12) to multi-stage stochastic multi-coverage.
+∗Supported in part by Israel Science Foundation grant 2233/19 and United States - Israel Binational Science
+Foundation grant 2018352.
+†Supported in part by NSF award number CCF-1918749, and by research awards from Amazon and Google.
+‡Part of this work done while the author was at Stanford University.
+
+Contents
+1
+Introduction
+1
+1.1
+Our Contributions
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+1.1.1
+Extensions and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+1.2
+Techniques
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+1.3
+Further related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2
+Preliminaries
+5
+2.1
+Negative association and strong Rayleigh properties . . . . . . . . . . . . . . . . . . .
+5
+3
+Online Level-Set Rounding
+6
+3.1
+Offline level-set rounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+3.2
+The online level-set rounding algorithm
+. . . . . . . . . . . . . . . . . . . . . . . . .
+7
+3.3
+Coupling the online and offline algorithms . . . . . . . . . . . . . . . . . . . . . . . .
+8
+4
+Rounding b-Matchings Online
+8
+4.1
+The improved ODRS: Overview and intuition . . . . . . . . . . . . . . . . . . . . . .
+9
+4.2
+The core of the improved ODRS
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+4.2.1
+The improved ODRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+4.2.2
+The b-matching ODRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+5
+Applications
+16
+5.1
+Applications and extensions of the matching ODRS . . . . . . . . . . . . . . . . . . .
+16
+5.1.1
+Online edge coloring (multigraphs) . . . . . . . . . . . . . . . . . . . . . . . .
+16
+5.1.2
+Stochastic extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+5.2
+Application of the level-set ODRS
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+23
+5.2.1
+Multi-stage stochastic optimization . . . . . . . . . . . . . . . . . . . . . . . .
+23
+5.2.2
+Negative association, fairness, diversity, and streaming . . . . . . . . . . . . .
+24
+6
+Lower Bounds
+26
+7
+Summary and Open Questions
+28
+A Additional Preliminaries
+28
+A.1 Contention Resolution Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+28
+A.2 Negative correlation properties
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+29
+A.3 Odds and Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+30
+B Deferred Proofs of Section 3
+30
+B.1
+Direct analysis of Algorithm 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+31
+B.2
+Coupling the offline and online algorithms . . . . . . . . . . . . . . . . . . . . . . . .
+32
+C Deferred Proofs of Section 4
+34
+C.1 Polytime implementation
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+34
+C.2 Deferred proofs of the b-matching ODRS . . . . . . . . . . . . . . . . . . . . . . . . .
+35
+D Deferred Proofs of Section 6
+36
+
+1
+Introduction
+Online bipartite b-matching is a prototypical online problem, tracing its origin to the seminal work
+of Karp et al. [51]. Here, nodes on one side of a bipartite graph are revealed over time, and each
+node v can be matched at most bv times, with arriving nodes matched immediately and irrevocably.
+This influential problem has been widely studied in various settings (e.g., fractional/integral, sim-
+ple/capacitated, unweighted/vertex-weighted/edge-weighted) [2, 11, 12, 29, 36, 39, 41, 50, 51, 60]
+and has been a cornerstone of the online algorithms literature.
+We study the abstract problem of online randomized rounding of fractional b-matchings. Here,
+both online nodes and their edges’ associated fractions in a fractional b-matching are revealed online.
+(The fractional matching may be provided by a online fractional algorithm, or some machine-learned
+advice—see discussion in Section 1.1.1.) An online dependent rounding scheme (ODRS) must decide
+immediately and irrevocably, upon a node’s arrival, which of its edges to match.1
+In offline settings, where all fractions and nodes are given upfront, numerous dependent rounding
+schemes for b-matching are known with the following desirable properties: they (1) match each
+edge with marginal probability equal to its fraction in the fractional b-matching (thus preserving
+any linear objective, while being oblivious to this objective), and (2) guarantee strong negative
+correlation properties (thus guaranteeing concentration and preservation of submodular objectives)
+[19, 22, 23, 44, 66]. These dependent rounding schemes have been highly influential, and yielded
+numerous applications in the approximation algorithms literature.
+Unfortunately for online applications, not only are all above offline rounding schemes not imple-
+mentable online, but online schemes satisfying Property (1) are provably impossible [17, 28, 29]. For
+example, in recent work [17], Buchbinder et al. asked what additional constraints can be imposed
+on the fractional matching to allow for Property (1) to be satisfied exactly. In this work, guided by
+the observation that approximate satisfaction of Property (1) would still allow us to approximately
+preserve linear objectives (with the ensuing applications), we study the natural complementary
+question: How well can one approximate Property (1)? Capturing this objective, we introduce the
+notion of an oblivious online rounding ratio, defined generally as follows.
+Definition 1.1. An ODRS A for a maximization problem has (oblivious) rounding ratio α ∈ [0, 1]
+if, when (batches of) elements e1, . . . , en are revealed online with their associated fractions x1, . . . , xn,
+with ⃗x a valid fractional solution, A outputs an integral solution I satisfying
+Pr[ei ∈ I] ≥ α · xi
+∀i ∈ [n].
+This question has been intensely studied for matchings in the context of one algorithmic paradigm:
+online (batched) contention resolution schemes (OCRSes), e.g., [37, 38, 43, 58]. There, each arriving
+element ei is active independently with probability xi,2 and an OCRS outputs a feasible subset of
+active elements. An ODRS can be obtained from an OCRS directly by simulating the activation
+process for arriving elements. However, even for simple b-matching problems such as rank-1 uniform
+matroids (selecting a single element), no rounding ratio beyond 0.5 is possible, via standard connec-
+tions between OCRSes and the prophet inequality problem [38, 43, 54]. Better oblivious rounding
+ratios are known (via different approaches) for simple bipartite matchings [28, 62, 65, 73], with the
+current best ratio standing at 0.526 [65].
+Can we improve on the previous best oblivious rounding ratios (for b-matchings more generally)?
+Can we achieve the ratio of 1 − 1
+e—ubiquitous in competitive ratios of online matching algorithms?
+Can we beat this bound, at least for key special cases?
+1As in the offline dependent rounding schemes of [44], the term dependent here underscores the need for random
+choices to depend on each other, since independent coin tosses do not yield a valid high-value (b-)matching.
+2Batched OCRSes [38] only require independence between batches of elements (e.g., edges of an arriving node).
+1
+
+1.1
+Our Contributions
+We provide positive answers to the above questions, and give multiple applications of our new
+ODRSes. We focus on the online rounding questions in this section.
+One special case we consider is rounding b-matchings in a star graph with a single offline node,
+also known as rounding uniform matroids, or level sets. Herein, a sequence of fractions arrive online
+over time and the goal is to round them to {0, 1} while preserving their marginals and the sum of
+every prefix (up to floors or ceilings). As we show later, this is a useful subroutine for rounding
+b-matchings in arbitrary bipartite graphs (and other applications).
+Our first contribution is an online level-set rounding algorithm breaking the ratio of 1 − 1
+e
+that is best possible via (even offline) CRSes [24, 58], attaining an oblivious rounding ratio of
+one. We thus provide an online counterpart to the well-known offline algorithm of Srinivasan [66].
+Indeed, as we show, our algorithm’s output distribution matches the offline algorithm’s output
+distribution, together with its useful properties, including strong negative correlations (“Strong
+Rayleigh" property, see Section 2) implying e.g., preservation of monotone submodular objectives.
+Theorem 1.2 (Informal). There exists an online level-set rounding algorithm with rounding
+ratio of one, and strong negative correlation properties.
+In arbitrary bipartite graphs, an ODRS with rounding ratio of one is impossible, even for simple
+matching, as discussed above. We further rule out any better than 2
+√
+2 − 2 ≈ 0.828 rounding ratio.
+On the positive side, we show that combining our online level-set algorithm with the single-item
+offline contention resolution scheme (CRS) of Feige and Vondrák [40] yields a simple ODRS with
+rounding ratio of 1 − 1/e ≈ 0.632. Our main result breaks this ubiquitous bound.
+Theorem 1.3. There exist bipartite matching (b-matching) ODRSes with rounding ratio 0.652
+(0.646). No such ODRS has rounding ratio greater than 2
+√
+2 − 2 ≈ 0.828.
+1.1.1
+Extensions and Applications
+Our ODRSes and the techniques underlying them have a number of applications and extensions.
+We briefly discuss these here, deferring detailed discussions to Section 5.
+Online edge coloring. Our ability to round fractional matchings has implications to online edge
+coloring. For this problem, it is known that the greedy algorithm’s competitive ratio of 2 is tight
+for graphs of low maximum degree ∆ = O(log n) [8], while better competitive ratios are known for
+high-degree graphs under various arrival models, mostly for simple graphs [3, 6, 10, 27, 55, 65]. For
+multigraphs, positive results (competitive ratios smaller than 2) were only known under random-
+order arrivals [3]. Addressing this gap between simple graphs and multigraphs, in Section 5.1.1 we
+extend known reductions from α-competitive edge coloring (α ≥ 1) to computing online matchings
+that match each (in this case, parallel) edge with probability
+1
+α∆ [27]. We thus show that a rounding
+ratio of 1/α implies a competitive ratio of α + o(1) for edge coloring multigraphs (and is, in fact,
+equivalent to it). Combined with our rounding algorithms, this yields the first positive result for
+online edge-coloring in multigraphs under adversarial arrivals, achieving a competitive factor of
+e/(e − 1) − Ω(1) in bipartite multigraphs, answering a question of Schrijver [28, Sec. VI].
+Stochastic optimization. Another application of our techniques concerns the online stochastic
+bipartite weighted matching problem, much studied in the online Prophet-Inequality literature
+[35, 38, 42, 54]. (See Section 5.1.2 for formal definition.) Recently, Papadimitriou et al. [62] initiated
+2
+
+the study of efficiently approximating the optimum online algorithm for this problem. They showed
+that it is pspace-hard to approximate the optimum algorithm within some universal constant β < 1,
+and provided a polynomial-time online algorithm that 0.51-approximates this online optimum. (In
+contrast, the best competitive ratio, or approximation of the offline optimum, is 0.5 for this problem
+[38].) The bound of [62] can be improved to 0.526 using ideas in [65] and was significantly improved
+to 1 − 1/e ≈ 0.632 by Braverman et al. [16]. All three results are achieved via online rounding.
+Extending our online rounding approach of Theorem 1.3 to such stochastic settings, we improve on
+the recent bound of [16] and break the ubiquitous bound of 1 − 1/e, providing an online polytime
+0.652-approximation of the optimal (computationally-unbounded) online algorithm.
+Algorithms with predictions. Our ODRS of Theorem 1.3 finds a direct application in the bur-
+geoning area of online algorithms with advice (see the survey [61] and site [1]). Here, an online
+algorithm is equipped with machine-learned (ML) predictions concerning the input. For example,
+many works show that for related problems, high-quality fractional solutions are efficiently learnable
+and can guide integral algorithms, via online rounding schemes [56, 57, 59]. Our ODRS immediately
+yields the first online edge-weighted bipartite matching algorithm with predictions, getting a com-
+petitive ratio of (1−1/e+Ω(1)) with sufficiently good predictions. In contrast, with no predictions,
+only recently was it shown how to break the barrier of 1/2 for this problem (using free disposal)
+[12, 39, 45], and 1 − 1/e + Ω(1) is impossible to achieve even in unweighted graphs [51].
+Multi-Stage stochastic optimization.
+In multi-stage stochastic optimization, uncertain pa-
+rameters are stochastic but their distributions get refined over time: actions in earlier stages are
+cheaper but perhaps less accurate due to the stochasticity, while actions in later stages are more
+expensive but more accurate. (See the survey [68].) A basic example is the (hyper-)graph covering
+problem, where sets (hyperedges) are revealed in k stages, and must be covered by bought elements
+(nodes). Swamy and Shmoys [69] presented a 2k-approximation for this problem, later improved to
+2, matching the offline hardness of the non-stochastic single-stage problem, in [67]. Building on the
+framework of [67, 69], in Section 5.2.1 we generalize this result to allow for sets with muti-coverage
+demands. Our algorithms have approximation ratio of 2 (and tending to one in some settings), and
+these ratios hold w.h.p., building on our algorithm’s strong negative correlation properties.
+Fairness and diversity in allocation.
+Building on the previous sentence, an even-stronger
+negative correlation property—negative association—helps our rounding guarantee that its resultant
+intersectional unfairness [18] (if any) is limited. Continuing on the theme of ML advice, we prove that
+our rounding algorithm produces online allocations (e.g., of vaccines as they are produced) from ML
+advice such that these allocations do not impact multiple intersectional groups unfairly, with high
+probability. Furthermore, our online-rounding algorithm only needs O(log n) space. In Section 5.2.2
+we leverage this and the fact that diversity/fairness models often lead to monotone submodular
+objectives which work very well with our rounding scheme, in order to develop logarithmic-space
+streaming algorithms that act on ML advice, in order to produce provably-diverse search results.
+1.2
+Techniques
+In this section we highlight some of the key ideas that allow us to achieve our main results, namely
+Theorems 1.2 and 1.3, deferring detailed discussions of applications to later sections.
+For level-set rounding, we provide a simple ODRS using limited memory (i.e., both online and
+streaming) – only remembering how many edges/elements were matched/taken – that achieves a
+rounding ratio of one. Surprisingly, we show via a coupling argument that this online algorithm’s
+distribution is identical to that of (offline) pivotal sampling, a.k.a. Srinivasan sampling [66]. This
+allows us to inherit this well-studied algorithm’s concentration properties [15, 31, 66]. In particular,
+3
+
+this implies that our online algorithm’s output distribution satisfies the Strong Rayleigh property (see
+Section 2), which in turn implies a slew of negative correlation properties, useful for our applications.
+For our impossibility result, we rely on a Ramsey-theoretic probabilistic lemma, whereby in any
+sufficiently large set of binary variables, some (near-)positive correlation is inevitable. An extension
+of this lemma (Lemma 6.3), of possible independent interest, rules out algorithmic approaches based
+on guaranteeing strict negative correlation between offline nodes’ matched statuses.
+ODRSes for b-matching. Our first ODRS for b-matchings more broadly is obtained by interleav-
+ing our level-set algorithm with repeated invocations of the offline single-item contention resolution
+scheme (CRS) of Feige and Vondrák [40]. (This is, to the best of our knowledge, the first online al-
+gorithm to use offline CRSes in such a black-box manner.) Specifically, we run independent copies of
+our level-set algorithm, one per offline node, thus having each offline node i “bid” for arriving online
+node t independently of other offline nodes with probability xi,t, while preserving the b-matching
+constraints of the offline nodes. To preserve online nodes’ matching constraints, we then use the
+single-item offline CRS of [40] at each time t: this CRS guarantees for such product distributions
+that t is allocated to at most one buyer, with every buyer i being allocated item t (i.e., (i, t) being
+matched) with probability at least (1 − 1/e) · xi,t, thus yielding a rounding ratio of 1 − 1/e.
+To improve on the above, we first note that the (1 − 1/e) · xit bound is loose if any of the xit
+fractions are bounded away from 0. Indeed, this 1 − 1/e factor is of the form
+�
+1 −
+�
+j
+(1 − xit)
+�� �
+i
+xi,t ≥ 1 − 1/e,
+where the inequality is loose if any xit is large. To beat this 1−1/e bound, we therefore simulate the
+existence of a large xit, by grouping offline nodes via a bin-packing heuristic, and letting at most one
+node per group (bin) B bid for t. The obtained distribution over biding offline nodes is no longer
+a product distribution, but it is strongly negatively correlated, and even negatively associated (see
+Section 2). This allows us to achieve the same 1 − 1/e bound using CRSes for negatively correlated
+distributions of Bansal and Cohen [7]. Moreover, this grouping allows us to beat the 1−1/e ratio in
+some cases: In a very concrete sense, it results in effectively a single aggregated offline node bidding
+with large probability �
+i∈B xi,t, thus allowing us to beat the bound of 1 − 1/e, if much grouping
+occurs. Furthermore, thinking of bids as “costing” the offline node future matching opportunities, we
+can even beat the bound of 1−1/e after giving nodes in a group a “group discount”, and having them
+bid with lower probability than xi,t. This discounting then leaves offline nodes with more budget
+(probability of not being matched), thus allowing us to charge these nodes more at later times: in
+particular, when these offline nodes are not grouped, we charge them an “individual markup”, by
+having them bid with higher probability than xi,t. This then increases the matching probability of
+those t for which little grouping occurs to beyond 1 − 1
+e, and the improved rounding ratio follows.
+1.3
+Further related work
+A natural approach to round a feasible fractional solution ⃗x is to activate each element e indepen-
+dently w.p. xe and then pick a feasible subset I of active elements, such that Pr[e ∈ I] ≥ α · xe.
+The latter is obtained by contention resolution schemes (CRSes), first studied in offline settings for
+single-item problems (rank-one matroids) by Feige and Vondrák [40] (see Section 2), and later for
+arbitrary matroids by Chekuri et al. [22]. This approach was then extended to online settings by
+Feldman et al. [43], and has since become a bedrock of stochastic online problems, from prophet
+inequalities [43], secretary problems [33, 34], posted-price mechanisms [21, 52] and sequential pricing
+[64], algorithmic contract design [9] and more. Unfortunately, for bipartite matching ODRSes, using
+4
+
+OCRSes (letting each i bid for t with probability xi,t independently of all prior bids, including i’s) is
+of limited use: such OCRS-based OCRSes have rounding ratio better at most 0.5 even for rank-one
+matroids, by connections to the prophet-inequality problem [38, 54]. Going beyond independent
+bids for each pair (i, t) is therefore crucial to obtain high oblivious rounding ratios.
+Another online matching rounding approach is provided by the recent exciting literature on
+Online Correlated Selection (OCS), first introduced in the groundbreaking paper of Fahrbach et al.
+[39] (and its predecessor by Huang and Tao [48]). Specifically, the OCSes of Gao et al. [45] underlie
+many rounding-based online algorithms for edge-weighted matching [45], fair matching [46] and i.i.d
+matching [47, 70]. This powerful machinery, however, does not provide oblivious rounding guaran-
+tees, but rather per-offline-vertex guarantees, and so is inapplicable for our problem applications.
+2
+Preliminaries
+Problem Definition. In the online b-matching rounding problem, an a priori unknown bipartite
+graph G = (L, R, E) is revealed online, together with fractions on the edges incident to the arriving
+edge’s vertices that satisfy the b-matching constraints. In more detail, at each time t an online vertex
+t ∈ L arrives, together with fractions xi,t assigned to its edges to offline neighbors i ∈ R = [n]. The
+fractions xi,t ∈ [0, 1], are promised to satisfy the (fractional) b-matching constraints, namely each
+vertex v ∈ (L ∪ R) has fractional degree �
+e∋v xe at most bv. By splitting each online node t with
+capacity bt into bt online unit-capacity nodes with identical edge sets and fractions xit/bt for each
+copy of edge (i, t), we can assume WLOG that all online nodes have unit capacity. Following arrival
+t, we must decide, immediately and irrevocably, which edges of t to add to the output b-matching
+M, while satisfying the (integral) b-matching constraints of matching each vertex v no more than
+bv times, and striving for a large oblivious rounding ratio.
+Our ODRSes for b-matchings will repeatedly invoke offline contention resolution schemes (CRSes),
+whose guarantees for product distributions, due to Feige and Vondrák [40], and arbitrary distribu-
+tions, due to Bansal and Cohen [7], are given below. (For completeness, we provide a self-contained
+proof for the general case in Appendix A.)
+Lemma 2.1. Let D : 2[n] → R≥0 be a distribution over subsets of [n], with its support denoted by
+supp(D) = {S ⊆ [n] | PrR∼D[R = S] ̸= 0}. Then, there exists a randomized algorithm CRS(R,⃗v)
+which on input set R ∼ D and vector ⃗v ∈ Rn, outputs a subset O ⊆ R of size |O| ≤ 1 satisfying
+Pr[i ∈ O] = vi · min
+S⊆[n]
+Pr[R ∩ S ̸= ∅]
+�
+i∈S vi
+∀i ∈ [n].
+The algorithm runs in time polynomial in n if D is a product distribution. Otherwise, it runs in
+time poly(n, T), where T ≥ |supp(D)| is the time to compute and write down the support of D.
+2.1
+Negative association and strong Rayleigh properties
+Definition 2.2. Random variables X1, . . . , Xn are negatively associated (NA) if for any two disjoint
+index sets I, J ⊆ [n], I∩J = ∅, and two functions f : R|I| → R and g : R|J| → R both non-decreasing,
+E[f(Xi : i ∈ I) · g(Xj : j ∈ J)] ≤ E[f(Xi : i ∈ I)] · E[g(Xj : j ∈ J)].
+By a simple inductive argument, negative association implies negative cylinder dependence [49].
+Lemma 2.3. For any real NA r.v.s X1, . . . , Xn and reals x1, . . . , xn, it holds that
+Pr
+� �
+i
+(Xi ≥ xi)
+�
+≤
+�
+i
+Pr[Xi ≥ xi]
+and
+Pr
+� �
+i
+(Xi ≤ xi)
+�
+≤
+�
+i
+Pr[Xi ≤ xi].
+5
+
+The following family of NA distributions is due to Dubhashi and Ranjan [32].
+Lemma 2.4. If X1, . . . , Xn are binary r.v.s with �
+i Xi ≤ 1 always, then they are NA.
+More elaborate NA distributions can be obtained via simple NA-preserving operations [49].
+Lemma 2.5. NA is closed under products and under disjoint non-decreasing function composition.
+That is, if ⃗X = (X1, . . . , Xn) is NA, then:
+1. if ⃗Y = (Y1, . . . , Ym) is NA and ⃗Y is independent of ⃗X, then (X1, . . . , Xn, Y1, . . . , Ym) are NA;
+2. (f1(Xi : i ∈ I1), . . . , fk(Xi : i ∈ Ik)) are NA, for any concordant functions (i.e., all non-
+decreasing or all non-increasing) f1, . . . , fk and disjoint sets I1, . . . , Ik ⊂ [n], Ij∩Ik = ∅ ∀j ̸= k.
+Negative association implies many more useful properties, most notably the applicability of
+Chernoff-Hoeffding type tail bounds [32] and submodular stochastic dominance (see Appendix A).
+An even stronger negative correlation property than NA is the strong Rayleigh property (SRP)
+(see Appendix A for a precise definition). This property implies NA (even under conditioning), as
+well as concentration of any Lipschitz function [63], and numerous other useful properties.
+3
+Online Level-Set Rounding
+In this section we introduce our online level-set rounding algorithm. As its analysis will involve
+coupling with the offline level-set rounding algorithm of Srinivasan [66], we first start by revisiting
+this algorithm and its properties.
+3.1
+Offline level-set rounding
+Here we consider an offline procedure due to [66].
+This algorithm, on input ⃗x ∈ [0, 1]n with
+integer sum �
+i xi ∈ Z (possibly by adding a dummy value) and partial sums sj := �
+i≤j xj,
+outputs a set S ⊆ [n] with characteristic vector ⃗X given by Xi := 1[i ∈ S] with partial sums
+Sj := �
+i≤j Xi = |S ∩ [j]| satisfying the following properties for all i.
+(P1) (Marginals) E[Xi] = xi.
+(P2) (Rounding) Si ∈ {⌊si⌋, ⌈si⌉}.
+Property (P1) and linearity of expectation imply that E[Si] = si.
+Property (P2) requires that
+Si always equals this expectation, “up to rounding”. In addition, we require the following strong
+negative correlation property.
+(P3) (Strongly Rayleigh) The joint distribution ⃗X is strongly Rayleigh (and thus, also NA).
+Our objective will be to design an online algorithm satisfying the above, but for now we revisit
+the offline algorithm of [66] and prove that when run with a certain specific way of choosing the key
+indices i1 and i2 (see Algorithm 1), then it satisfies the above three properties.
+As usual, “min(S)" in Algorithm 1 refers to the minimum element of a nonempty set S; the
+index i2 in Algorithm 1 is always well-defined since �
+i xi ∈ Z and, as we shall see, �
+i yi = �
+i xi
+always. We emphasize that the algorithm of [66] allows much liberty in the choice of i1 and i2, and
+that our specific choice of i1 and i2 is required to satisfy Property (P2). Also, the random choice in
+each call to STEP is independent of the random choices of the past.
+6
+
+Algorithm 1 Offline Level-Set Rounding
+1: Initialize ⃗y ← ⃗x
+2: while frac(y) := {i | yi ̸∈ {0, 1}} ̸= ∅ do
+3:
+Let i1 ← min(frac(y))
+4:
+Let i2 ← min(frac(y) \ {i1})
+5:
+STEP(yi1, yi2)
+6: Output S := {i | yi = 1}
+Algorithm 2 STEP(A, B)
+⊲ modifies inputs
+1: if A + B < 1 then
+2:
+(A, B) ←
+�
+(A + B, 0)
+w.p.
+A
+A+B
+(0, A + B)
+w.p.
+B
+A+B
+3: else
+⊲ A + B ∈ [1, 2)
+4:
+(A, B) ←
+�
+(1, A + B − 1)
+w.p.
+1−B
+2−A−B
+(A + B − 1, 1)
+w.p.
+1−A
+2−A−B
+That Algorithm 1 terminates and outputs a random set satisfying Property (P1) is known [66].
+Nonetheless, for completeness, we provide a proof of this fact in Appendix B. In the same appendix
+we prove by induction on the number of invocations of STEP() that every prefix sum is preserved
+with probability one, up to rounding, and so Algorithm 1 satisfies Property (P2).
+Lemma 3.1. Let sj := �
+i≤j xi and Y t
+j := �
+i≤j yt
+i, where yt
+i is the value yi after t applications of
+STEP in Algorithm 1. Then, Y t
+j ∈ [⌊sj⌋, ⌈sj⌉] for all j and t.
+Strong negative correlation (Property (P3)) was proven in [15] as long as, e.g., we use some fixed
+choice of i1 and i2 as in Algorithm 1:
+Lemma 3.2. Let Xi := 1[i ∈ S] be the indicators for i being in the output set of Algorithm 1.
+Then, ⃗X = (X1, X2, . . . , Xn) is a family of strongly Rayleigh random variables.
+To conclude, we have the following.
+Lemma 3.3. Algorithm 1 satisfies properties (P1) (P2) and (P3).
+3.2
+The online level-set rounding algorithm
+We now turn to providing an online counterpart to Algorithm 1. Here, the input is a sequence of
+reals, x1, x2, · · · ∈ [0, 1], revealed in an online fashion. Algorithm 3 maintains a set S ⊆ N (initially
+empty), and decides at time t (when xt is revealed) whether to add t to its output set S, immediately
+and irrevocably. Let Xt := 1[t ∈ S], and let St := �
+t′≤t Xt′ denote the cardinality of S right after
+time t, i.e., St = |S ∩ [t]|. Finally, let st := �
+t′≤t xt′. We will show that Algorithm 3 satisfies
+properties (P1), (P2) and (P3) given by its offline counterpart, Algorithm 1.
+Algorithm 3 Online Level-Set Rounding
+1: Initialize S ← ∅
+2: for arrival xt (at time t ≥ 1) do
+3:
+add t to S with probability pt :=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+0
+|S| = ⌈st⌉
+1
+|S| < ⌊st⌋
+xt
+⌊st−1⌋+1−st−1
+|S| = ⌊st⌋ = ⌊st−1⌋
+st−⌊st⌋
+st−1−⌊st−1⌋
+|S| = ⌊st⌋ > ⌊st−1⌋ and st−1 ̸= ⌊st−1⌋
+0
+else.
+4: Output S
+The random bits used to make the random choice in each step of Algorithm 3 are independent
+of the random bits used in the past: i.e., while pt depends on St−1, the random bits used to generate
+the event of probability pt in step t, are independent of the random bits of past steps.
+7
+
+3.3
+Coupling the online and offline algorithms
+Here we prove that Algorithm 3 satisfies properties (P1), (P2) and (P3). Proving properties (P1),
+(P2) directly is not too hard (see Lemma B.2).3
+Proving Property (P3), on the other hand, is
+slightly more involved; we do so now, using that the much-simpler properties (P1) and (P2) hold
+for both the offline and online algorithms. To prove that the last property holds, we show that both
+algorithms induce the same distribution over outputs. We outline a proof here, deferring a complete
+proof to Appendix B.
+Theorem 3.4. Fix a vector ⃗x ∈ [0, 1]n. Let D and D′ denote the probability distributions on {0, 1}n
+induced by the online and offline algorithms run on ⃗x, respectively. Then, D = D′.
+Proof (Sketch). We prove by induction on t ∈ [n] that the following holds:
+∀(b1, . . . , bt−1) ∈ {0, 1}t−1, Pr
+D [Xt = 1
+�� (∀i < t, Xi = bi)] = Pr
+D′[Xt = 1
+�� (∀i < t, Xi = bi)].
+(1)
+The (strong) inductive hypothesis, whereby Equation (1) holds for all t′ < t, clearly implies that for
+all (b1, . . . , bt) ∈ {0, 1}t−1 we have that PrD[(∀i < t, Xi = bi)] = PrD′[(∀i < t, Xi = bi)]. Similarly,
+Equation (1) holds for all t ≤ n iff the two distributions are equal. The proof then proceeds by
+considering the different possibilities for the prefix sums st−1 and st, which for the online algorithm
+directly determine the above conditional probability. These conditions also determine the offline
+algorithm’s performance up to the (t − 1)st invocation of STEP(), from which a careful analysis
+implies that the conditional probabilities of the offline and online algorithms are equal.
+Theorem 3.4 combined with Lemma 3.3 immediately implies the desired properties of our Algo-
+rithm 3, as these properties are determined by the distribution over output sets.
+Lemma 3.5. Algorithm 3 is an online algorithm satisfying properties (P1), (P2) and (P3).
+4
+Rounding b-Matchings Online
+In this section we address the ODRS problem in bipartite b-matching instances more broadly.
+We start with a simple algorithm, combining our ODRS for uniform matroids (Algorithm 3)
+with repeated invocations of an offline CRS (Lemma 2.1).
+Warm-up: 1 − 1/e rounding ratio. We consider the following simple algorithm: Run, for each
+offline node i, an instance of Algorithm 3, applied to xi1, xi2, . . . . When the i-th copy of Algorithm 3
+fixes Xit = 1, we say that i bids for online node t. Out of the set of bidders (potential buyers) Pt,
+we then pick at most one i ∈ Pt to match t to, using the offline contention resolution scheme of
+Lemma 2.1, guaranteeing each node i a probability of xi,t · minS⊆[n]
+Pr[S∩Pt̸=∅]
+�
+i∈S xi,t
+of being matched.
+This step runs in polytime, by independence of the bids. Since Algorithm 3 is an online algorithm,
+so is the resultant algorithm. Moreover, since the output of Algorithm 3 satisfies the b-matching
+constraints, and t is matched to at most one neighbor by the CRS, this algorithm’s output satisfies
+the b-matching constraints. Finally, for each offline node set S ⊆ [n], by independence of the bids,
+the Taylor expansion of exp(−x) and convexity (see Claim A.5), we have
+Pr[S ∩ Pt ̸= ∅] = 1 −
+�
+i∈S
+(1 − xit) ≥ 1 −
+�
+i∈S
+exp(−xit) ≥
+�
+1 − 1
+e
+�
+·
+�
+i∈S
+xit.
+(2)
+Therefore, by Lemma 2.1, this ODRS matches each edge (i, t) with probability at least (1−1/e)·xi,t.
+3In fact, Algorithm 3 can be shown to be the only online algorithm satisfying both these simple properties if we
+further restrict it to only storing in memory the partial sums st−1 and St−1 at the beginning of step t.
+8
+
+Lemma 4.1. There exists a polytime ODRS for bipartite b-matching with rounding ratio 1 − 1/e.
+One natural approach to improve the above algorithm’s rounding ratio would be to attempt to
+negatively correlate the probability of different offline nodes to have previously bid, thus increasing
+Pr[S ∩ Pt ̸= ∅] for all offline sets S ⊆ [n]. Unfortunately, for a sufficiently large number of offline
+nodes, (near-)positive correlation is, generally, unavoidable: see Lemma 6.3 for a proof of such a
+Ramsey-theoretic statement. Instead, to achieve a better rounding ratio, we explicitly negatively
+correlate the bids of previously-non-bidding offline nodes when they do bid.
+4.1
+The improved ODRS: Overview and intuition
+In this section we outline our improved ODRS. We focus here and in most subsequent sections on
+simple matchings, for clarity’s sake, deferring an extension to b-matchings to Section 4.2.2.
+A bad example, and a false start. Consider a graph with a single online node t with xit = 1
+n for
+all n offline nodes. In this case, under independent bids, t gets no bids with probability (1− 1
+n)n ≈ 1
+e,
+and so one of the n offline nodes must get matched with probability no greater than (1− 1
+e) · 1
+n. For
+this simple example a rounding ratio of one is possible, by correlating bids for t: if we pick exactly
+one neighbor i to bid with probability xit, then each offline node both bids and buys (is matched
+to) t with probability xit. Unfortunately, this approach is impossible to implement in general: if
+we let each offline node i bid for at most one time t (recall that this is a simple matching instance)
+with probability xi,t, then, for sit := �
+t′<t xit′ the fractional degree of i at time t, we must have i
+bid for t with probability pit := xit/(1 − sit), and yet �
+i pit may exceed one.
+Bin-packing low-degree nodes. Our first step is to adopt a partial implementation of the above
+approach, by grouping offline nodes into groups B for which �
+i∈B pit ≤ 1, and using one Uni(0, 1)
+variable to offer at most one bid per group. Specifically, if we denote the set of low-degree neighbors
+of t by Lt := {i ∈ N(t) | sit ≤ θ} for some threshold θ ∈ [0, 1] that we will optimize later, we will
+pack low-degree neighbors into bins using a greedy bin-packing heuristic such that each bin B has
+�
+i∈B pi,t ≤ 1. Moreover, each such bin B (but one) has �
+i∈B pi,t ≥ 1/2 and hence has high total
+x-value of �
+i∈B xi,t ≥ 1−θ
+2 . (Here we use that pi,t = xi,t/(1 − si,t) ≤ xi,t/(1 − θ).) Using negative
+association arguments (see Section 2), we can show that this approach essentially replaces multiple
+bids of small x-value with bids of x-value equal to the small bids’ sum. In particular, if the total
+x-value of low-degree neighbors i ∈ Lt is high, this results in minS⊆[n]
+Pr[S∩Pt̸=∅]
+�
+i∈S xi,t
+≥ 1 − 1
+e + Ω(1),
+and a rounding ratio greater than 1 − 1
+e, by Lemma 2.1.
+Group discounting and individual markup. It remains to address the challenge of increasing
+Pr[S ∩Pt ̸= ∅] if the total x-value of low-degree nodes Lt is small, and so very little grouping occurs.
+For this, we note that from the perspective of every offline node i, a time step t has: (1) a value
+(matching probability), and (2) a cost (bidding probability). The latter is indeed a cost, since we
+do not allow offline nodes to bid more than once. The value at each time is monotone increasing in
+offline nodes’ costs. Now, when we use grouping we decrease the contention faced by the CRS, as it
+now faces fewer bidding nodes, due to each bin only providing one bid (or none). We can therefore
+safely decrease the bidding probability (cost) in the case that much grouping occurs, giving the
+grouped nodes a group discount, while still obtaining a rounding ratio greater than 1 − 1/e. The
+upshot of the group discount is that offline nodes now have a higher chance of not having bid before
+later time steps where they are not grouped, in which case such individual bidders can pay an
+individual markup, allowing us to guarantee that in this case too minS⊆[n]
+Pr[S∩Pt̸=∅]
+�
+i∈S xi,t
+≥ 1− 1
+e +Ω(1),
+with the ensuing 1 − 1
+e + Ω(1) rounding ratio following from the offline CRS of Lemma 2.1.
+9
+
+4.2
+The core of the improved ODRS
+In this section we introduce our improved ODRS’ core subroutine, Algorithm 4. (The parameter
+θ ∈ [0, 1] will be optimized later, and for now it is safe to think of the input vectors x, v as both
+equaling the input fractional matching.) At each time step t, the algorithm groups low-fractional-
+degree offline nodes, and then all remaining nodes, into buckets, using the classic first-fit bin-packing
+heuristic,4 with each offline node i having a size of
+xi,t
+1−si,t . This is the conditional probability with
+which i should bid for t if i is free, i.e., has not bid before, to guarantee a marginal bidding probability
+of xi,t. (This follows from our online level-set rounding algorithm’s probabilities.) Items are grouped
+into bins of size one, and then at most one offline node bids per bin (lines 9-12), resulting in the
+set of candidates, Ct. The free candidates Pt = Ct ∩ F then bid for t, who is then matched to (at
+most) one bidding neighbor i ∈ Pt, chosen by a CRS.
+Algorithm 4 ODRS-core(θ, x, v)
+1: M ← ∅
+2: F ← [n]
+⊲ Free offline nodes
+3: for each time t do
+4:
+for each i ∈ [n] do
+5:
+si,t ← �
+t′<t xi,t′
+6:
+Bt ← FirstFit
+���
+i,
+xi,t
+1−si,t
+� ��� i ∈ [n], si,t ≤ θ
+��
+⊲ Bucketing low-degree nodes
+7:
+Bt ← Bt ∪ FirstFit
+���
+i,
+xi,t
+1−si,t
+� ��� i ∈ [n], si,t > θ
+��
+⊲ Bucketing high-degree nodes
+8:
+Ct ← ∅
+⊲ Candidates
+9:
+for each B ∈ Bt do
+10:
+U ∼ Uni[0, 1]
+11:
+if U ≤ �
+i∈B
+xi,t
+1−si,t then
+12:
+Ct ← Ct ∪ mini
+�
+i ∈ [k]
+���� U ≤ �
+j∈B
+j≤i
+xj,t
+1−sj,t
+�
+13:
+Pt ← F ∩ Ct
+⊲ Bidders = free candidates
+14:
+F ← F \ Pt
+⊲ Bidders cease being free
+15:
+O ← CRS(Pt, {vi,t}i)
+⊲ Contention Resolution
+16:
+if |O| = 1 then
+17:
+M ← M ∪ {(i, t) | i ∈ O}
+18: Output M
+First, we note that Algorithm 4—including the bin-packing steps that require
+xi,t
+1−si,t ≤ 1—is
+well-defined provided �
+t xi,t ≤ 1 for all i, t, since this implies that xi,t ≤ 1 − �
+t′≤t xi,t′ = 1 − si,t.
+Next, by construction, each offline node bids (and is thus matched) at most once, while the CRS
+guarantees that each online node is matched at most once. To summarize, we have the following.
+Observation 4.2. Algorithm 4 is well-defined and outputs a matching if �
+t xi,t ≤ 1 ∀i ∈ [n].
+We turn to analyzing the algorithm’s rounding ratio, starting with the following lemma asserting
+that if much grouping occurs, then most bins have high x-value.
+Fact 4.3. For each time t, at most one bin B ∈ Bt at Line 6 has �
+i∈B xi,t < 1−θ
+2 .
+4This algorithm maintains a sorted list of bins with each bin B containing items of overall size at most one,
+�
+i∈B si ≤ 1, and for each item i in order, adds i to the first (possibly newly-opened) bin that has enough room left.
+10
+
+Proof. A well-known property of the First-Fit bin packing heuristic is that at most one bin in the
+output of this algorithm is less than 1
+2 full (see, e.g., [72, 74]). Therefore, since si,t ≤ θ for each
+offline node i in a bin B ∈ Bt computed in Line 6, each such bin B (barring perhaps one) has
+1
+2 ≤
+�
+i∈B
+xi,t
+1 − si,t
+≤
+�
+i∈B
+xi,t
+1 − θ.
+The following lemma, combined with Fact 4.3, provides a guarantee no worse than the independent-
+proposals approach. Indeed, as we will show shortly, it implies a better rounding ratio if much
+grouping of low-degree neighbors occurs.
+Lemma 4.4. For any time t and set of offline nodes S ⊆ [n], Algorithm 4 satisfies
+Pr[S ∩ Pt ̸= ∅] ≥ 1 −
+�
+B∈Bt
+�
+1 −
+�
+i∈B∩S
+xit
+�
+.
+Proof. Let Ft denote F at time t, and let Ct and Pt = Ct ∩ Ft be as in Algorithm 4. Next, let
+Ci,t = 1[i ∈ Ct] indicate whether i is a candidate at time t, and define Fi,t and Pi,t similarly.
+By independence of Ci,t and Fi,t and since Pr[Ci,t] =
+xi,t
+1−si,t and Pr[Fi,t] = 1 − si,t (provable by a
+simple induction on t ≥ 0), we have that Pr[Pi,t] = xi,t. Therefore, by linearity of expectation and
+�
+i∈B Ci,t ≤ 1, we have that Pr[Pt ∩ S ∩ B = ∅] = 1 − �
+i∈B∩S xi,t. It remains to show that the
+events [Pt ∩S ∩B = ∅] for different B are negative cylinder dependent, giving the desired inequality
+Pr[Pt ∩ S = ∅] = Pr
+� �
+B∈Bt
+(Pt ∩ S ∩ B = ∅)
+�
+≤
+�
+B∈Bt
+Pr[Pt ∩ S ∩ B = ∅] =
+�
+B∈Bt
+�
+1 −
+�
+i∈B∩S
+xi,t
+�
+.
+To this end, we will show that the events [Pt ∩ S ∩ B = ∅] are negatively associated (NA),
+which by Lemma 2.3 yields the required above inequality, and hence yields the lemma. First, by
+Lemma 2.4, for any bin B′ at time t′ < t, the binary variables Ci,t′ for all nodes i ∈ B′ ∩ S, whose
+sum is at most one, are NA. Moreover, the distributions over different bins and different time steps
+are independent, and so by closure of NA under products (Lemma 2.5), all Ci,t′ are NA. Therefore,
+the variables Fi,t = 1 − maxt′<t Ci,t′ are non-increasing functions of disjoint NA variables Ci,t′, and
+so by closure of NA under such functions (Lemma 2.5), the variables Fi,t are NA. Moreover, the
+variables Ci,t are independent from the variables Fi,t (who are functions only of Ci,t′ for all t′ < t),
+and consequently, by another application of closure of NA under products (Lemma 2.5), all variables
+{Fi,t, Ci,t | i ∈ S} are NA. Finally, since 1[Pt ∩ S ∩ B = ∅] = 1 − �
+i∈B∩S Ci,t · Fi,t are monotone
+decreasing functions of disjoint NA variables (different bins B), then by closure of NA under such
+functions (Lemma 2.5), the variables 1[Pt ∩ S ∩ B = ∅] are NA, as desired.
+4.2.1
+The improved ODRS
+Lemma 4.4 and Fact 4.3 allow us to argue that if much of t’s fractional degree is contributed by
+neighbors i with low fractional degree, si,t ≤ θ, then Pr[S ∩ Pt ̸= ∅] ≥ 1 − 1
+e + Ω(1). By Lemma 2.1,
+this allows to match edges of such nodes t with probability greater than (1 − 1
+e) · xi,t. Indeed,
+we can afford to decrease the fractions xi,t of low-degree nodes (that may be grouped effectively)
+and still retain a similar 1 − 1
+e + Ω(1) rounding ratio for such t.
+In return, we can afford to
+increase the fractions xi,t of nodes of high degree (who might not be grouped). More precisely,
+we consider the following transformation to x, capturing this aforementioned group discount and
+11
+
+individual markup. For θ =
+δ
+ǫ+δ, with ǫ, δ ∈ [0, 1] to be optimized shortly, we consider the following
+assignment ˆx : E → R≥0.
+ˆxi,t = xi,t · (1 − ǫ) +
+� max(θ, si,t+xi,t)
+z=max(θ,si,t)
+(ǫ + δ) dz.
+(3)
+Intuitively, ˆx decreases/increases the x-value of (parts of) edges of i until/after i has fractional
+degree θ, by a (1 − ǫ) and (1 + δ) factor, respectively. So, for example, ˆxi,t ≥ xi,t · (1 − ǫ) if si,t ≤ θ
+and ˆxi,t = xi,t · (1 + δ) if si,t > θ. While ˆx is not a fractional matching, since online nodes t may
+have �
+i ˆxi,t > 1, this assignment does satisfy the fractional degree constraints of offline nodes.
+Fact 4.5. For every offline node i ∈ [n] and time t, we have that ˆsi,t := �
+t′<t ˆxi,t′ ≤ si,t ≤ 1.
+Proof. If si,t ≤ θ, trivially ˆsi,t = si,t · (1 − ǫ) ≤ si,t. Otherwise, by our choice of θ =
+δ
+ǫ+δ, we have
+ˆsi,t = si,t · (1 − ǫ) + (si,t − θ)+ · (ǫ + δ) = si,t · (1 + δ) − δ ≤ si,t.
+Our ODRS, given in Algorithm 5, first scales the fractions as in Equation (3) (easily computable
+online, as these ˆxi,t only depend on information known by time t) and feeds the resultant vectors ˆx, x
+(playing the roles of x and v, respectively) into our core ODRS subroutine, Algorithm 4. By Fact 4.5
+and Observation 4.2, we have that Algorithm 5 is a well-defined online algorithm, outputting a valid
+matching. It remains to analyze this algorithm’s rounding ratio, starting with the following lemma.
+Algorithm 5 ODRS(ǫ, δ, x)
+1: θ ←
+δ
+ǫ+δ
+2: Init ODRS-core(θ · (1 − ǫ), ˆx, x)
+3: for each time t do
+4:
+for each i ∈ [n] do
+5:
+si,t ← �
+t′<t xi,t′
+6:
+Compute ˆxi,t from (xi,t, si,t, ǫ, δ) as in Equation (3)
+7:
+Run next step of ODRS-core(θ · (1 − ǫ), ˆx, x)
+8: Output M computed by ODRS-core(θ · (1 − ǫ), ˆx, x)
+Lemma 4.6. If ǫ, δ ∈ [0, 1] guarantee that for all z ≥ 1−θ
+2
+=
+ǫ
+2(ǫ+δ),
+fǫ,δ(z) := exp(−z · (1 + δ)) − (1 − z · (1 − ǫ)) ≥ 0,
+(4)
+then Algorithm 5 with parameters ǫ and δ has rounding ratio at least
+1 − exp
+�
+−1 − δ + ǫ + δ
+1 − ǫ
+�
+· 1 − ǫ
+1 + δ.
+Proof. By construction of ˆxi,t, we have that ˆsi,t ≤ θ · (1− ǫ) iff si,t ≤ θ. Define Lt := {i | si,t ≤ θ} =
+{i | ˆsi,t ≤ θ · (1 − ǫ)}. By Lemma 4.4 and the Taylor expansion of 1 − exp(−z), for any offline node
+set S ⊆ [n] and time t, the set of bidders Pt computed by ODRS-core(θ · (1 − ǫ), ˆx, x) satisfies
+Pr[S ∩ Pt ̸= ∅] ≥ 1 −
+�
+B̸⊆Lt
+�
+1 −
+�
+i∈B∩S
+ˆxit
+�
+·
+�
+B⊆Lt
+�
+1 −
+�
+i∈B∩S
+ˆxit
+�
+≥ 1 − exp
+
+−
+�
+i∈S\Lt
+xit · (1 + δ)
+
+ ·
+�
+B⊆Lt
+�
+1 −
+�
+i∈B∩S
+xit · (1 − ǫ)
+�
+.
+12
+
+We now lower bound the ratio Pr[S∩Pt̸=∅]
+�
+i∈S xit . To do so, we note that our lower bound for Pr[S ∩Pt ̸= ∅]
+is a concave function in �
+i∈S xit in the range [0, 1], and so our lower bound for the above ratio
+is minimized by virtually increasing �
+i∈S xit until �
+i∈S xit = 1, by adding all i /∈ S to the set
+and possibly adding dummy nodes J with sj,t = θ for all j ∈ J and �
+J xjt = 1 − �
+i xit ≥ 0
+(see Claim A.5).5 On the other hand, by Fact 4.3, every bin B ⊆ Lt but at most one bin B∗ has
+�
+i∈B ˆxi,t ≥ 1−θ(1−ǫ)
+2
+≥ (1−θ)(1−ǫ)
+2
+, and hence �
+i∈B xi,t ≥ 1−θ
+2 . Combining the above, we get
+Pr[S ∩ Pt ̸= ∅]
+�
+i xit
+≥ 1 − exp
+
+−
+
+1 −
+�
+i̸∈Lt
+xit
+
+ · (1 + δ)
+
+ ·
+�
+B⊆Lt
+�
+1 −
+�
+i∈B
+xit · (1 − ǫ)
+�
+≥ 1 − exp
+�
+−
+�
+1 −
+�
+i∈B∗
+xit
+�
+· (1 + δ)
+�
+·
+�
+1 −
+�
+i∈B∗
+xit · (1 − ǫ)
+�
+≥ 1 − exp
+�
+−1 − δ + ǫ + δ
+1 − ǫ
+�
+· 1 − ǫ
+1 + δ .
+(5)
+Above, the second inequality follows from the lemma’s hypothesis, in Equation (4), and the last
+inequality follows from the following claim, whose proof by inspection of the function g is deferred
+to Appendix C.
+Claim 4.7. If ǫ, δ ∈ [0, 1], the function g(y) := 1−exp (− (1 − y) · (1 + δ))·(1 − y · (1 − ǫ)) satisfies
+∀y ∈ R :
+g(y) ≥ g
+�
+ǫ + δ
+(1 − ǫ)(1 + δ)
+�
+= 1 − exp
+�
+−1 − δ + ǫ + δ
+1 − ǫ
+�
+· 1 − ǫ
+1 + δ .
+Finally, the rounding ratio of the ODRS then follows from Lemma 2.1 and Equation (5).
+Lemma 4.6 provides a sufficient condition for ǫ, δ to (possibly) allow for greater-than-(1 − 1/e)
+rounding ratios, though this condition requires satisfying infinitely many non-linear constraints.
+The following lemma, which follows from convexity of fǫ,δ and is proven in Appendix C, provides a
+simpler condition that is sufficient to guarantee all the above infinitely-many constraints.
+Lemma 4.8. Let fǫ,δ(z) be as in Lemma 4.6. If fǫ,δ(a) ≥ 0 and f ′
+ǫ,δ(a) ≥ 0, then fǫ,δ(b) ≥ 0 ∀b ≥ a.
+Combining the above, we are finally ready to optimize the rounding ratio of Algorithm 5.
+Theorem 4.9. Algorithm 5 run with (ǫ, δ) ≈ (0.0480, 0.0643) achieves a rounding ratio of 0.652.
+Proof. By lemmas 4.6 and 4.8, the rounding ratio of Algorithm 4 with parameters (ǫ, δ) is at least
+as high as 1 − exp
+�
+−1 − δ + ǫ+δ
+1−ǫ
+�
+· 1−ǫ
+1+δ, provided that fǫ,δ
+� 1−θ
+2
+�
+≥ 0 and f ′
+ǫ,δ
+�1−θ
+2
+�
+≥ 0, where
+1−θ
+2
+=
+ǫ
+2(ǫ+δ) and fǫ,δ is as in Lemma 4.6. Thus, the algorithm’s rounding ratio is at least as high as
+the value of the following program, when run with parameters given by its optimal solution (ǫ, δ).
+Off-the-shelf numerical solvers yield a maximum value of α = 0.652 at (ǫ, δ) ≈ (0.0480, 0.0643).
+max
+α = 1 − exp
+�
+−1 − δ + ǫ + δ
+1 − ǫ
+�
+· 1 − ǫ
+1 + δ
+s.t.
+exp
+�
+−
+ǫ
+2(ǫ + δ) · (1 + δ)
+�
+− 1 +
+ǫ
+2(ǫ + δ) · (1 − ǫ) ≥ 0
+− (1 + δ) · exp
+�
+−
+ǫ
+2(ǫ + δ) · (1 + δ)
+�
++ 1 − ǫ ≥ 0
+ǫ, δ ∈ [0, 1].
+5Concretely: adding any value to �
+i∈S xi,t corresponding to adding (part of) another (dummy) element to S
+decreases the value, so concavity of the resulting function together with Claim A.5 completes the proof of this claim.
+13
+
+Computational considerations. The algorithm implied by Theorem 4.9, as stated, takes expo-
+nential time, due to its reliance on the CRS of Lemma 2.1 for non-independent bid distributions.
+As we show in Appendix C.1, a slight downscaling of the x-values allows us to achieve essentially
+the same rounding ratio, while also guaranteeing that the number of bidders at each time step is
+small. This allows us to compute the support of the distribution of Pt in polynomial time, which
+by Lemma 2.1 then gives us the following polytime counterpart to Theorem 4.9.
+Theorem 4.10. For any γ > 0, there exists an nO(1/γ)-time bipartite matching ODRS with rounding
+ratio of 0.652 − γ.
+4.2.2
+The b-matching ODRS
+In this section we extend our ODRS from matchings to b-matchings. To avoid repetition, we only
+outline the changes needed in the algorithm and analysis, and defer the proofs to Appendix C.
+The change to the core algorithm. As we did (implicitly) in Algorithm 4, we let the online
+level-set algorithm dictate the marginal probabilities with which an offline node i should bid to t,
+based on the number of times i has bid by time t, which we denote by Si,t. (This corresponds
+to St in Algorithm 3 run from the perspective of the offline node i.) We process offline nodes at
+different times t as follows, with the choice of probabilities guided by Algorithm 3 to guarantee that
+E[Si,t+1 − Si,t] = xi,t, by Property (P1), and Si,t ∈ [⌊si,t⌋, ⌈si,t⌉], by Property (P2).
+For nodes with ⌈st+1⌉ = ⌈st⌉ — whose fractional degrees both before and after time t lie in the
+range [⌊si,t⌋, ⌈si,t⌉] — we group these offline nodes and have them bid (at most one bid per bin) as
+in Algorithm 4, with the following changes: si,t is replaced by its fractional part, si,t − ⌊si,t⌋: this is
+done both in the bidding probability, so these items have size
+xi,t
+1−(si,t−⌊si,t⌋) when bin packing, and
+in the condition si,t ≤ θ, which are now replaced by the condition si,t − ⌊si,t⌋ ≤ θ (and similarly
+for the condition si,t > θ), with θ to be chosen later. Moreover, we add a candidate offline node
+i ∈ Ct to the list of potential matches Pt (the bidders) if Si,t < ⌊si,t⌋ (corresponding to i ∈ F ∩ Ct
+in Algorithm 4 for simple matchings).
+For all offline nodes with ⌊si,t+1⌋ = ⌊si,t⌋+1, we place these nodes in singleton bins, and we have
+them bid with probability if Si,t = ⌊si,t⌋, or with probability si,t+1−⌊si,t+1⌋
+si,t−⌊si,t⌋
+if Si,t = ⌈si,t⌉ = ⌊si,t+1⌋.
+Change to core ODRS’ analysis: Fact 4.3 and its proof hold unchanged for our extension of
+the core ODRS. Lemma 4.4, too, holds unchanged, although the proof, deferred to Appendix C.2,
+is now slightly more involved. Specifically, the key ingredient, proving that the events [Pt ∩ S ∩ B]
+over different bins B are negatively associated (NA) requires a proof that the variables Si,t are NA.
+Change to the improved ODRS. In our ODRS for b-matchings, since offline nodes i for which
+⌊si,t+1⌋ = ⌊si,t⌋ + 1 are grouped into singelton bins, we wish to apply a markup to these nodes’
+xi,t-values, if xi,t is small. We therefore consider the following assignment, with 0 ≤ θ1 ≤ θ2 ≤ 1
+satisfying θ2 − θ1 =
+δ
+ǫ+δ, to be chosen shortly.
+ˆxi,t =
+� si,t+1
+z=si,t
+(1 − ǫ) · 1 [⌊z⌋ ∈ [θ1, θ2)] + (1 + δ) · 1 [⌊z⌋ ̸∈ [θ1, θ2)] dz.
+(6)
+In words: this assignment decreases the part of xi,t corresponding to the parts of i’s fractional
+degree that are bounded away from 0 and 1 (specifically, those in the range [θ1, θ2), and increases
+the part of the fractional degree that is somewhat close to 0 or 1. Our new ODRS then runs the
+modified core ODRS with inputs θ := θ2 · (1− ǫ) + θ1 · (δ + ǫ) and ˆx, x. The choice of the parameter
+θ is taken to have ˆsi,t − ⌊ˆsi,t⌋ ≤ θ if and only if si,t − ⌊si,t⌋ ≤ θ2.
+14
+
+Change to the improved ODRS’ analysis. First, Fact 4.5 generalizes as follows, arguing that
+modified fractional degrees ˆsi,t = �
+t′<t ˆxi,t′ fall within the same integer range for all (i, t).
+Fact 4.11. For every offline node i ∈ [n] and time t, we have that ⌊ˆsi,t⌋ = ⌊si,t⌋ and ⌈ˆsi,t⌉ = ⌈si,t⌉.
+Fact 4.11 and Property (P2) of Algorithm 3 imply that offline node i makes some Si,t ∈
+[⌊si,t⌋, ⌈si,t⌉] many bids by time t. As i is not matched more time than it bids, this implies that the
+extended ODRS outputs a valid b-matching.
+Next, the key lemma in the analysis of Algorithm 5, namely Lemma 4.6, needs to be extended
+slightly, and in particular the condition of Equation (4) needs to hold for all z ≥ max
+�
+1−θ2
+2(1+δ), θ1
+�
+.
+To make this constraint as less restrictive as possible, we take θ1 and θ2 to guarantee the needed
+θ2 − θ1 =
+δ
+ǫ+δ, and also satisfy
+1−θ2
+2(1+δ) = θ1, resulting in θ1 =
+ǫ
+(3+2δ)(ǫ+δ) and θ2 =
+ǫ+3δ+2δ2
+(3+2δ)(ǫ+δ). This
+modification of the range for which we require Condition (4) to hold will again allow us to argue that
+for all bins B except for at most one bin B∗ (to be characterized shortly), we have the following:
+1 −
+�
+i∈B
+ˆxi,t ≤ exp
+�
+−
+�
+i∈B
+xi,t(1 + δ)
+�
+(7)
+Specifically, for bins containing a single offline node i with ⌊si,t+1⌋ = ⌊si,t⌋+1, we have the following:
+If si,t+1 − ⌊si,t+1⌋ ≥ θ1, then
+xi,t = si,t+1 − si,t ≥ si,t+1 − ⌊si,t+1⌋ ≥ θ1.
+Similarly, if si,t − ⌊si,t+1⌋ ≤ θ2, then
+xi,t = si,t+1 − si,t ≥ ⌊si,t+1⌋ − si,t = 1 + ⌊si,t⌋ − si,t ≥ 1 − θ2 ≥
+1 − θ2
+2(1 + δ).
+In both cases, Condition (4) holding for all z ≥ max
+�
+1−θ2
+2(1+δ), θ1
+�
+implies Equation (7). In the
+alternative scenario where si,t+1 −⌊si,t+1⌋ ≤ θ1 and si,t−⌊si,t⌋ ≤ θ2, we have that ˆxi,t = (1+δ)·xi,t,
+and Equation (7) follows from the basic inequality 1 − z ≤ exp(−z). Finally, by Fact 4.3, for all
+bins B containing offline nodes i with ⌊si,t⌋ = ⌊si,t+1⌋ and ˆsi,t ≤ θ (i.e., si,t ≤ θ2), except for at
+most one bin B∗, we have that
+�
+i∈B
+xi,t · (1 + δ) ≥
+�
+i∈B
+ˆxi,t ≥ 1 − θ
+2
+≥ 1 − θ2
+2
+,
+where the last inequality, whereby θ ≤ θ2, is proven similarly to Fact 4.11.
+So, Condition (4)
+holding for all z ≥
+1−θ2
+2(1+δ) implies Equation (7) for all remaining bins other than B∗. The analysis
+then proceeds as in Lemma 4.6, yielding the following.
+Lemma 4.12. If ǫ, δ ∈ [0, 1] guarantee that for all z ≥ max
+�
+1−θ2
+2 , θ1
+�
+=
+ǫ
+(3+2δ)·(ǫ+δ),
+fǫ,δ(z) := exp(−z · (1 + δ)) − (1 − z · (1 − ǫ)) ≥ 0,
+(8)
+then the extension of Algorithm 5 to b-matchings with parameters ǫ and δ has rounding ratio at least
+1 − exp
+�
+−1 − δ + ǫ + δ
+1 − ǫ
+�
+· 1 − ǫ
+1 + δ.
+Combined with Lemma 4.8, the above lemma then yields our b-matching ODRS’ rounding ratio,
+after solving the appropriate modification of the mathematical program used to prove Theorem 4.9.
+Theorem 4.13. The extension of Algorithm 5 to b-matchings, run with (ǫ, δ) ≈ (0.0347, 0.0425),
+achieves a rounding ratio of 0.646.
+15
+
+5
+Applications
+In this section we provide applications and extensions of our online rounding algorithms. Mirroring
+the exposition in Section 1.1.1, we start with the application of our online (b-)matching algorithms,
+omitting the edge-weighted matching application with ML predictions, which follows directly from
+our main result.
+5.1
+Applications and extensions of the matching ODRS
+5.1.1
+Online edge coloring (multigraphs)
+We show here how to obtain the first online algorithm that achieves a better-than-2 competitive
+factor for online edge-coloring in bipartite multigraphs under adversarial arrivals, specifically under
+one-sided node arrivals, as studied in [27]. We assume that upon arrival of each node, its neighboring
+edges are revealed and must be colored. As already mentioned, for online edge coloring, it is known
+that the greedy algorithm’s competitive ratio of 2 is tight for graphs of low maximum degree
+∆ = O(log n) [8], while better competitive ratios are known for high-degree graphs under various
+arrival models, all for simple graphs [6, 10, 27, 55, 65], with the exception of the random-order result
+of [3].
+We obtain our improved bounds for multigraphs by extending a reduction from online edge
+coloring to “fair” matching, suggested by [27] for simple graphs (see also Saberi and Wajc [65,
+Appendix B]), given in Lemma 5.2.
+Definition 5.1. We say that a random matching M in a multigraph with maximum degree ∆ is
+α-fair if for each (parallel) edge e we have that Pr[e ∈ M] ≥
+1
+α∆.
+An α-fair algorithm with α = 1 gives the highest uniform lower bound on edges’ matching
+probability. An α-fair algorithm with other α (α > 1) therefore α-approximates this maximally-fair
+algorithm.
+With the above terminology, we can now state the reduction.
+Lemma 5.2. An online α-fair matching algorithm yields an online (α + o(1))∆-edge-coloring algo-
+rithm for n-node bipartite multigraphs with maximum degree ∆ = ω(log n).
+We defer a brief proof sketch of the above to the end of the section, and turn to discussing its
+consequences and converses.
+Corollary 5.3. An ODRS with rounding ratio 1/α yields an online (α + o(1))∆-edge-coloring
+algorithm for n-node multigraphs with maximum degree ∆ = ω(log n).
+Proof. Consider the fractional matching x assigning value xe = κ(e)
+∆
+to the simple edge e with κ(e)
+parallel copies in the multigraph.
+(Note that �
+e∋v κ(e) ≤ ∆ by definition, so this is indeed a
+fractional matching.) Applying an ODRS with rounding ratio 1/α to this fractional matching x
+yields a randomized matching M that matches every (parallel) edge in the graph with probability
+1
+α∆, i.e., it yields an α-fair matching.
+We note that there is in fact an equivalence between α-fair matching algorithms and α-competitive
+edge-coloring algorithms, as “the converse” also holds.
+Observation 5.4. An online α∆-edge-coloring algorithm A for n-node multigraphs with maximum
+degree ∆ yields an ODRS with rounding ratio 1/α for fractional matchings x with xe · ∆ integral
+for all e.
+16
+
+Proof. Given a fractional matching x ∈ QE in a simple graph G = (V, E), using that xe·∆ is integral
+for each edge e, we provide to A a multigraph H = (V, F) with each edge e having multiplicity
+κ(e) = xe · ∆ in H, and then randomly sample one of the α∆ edge colors (matchings) M computed
+by A. This yields an ODRS with rounding ratio of 1/α, since for each edge e ∈ E we have that
+Pr[e ∈ M] = κ(e) ·
+1
+α∆ = xe
+α .
+By the equivalence between edge coloring multigraphs and ODRSes for matchings, together
+with our (upper and lower) bounds on the rounding ratio of such ODRSes, we obtain the following
+bounds on the number of colors needed to edge-color multigraphs online.
+Theorem 5.5. In n-node bipartite multigraphs with maximum degree ∆ under adversarial one-sided
+vertex arrivals, there exists an online 1.533∆-edge-coloring algorithm, assuming ∆ = ω(log n). In
+contrast, no (1/(2
+√
+2 − 2 + ǫ))∆ ≈ (1.207 − O(ǫ))∆-edge-coloring exists even for ∆ = 2.
+We note in passing that all known positive results for online edge coloring under adversarial
+arrivals follow from Lemma 5.2, and indeed from Corollary 5.3 [27, 55, 65]. The reason most of
+these results do not extend to multigraphs is that their ODRSes required their input fractional
+matchings x to be of the form xe = 1
+∆, or more generally for xe = o(1) for each edge e (including its
+multiplicities). The work of Saberi and Wajc [65] is a lone exception to this rule, and combining its
+ODRS with Lemma 5.2 also yields an algorithm for multigraphs, but using as many as 1.9∆ colors.
+We conclude the section with a brief proof sketch of Lemma 5.2.
+Proof sketch of Lemma 5.2. We describe the algorithm in an offline setting first. Let C be such
+that C = ω(log n) and C = o(∆). (Such C exist, by our necessary assumption that ∆ = ω(log n).)
+Initially, the entire multigraph is uncolored. For some ∆/C many rounds, compute α · C colors
+as follows: In the uncolored subgraph U, compute α · C many α-fair matchings M1, . . . , MαC,
+and let these occupy the next αC colors. Since every vertex has at most αC = o(∆) many edges
+colored during a round, any vertex with degree close to ∆ at the beginning of the round has degree
+∆ − o(∆) by the end of the round. Therefore, since these Mi are α-fair matchings, and so match
+each (parallel) edge with probability at least
+1
+α·∆, by standard concentration bounds, all vertices
+that have degree close to ∆ have some C · (1 − o(1)) many edges colored during a round (with
+high probability). This allows us to compute a high-probability upper bound of on the resultant
+uncolored multigraph at the end of the round (possibly useful to compute α-fair matchings), and
+moreover implies that after ∆/C many rounds (and so with (∆/C) · α · C = α · ∆ colors used), the
+uncolored multigraph has maximum degree ∆ − (∆/C) · C · (1 − o(1)) = o(∆), and can be colored
+greedily using a further 2 · o(∆) = o(∆) many colors, for (α + o(1))∆ colors overall.
+While the above algorithm was described in an offline setting, if one can compute an α-fair
+algorithm online (possibly with information regarding the high-probability upper bound on ∆ in each
+round), then the entire algorithm can be made to run online: for each (vertex/edge) arrival, perform
+the next steps of the α∆ many α-fair matching algorithms outputting color classes M1, . . . , Mα∆,
+where for the i-th such algorithm, we simulate only the arrival of the parts of the subgraph that
+are not contained M1, . . . , Mi−1. The same approach is used to simulate the final greedy coloring
+steps, using the fact that the greedy (2∆ − 1)-edge-coloring algorithm is an online algorithm.
+5.1.2
+Stochastic extension
+In this section, we generalize our matching ODRS and its analysis to the following stochastic online
+bipartite matching problem. At each time t, an online node arrives with weight vector ⃗wt ∼ Dt,
+17
+
+where D1, D2, . . . are a priori known independent distributions. A simple example which we focus
+on for notational simplicity (though our approach generalizes to this problem in full generality)
+is as follows: each online node t arrives according to a Bernoulli event At ∼ Ber(pt), each edge
+(i, t) having some intrinsic value wit, but realized weight vit = wit · At. This problem (in its full
+generality) is an online Bayesian selection problem, and a number of algorithms with competitive
+ratio of 1
+2 are known for it [35, 38, 42]. By the classic lower bound of Krengel and Sucheston for the
+single-item prophet inequality problem [54], the above ratio is best-possible when comparing with
+the offline optimum algorithm.
+In [62], Papadimitriou et al. initiate the study of this problem in terms of the (polytime) approx-
+imability of the optimum online algorithm. Perhaps surprisingly, they show that while for simple
+cases of the problem (e.g., the single-offline-node problem), the optimum online algorithm is com-
+putable using a simple poly-sized dynamic program, in general it is pspace-hard to approximate the
+optimum online algorithm within some absolute constant α < 1 (say, α ≈ 0.999999), and showed
+that a better-than-0.5 approximation of the optimum online algorithm can be obtained by polytime
+online algorithms. This positive result was subsequently improved to 0.526 implicitly [65] and to
+1 − 1/e ≈ 0.632 explicitly [16]. All these results are achieved by rounding the following LP, which
+can be shown to upper bound the optimum online algorithm’s value [62, 71].
+max
+�
+i,t
+wi,t · xi,t
+(LP OPTon)
+s.t.
+�
+t,r
+xi,t ≤ 1
+∀i ∈ [n]
+(9)
+�
+i
+xi,t ≤ pt
+∀t
+(10)
+xi,t ≤ pt ·
+�
+1 −
+�
+t′<t
+xi,t′
+�
+∀i, t
+(11)
+xi,t ≥ 0
+∀i, t.
+(12)
+Lemma 5.6. The optimum of the above LP upper bounds the expected gain of any online algorithm.
+Most of these constraints are evident, and follow from (expected) matching constraints, which
+apply also to offline algorithms. The one constraint which applies to online algorithms but not to
+offline ones is Constraint (11). This constraint follows from the fact that if offline node i is matched
+to t, then t has already arrived, and that i is unmatched before time t — two events which are
+independent for online algorithms.
+We note that in their paper’s full version, Papadimitriou et al. [62] (private communication) show
+that the above LP has a “gap” of 1− 1
+2e, in the sense that no online algorithm (even computationally-
+unbounded ones) can get a better-than-(1 − 1
+2e) fraction of this LP’s value. So, an approximation
+ratio greater than 1− 1
+2e is unachievable using this LP, and a ratio of 1− 1
+e is achievable [16]. Is the
+latter bound tight? In what follows, who show how the approach underlying our ODRSes, combined
+with (and simplifying) the high-level analysis of [16], allows one to break this bound.
+ODRS for the stochastic setting. As in the non-stochastic setting of Section 4, specifically
+Algorithm 4, we group offline nodes and have them coordinate a single bid per group (bin). Similarly,
+our stochastic counterpart to Algorithm 5 applies group discounting and individual markup to the
+target marginal bidding probability, and then calls the (soon-to-be) modified Algorithm 4 with
+18
+
+parameter θ · (1 − ǫ) and vector ˆxi,t given by Equation (3), restated below for ease of reference.
+ˆxi,t = xi,t · (1 − ǫ) +
+� max(θ, si,t+xi,t)
+z=max(θ,si,t)
+(ǫ + δ) dz.
+The three main differences compared to Section 4 are as follows: First, to account for the probability
+pt of t to arrive, the conditional probability for i to bid if i is free and t arrives used in Line 6-Line 12
+is now
+ˆxi,t
+pt·(1−ˆsi,t), for ˆsi,t := �
+t′<t ˆxi,t. Second, we allow offline nodes to bid more than once, and
+denote them as free (i.e., belonging to F) as long as they are not matched. This avoids offline nodes
+becoming positively correlated due to an unlikely online node’s arrival. The offline nodes’ matching
+constraints are trivially satisfied. Finally, satisfying the online nodes’ matching constraints is even
+easier than in Algorithm 4 for the non-stochastic setting. Rather than using a CRS, we have an
+arriving online node t that receives bids simply greedily match to the bidding node i maximizing
+the value wi,t. See the pseudocode in Algorithm 6.
+Algorithm 6 Stochastic Matching Algorithm
+1: M ← ∅
+2: x ← solution to (LP OPTon)
+3: for each time t do
+4:
+for each i ∈ [n] do
+5:
+compute ˆxi,t as in Equation (3), and ˆsi,t = �
+t′<t ˆxi,t.
+6:
+Bt ← FirstFit
+���
+i,
+ˆxi,t
+pt·(1−ˆsi,t)
+� ��� i ∈ [n], si,t ≤ θ
+��
+⊲ Bucketing high-degree nodes
+7:
+Bt ← Bt ∪ {{i} | i ∈ [n], si,t > θ}
+⊲ Bucketing low-degree nodes
+8:
+Ct ← ∅
+⊲ Candidates
+9:
+for each B ∈ Bt do
+10:
+U ∼ Uni[0, 1]
+11:
+if U ≤ �
+i∈B
+ˆxi,t
+pt·(1−ˆsi,t) then
+12:
+Ct ← Ct ∪ mini
+�
+i ∈ [k]
+��� U ≤ �
+j∈B, j≤i
+ˆxj,t
+pt·(1−ˆsj,t)
+�
+13:
+Pt ← Ct \ V (M)
+⊲ Bidders = free candidates
+14:
+if Pt ̸= ∅ and t arrived then
+15:
+pick some i ∈ arg max{wi,t | i ∈ Pt}
+16:
+M ← M ∪ {(i, t)}
+17: Output M
+Fact 5.7. Algorithm 6 is well-defined. In particular,
+ˆxi,t
+pt·(1−ˆsi,t) ≤ 1 for every pair (i, t).
+Proof. By Fact 4.5, we have that ˆsi,t = si,t · (1 − ǫ) + (si,t − θ)+ · (ǫ + δ) ≤ si,t. Therefore, by our
+choice of θ =
+δ
+ǫ+δ and LP Constraint (11), we have that, indeed
+ˆxi,t
+pt · (1 − ˆsi,t) ≤
+� xi,t·(1−ǫ)
+pt·(1−si,t) ≤
+xi,t
+pt·(1−si,t) ≤ 1
+si,t ≤ θ
+xi,t·(1+δ)
+pt·(1−si,t(1+δ)+δ) =
+xi,t
+pt·(1−si,t)) ≤ 1
+si,t > θ.
+Algorithm 6 is clearly a polytime algorithm. It remains to analyze its approximation ratio.
+The following lemma makes explicit a fact implicit in the proof of the main theorem of [16]:
+specifically, that a per-online-node “approximation ratio”, relating w(M(t), t)—the weight obtained
+by matching t—to the LP solution’s value from t, implies a global approximation ratio of the same
+value. We provide a short self-contained proof of this fact for completeness.
+19
+
+Lemma 5.8. If for each online node t and z ≥ 0 we have that Pr[w(M(t), t) ≥ z] ≥ �
+i:wi,t≥z α·xi,t,
+then the algorithm is an α-approximation of the optimum online algorithm.
+Proof. Denote by wt := w(M(t), t) the weight of the matched edge of t, with wt = 0 if t is not
+matched (possibly due to non-arrival). Then, the weight of the output matching M satisfies
+�
+t
+E[wt] =
+�
+t
+� ∞
+z=0
+Pr[wt ≥ z] dz ≥
+�
+t
+� ∞
+z=0
+α ·
+�
+i:wi,t≥z
+xi,t dz = α ·
+�
+i,t
+wi,t · xi,t.
+The proof is then completed by Lemma 5.6, implying that the RHS upper bounds α times the value
+of the optimum online algorithm’s gain.
+The objective of our analysis will therefore be to lower bound Pr[w(M(t)) ≥ z] for all z ≥ 0.
+In our analysis, we let Ft = [n] \ V (M) be the set of free offline nodes at time t, and let
+Fi,t = 1[i ∈ Ft] = 1 − �
+t′<t 1[(i, t′) ∈ M] indicate whether i is free by time t. Similarly, we let
+Ei,t = 1 − Fi,t indicate whether i was matched (“engaged”) before time t.
+Our first observation is that our modification to the notion of freedom results in the following
+lower bound on the probability of a node being free at time t.
+Observation 5.9. For each time t and offline i ∈ [n], we have that Pr[Fi,t] ≥ 1 − ˆsi,t.
+Proof. Let B1
+i,t indicate that i bid for the first before time t.
+We prove that E[B1
+i,t] = ˆsi,t, by
+induction on t ≥ 1. The base case, is trivial, as Fi,1 ≡ 1 and ˆsi,1 ≡ 0. Now, by induction, we have
+E[B1
+i,t+1 − B1
+i,t] = pt ·
+ˆxi,t
+pt · (1 − ˆsi,t) · (1 − E[B1
+i,t]) = ˆxi,t,
+where the first equality relies on independence of t’s arrival, i becoming a candidate at time t, and
+i not having yet bid, and the second equality follows from the inductive hypothesis. By linearity,
+we then have that E[B1
+i,t+1] = ˆsi,t+1. Since 1 − Fi,t = Ei,t ≤ B1
+i,t, the claim follows.
+Next, for any time t and offline node set S ⊆ [n], we let FS,t = �
+i∈S Fi,t and ES,t = �
+i∈S Ei,t
+indicate whether all of S is free (respectively, engaged) by time t. So far, we have established that
+Pr[Ei,t] ≤ ˆsi,t. The following lemma, whose statement and proof mirror that of [16, Lemma 3.3] (who
+use xi,t and si,t, as opposed to ˆxi,t and ˆsi,t), asserts that this upper bound is “sub-multiplicative”.
+Lemma 5.10. For each time t and set of offline nodes S ⊆ [n], we have that
+Pr[ES,t] ≤
+�
+i∈S
+ˆsi,t.
+To prove the above, we need the following claim from [16]. For completeness, we provide a short
+probabilistic proof of this lemma (see [16, Lemma 3.2] for a longer algebraic proof).
+Lemma 5.11. For all i ∈ [n], let qi ∈ [0, 1]. Then,
+�
+A⊆S
+Pr[EA,t, FS\A,t] ·
+�
+i∈S\A
+qi =
+�
+A⊆S
+Pr[EA,t] ·
+�
+i∈A
+(1 − qi)
+�
+i∈S\A
+qi.
+Proof. Consider a probability space where at time t all offline nodes toss a coin with success prob-
+ability qi, independently. The LHS corresponds to all free nodes having their coin come up heads.
+The RHS corresponds to the same event, i.e., all nodes whose coin came up tails were engaged.
+20
+
+Proof of Lemma 5.10. The proof is by induction on t ≥ 1 for all sets S. The base case t = 1, for
+which sit ≡ 0 for all i, is trivial. For the inductive step, denote the marginal probability of i bidding
+at time t (requiring t’s arrival) by
+mi,t := Pr[i ∈ Pt] =
+ˆxi,t
+1 − ˆsi,t
+.
+We note that with this terminology, we have that
+ˆsi,t+1 = ˆsi,t + ˆxi,t = ˆsi,t + mi,t · (1 − ˆsi,t) = ˆsi,t · (1 − mit) + mit.
+(13)
+The inductive step follows by noting that at most one of the nodes i ∈ S may be matched at time
+t, and this requires that the single free node i ∈ S bid for t, which happens with probability mi,t
+(conditioned on any event determined by randomness up to time t that implies that the node is
+free), as follows.
+Pr[ES,t+1] ≤
+�
+A⊆S
+|S\A|≤1
+Pr[EA,t, FS\A,t] ·
+�
+i∈S\A
+mi,t
+=
+�
+A⊆S
+|S\A|≤1
+Pr[EA,t] ·
+�
+i∈A
+(1 − mi,t)
+�
+i∈S\A
+mi,t
+Lem 5.11
+≤
+�
+A⊆S
+Pr[EA,t] ·
+�
+i∈A
+(1 − mi,t)
+�
+i∈S\A
+mi,t
+≤
+�
+A⊆S
+�
+i∈A
+Ui,t ·
+�
+i∈A
+(1 − mi,t)
+�
+i∈S\A
+mi,t
+I.H.
+=
+�
+i∈S
+(ˆsi,t · (1 − mit) + mit)
+binomial expansion
+≤
+�
+i∈S
+ˆsi,t+1
+Eqn. (13).
+The preceding lemma allows us to prove the following counterpart to Lemma 4.4, key to our
+improved approximation ratios.
+Lemma 5.12. For any time t and set of offline nodes S ⊆ [n], Algorithm 6 satisfies
+Pr[S ∩ Pt ̸= ∅] ≥ 1 −
+�
+B∈Bt
+�
+1 −
+�
+i∈B∩S
+ˆxit/pt
+�
+.
+Proof. Let Ct and Pt be as in Algorithm 6. Since Pt = Ct \{i | Ei,t = 1}, there are no bids from set
+S (S ∩ Pt = ∅) iff all i ∈ A = Ct ∩ S are no longer free, i.e., iff EA,t holds. So, by total probability
+21
+
+over Ct ∩ S, we have that (⋆) := Pr[t arrives and S ∩ Pt = ∅] satisfies
+(⋆) =
+�
+A⊆S
+Pr[Ct ∩ S = A] · Pr[EA,t]
+=
+�
+A⊆S
+�
+B:|B∩A|=0
+Pr[Ct ∩ B ∩ A = ∅]
+�
+B:|B∩A|=1
+�
+i∈B∩A
+Pr[Ct ∩ B ∩ A = {i}] · Pr[EA,t]
+≤
+�
+A⊆S
+�
+B:|B∩A|=0
+Pr[Ct ∩ B ∩ A = ∅]
+�
+B:|B∩A|=1
+�
+i∈B∩A
+Pr[Ct ∩ B ∩ A = {i}] · Pr[Ei,t]
+=
+�
+B∈Bt
+�
+Pr[Ct ∩ B ∩ S = ∅] +
+�
+i∈B∩S
+Pr[Ct ∩ B = {i}] · Pr[Ei,t]
+�
+=
+�
+B∈Bt
+�
+1 −
+�
+i∈B∩S
+Pr[Ct ∩ B = {i}] · Pr[Fi,t]
+�
+≤
+�
+B∈B
+�
+1 −
+�
+i∈B∩S
+ˆxi,t/pt
+�
+.
+Above, the second and third equality follow from independence of the sets {Ct∩B | B ∈ Bt} and from
+|Ct ∩B| ≤ 1 always. The first inequality follows from Lemma 5.10. The fourth equality follows from
+|Ct∩B| ≤ 1, and Fi,t = 1−Ei,t. Finally, the last inequality follows from Pr[Ct∩B = {i}] =
+ˆxi,t
+pt·(1−ˆsi,t)
+and Observation 5.9, implying that Pr[Ct ∩ B = {i}] · Pr[Fi,t] ≥
+ˆxi,t
+pt·(1−ˆsi,t) · (1 − ˆsi,t) ≥ ˆxi,t/pt.
+The following lemma, which is a counterpart to Lemma 4.6 for the non-stochastic setting, allows
+us to obtain a concrete lower bound from Lemma 5.12.
+Lemma 5.13. If ǫ, δ ∈ [0, 1] guarantee that for all z ≥ 1−θ
+2
+=
+ǫ
+2(ǫ+δ),
+fǫ,δ(z) := exp(−z · (1 + δ)) − (1 − z · (1 − ǫ)) ≥ 0,
+(14)
+then Algorithm 6 with parameters ǫ and δ satisfies for each time t and set S ⊆ [n]
+Pr[S ∩ Pt ̸= ∅] ≥
+�
+1 − exp
+�
+−1 − δ + ǫ + δ
+1 − ǫ
+�
+· 1 − ǫ
+1 + δ
+�
+·
+�
+i∈S
+xi,t/pt.
+The proof is near identical to that of Lemma 4.6, and is therefore omitted.6
+Finally, the same optimization over ǫ, δ as in Section 4 yields our main result of this section.
+Theorem 5.14. Algorithm 6 with parameters (ǫ, δ) ≈ (0.0480, 0.0643) is a polytime online stochastic
+weighted bipartite matching algorithm that 0.652-approximates the optimum online algorithm.
+Proof. Let α = 1−exp
+�
+−1 − δ + ǫ+δ
+1−ǫ
+�
+· 1−ǫ
+1+δ ≥ 0.652. By our algorithm’s greedy matching choice, t
+gets matched to a node i with wi,t ≥ w iff t arrives and gets a bid from the set St,w := {i | wi,t ≥ w}.
+So, by Lemma 5.13, as (ǫ, δ) above satisfy Equation (14) for all z ≥ 0 (see the proof of Theorem 4.9),
+Pr[w(M(t), t) ≥ w] = Pr[t arrives ∧ Pt ∩ St,w ̸= ∅] ≥ pt · α ·
+�
+i|wi,t≥w
+xi,t/pt = α ·
+�
+i|wi,t≥w
+xi,t.
+The approximation ratio of Algorithm 6 then follows from Lemma 5.8, while the algorithm’s poly-
+nomial running time is immediate.
+6The only changes to the proof are (1) the syntactic generalization of Fact 4.3 implies that all but one bin B ∈ Bt
+computed in Line 6 has �
+i ˆxi,t/pt ≥ 1−θ
+2 , and (2) the convexity argument requires that we add dummy nodes J
+with sj,t = θ for all j ∈ J and �
+j∈J xj,t = pt − �
+i xi,t, where this sum is non-negative, by LP Constraint 10.
+22
+
+5.2
+Application of the level-set ODRS
+We now turn to discussing applications of our level-set algorithm, and its strong negative correlation
+properties.
+5.2.1
+Multi-stage stochastic optimization
+We consider an application to multi-stage stochastic optimization, building on the framework of [67].
+We recall briefly that in such multi-stage problems, various parameters of an optimization problem
+materialize stochastically over some number k of stages, where each stage contains stochastic in-
+formation about the following stage(s) and wherein we can take actions—with actions cheaper in
+earlier stages, but possibly inaccurate since we only know the future stochastically.
+Motivated by the results of [69], a broad class of multi-stage stochastic covering problems can
+be cast as the following family of online problems: see Section 2 of [67], from which we quote the
+framework almost verbatim next. There is a hidden covering problem “minimize cT · x subject to
+Ax ≥ b and with all variables in x being non-negative integers” that is revealed online as follows.We
+let m denote the number of rows of A; also, the variables in x are indexed as xj,ℓ, where 1 ≤ j ≤ n
+and 1 ≤ ℓ ≤ k. (We interchange m and n from [67].) Such a covering problem, as well as a feasible
+fractional solution x∗ for it (where each x∗
+j,ℓ is allowed to be a non-negative real), are revealed to us
+in k stages as follows. In each stage ℓ where 1 ≤ ℓ ≤ k, we are given the ℓth-stage fractional values
+(x∗
+j,ℓ :
+1 ≤ j ≤ n) of the variables along with their respective columns in the coefficient matrix
+A, and their respective coefficients in the objective-function vector c. Given this setup, we need to
+round the variables (x∗
+j,ℓ : 1 ≤ j ≤ n) immediately and irrevocably at stage ℓ, using randomization
+if necessary. (Note the direct connection to online computation.) The goal is to develop such a
+rounded vector (sj,ℓ :
+1 ≤ j ≤ n, 1 ≤ ℓ ≤ k) that satisfies the constraints Ay ≥ b, and whose
+(expected) approximation ratio E[cT · y]/cT · x∗ is small, where the expectation is only over any
+random choices made by our online algorithm in this arrival model. This completes the description
+of the framework from [67].
+We make two contributions in this framework. A motivating family of problems to consider is
+vertex-(multi-)cover on graphs and hypergraphs. The multi-stage stochastic version of the classical
+vertex-cover problem on a given graph G = (V, E) is basically as follows: the coverage constraint
+corresponding to an edge e = (u, v) ∈ E is now
+� k
+�
+ℓ=1
+xu,ℓ
+�
++
+� k
+�
+ℓ=1
+xv,ℓ
+�
+≥ 1.
+(15)
+Improving on the 2k-approximation of Swamy and Shmoys [69] for this problem—in the “online
+rounding” model mentioned in the previous paragraph—a 2-approximation is developed in [67].
+Our first contribution is the following generalization of the 2-approximation for multi-stage
+stochastic vertex cover from [67]. Consider the following more-general problem in d-regular hy-
+pergraphs H = (V, E) (or hypergraphs with all edge-sizes lower bounded by d): for some integer
+parameter t, we want the vertices in each edge e ∈ E to be covered in total to an extent of at
+least t, where vertices can be multi-covered.
+That is, generalizing (15), the coverage constraint
+corresponding to an edge e = {v1, v2, . . . , vd} ∈ E is
+d
+�
+i=1
+k
+�
+ℓ=1
+xvi,ℓ ≥ t,
+(16)
+where each xvi,ℓ is allowed to be any non-negative integer; note that the above-seen stochastic vertex
+cover is the special case where d = 2 and t = 1. Recalling our online arrival model, we need to
+23
+
+round the LP solution (x∗
+v,ℓ : v ∈ V ) right away at stage ℓ, for each ℓ ∈ [k]. We proceed as follows
+to obtain an α := (d + t − 1)/t–approximation in this model, generalizing the 2-approximation for
+d = 2 and t = 1. In particular, if t is large, this is essentially a 1-approximation.
+Our algorithm works as follows. For each vertex v ∈ V , consider the scaled vector z(v) :=
+(α · x∗
+v,ℓ : ℓ ∈ [k]) and independently for all v, run Algorithm 3 on z(v) to obtain the final rounded
+values Xv,ℓ : ℓ ∈ [k]). (As usual, we imagine padding this vector with a dummy final element so
+that the sum of the entries is an integer.) We first verify that (16) is satisfied with probability one
+for each e = {v1, v2, . . . , vd} ∈ E:
+d
+�
+i=1
+k
+�
+ℓ=1
+Xvi,ℓ ≥
+d
+�
+i=1
+� k
+�
+ℓ=1
+α · x∗
+vi,ℓ
+�
+(by Property (P2))
+(17)
+>
+d
+�
+i=1
+�� k
+�
+ℓ=1
+α · x∗
+vi,ℓ
+�
+− 1
+�
+(⌊β⌋ > β − 1 for all β)
+(18)
+≥ α · t − d
+(since x∗ satisfies (16))
+= t − 1,
+which implies that �d
+i=1
+�k
+ℓ=1 Xvi,ℓ ≥ t, as required, since the LHS here is an integer and because
+of the strict inequality connecting (17) and (18). We thus obtain an online solution to the rounding
+problem which satisfies all constraints with probability one, and whose expected objective function
+value is at most α(cT · x∗). We observe that the work of [67] also runs an online rounding algo-
+rithm, but that the problem considered there is much simpler since all we require there (in place of
+Property (P2)) is that if the entries of the given input vector sum to at least one, then at least one
+entry is rounded to one with probability one. In particular, this does not address the multi-coverage
+constraint we satisfy here.
+Our second contribution is that because of Property (P3), the objective function is strongly
+concentrated around its mean (given by the Chernoff bound). This is an especially useful property
+in areas like stochastic optimization, where the algorithm can be run only once. This is in comparison
+to offline minimization problems with a non-negative objective where we can repeat the algorithm
+O(1/ǫ) times and choose the best solution obtained, and apply Markov’s inequality to each iteration
+to show that the best solution is at most (1 + ǫ) times the expected value with high probability.
+Thus, the fact that Algorithm 3 yields sharp concentration as implied by Property (P3)—and does
+not just guarantee Properties (P1) and (P2)—is very useful in the stochastic-optimization context.
+Summarizing the preceding discussion (and formalizing the preceding paragraph), we obtain the
+following result.
+Theorem 5.15. For any k ≥ 1, there exists a k-stage stochastic hypergraph vertex cover algorithm
+for d-regular hypergraphs where each hyper-edge must be covered t times, with approximation ratio
+that is α = d+t−1
+t
+in expectation and is α(1 + o(1)) w.h.p. if costs are polynomially bounded and
+OPT = ω(log n).
+Proof. Our algorithm satisfies the (multi-)coverage constraint, by the above, and has expected ap-
+proximation ratio α by construction. The high-probability bound follows from standard Chernoff
+bounds for negatively-associated variables, relying on Property (P3) and closure of NA under prod-
+ucts (Lemma 2.5), relevant since we run independent copies of Algorithm 3 for each vertex.
+5.2.2
+Negative association, fairness, diversity, and streaming
+While Section 5.2.1 only required the fact that Property (P3) yields sharp tail bounds for sums of
+the Xi with non-negative weights, we go further now and use the negative association guaranteed by
+24
+
+Property (P3), which helps us correlate multiple such sums as well as handle submodular objectives
+well. The submodular-objective application further benefits from the fact that Algorithm 3 can
+also be viewed as a data-stream algorithm: note that at when xt arrives, the algorithm need only
+remember st−1 and St, and hence can be implemented with O(log n) space.
+Intersectionality is a key notion in the study of fairness, where the intersection of multiple
+attributes (e.g., race, gender, age) can impact the outcomes of a person in a more fine-grained
+manner than does standard group-fairness (where we typically take a single attribute). Its study
+has naturally impacted AI/ML fairness as well—see, e.g., [18] and its followups—and is a topic
+of much debate in terms of precise definitions (see, e.g., [53]). We make a contribution to this
+nascent area, while being aware that many such formulations/contributions in the still-early stages
+of research in AI/ML fairness are speculative.
+Consider people arriving online and requesting a resource whose availability expands over time:
+e.g., vaccines for a new pathogen or a popular new model of car. It is anticipated that by time t, at
+most at units of this resource will be available. An ML system that adapts to the changing realities
+on the ground (e.g., how the pathogen is spreading and which communities seem to be impacted the
+most) decides online, the probability xt with which to allocate the resource to the request arriving
+at time t; we naturally require �t
+i=1 xi ≤ at for all t. We respond to this in real time by constructing
+Xt ∈ {0, 1} using Algorithm 3; Property (P2) ensures that we never exceed the supply-limits at.
+(In reality, a batch of requests will arrive at time t, which of course Algorithm 3 naturally extends
+to.) The ML system outputting the vector x may be (required to be) unware of protected attributes
+that characterize our underlying demographic groups and their intersections: given x, can we show
+that while intersectional unfairness may be present in x (in which case the ML system needs to be
+refined), our rounding algorithm tries to at least limit the scope of such intersectional unfairness in
+some way?
+We formulate this problem as follows. Suppose each person requesting the resource has three
+attributes A1, A2, and A3: the “three" here is just for simplicity, and it is easy to see that the
+framework below generalizes to any number of attributes. Suppose U1, U2, . . . , Uα is some given
+partition of the population according to their values for A1 (e.g., if A1 is age, the Ui’s may be
+disjoint age-intervals). Similarly, suppose we have a partition of the population according to A2
+as V1, V2, . . . , Vβ and according to A3 as W1, W2, . . . , Wγ. In order to model intersectionality, let
+us define the subgroup Gi,j,k := Ui ∩ Vj ∩ Wk. Let si,j,k = �
+r∈Gi,j,k Xr be the random variable
+denoting the amount of resource received in total by members of Gi,j,k. We ask: “While intersectional
+unfairness may be present, say inherently in the vector x, does our rounding at least limit the extent
+of it?" That this is indeed true follows from Property (P3) which guarantees negative association,
+as follows. For any sequence of thresholds (ti,j,k), the probabilities of different subgroups Gi,j,k
+receiving intersectional unfairness in the sense of si,j,k ≤ ti,j,k, can be bounded as well as if the Xr’s
+had been independent:
+• ∀S ⊆ ([α] × [β] × [γ]) , Pr[�
+(i,j,k)∈S(si,j,k ≤ ti,j,k)] ≤ �
+(i,j,k)∈S Pr[si,j,k ≤ ti,j,k]; and, more
+generally,
+• in disease-spread and meme-spread-like contexts, we are often interested in monotone non-
+linear functions that model phase transitions: e.g., if a heavily-interacting subgroup receives
+fewer than a threshold of vaccines, an explosive epidemic could happen within that subgroup—
+with monotone behavior away from this threshold. Given any sequence of monotone functions
+(all increasing or all decreasing) (fi,j,k), we have
+∀S ⊆ ([α] × [β] × [γ]) , E
+
+
+�
+(i,j,k)∈S
+fi,j,k(si,j,k)
+
+ ≤
+�
+(i,j,k)∈S
+E [fi,j,k(si,j,k)] .
+25
+
+As is well-known, letting χ(·) be the indicator function, this implies that �
+(i,j,k)∈S χ(si,j,k ≤
+ti,j,k) has Chernoff-like concentration around its mean; in particular, while some subgroups in
+S may face intersectional unfairness (that needs to be separately analyzed by inspecting x),
+it is unlikely that many do.
+We next present an application to diversity in search results. In addition to the classical interest
+in this problem in information retrieval [14], there has been much recent work in search-result
+diversification, motivated, e.g., by the fact that the document-set retrieved should account for the
+interests of the user population [20, 26]. By developing a model of knowledge about the topics the
+query or the documents may refer to, the work of [4] presents an (1 − 1/e)–approximation for their
+objective of maximizing average-user satisfaction with the search results. Given a search query q, a
+corpus of N documents D, and a bound k on the number of documents to retrieve from D for q, a
+monotone submodular function f : 2D → [0, 1] is developed in [4], where f(T) is the (approximate)
+mix of relevance and diversity if the set T with |T| ≤ k is then returned to the user. The problem
+is thus to approximately maximize f(T) subject to |T| ≤ k. Consider the setting where D is very
+large, we only get streaming access to it, and where, after encountering document d ∈ D, we get
+ML advice on the probability xd with which to choose d. Algorithm 3 is tailor-made for this, as
+it needs only O(log N) space. Furthermore, by the already-known Theorem A.4, we obtain that
+our final expected value F(X1, X2, . . . , XN) is at least as large as if we had instead chosen the Xi’s
+independently with marginals xi. (Also see [5, 30] for applications of submodular functions toward
+diversity in bipartite matching.)
+6
+Lower Bounds
+In this section we present some impossibility results for ODRSes.
+A simple example in [29] rules out a rounding ratio of one, or even just greater than 7
+8 = 0.875.
+In this section we improve this bound to (2
+√
+2 − 2) ≈ 0.828 with a similarly simple example.
+Before providing our lower bound, we provide an even simpler example ruling out a rounding
+ratio of one.
+This example highlights the need of ODRSes with high rounding ratio to create
+some form of negative correlation between all subsets of offline nodes, motivating our lower bound
+example. (We are also hopeful that this insight may inform future ODRSes.)
+Example. Suppose an ODRS A with rounding ratio of one exists, and apply it to the following
+input on n = 3 offline nodes and 4 online nodes: For k = 1, 2, 3, online node k only neighbors offline
+node k, and xkk = 1/2. So, each offline node’s matched status so far is a Ber(1/2) random variable.
+Finally, online node t = 4 arrives, neighboring some two offline nodes, i, j, and xit = xjt = 1/2. The
+above is clearly a feasible fractional matching, but since we assumed that A has a rounding ratio
+of one, it must match t with probability one. Since t cannot be matched if both i are j previously
+matched, and both are matched with probability 1/2 before time t, this requires i’s and j’s matched
+status before time t to be perfectly negatively correlated. That is, i is unmatched iff j is matched,
+and vice versa. Since i, j could be any two offline nodes, this requires perfect negative correlation
+between any two of these three variables, which is impossible.7 Thus, we reach a contradiction, and
+an ODRS with rounding ratio of one is impossible.
+To generalize the above and obtain a concrete numeric bound, we need the following fact,
+whereby (near-)positive pairwise correlation is unavoidable in a large set of Bernoulli random vari-
+ables. (See Lemma 6.3 for a potentially useful generalization to k-wise correlations.)
+7Perfect negative pairwise correlation between three binary variables X1, X2, X3 ∈ {0, 1} can be stated succinctly
+as X1 + X2 = X1 + X3 = X2 + X3 = 1, but this linear system only has a fractional solution X1 = X2 = X3 = 1/2.
+26
+
+Fact 6.1. Let Yi ∼ Ber(p) for i = 1, . . . , n be (possibly dependent) Bernoulli variables. Then,
+max
+i,j Cov(Yi, Yj) ≥ −2p/(n − 1).
+Proof. This bound follows from non-negativity of variance and the following, after rearranging.
+0 ≤ Var
+��
+i
+Yi
+�
+=
+�
+i
+Var(Yi) +
+�
+i,j
+Cov(Yi, Yj) ≤ n · p +
+�n
+2
+�
+max
+ij
+Cov(Yi, Yj).
+Lemma 6.2. Any online rounding algorithm for bipartite fractional matchings has some fractional
+matching x = 1
+2 · 1E on graph G = (V, E) such that for some edge e ∈ E, the output matching M
+satisfies
+Pr[e ∈ M] ≤ (2
+√
+2 − 2) · xe ≈ 0.828 · xe.
+Proof. Consider an input with xe = 1/2 for each edge, with the first n online nodes neighboring
+two distinct offline nodes, with one last online node neighboring two (judiciously chosen) offline
+nodes, from two prior online nodes’ neighborhoods. We wish to show that for some instantiation of
+this family of x and G, some online node (and hence some edge) is matched with low probability.
+Specifically, letting p = 2
+√
+2 − 2, we wish to show that for some such x and G,
+∃t ∈ [n + 1] such that Pr[t ∈ V (M)] ≤ p +
+p
+2(n − 1),
+(19)
+Since each online node t has �
+i xit = 1, this implies that some edge e is matched with probability
+at most (2
+√
+2 − 2 +
+p
+2(n−1)) · xe, by linearity of expectation. Taking n to be sufficiently large rules
+out any (2
+√
+2 − 2 + ǫ)-approximate rounding. We turn to prove Equation (19).
+If any of the first online nodes is matched with probability less than p, we are done. Otherwise,
+let Yt = 1[t ∈ V (M)] be indicators for online node t being matched. We couple these with variables
+Zt ≥ Ber(p) such that Yt ≥ Zt always.
+By Fact 6.1, there exist two online nodes t ̸= t′ with
+Cov(Zt, Zt′) ≥ −2p/(n − 1). Let N(t) = {i1, i2} and N(t′) = {i3, i4} be the neighborhoods of t and
+t′, respectively. By a simple averaging argument, some pair of nodes (i, j) ∈ N(t) × N(t′) are both
+matched before the online last arrival with probability at least
+Pr[Yt, Yt′]
+4
+≥ Pr[Zt, Zt′]
+4
+≥ Pr[Zt] · Pr[Zt′] − 2p/(n − 1)
+4
+≥ p2 − 2p/(n − 1)
+4
+.
+So, if the last online node neighbors {i, j}, it is matched with probability at most 1 − p2−2p/(n−1)
+4
+.
+However, as p = 2
+√
+2−2 is a root of 1−y− y2
+4 , this probability is precisely 1− p2
+4 +
+p
+2(n−1) = p+
+p
+2(n−1),
+as desired.
+Inevitable near-positive cylinder dependence.
+A useful extension of Fact 6.1 of possible
+independent interest argues that for any (sufficiently large) subset of Bernoulli variables, some
+subset of these variables must be (nearly) positively cylinder dependent. In Appendix D we prove
+this Ramsey-theoretic lemma.
+Lemma 6.3. Let r ∈ N and 0 ≤ ǫ ≤ p ≤ 1. Then, there exists some n = nr(p, ǫ) such that any
+Bernoulli variables Y1, . . . , Yn ∼ Ber(p) contains some subset I ⊆ [n] of size |I| = 2r such that
+E
+��
+i
+Yi
+�
+≥
+�
+i
+E[Yi] − ǫ.
+27
+
+7
+Summary and Open Questions
+In this work we provided a methodical study of online dependent rounding schemes (ODRSes)
+for bipartite b-matching. We obtained optimal rounding ratios with strong concentration for star
+graphs (uniform-matroid rounding) and improved bounds beyond a ratio of 1− 1
+e for (b-)matchings,
+including the first results for b-matchings not obtained via OCRSes. (Consequently, ours is the only
+ratio beyond 1/2 known for this problem.) We furthermore provided a number of applications of our
+ODRSes and their algorithmic approach. Beyond the natural question of improving our rounding
+ratios for b-matchings (and the approximation/competitive ratios of their derived applications), our
+work raises a number of directions for future research.
+Online applications. We provided a number of online applications of our ODRSes. What further
+applications do these schemes have? The bipartite setting seems particularly relevant for online
+scheduling problems, with machines and jobs represented by offline and online nodes, respectively.
+For minimization problems, it may prove useful to observe that our ODRSes for b-matching have
+modest two-sided error: edges (i, t) are matched with probability at most the probability that i bids
+for t, namely ˆxi,t ≤ (1 + δ)xi,t ≈ 1.064 · xi,t. Moreover, these latter events satisfy strong upper-tail
+bounds, by our ODRS’ strong negative correlation properties (see Property (P3) and Lemma C.1).
+Streaming applications. We note that our ODRSes are not only online algorithms, but moreover
+are streaming (for uniform matroids) or semi-streaming algorithms (for arbitrary b-matchings).
+Indeed, they require us to only remember O(log n)-bit partial sums and a single number per offline
+node. Does this property have further applications in streaming or graph-analytics contexts?
+Richer ODRSes. Finally, we recall that online contention resolution schemes (OCRSes) yield
+ODRSes, though this connection does not yield optimal rounding ratios. Recent years have seen an
+explosion of work on OCRSes for increasingly-rich families of arrival models and combinatorial con-
+straints. These have been fueled in large part by connections to various other online and economic
+applications, most notably the prophet-inequality problem under more involved combinatorial con-
+straints (see discussion in the influential work of Kleinberg and Weinberg [52]). Can a similarly
+rich theory of online dependent rounding more broadly be developed, capturing other combinatorial
+constraints beyond matchings?
+Acknowledgements.
+We thank Uri Feige for pointing out the connection of our work to offline CRSes, and thank Sahil
+Singla for discussions about (lack of prior) examples of black-box application of offline CRSes to
+online problems. We thank Manish Purohit for asking about the two-sided error of our ODRSes,
+which we anticipate may be of use in future applications for online minimization problems.
+APPENDIX
+A
+Additional Preliminaries
+In this section we provide omitted proofs and additional preliminaries not covered in Section 2.
+A.1
+Contention Resolution Schemes
+We start with a brief, self-contained proof of Lemma 2.1, restated below for ease of reference.
+28
+
+Lemma 2.1. Let D : 2[n] → R≥0 be a distribution over subsets of [n], with its support denoted by
+supp(D) = {S ⊆ [n] | PrR∼D[R = S] ̸= 0}. Then, there exists a randomized algorithm CRS(R,⃗v)
+which on input set R ∼ D and vector ⃗v ∈ Rn, outputs a subset O ⊆ R of size |O| ≤ 1 satisfying
+Pr[i ∈ O] = vi · min
+S⊆[n]
+Pr[R ∩ S ̸= ∅]
+�
+i∈S vi
+∀i ∈ [n].
+The algorithm runs in time polynomial in n if D is a product distribution. Otherwise, it runs in
+time poly(n, T), where T ≥ |supp(D)| is the time to compute and write down the support of D.
+Proof. The lemma for product distributions follows from the work of Feige and Vondrák [40]. We
+focus on the more general case, addressed by Bansal and Cohen [7], but proven here for completeness.
+Let α := minS⊆[n]
+Pr[R∩S̸=∅]
+�
+i∈S vi .
+We prove the existence of conditional probabilities pi,S with
+pi,S = 0 if i ̸∈ S and �
+i pi,S = 1 for all S such that setting Pr[O = {i} | R = S] = pi,S yields
+Pr[O = {i}] = �
+S Pr[O = {i} | R = S] · Pr[R = S] ≥ α · vi. We prove the existence of these
+probabilities using the max-flow/min-cut theorem applied to the following directed flow network
+that has a source vertex src, a sink vertex sink, and two other disjoint sets of vertices A and B.
+Vertices on side A correspond to subsets of elements in [n], with an edge of capacity Pr[R = S] from
+src to the node in A representing set S. Vertices on side B correspond to elements i ∈ [n], with an
+edge of capacity α·vi leaving this vertex to sink. Infinite-capacity directed edges go from each vertex
+in A corresponding to some set S, to vertices in B corresponding to each i ∈ S. Each vertex i ∈ S
+in B, has a directed edge to sink with capacity α · vi. This network is reminiscent of the network
+used to prove Hall’s Theorem via the max-flow/min-cut theorem. Indeed, minS⊆[n]
+Pr[R∩S̸=∅]
+�
+i∈S vi
+≥ α is
+a (scaled) version of Hall’s condition, and implies by the same argument the existence of a “perfect”
+flow f of value α·�
+i vi, implying that each edge of the form (i, sink) has its capacity α·vi saturated
+by this flow. This in turn implies the existence of the probabilities pi,S as desired, where if the flows
+through the edges (src, S) and (S, i) are fsrc,S and fS,i respectively, then pi,S = fS,i/ Pr[R = S] (if
+Pr[R = S] = 0, we take arbitrary non-negative pi,S with pi,S = 0 if i ̸∈ S and �
+i:i∈S pi,S = 1).
+Thus, the above algorithm outputs a set O ⊆ R of size at most one with each element belonging to
+O with the requisite probability. The running time of the algorithm for general distributions follows
+by polytime solubility of the maximum flow problem.
+A.2
+Negative correlation properties
+Here we formalize the strong notion of negative correlation that our online level-set algorithm enjoys.
+Definition A.1. A distribution µ : 2[n] → R≥0 is strongly Rayleigh if its generating polynomial,
+gµ(z) =
+�
+S⊆[n]
+µ(S)
+�
+i∈S
+zi,
+is real stable, i.e., it has no root z ∈ Cn satisfying ℑ(zi) > 0 for all i ∈ [n].
+As stated in Section 2 and proven in [13], the strong Rayleigh property implies NA.
+Lemma A.2. If (X1, . . . , Xn) are strongly Rayleigh binary r.v.s, then they are NA.
+Moreover, since SRP is closed under conditioning [13], we find that SRP distributions are NA
+even after conditioning.
+The strong Rayleigh property, central to the area of geometry of polynomials, has been highly
+influential in recent years.
+To the best of our knowledge, aside from the classic balls-and-bins
+29
+
+process, its variants and conditional counterparts (see [15]), our online level-set algorithm is the
+only online algorithm known to satisfy such a strong negative-correlation property. We believe that
+this property of our algorithm will find further applications outside of this work.
+Submodular dominance. As stated in Section 2, negative association implies stochastic dom-
+inance in the submodular order. To formally define the above, we briefly recall the definition of
+submodular functions, which are set functions capturing the notion of diminishing returns.
+Definition A.3. A set function f : 2[n] → R is submodular if for all sets A ⊆ B ⊆ [n] and element
+x ∈ [n] \ B, we have that
+f(A ∪ {x}) − f(A) ≥ f(B ∪ {x}) − f(B).
+The following submodular dominance result of Christofides and Vaggelatou [25] implies that
+our online level-set algorithm not only preserves weighted objectives losslessly, but also preserves
+submodular objectives. (See Appendix B.)
+Theorem A.4. Let X1, . . . , Xn be NA random variables and let X∗
+1, . . . , X∗
+n be independent random
+variables that are place-wise equal to the X1, . . . , Xn respectively in distribution. Then, for any
+monotone submodular function f,
+E[f(X1, . . . , Xn)] ≥ E[f(X∗
+1, . . . , X∗
+n)].
+A.3
+Odds and Ends
+We will also make use of the following direct corollary of convexity.
+Claim A.5. Let f be a non-negative concave function. Then, for any 0 ≤ t ≤ α,
+f(t)
+t
+≥ f(α)
+α
+.
+Proof. Writing t as a convex combination of α and 0, the claim follows from convexity, as follows.
+f(t) = f
+� t
+α · α +
+�
+1 − t
+α
+�
+· 0
+�
+≥ t
+α · f(α) +
+�
+1 − t
+α
+�
+· f(0) ≥ t
+α · f(α).
+B
+Deferred Proofs of Section 3
+We start by briefly proving that Algorithm 1 terminates and preserves marginals.
+Fact B.1. Algorithm 1 terminates, and satisfies Pr[i ∈ S] = xi for all i ∈ [n] (Property (P1)).
+Proof. It is easy to see that STEP preserves the sum of its inputs/outputs while decreasing the
+number of fractional such inputs/outputs.
+Consequently, �
+i yi is integral during the execution
+of Algorithm 1, while the number of fractional yi decreases and cannot be one, so this algorithm
+terminates with all yi binary. Moreover, STEP is easily seen to preserve these values’ expectations,
+and so by induction we have that E[yi] = xi for all i, i.e., Algorithm 1 satisfies Property (P1).
+Lemma 3.1. Let sj := �
+i≤j xi and Y t
+j := �
+i≤j yt
+i, where yt
+i is the value yi after t applications of
+STEP in Algorithm 1. Then, Y t
+j ∈ [⌊sj⌋, ⌈sj⌉] for all j and t.
+30
+
+Proof. We prove this for all j by induction on t ≥ 0. The base case is trivial. Fix a time t and let
+i1 and i2 be as in Algorithm 1 at time t + 1. If i1, i2 ̸∈ [j] or i1, i2 ∈ [j], then, since STEP(yi1, yi2)
+does not affect Y t
+j in the former case and preserves the sums yi1 + yi2 and Yj in the latter case,
+the inductive hypothesis implies Y t+1
+j
+= Y t
+j ∈ [⌊sj⌋, ⌈sj⌉]. Conversely, if i1 ∈ [j] and i2 ̸∈ [j] (and
+therefore i1 = j), we have that yt+1
+i
+= yt
+i ∈ {0, 1} for all i ∈ [j − 1], while yt+1
+i1
+∈ [0, 1]. Since by the
+inductive hypothesis we have that Y t
+j = Y t
+j−1 + yt
+j ∈ [⌊sj⌋, ⌈sj⌉] and Y t
+j−1 = Y t+1
+j−1 is the sum of 0-1
+terms, while yt
+j ∈ (0, 1), this implies that Y t+1
+j
+= Y t+1
+j−1 + yt+1
+j
+= Y t
+j−1 + yt+1
+j
+∈ [⌊sj⌋, ⌈sj⌉].
+B.1
+Direct analysis of Algorithm 3
+In this section we provide a simple, self-contained proof that Algorithm 3 satisfies the first two
+properties we desire of it.
+In the main paper body, we use this direct proof to prove that our
+algorithm’s output distribution is the same as its offline counterpart, from which we also obtain the
+third desideratum, namely Property (P3).
+Lemma B.2. Algorithm 3 is well-defined (i.e., pt ∈ [0, 1] for all times t and any realization of the
+randomness) and it satisfies properties (P1) and (P2).
+Proof. We prove the above three properties (including pt ∈ [0, 1]) by strong induction on t ≥ 0.
+The base case, t = 0, is trivial. As our inductive hypothesis (I.H.), we assume that these properties
+hold for all t′ ≤ t − 1 and prove them for all t′ ≤ t, starting with Property (P2). By the I.H.,
+St−1 ≤ ⌈st−1⌉ ≤ ⌈st⌉; so, since pt = 0 if St−1 = ⌈st⌉, we trivially have that St ≤ ⌈st⌉. Moreover,
+by the I.H., St−1 ≥ ⌊st−1⌋ = ⌊st − xt⌋ ≥ ⌊st⌋ − 1; so, since pt = 1 if St−1 < ⌊st⌋, in which case
+St−1 = ⌊st⌋ − 1, we have that St ≥ ⌊st⌋. Thus, St ∈ {⌊st⌋, ⌈st⌉}, proving Property (P2).
+Next we prove that Algorithm 3 is well-defined, as pt ∈ [0, 1] (regardless of prior randomness).
+The first non-trivial case is when St−1 = ⌊st⌋ = ⌊st−1⌋. Non-negativity of pt—a ratio of non-negative
+terms—is immediate, while
+pt =
+xt
+⌊st−1⌋ + 1 − st−1
+=
+st − st−1
+⌊st−1⌋ + 1 − st−1
+≤
+⌊st⌋ + 1 − st−1
+⌊st−1⌋ + 1 − st−1
+= ⌊st−1⌋ + 1 − st−1
+⌊st−1⌋ + 1 − st−1
+= 1.
+The second non-trivial case is when St−1 = ⌊st⌋ > ⌊st−1⌋, in which case ⌊st⌋ = ⌊st−1⌋ + 1. Again,
+non-negativity of pt is immediate, while
+pt =
+st − ⌊st⌋
+st−1 − ⌊st−1⌋ = st−1 + xt − ⌊st−1⌋ − 1
+st−1 − ⌊st−1⌋
+≤ st−1 − ⌊st−1⌋
+st−1 − ⌊st−1⌋ = 1.
+To prove Property (P1), we will need a closed form for qt := Pr[St = ⌊st⌋], useful when discussing
+Pr[St−1 = ⌊st⌋], the probability that t may be added to S, and Pr[St−1 < ⌊st⌋], the probability that
+t must be added to S (to satisfy Property (P2)). Note that St ∼ ⌊st⌋ + Ber(1 − qt) by Property
+(P2), while E[St] = �
+t′≤t E[Xt′] = �
+t′≤t xt′ = st, by Property (P1) (for all t′ ≤ t) and linearity of
+expectation. Combining these observations, we obtain st = E[St] = ⌊st⌋ + 1 − qt, and therefore,
+qt = ⌊st⌋ + 1 − st.
+(20)
+We now use the closed form for qt−1 to prove Property (P1) for time t. The easy case is when
+⌊st⌋ = ⌊st−1⌋. Here we have that E[Xt | St−1 = ⌊st⌋] =
+xt
+qt−1, while trivially E[Xt | St−1 = ⌈st⌉] = 0.
+Consequently, since St−1 ∈ {⌊st−1⌋, ⌈st−1⌉} = {⌊st⌋, ⌈st⌉} in this case, then by total probability we
+have that E[Xt] = qt−1 ·
+xt
+qt−1 + 0 = xt, as desired.
+Finally, we consider the case where ⌊st⌋ > ⌊st−1⌋, i.e., ⌊st⌋ = ⌊st−1⌋ + 1. If in addition st−1 =
+⌊st−1⌋ (i.e., st−1 is integral, implying that st = st−1 + 1, and hence xt = st − st−1 = 1), we have by
+31
+
+Property (P2) that St−1 = st−1 < ⌊st⌋ (with probability one), and so we add t to S with probability
+one, resulting in E[Xt] = 1 = xt, as desired. If, conversely, we have that st−1 ̸= ⌊st−1⌋ < ⌊st⌋, then
+by total probability, we obtain the desired equality as follows.
+E[Xt] = Pr[St−1 = ⌊st−1⌋] · 1 + Pr[St−1 ̸= ⌊st−1⌋] ·
+st − ⌊st⌋
+st−1 − ⌊st−1⌋
+= qt−1 + (1 − qt−1) · xt + st−1 − (⌊st−1⌋ + 1)
+st−1 − ⌊st−1⌋
+�
+st = xt + st−1
+⌊st⌋ = ⌊st−1⌋ + 1
+= qt−1 + (1 − qt−1) · xt − qt−1
+1 − qt−1
+Equation (20)
+= xt.
+B.2
+Coupling the offline and online algorithms
+Here we prove that our online level-set algorithm, Algorithm 3, induces the same output distribution
+as the offline algorithm of [66], Algorithm 1
+Theorem 3.4. Fix a vector ⃗x ∈ [0, 1]n. Let D and D′ denote the probability distributions on {0, 1}n
+induced by the online and offline algorithms run on ⃗x, respectively. Then, D = D′.
+Proof. We prove by induction on t ∈ [n] that the following holds:
+∀(b1, . . . , bt−1) ∈ {0, 1}t−1, Pr
+D [Xt = 1
+�� (∀i < t, Xi = bi)] = Pr
+D′[Xt = 1
+�� (∀i < t, Xi = bi)].
+(21)
+The (strong) inductive hypothesis, whereby Equation (21) holds for all t′ < t clearly implies that for
+all (b1, . . . , bt) ∈ {0, 1}t−1 we have that PrD[(∀i < t, Xi = bi)] = PrD′[(∀i < t, Xi = bi)]. Similarly,
+Equation (21) holds for all t ≤ n iff the two distributions are the same.
+The base case t = 1 follows from Property (P1). So suppose t > 1. Fix b1, . . . , bt−1 ∈ {0, 1},
+and let E denote the event “∀i < t, Xi = bi" (measured w.r.t. D or w.r.t. D′). Then, by the strong
+induction hypothesis, PrD[E] = PrD′[E], and in particular, PrD[E] ̸= 0 iff PrD′[E] ̸= 0; thus we
+may assume that the events conditioned on in the LHS and in the RHS of (21) both have nonzero
+(identical) probabilities. This is the only place where we will need the induction hypothesis.
+Recall that st−1 = �t−1
+i=1 xi. Furthermore, although St−1 technically depends on the rounding
+algorithm, it is natural to take it to be �t−1
+i=1 bi here. Note that since both the offline and online
+algorithms satisfy Property (P2), we have that St−1 ∈ {⌊st−1⌋, ⌈st−1⌉}.
+Case I: st−1 ∈ Z. This is an easy case. It is immediate that PrD[Xt = 1
+�� E] = xt here. This is
+also not hard to see this for the offline algorithm. Recall that the offline algorithm starts with a
+copy y of x. Since st−1 ∈ Z, the offline algorithm would set Xi for exactly st−1 indices i ∈ [t − 1] to
+one (and the rest in [t − 1] to zero), and would then recursively run on the suffix (yt, yt+1, . . . , yn)
+of y with no correlation with the rounding thus far. Furthermore, yj = xj for all j ≥ t. Thus, by
+Property (P1) applied to this suffix, we have that PrD′[Xt = 1
+�� E] = xt also.
+We may thus assume from now on that z .= st−1 − ⌊st−1⌋ lies in (0, 1).
+Case II: st−1 ̸∈ Z and z + xt ≤ 1. An easy sub-case here is that St−1 = ⌈st−1⌉; here, since both
+the offline and online algorithms satisfy Property (P2), it is immediate that in this sub-case
+Pr
+D [Xt = 1
+�� E] = Pr
+D′[Xt = 1
+�� E] = 0.
+We may thus assume that St−1 = ⌊st−1⌋. By the definition of the online algorithm 3 we have that
+Pr
+D [Xt = 1
+�� E] =
+xt
+1 − z .
+(22)
+32
+
+We now analyze PrD′[Xt = 1
+�� E]. Consider the offline algorithm just before it involves xt in STEP:
+at this point in time, it has:
+• set ⌊st−1⌋ of the variables (Xi : i < t) to one;
+• set t − 2 − ⌊st−1⌋ of the variables (Xi : i < t) to zero; and, recalling again that the offline
+algorithm starts with a copy y of x,
+• it has assigned the current value z to one variable yi∗ where i∗ ∈ [t − 1] is a random variable
+from some distribution.
+At this point, the offline algorithm is basically recursively run on the sequence (yi∗, yt, yt+1, . . . , yn),
+where yi∗ = z and yj = xj for all j ≥ t. Crucially, since St−1 = ⌊st−1⌋, the conditioning on E says
+that the variable yi∗ is eventually rounded to 0. Thus, we can compute PrD′[Xt = 1
+�� E] as follows.
+Suppose the offline algorithm is run on the sequence (u1, u2, . . . , un−t+2) .= (yi∗, yt, yt+1, . . . , yn), to
+output a random bit-vector (U1, U2, . . . , Un−t+2); then,
+Pr
+D′[Xt = 1
+�� E] = Pr
+D′[U2 = 1
+�� U1 = 0].
+(23)
+But since u1 + u2 = z + xt ≤ 1, the first STEP applied to u1 and u2 ensures that at most one of u1
+and u2 gets rounded to one. This yields
+Pr
+D′[U2 = 1
+�� U1 = 0] = PrD′[(U2 = 1) ∧ (U1 = 0)]
+PrD′[U1 = 0]
+= PrD′[U2 = 1]
+PrD′[U1 = 0] =
+xt
+1 − z ,
+where the second equality holds since at most one of U1 and U2 is one; we are thus done by (22).
+Case III: st−1 ̸∈ Z and z + xt > 1.
+This is handled similarly.
+The easy sub-case here is
+that St−1 = ⌊st−1⌋; here, since both the offline and online algorithms satisfy Property (P2), it
+is immediate that
+Pr
+D [Xt = 1
+�� E] = Pr
+D′[Xt = 1
+�� E] = 1.
+We may thus assume that St−1 = ⌈st−1⌉. By the definition of the online algorithm 3 we have that
+Pr
+D [Xt = 1
+�� E] = xt + z − 1
+z
+.
+(24)
+Now, the offline algorithm is, like in Case II, essentially run on the sequence (u1, u2, . . . , un−t+2) .=
+(yi∗, yt, yt+1, . . . , yn)—where yi∗ is initially z—to output a random bit-vector (U1, U2, . . . , Un−t+2),
+with the key difference from Case II being that U1 will eventually be set to 1. So,
+Pr
+D′[Xt = 1
+�� E] = Pr
+D′[U2 = 1
+�� U1 = 1].
+(25)
+Now, since u1 + u2 = z + xt > 1, the first STEP applied to u1 and u2 ensures that at least one of
+u1 and u2 gets rounded to one. This implies
+Pr
+D′[U2 = 1
+�� U1 = 1]
+=
+PrD′[U2 = 1] − Pr[(U2 = 1) ∧ (U1 = 0)]
+PrD′[U1 = 1]
+=
+PrD′[U2 = 1] − PrD′[U1 = 0]
+PrD′[U1 = 1]
+=
+xt + z − 1
+z
+,
+where the second equality holds since at least one of U1 and U2 is one; we are thus done by (24).
+This concludes the proof of the inductive step.
+33
+
+C
+Deferred Proofs of Section 4
+In this section we provide deferred proofs of claims from Section 4, restated below for ease of
+reference.
+Claim 4.7. If ǫ, δ ∈ [0, 1], the function g(y) := 1−exp (− (1 − y) · (1 + δ))·(1 − y · (1 − ǫ)) satisfies
+∀y ∈ R :
+g(y) ≥ g
+�
+ǫ + δ
+(1 − ǫ)(1 + δ)
+�
+= 1 − exp
+�
+−1 − δ + ǫ + δ
+1 − ǫ
+�
+· 1 − ǫ
+1 + δ .
+Proof of Claim 4.7. Taking the derivative of g(y), we get
+g′(y) = exp(−(1 − y) · (1 + δ)) · ((1 − ǫ) − (1 − y · (1 − ǫ)) · (1 + δ)) ,
+which vanishes at y∗ =
+ǫ+δ
+(1−ǫ)(1+δ). Next, we consider the second derivative of g.
+g′′(y) = g′(y) · (1 + δ) + exp(−(1 − y) · (1 + δ)) · (1 − ǫ) · (1 + δ).
+The first summand, g′(y) · (1 + δ), vanishes at y∗, by the above, while the second summand is
+positive, since ǫ ≤ 1 and δ ≥ 0. We conclude that y∗ is the global minimum of g.
+Lemma 4.8. Let fǫ,δ(z) be as in Lemma 4.6. If fǫ,δ(a) ≥ 0 and f ′
+ǫ,δ(a) ≥ 0, then fǫ,δ(b) ≥ 0 ∀b ≥ a.
+Proof. Fix ǫ, δ. For notational simplicity, let f = fǫ,δ and let z∗ = z∗
+ǫ,δ. Since f is the sum of twice-
+differentiable convex functions (namely an exponential and a linear term), we have that f ′′(z) ≥ 0
+for all z. Therefore, if f ′(z∗) ≥ 0, then for all z ≥ z∗ we have that
+f ′(z) = f ′(z∗) +
+� z
+z∗ f ′′(x) dx ≥ f ′(z∗) ≥ 0.
+Thus, if moreover f(z∗) ≥ 0, then the desired statement follows, i.e., for all x ≥ z∗ we have that
+f(z) = f(z∗) +
+� z
+z∗ f ′(x) dx ≥ f(z∗) ≥ 0.
+C.1
+Polytime implementation
+So far, we ignored computational aspects of our ODRS of Theorem 4.9. In this section we show
+how our grouping method used to increase our rounding ratio beyond 1 − 1
+e moreover allows us to
+compute Pr[Pt = S] for all sets S ⊆ [n] in polynomial time — and thus obtain a polytime ODRS
+— while only decreasing our rounding ratio by an arbitrarily small constant γ > 0.
+Theorem 4.10. For any γ > 0, there exists an nO(1/γ)-time bipartite matching ODRS with rounding
+ratio of 0.652 − γ.
+Proof. First, we scale down all the fractions xi,t by a (1− γ) factor (this results in a valid fractional
+matching). As this additional downscaling is linear, this results in each offline node having fractional
+degree ˆsi,t ≤ 1−γ in the assignment ˆx, by Fact 4.5. Consequently, by the same argument as Fact 4.3,
+each bin but at most one in B ∈ Bt has �
+i∈B ˆxi,t ≤ γ
+2. But since �
+i ˆxi,t ≤ �
+i xi,t ·(1+δ) ≤ (1+δ),
+this implies that each bin but one has total x-value at least �
+i∈B xit ≥ γ/2(1+δ). Consequently, by
+the fractional matching constraint, whereby �
+i xi,t ≤ 1, there are at most 2(1 + δ)/γ + 1 = O(1/γ)
+such non-empty bins.
+34
+
+The benefit of few bins is that now at each time t the set of bidders is a set of at most O(1/γ)
+offline nodes, and so there are at most nO(1/ǫ) possible sets S ⊆ [n] in the support of the distribution
+over Pt. Moreover, for each such set S ⊆ [n], it is straightforward to compute the probability that
+S is the set of first-time proposers at time t in time nO(1/γ). (Briefly, for each subset S, we guess for
+each of the O(1/γ) nodes in S at what time they made their first bid. The probability of each of these
+nO(1/γ) guessed events is then easy to compute in polynomial time, by computing the probability
+that these nodes did bid in these guessed time steps and no other times.) All in all, this allows to
+compute, exactly, the support of the distribution of Pt, in time nO(1/γ). Therefore, by Lemma 2.1,
+the CRS invocations take time poly(nO(1/γ) = nO(1/γ) per time step. On the other hand, having
+scaled down the values xi,t by a factor of (1 − γ) trivially incurs a multiplicative loss of (1 − γ) in
+the rounding ratio, and so the obtained ODRS has rounding ratio of 0.652(1 − γ) ≥ 0.652 − γ.
+C.2
+Deferred proofs of the b-matching ODRS
+In this section we prove that our extension of Algorithm 4 from matching to b-matchings results in
+similar benefits when much grouping occurs. We start by introducing some useful notation.
+Our online level-set rounding algorithm, Algorithm 3, makes at most one random choice when
+deciding whether to allocate another online node to the center of the star. Our b-matching ODRS
+follows the same logic (from the perspective of the offline node, playing the role of the star’s center).
+Denote by Ci,t an indicator for the relevant coin for i coming up heads. Our extended core ODRS,
+Algorithm 4, correlates these coins for different offline nodes i at time t. Let Si,t ∈ {⌊si,t⌋, ⌈si,t⌉} be
+the number of times i bids by time t. By our algorithm’s definition, we have the following expression
+for [Si,t ̸= ⌈si,t⌉].
+[Si,t ̸= ⌈si,t⌉] =
+�
+[Si,t−1 ̸= ⌈si,t−1⌉] · (1 − Ci,t−1)
+if ⌈si,t⌉ = ⌈si,t−1⌉
+1 − [Si,t−1 = ⌈si,t−1⌉] · Ci,t−1
+if ⌈si,t⌉ > ⌈si,t−1⌉.
+Lemma C.1. For any time t, the random variables {Si,t}i are NA.
+Proof. Proof by induction on t ≥ 1. The base case is trivial, as Si,1 = 0 deterministically for all i. We
+turn to the inductive step for t, assuming that the variables Si,t−1 are NA. First, by Lemma 2.4, for
+any bin B ∈ Bt−1 at time t−1, the binary variables Ci,t−1 for all nodes i ∈ B, whose sum is at most
+one, are NA. Moreover, the distributions over different bins are independent, and so by closure of
+NA under products (Lemma 2.5), all Ci,t−1 are NA. Moreover, the variables Ci,t−1 are independent
+from the variables Si,t−1 (who are functions only of Ci,t′ for all t′ < t − 1), and consequently, by
+our inductive hypothesis and another application of closure of NA under products (Lemma 2.5),
+all the variables {Si,t−1, Ci,t−1}i are NA. Therefore, since Si,t−1 ∈ [⌊si,t−1⌋, ⌈si,t−1⌉], we find that
+[Si,t ̸= ⌈si,t⌉] is a decreasing function of Si,t−1 and Ci,t−1, i.e., they are decreasing functions of disjoint
+NA variables. Therefore, by yet another application of closure of NA under products (Lemma 2.5),
+the events {[Si,t ̸= ⌈si,t⌉]}i are NA. Finally, since Si,t = ⌈si,t⌉−[Si,t ̸= ⌈si,t⌉] are monotone decreasing
+functions of NA variables, the variables Si,t are themselves NA, as desired.
+We are now ready to prove that our generalization of Algorithm 4 results in bins essentially
+simulating offline nodes bidding with a probability equal to their constituent nodes’ bids. That is,
+we are ready to prove the following generalization of Lemma 4.4.
+Lemma C.2. For any time t and set of offline nodes S ⊆ [n], the b-matching extension of Algo-
+rithm 4 satisfies
+Pr[S ∩ Pt ̸= ∅] ≥ 1 −
+�
+B∈Bt
+�
+1 −
+�
+i∈B∩S
+xit
+�
+.
+35
+
+Proof. We denote by B′
+t ⊆ Bt the set of singleton bins B = {i} for which ⌈si,t+1⌉ > ⌈si,t⌉. For such
+bins B ∈ B′
+t, B = {i}, we have that Pr[Pt ∩ S ∩ B = ∅] = 1 − xi,t. For other bins B ∈ Bt \ B′
+t, we
+have that Pr[i ∈ Pt] = xi,t, by independence of Ci,t and Si,t and since Pr[Ci,t] =
+xi,t
+1−(si,t−⌊si,t⌋) and
+Pr[Si,t = ⌊si,t⌋] = si,t − ⌊si,t⌋ (the latter following from our online level-set algorithm’s analysis).
+Therefore, by �
+i∈B Ci,t ≤ 1 implying disjointness of the events [i ∈ Pt] for i ∈ B, we have that
+Pr[Pt ∩ S ∩ B = ∅] = 1 − �
+i∈B∩S xi,t. It remains to show that the events [Pt ∩ S ∩ B = ∅] for
+different B are negative cylinder dependent, giving the desired inequality,
+Pr[Pt ∩ S = ∅] = Pr
+� �
+B∈Bt
+(Pt ∩ S ∩ B = ∅)
+�
+≤
+�
+B∈Bt
+Pr[Pt ∩ S ∩ B = ∅] =
+�
+B∈Bt
+�
+1 −
+�
+i∈B∩S
+xi,t
+�
+.
+Now, for bins B = {i} ∈ B′
+t, we have that 1[Pt ∩ S ∩ B = ∅] = 1 − [Si,t = ⌈si,t⌉] · Ci,t.
+For
+non-singleton bins B ∈ Bt, we have that 1[Pt ∩S ∩B = ∅] = 1−�
+i∈B∩S Ci,t ·[Si,t = ⌊si,t⌋]. In both
+cases, the indicators 1[Pt ∩S ∩B = ∅] are NA, as they are monotone decreasing functions of disjoint
+NA variables (here, we need to first apply decreasing functions that map Si,t to ⌈si,t⌉ − Si,t for all
+i ∈ B ∈ Bt \B′
+t and all other variables to themselves). Hence, by closure of NA under such functions
+(Lemma 2.5), the variables 1[Pt ∩ S ∩ B = ∅] are NA. Therefore, by Lemma 2.3, these variables are
+negative cylinder dependent, yielding the required inequality above. The lemma follows.
+D
+Deferred Proofs of Section 6
+In this section we prove that any sufficiently many binary variables must contain some 2r with some
+(near-)positive 2r-wise correlation, generalizing the special case of r = 1 given by Fact 6.1.
+Lemma 6.3. Let r ∈ N and 0 ≤ ǫ ≤ p ≤ 1. Then, there exists some n = nr(p, ǫ) such that any
+Bernoulli variables Y1, . . . , Yn ∼ Ber(p) contains some subset I ⊆ [n] of size |I| = 2r such that
+E
+��
+i
+Yi
+�
+≥
+�
+i
+E[Yi] − ǫ.
+Proof. We prove this claim for all 0 ≤ ǫ ≤ p ≤ 1 by induction on r ≥ 1, and note that the claim is
+trivial if p2r ≤ ǫ, in which case the RHS is negative, so nr(p, ǫ) = 2r suffices. Fact 6.1 proves the
+base case, showing that n1(p, ǫ) ≤ ⌈2p
+ǫ + 1⌉ suffices. To prove the inductive step, consider a set of
+k := n1(p,
+ǫ
+22r ) + 2m variables Y1, . . . , Yk ∼ Ber(p), for m to be determined shortly. Then, we find
+a sequence of disjoint pairs I1, I2, . . . , Im ⊆ [n], where for each Is = {i, j}, s ∈ [m], we have that
+Cov(Yi, Yj) ≥ − ǫ
+22r . The existence of such pairs follows by repeatedly invoking Fact 6.1 applied
+to the variables {Yi | i ∈ [n] \ �s
+ℓ=1 Iℓ} to obtain the pair Is+1. Now, for each pair Is = {i, j} as
+above, define a new variable Zs := Yi · Yj. By Cov(Yi, Yj) ≥ − ǫ
+22r , we have that Pr[Zs] ≥ p2 −
+ǫ
+22r .
+Next, couple these combined variables with equiprobable variables As ∼ Ber(p2 −
+ǫ
+22r ) with Zs ≥ As
+always. Then, if we take m = nr−1(p2 −
+ǫ
+22r , ǫ
+2), the inductive hypothesis applied to the m variables
+As implies the existence of a subset S ⊆ [m] of 2r−1 of these variables satisfing the following.
+E
+
+
+�
+i∈�
+s∈S Is
+Yi
+
+ = E
+��
+s∈S
+Zs
+�
+≥ E
+��
+s∈S
+As
+�
+≥
+�
+p2 − ǫ
+22r
+�2r−1
+− ǫ
+2 ≥ p2r − 22r−1 · ǫ
+22r − ǫ
+2 ≥ p2r − ǫ.
+Above, the first inequality follows by the coupling Zs ≥ As, the second inequality follows from the
+inductive hypothesis, the third inequality follows from (a − b)k ≥ ak − 2k · b for any a, b ∈ [0, 1] and
+k ∈ N (as seen by expanding (a − b)k), and the final inequality follows from 22r−1 ≥ 22r−1 for r ≥ 1.
+We conclude that {Yi | i ∈ �
+s∈S Is} is the desired subset of 2r variables in Y1, . . . , Yn.
+36
+
+Remark D.1. We did not attempt to optimize the minimum possible nr(p, ǫ) above. Nonetheless,
+for concrete bounds, we note that if p2r ≥ ǫ, then our proof yields bounds satisfying the following
+recurrence.
+n1(p, ǫ) ≤
+�2p
+ǫ + 1
+�
+≤ 4p
+ǫ ,
+nr(p, ǫ) ≤ n1
+�
+p, ǫ
+22r
+�
++ 2nr−1
+�
+p2 − ǫ
+22r , ǫ
+2
+�
+≤ 4 · 22rp
+ǫ
++ 2nr−1
+�
+p2 − ǫ
+22r , ǫ
+2
+�
+.
+Noting that this recurrence is monotone increasing in p, we have that nr(p, ǫ) = O
+�
+22r · p
+ǫ
+�
+.
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diff --git a/K9FAT4oBgHgl3EQfwR5T/content/tmp_files/load_file.txt b/K9FAT4oBgHgl3EQfwR5T/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,2808 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf,len=2807
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='08680v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='DS]  20 Jan 2023  Online Dependent Rounding Schemes  Joseph (Seffi) Naor∗1, Aravind Srinivasan†2, and David Wajc‡3  1Technion – Israel Institute of Technology  2University of Maryland, College Park  3Google Research  January 23, 2023  Abstract  We study the abstract problem of rounding fractional bipartite b-matchings online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The  input to the problem is an unknown fractional bipartite b-matching, which is exposed node-by-  node on one side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The objective is to maximize the rounding ratio of the output matching M,  which is the minimum over all fractional b-matching x, and edges e, of the ratio Pr[e ∈ M]/xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In offline settings, dependent rounding schemes achieving a ratio of one and strong negative  correlation properties are known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', Gandhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', FOCS’02, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='ACM’06 and Chekuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=',  FOCS’10), and have found numerous applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Motivated by online applications, we present  online dependent-rounding schemes (ODRSes) for b-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  For the basic case of a single offline node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', uniform matroids (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' level sets), we achieve  an online rounding ratio of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover, we show that our ODRS yields the same distribution  as (offline) pivotal sampling (Srinivasan, FOCS’01, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='ACM’06), and so inherits the latter’s known  strong correlation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In arbitrary bipartite graphs, for which online rounding ratios of  one are impossible, we show that a combination of our level-set ODRS with repeated invocations  of offline contention resolution schemes (CRSes) yields a rounding ratio of 1 − 1  e ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Our  main technical contribution is an ODRS breaking this pervasive bound, yielding rounding ratios  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='646 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='652 for b-matchings and simple matchings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We obtain these results  by grouping nodes and using CRSes for negatively-correlated distributions, together with a new  method we call group discount and individual markup, analyzed using the theory of negative  association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Extending and applying our ODRSes, we obtain a number of new results, including (directly)  the first online bipartite edge-weighted matching algorithms with ML predictions, and:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The first results for online edge coloring multigraphs under adversarial arrivals, adding to  the recent exciting work on online edge coloring, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', Kulkarni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (STOC’22), Bhat-  tacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (SODA’21) and Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (FOCS’19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Improved polytime online approximation of the optimum (computationally-unbounded)  online algorithm for online stochastic weighted bipartite matching, improving on recent  works of Papadimitriou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (EC’21) and Braverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (EC’22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' A generalization of a result of Srinivasan (SODA’07) that improved a previous result of  Swamy and Shmoys (FOCS’05, SICOMP’12) to multi-stage stochastic multi-coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  ∗Supported in part by Israel Science Foundation grant 2233/19 and United States - Israel Binational Science  Foundation grant 2018352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  †Supported in part by NSF award number CCF-1918749, and by research awards from Amazon and Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  ‡Part of this work done while the author was at Stanford University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Contents  1  Introduction  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Our Contributions  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Extensions and Applications .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='2  Techniques  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='1  Offline level-set rounding .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  The b-matching ODRS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  14  5  Applications  16  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Online edge coloring (multigraphs) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  Stochastic extension .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  23  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Multi-stage stochastic optimization .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  23  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  Negative association, fairness, diversity, and streaming .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  24  6  Lower Bounds  26  7  Summary and Open Questions  28  A Additional Preliminaries  28  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1 Contention Resolution Schemes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  28  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2 Negative correlation properties  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='3 Odds and Ends .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='1  Direct analysis of Algorithm 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='2  Coupling the offline and online algorithms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='1 Polytime implementation  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  35  D Deferred Proofs of Section 6  36  1  Introduction  Online bipartite b-matching is a prototypical online problem, tracing its origin to the seminal work  of Karp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Here, nodes on one side of a bipartite graph are revealed over time, and each  node v can be matched at most bv times, with arriving nodes matched immediately and irrevocably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  This influential problem has been widely studied in various settings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', fractional/integral, sim-  ple/capacitated, unweighted/vertex-weighted/edge-weighted) [2, 11, 12, 29, 36, 39, 41, 50, 51, 60]  and has been a cornerstone of the online algorithms literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We study the abstract problem of online randomized rounding of fractional b-matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Here,  both online nodes and their edges’ associated fractions in a fractional b-matching are revealed online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (The fractional matching may be provided by a online fractional algorithm, or some machine-learned  advice—see discussion in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') An online dependent rounding scheme (ODRS) must decide  immediately and irrevocably, upon a node’s arrival, which of its edges to match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  In offline settings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' where all fractions and nodes are given upfront,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' numerous dependent rounding  schemes for b-matching are known with the following desirable properties: they (1) match each  edge with marginal probability equal to its fraction in the fractional b-matching (thus preserving  any linear objective,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' while being oblivious to this objective),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' and (2) guarantee strong negative  correlation properties (thus guaranteeing concentration and preservation of submodular objectives)  [19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' These dependent rounding schemes have been highly influential, and yielded  numerous applications in the approximation algorithms literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Unfortunately for online applications, not only are all above offline rounding schemes not imple-  mentable online, but online schemes satisfying Property (1) are provably impossible [17, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For  example, in recent work [17], Buchbinder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' asked what additional constraints can be imposed  on the fractional matching to allow for Property (1) to be satisfied exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In this work, guided by  the observation that approximate satisfaction of Property (1) would still allow us to approximately  preserve linear objectives (with the ensuing applications), we study the natural complementary  question: How well can one approximate Property (1)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Capturing this objective, we introduce the  notion of an oblivious online rounding ratio, defined generally as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' An ODRS A for a maximization problem has (oblivious) rounding ratio α ∈ [0, 1]  if, when (batches of) elements e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , en are revealed online with their associated fractions x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , xn,  with ⃗x a valid fractional solution, A outputs an integral solution I satisfying  Pr[ei ∈ I] ≥ α · xi  ∀i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  This question has been intensely studied for matchings in the context of one algorithmic paradigm:  online (batched) contention resolution schemes (OCRSes), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', [37, 38, 43, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' There, each arriving  element ei is active independently with probability xi,2 and an OCRS outputs a feasible subset of  active elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' An ODRS can be obtained from an OCRS directly by simulating the activation  process for arriving elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' However, even for simple b-matching problems such as rank-1 uniform  matroids (selecting a single element), no rounding ratio beyond 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5 is possible, via standard connec-  tions between OCRSes and the prophet inequality problem [38, 43, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Better oblivious rounding  ratios are known (via different approaches) for simple bipartite matchings [28, 62, 65, 73], with the  current best ratio standing at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='526 [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Can we improve on the previous best oblivious rounding ratios (for b-matchings more generally)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Can we achieve the ratio of 1 − 1  e—ubiquitous in competitive ratios of online matching algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Can we beat this bound, at least for key special cases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  1As in the offline dependent rounding schemes of [44], the term dependent here underscores the need for random  choices to depend on each other, since independent coin tosses do not yield a valid high-value (b-)matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  2Batched OCRSes [38] only require independence between batches of elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', edges of an arriving node).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Our Contributions  We provide positive answers to the above questions, and give multiple applications of our new  ODRSes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We focus on the online rounding questions in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  One special case we consider is rounding b-matchings in a star graph with a single offline node,  also known as rounding uniform matroids, or level sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Herein, a sequence of fractions arrive online  over time and the goal is to round them to {0, 1} while preserving their marginals and the sum of  every prefix (up to floors or ceilings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' As we show later, this is a useful subroutine for rounding  b-matchings in arbitrary bipartite graphs (and other applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Our first contribution is an online level-set rounding algorithm breaking the ratio of 1 − 1  e  that is best possible via (even offline) CRSes [24, 58], attaining an oblivious rounding ratio of  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We thus provide an online counterpart to the well-known offline algorithm of Srinivasan [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Indeed, as we show, our algorithm’s output distribution matches the offline algorithm’s output  distribution, together with its useful properties, including strong negative correlations (“Strong  Rayleigh" property, see Section 2) implying e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', preservation of monotone submodular objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2 (Informal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' There exists an online level-set rounding algorithm with rounding  ratio of one, and strong negative correlation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In arbitrary bipartite graphs, an ODRS with rounding ratio of one is impossible, even for simple  matching, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We further rule out any better than 2  √  2 − 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='828 rounding ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  On the positive side, we show that combining our online level-set algorithm with the single-item  offline contention resolution scheme (CRS) of Feige and Vondrák [40] yields a simple ODRS with  rounding ratio of 1 − 1/e ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Our main result breaks this ubiquitous bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' There exist bipartite matching (b-matching) ODRSes with rounding ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='652  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='646).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' No such ODRS has rounding ratio greater than 2  √  2 − 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Extensions and Applications  Our ODRSes and the techniques underlying them have a number of applications and extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We briefly discuss these here, deferring detailed discussions to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Online edge coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Our ability to round fractional matchings has implications to online edge  coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For this problem, it is known that the greedy algorithm’s competitive ratio of 2 is tight  for graphs of low maximum degree ∆ = O(log n) [8], while better competitive ratios are known for  high-degree graphs under various arrival models, mostly for simple graphs [3, 6, 10, 27, 55, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For  multigraphs, positive results (competitive ratios smaller than 2) were only known under random-  order arrivals [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Addressing this gap between simple graphs and multigraphs, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1 we  extend known reductions from α-competitive edge coloring (α ≥ 1) to computing online matchings  that match each (in this case, parallel) edge with probability  1  α∆ [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We thus show that a rounding  ratio of 1/α implies a competitive ratio of α + o(1) for edge coloring multigraphs (and is, in fact,  equivalent to it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Combined with our rounding algorithms, this yields the first positive result for  online edge-coloring in multigraphs under adversarial arrivals, achieving a competitive factor of  e/(e − 1) − Ω(1) in bipartite multigraphs, answering a question of Schrijver [28, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' VI].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Another application of our techniques concerns the online stochastic  bipartite weighted matching problem, much studied in the online Prophet-Inequality literature  [35, 38, 42, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (See Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2 for formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') Recently, Papadimitriou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' [62] initiated  2  the study of efficiently approximating the optimum online algorithm for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' They showed  that it is pspace-hard to approximate the optimum algorithm within some universal constant β < 1,  and provided a polynomial-time online algorithm that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='51-approximates this online optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (In  contrast, the best competitive ratio, or approximation of the offline optimum, is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5 for this problem  [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') The bound of [62] can be improved to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='526 using ideas in [65] and was significantly improved  to 1 − 1/e ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='632 by Braverman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' All three results are achieved via online rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Extending our online rounding approach of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3 to such stochastic settings, we improve on  the recent bound of [16] and break the ubiquitous bound of 1 − 1/e, providing an online polytime  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='652-approximation of the optimal (computationally-unbounded) online algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Algorithms with predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Our ODRS of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3 finds a direct application in the bur-  geoning area of online algorithms with advice (see the survey [61] and site [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Here, an online  algorithm is equipped with machine-learned (ML) predictions concerning the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For example,  many works show that for related problems, high-quality fractional solutions are efficiently learnable  and can guide integral algorithms, via online rounding schemes [56, 57, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Our ODRS immediately  yields the first online edge-weighted bipartite matching algorithm with predictions, getting a com-  petitive ratio of (1−1/e+Ω(1)) with sufficiently good predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In contrast, with no predictions,  only recently was it shown how to break the barrier of 1/2 for this problem (using free disposal)  [12, 39, 45], and 1 − 1/e + Ω(1) is impossible to achieve even in unweighted graphs [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Multi-Stage stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In multi-stage stochastic optimization, uncertain pa-  rameters are stochastic but their distributions get refined over time: actions in earlier stages are  cheaper but perhaps less accurate due to the stochasticity, while actions in later stages are more  expensive but more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (See the survey [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') A basic example is the (hyper-)graph covering  problem, where sets (hyperedges) are revealed in k stages, and must be covered by bought elements  (nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Swamy and Shmoys [69] presented a 2k-approximation for this problem, later improved to  2, matching the offline hardness of the non-stochastic single-stage problem, in [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Building on the  framework of [67, 69], in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1 we generalize this result to allow for sets with muti-coverage  demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Our algorithms have approximation ratio of 2 (and tending to one in some settings), and  these ratios hold w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', building on our algorithm’s strong negative correlation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Fairness and diversity in allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Building on the previous sentence, an even-stronger  negative correlation property—negative association—helps our rounding guarantee that its resultant  intersectional unfairness [18] (if any) is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Continuing on the theme of ML advice, we prove that  our rounding algorithm produces online allocations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', of vaccines as they are produced) from ML  advice such that these allocations do not impact multiple intersectional groups unfairly, with high  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Furthermore, our online-rounding algorithm only needs O(log n) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  we leverage this and the fact that diversity/fairness models often lead to monotone submodular  objectives which work very well with our rounding scheme, in order to develop logarithmic-space  streaming algorithms that act on ML advice, in order to produce provably-diverse search results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  Techniques  In this section we highlight some of the key ideas that allow us to achieve our main results, namely  Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3, deferring detailed discussions of applications to later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  For level-set rounding, we provide a simple ODRS using limited memory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', both online and  streaming) – only remembering how many edges/elements were matched/taken – that achieves a  rounding ratio of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Surprisingly, we show via a coupling argument that this online algorithm’s  distribution is identical to that of (offline) pivotal sampling, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Srinivasan sampling [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This  allows us to inherit this well-studied algorithm’s concentration properties [15, 31, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In particular,  3  this implies that our online algorithm’s output distribution satisfies the Strong Rayleigh property (see  Section 2), which in turn implies a slew of negative correlation properties, useful for our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  For our impossibility result, we rely on a Ramsey-theoretic probabilistic lemma, whereby in any  sufficiently large set of binary variables, some (near-)positive correlation is inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' An extension  of this lemma (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3), of possible independent interest, rules out algorithmic approaches based  on guaranteeing strict negative correlation between offline nodes’ matched statuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  ODRSes for b-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Our first ODRS for b-matchings more broadly is obtained by interleav-  ing our level-set algorithm with repeated invocations of the offline single-item contention resolution  scheme (CRS) of Feige and Vondrák [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (This is, to the best of our knowledge, the first online al-  gorithm to use offline CRSes in such a black-box manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') Specifically, we run independent copies of  our level-set algorithm, one per offline node, thus having each offline node i “bid” for arriving online  node t independently of other offline nodes with probability xi,t, while preserving the b-matching  constraints of the offline nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' To preserve online nodes’ matching constraints, we then use the  single-item offline CRS of [40] at each time t: this CRS guarantees for such product distributions  that t is allocated to at most one buyer, with every buyer i being allocated item t (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', (i, t) being  matched) with probability at least (1 − 1/e) · xi,t, thus yielding a rounding ratio of 1 − 1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  To improve on the above, we first note that the (1 − 1/e) · xit bound is loose if any of the xit  fractions are bounded away from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Indeed, this 1 − 1/e factor is of the form  �  1 −  �  j  (1 − xit)  �� �  i  xi,t ≥ 1 − 1/e,  where the inequality is loose if any xit is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' To beat this 1−1/e bound, we therefore simulate the  existence of a large xit, by grouping offline nodes via a bin-packing heuristic, and letting at most one  node per group (bin) B bid for t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The obtained distribution over biding offline nodes is no longer  a product distribution, but it is strongly negatively correlated, and even negatively associated (see  Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This allows us to achieve the same 1 − 1/e bound using CRSes for negatively correlated  distributions of Bansal and Cohen [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover, this grouping allows us to beat the 1−1/e ratio in  some cases: In a very concrete sense, it results in effectively a single aggregated offline node bidding  with large probability �  i∈B xi,t, thus allowing us to beat the bound of 1 − 1/e, if much grouping  occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Furthermore, thinking of bids as “costing” the offline node future matching opportunities, we  can even beat the bound of 1−1/e after giving nodes in a group a “group discount”, and having them  bid with lower probability than xi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This discounting then leaves offline nodes with more budget  (probability of not being matched), thus allowing us to charge these nodes more at later times: in  particular, when these offline nodes are not grouped, we charge them an “individual markup”, by  having them bid with higher probability than xi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This then increases the matching probability of  those t for which little grouping occurs to beyond 1 − 1  e, and the improved rounding ratio follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3  Further related work  A natural approach to round a feasible fractional solution ⃗x is to activate each element e indepen-  dently w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' xe and then pick a feasible subset I of active elements, such that Pr[e ∈ I] ≥ α · xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The latter is obtained by contention resolution schemes (CRSes), first studied in offline settings for  single-item problems (rank-one matroids) by Feige and Vondrák [40] (see Section 2), and later for  arbitrary matroids by Chekuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This approach was then extended to online settings by  Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' [43], and has since become a bedrock of stochastic online problems, from prophet  inequalities [43], secretary problems [33, 34], posted-price mechanisms [21, 52] and sequential pricing  [64], algorithmic contract design [9] and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Unfortunately, for bipartite matching ODRSes, using  4  OCRSes (letting each i bid for t with probability xi,t independently of all prior bids, including i’s) is  of limited use: such OCRS-based OCRSes have rounding ratio better at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5 even for rank-one  matroids, by connections to the prophet-inequality problem [38, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Going beyond independent  bids for each pair (i, t) is therefore crucial to obtain high oblivious rounding ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Another online matching rounding approach is provided by the recent exciting literature on  Online Correlated Selection (OCS), first introduced in the groundbreaking paper of Fahrbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  [39] (and its predecessor by Huang and Tao [48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Specifically, the OCSes of Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' [45] underlie  many rounding-based online algorithms for edge-weighted matching [45], fair matching [46] and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='d  matching [47, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This powerful machinery, however, does not provide oblivious rounding guaran-  tees, but rather per-offline-vertex guarantees, and so is inapplicable for our problem applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  2  Preliminaries  Problem Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In the online b-matching rounding problem, an a priori unknown bipartite  graph G = (L, R, E) is revealed online, together with fractions on the edges incident to the arriving  edge’s vertices that satisfy the b-matching constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In more detail, at each time t an online vertex  t ∈ L arrives, together with fractions xi,t assigned to its edges to offline neighbors i ∈ R = [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The  fractions xi,t ∈ [0, 1], are promised to satisfy the (fractional) b-matching constraints, namely each  vertex v ∈ (L ∪ R) has fractional degree �  e∋v xe at most bv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By splitting each online node t with  capacity bt into bt online unit-capacity nodes with identical edge sets and fractions xit/bt for each  copy of edge (i, t), we can assume WLOG that all online nodes have unit capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Following arrival  t, we must decide, immediately and irrevocably, which edges of t to add to the output b-matching  M, while satisfying the (integral) b-matching constraints of matching each vertex v no more than  bv times, and striving for a large oblivious rounding ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Our ODRSes for b-matchings will repeatedly invoke offline contention resolution schemes (CRSes),  whose guarantees for product distributions, due to Feige and Vondrák [40], and arbitrary distribu-  tions, due to Bansal and Cohen [7], are given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (For completeness, we provide a self-contained  proof for the general case in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=')  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let D : 2[n] → R≥0 be a distribution over subsets of [n], with its support denoted by  supp(D) = {S ⊆ [n] | PrR∼D[R = S] ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, there exists a randomized algorithm CRS(R,⃗v)  which on input set R ∼ D and vector ⃗v ∈ Rn, outputs a subset O ⊆ R of size |O| ≤ 1 satisfying  Pr[i ∈ O] = vi · min  S⊆[n]  Pr[R ∩ S ̸= ∅]  �  i∈S vi  ∀i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The algorithm runs in time polynomial in n if D is a product distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Otherwise, it runs in  time poly(n, T), where T ≥ |supp(D)| is the time to compute and write down the support of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Negative association and strong Rayleigh properties  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Random variables X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Xn are negatively associated (NA) if for any two disjoint  index sets I, J ⊆ [n], I∩J = ∅, and two functions f : R|I| → R and g : R|J| → R both non-decreasing,  E[f(Xi : i ∈ I) · g(Xj : j ∈ J)] ≤ E[f(Xi : i ∈ I)] · E[g(Xj : j ∈ J)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  By a simple inductive argument, negative association implies negative cylinder dependence [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For any real NA r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='s X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Xn and reals x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , xn, it holds that  Pr  � �  i  (Xi ≥ xi)  �  ≤  �  i  Pr[Xi ≥ xi]  and  Pr  � �  i  (Xi ≤ xi)  �  ≤  �  i  Pr[Xi ≤ xi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  5  The following family of NA distributions is due to Dubhashi and Ranjan [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Xn are binary r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='s with �  i Xi ≤ 1 always, then they are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  More elaborate NA distributions can be obtained via simple NA-preserving operations [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' NA is closed under products and under disjoint non-decreasing function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  That is, if ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Xn) is NA, then:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' if ⃗Y = (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Ym) is NA and ⃗Y is independent of ⃗X, then (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Xn, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Ym) are NA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (f1(Xi : i ∈ I1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , fk(Xi : i ∈ Ik)) are NA, for any concordant functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', all non-  decreasing or all non-increasing) f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , fk and disjoint sets I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Ik ⊂ [n], Ij∩Ik = ∅ ∀j ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Negative association implies many more useful properties, most notably the applicability of  Chernoff-Hoeffding type tail bounds [32] and submodular stochastic dominance (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  An even stronger negative correlation property than NA is the strong Rayleigh property (SRP)  (see Appendix A for a precise definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This property implies NA (even under conditioning), as  well as concentration of any Lipschitz function [63], and numerous other useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  3  Online Level-Set Rounding  In this section we introduce our online level-set rounding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' As its analysis will involve  coupling with the offline level-set rounding algorithm of Srinivasan [66], we first start by revisiting  this algorithm and its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Offline level-set rounding  Here we consider an offline procedure due to [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  This algorithm, on input ⃗x ∈ [0, 1]n with  integer sum �  i xi ∈ Z (possibly by adding a dummy value) and partial sums sj := �  i≤j xj,  outputs a set S ⊆ [n] with characteristic vector ⃗X given by Xi := 1[i ∈ S] with partial sums  Sj := �  i≤j Xi = |S ∩ [j]| satisfying the following properties for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (P1) (Marginals) E[Xi] = xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (P2) (Rounding) Si ∈ {⌊si⌋, ⌈si⌉}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Property (P1) and linearity of expectation imply that E[Si] = si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Property (P2) requires that  Si always equals this expectation, “up to rounding”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In addition, we require the following strong  negative correlation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (P3) (Strongly Rayleigh) The joint distribution ⃗X is strongly Rayleigh (and thus, also NA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Our objective will be to design an online algorithm satisfying the above, but for now we revisit  the offline algorithm of [66] and prove that when run with a certain specific way of choosing the key  indices i1 and i2 (see Algorithm 1), then it satisfies the above three properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  As usual, “min(S)" in Algorithm 1 refers to the minimum element of a nonempty set S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' the  index i2 in Algorithm 1 is always well-defined since �  i xi ∈ Z and, as we shall see, �  i yi = �  i xi  always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We emphasize that the algorithm of [66] allows much liberty in the choice of i1 and i2, and  that our specific choice of i1 and i2 is required to satisfy Property (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Also, the random choice in  each call to STEP is independent of the random choices of the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  6  Algorithm 1 Offline Level-Set Rounding  1: Initialize ⃗y ← ⃗x  2: while frac(y) := {i | yi ̸∈ {0, 1}} ̸= ∅ do  3:  Let i1 ← min(frac(y))  4:  Let i2 ← min(frac(y) \\ {i1})  5:  STEP(yi1, yi2)  6: Output S := {i | yi = 1}  Algorithm 2 STEP(A, B)  ⊲ modifies inputs  1: if A + B < 1 then  2:  (A, B) ←  �  (A + B, 0)  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  A  A+B  (0, A + B)  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  B  A+B  3: else  ⊲ A + B ∈ [1, 2)  4:  (A, B) ←  �  (1, A + B − 1)  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  1−B  2−A−B  (A + B − 1, 1)  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  1−A  2−A−B  That Algorithm 1 terminates and outputs a random set satisfying Property (P1) is known [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Nonetheless, for completeness, we provide a proof of this fact in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In the same appendix  we prove by induction on the number of invocations of STEP() that every prefix sum is preserved  with probability one, up to rounding, and so Algorithm 1 satisfies Property (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let sj := �  i≤j xi and Y t  j := �  i≤j yt  i, where yt  i is the value yi after t applications of  STEP in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, Y t  j ∈ [⌊sj⌋, ⌈sj⌉] for all j and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Strong negative correlation (Property (P3)) was proven in [15] as long as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', we use some fixed  choice of i1 and i2 as in Algorithm 1:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let Xi := 1[i ∈ S] be the indicators for i being in the output set of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Then, ⃗X = (X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Xn) is a family of strongly Rayleigh random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  To conclude, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Algorithm 1 satisfies properties (P1) (P2) and (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  The online level-set rounding algorithm  We now turn to providing an online counterpart to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Here, the input is a sequence of  reals, x1, x2, · · · ∈ [0, 1], revealed in an online fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Algorithm 3 maintains a set S ⊆ N (initially  empty), and decides at time t (when xt is revealed) whether to add t to its output set S, immediately  and irrevocably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let Xt := 1[t ∈ S], and let St := �  t′≤t Xt′ denote the cardinality of S right after  time t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', St = |S ∩ [t]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Finally, let st := �  t′≤t xt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We will show that Algorithm 3 satisfies  properties (P1), (P2) and (P3) given by its offline counterpart, Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Algorithm 3 Online Level-Set Rounding  1: Initialize S ← ∅  2: for arrival xt (at time t ≥ 1) do  3:  add t to S with probability pt :=  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  0  |S| = ⌈st⌉  1  |S| < ⌊st⌋  xt  ⌊st−1⌋+1−st−1  |S| = ⌊st⌋ = ⌊st−1⌋  st−⌊st⌋  st−1−⌊st−1⌋  |S| = ⌊st⌋ > ⌊st−1⌋ and st−1 ̸= ⌊st−1⌋  0  else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  4: Output S  The random bits used to make the random choice in each step of Algorithm 3 are independent  of the random bits used in the past: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', while pt depends on St−1, the random bits used to generate  the event of probability pt in step t, are independent of the random bits of past steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3  Coupling the online and offline algorithms  Here we prove that Algorithm 3 satisfies properties (P1), (P2) and (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Proving properties (P1),  (P2) directly is not too hard (see Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3  Proving Property (P3), on the other hand, is  slightly more involved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' we do so now, using that the much-simpler properties (P1) and (P2) hold  for both the offline and online algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' To prove that the last property holds, we show that both  algorithms induce the same distribution over outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We outline a proof here, deferring a complete  proof to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Fix a vector ⃗x ∈ [0, 1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let D and D′ denote the probability distributions on {0, 1}n  induced by the online and offline algorithms run on ⃗x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, D = D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof (Sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We prove by induction on t ∈ [n] that the following holds:  ∀(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , bt−1) ∈ {0, 1}t−1, Pr  D [Xt = 1  �� (∀i < t, Xi = bi)] = Pr  D′[Xt = 1  �� (∀i < t, Xi = bi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (1)  The (strong) inductive hypothesis, whereby Equation (1) holds for all t′ < t, clearly implies that for  all (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , bt) ∈ {0, 1}t−1 we have that PrD[(∀i < t, Xi = bi)] = PrD′[(∀i < t, Xi = bi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Similarly,  Equation (1) holds for all t ≤ n iff the two distributions are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The proof then proceeds by  considering the different possibilities for the prefix sums st−1 and st, which for the online algorithm  directly determine the above conditional probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' These conditions also determine the offline  algorithm’s performance up to the (t − 1)st invocation of STEP(), from which a careful analysis  implies that the conditional probabilities of the offline and online algorithms are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4 combined with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3 immediately implies the desired properties of our Algo-  rithm 3, as these properties are determined by the distribution over output sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Algorithm 3 is an online algorithm satisfying properties (P1), (P2) and (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  4  Rounding b-Matchings Online  In this section we address the ODRS problem in bipartite b-matching instances more broadly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We start with a simple algorithm, combining our ODRS for uniform matroids (Algorithm 3)  with repeated invocations of an offline CRS (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Warm-up: 1 − 1/e rounding ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We consider the following simple algorithm: Run, for each  offline node i, an instance of Algorithm 3, applied to xi1, xi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' When the i-th copy of Algorithm 3  fixes Xit = 1, we say that i bids for online node t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Out of the set of bidders (potential buyers) Pt,  we then pick at most one i ∈ Pt to match t to, using the offline contention resolution scheme of  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1, guaranteeing each node i a probability of xi,t · minS⊆[n]  Pr[S∩Pt̸=∅]  �  i∈S xi,t  of being matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  This step runs in polytime, by independence of the bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Since Algorithm 3 is an online algorithm,  so is the resultant algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover, since the output of Algorithm 3 satisfies the b-matching  constraints, and t is matched to at most one neighbor by the CRS, this algorithm’s output satisfies  the b-matching constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Finally, for each offline node set S ⊆ [n], by independence of the bids,  the Taylor expansion of exp(−x) and convexity (see Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5), we have  Pr[S ∩ Pt ̸= ∅] = 1 −  �  i∈S  (1 − xit) ≥ 1 −  �  i∈S  exp(−xit) ≥  �  1 − 1  e  � �  i∈S  xit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (2)  Therefore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1, this ODRS matches each edge (i, t) with probability at least (1−1/e)·xi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  3In fact, Algorithm 3 can be shown to be the only online algorithm satisfying both these simple properties if we  further restrict it to only storing in memory the partial sums st−1 and St−1 at the beginning of step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  8  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' There exists a polytime ODRS for bipartite b-matching with rounding ratio 1 − 1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  One natural approach to improve the above algorithm’s rounding ratio would be to attempt to  negatively correlate the probability of different offline nodes to have previously bid, thus increasing  Pr[S ∩ Pt ̸= ∅] for all offline sets S ⊆ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Unfortunately, for a sufficiently large number of offline  nodes, (near-)positive correlation is, generally, unavoidable: see Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3 for a proof of such a  Ramsey-theoretic statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Instead, to achieve a better rounding ratio, we explicitly negatively  correlate the bids of previously-non-bidding offline nodes when they do bid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  The improved ODRS: Overview and intuition  In this section we outline our improved ODRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We focus here and in most subsequent sections on  simple matchings, for clarity’s sake, deferring an extension to b-matchings to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  A bad example, and a false start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Consider a graph with a single online node t with xit = 1  n for  all n offline nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In this case, under independent bids, t gets no bids with probability (1− 1  n)n ≈ 1  e,  and so one of the n offline nodes must get matched with probability no greater than (1− 1  e) · 1  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For  this simple example a rounding ratio of one is possible, by correlating bids for t: if we pick exactly  one neighbor i to bid with probability xit, then each offline node both bids and buys (is matched  to) t with probability xit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Unfortunately, this approach is impossible to implement in general: if  we let each offline node i bid for at most one time t (recall that this is a simple matching instance)  with probability xi,t, then, for sit := �  t′<t xit′ the fractional degree of i at time t, we must have i  bid for t with probability pit := xit/(1 − sit), and yet �  i pit may exceed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Bin-packing low-degree nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Our first step is to adopt a partial implementation of the above  approach, by grouping offline nodes into groups B for which �  i∈B pit ≤ 1, and using one Uni(0, 1)  variable to offer at most one bid per group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Specifically, if we denote the set of low-degree neighbors  of t by Lt := {i ∈ N(t) | sit ≤ θ} for some threshold θ ∈ [0, 1] that we will optimize later, we will  pack low-degree neighbors into bins using a greedy bin-packing heuristic such that each bin B has  �  i∈B pi,t ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover, each such bin B (but one) has �  i∈B pi,t ≥ 1/2 and hence has high total  x-value of �  i∈B xi,t ≥ 1−θ  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (Here we use that pi,t = xi,t/(1 − si,t) ≤ xi,t/(1 − θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') Using negative  association arguments (see Section 2), we can show that this approach essentially replaces multiple  bids of small x-value with bids of x-value equal to the small bids’ sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In particular, if the total  x-value of low-degree neighbors i ∈ Lt is high, this results in minS⊆[n]  Pr[S∩Pt̸=∅]  �  i∈S xi,t  ≥ 1 − 1  e + Ω(1),  and a rounding ratio greater than 1 − 1  e, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Group discounting and individual markup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' It remains to address the challenge of increasing  Pr[S ∩Pt ̸= ∅] if the total x-value of low-degree nodes Lt is small, and so very little grouping occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  For this, we note that from the perspective of every offline node i, a time step t has: (1) a value  (matching probability), and (2) a cost (bidding probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The latter is indeed a cost, since we  do not allow offline nodes to bid more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The value at each time is monotone increasing in  offline nodes’ costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Now, when we use grouping we decrease the contention faced by the CRS, as it  now faces fewer bidding nodes, due to each bin only providing one bid (or none).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We can therefore  safely decrease the bidding probability (cost) in the case that much grouping occurs, giving the  grouped nodes a group discount, while still obtaining a rounding ratio greater than 1 − 1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The  upshot of the group discount is that offline nodes now have a higher chance of not having bid before  later time steps where they are not grouped, in which case such individual bidders can pay an  individual markup, allowing us to guarantee that in this case too minS⊆[n]  Pr[S∩Pt̸=∅]  �  i∈S xi,t  ≥ 1− 1  e +Ω(1),  with the ensuing 1 − 1  e + Ω(1) rounding ratio following from the offline CRS of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  The core of the improved ODRS  In this section we introduce our improved ODRS’ core subroutine, Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (The parameter  θ ∈ [0, 1] will be optimized later, and for now it is safe to think of the input vectors x, v as both  equaling the input fractional matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') At each time step t, the algorithm groups low-fractional-  degree offline nodes, and then all remaining nodes, into buckets, using the classic first-fit bin-packing  heuristic,4 with each offline node i having a size of  xi,t  1−si,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This is the conditional probability with  which i should bid for t if i is free, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', has not bid before, to guarantee a marginal bidding probability  of xi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (This follows from our online level-set rounding algorithm’s probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') Items are grouped  into bins of size one, and then at most one offline node bids per bin (lines 9-12), resulting in the  set of candidates, Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The free candidates Pt = Ct ∩ F then bid for t, who is then matched to (at  most) one bidding neighbor i ∈ Pt, chosen by a CRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Algorithm 4 ODRS-core(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' v)  1: M ← ∅  2: F ← [n]  ⊲ Free offline nodes  3: for each time t do  4:  for each i ∈ [n] do  5:  si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t ← �  t′<t xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t′  6:  Bt ← FirstFit  ���  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t  1−si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t  � ��� i ∈ [n],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t ≤ θ  ��  ⊲ Bucketing low-degree nodes  7:  Bt ← Bt ∪ FirstFit  ���  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t  1−si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t  � ��� i ∈ [n],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t > θ  ��  ⊲ Bucketing high-degree nodes  8:  Ct ← ∅  ⊲ Candidates  9:  for each B ∈ Bt do  10:  U ∼ Uni[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 1]  11:  if U ≤ �  i∈B  xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t  1−si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t then  12:  Ct ← Ct ∪ mini  �  i ∈ [k]  ���� U ≤ �  j∈B  j≤i  xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t  1−sj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t  �  13:  Pt ← F ∩ Ct  ⊲ Bidders = free candidates  14:  F ← F \\ Pt  ⊲ Bidders cease being free  15:  O ← CRS(Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' {vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t}i)  ⊲ Contention Resolution  16:  if |O| = 1 then  17:  M ← M ∪ {(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' t) | i ∈ O}  18: Output M  First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' we note that Algorithm 4—including the bin-packing steps that require  xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t  1−si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t ≤ 1—is  well-defined provided �  t xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t ≤ 1 for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' since this implies that xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t ≤ 1 − �  t′≤t xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t′ = 1 − si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Next, by construction, each offline node bids (and is thus matched) at most once, while the CRS  guarantees that each online node is matched at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' To summarize, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Algorithm 4 is well-defined and outputs a matching if �  t xi,t ≤ 1 ∀i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We turn to analyzing the algorithm’s rounding ratio, starting with the following lemma asserting  that if much grouping occurs, then most bins have high x-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For each time t, at most one bin B ∈ Bt at Line 6 has �  i∈B xi,t < 1−θ  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  4This algorithm maintains a sorted list of bins with each bin B containing items of overall size at most one,  �  i∈B si ≤ 1, and for each item i in order, adds i to the first (possibly newly-opened) bin that has enough room left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  10  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' A well-known property of the First-Fit bin packing heuristic is that at most one bin in the  output of this algorithm is less than 1  2 full (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', [72, 74]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Therefore, since si,t ≤ θ for each  offline node i in a bin B ∈ Bt computed in Line 6, each such bin B (barring perhaps one) has  1  2 ≤  �  i∈B  xi,t  1 − si,t  ≤  �  i∈B  xi,t  1 − θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The following lemma, combined with Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3, provides a guarantee no worse than the independent-  proposals approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Indeed, as we will show shortly, it implies a better rounding ratio if much  grouping of low-degree neighbors occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For any time t and set of offline nodes S ⊆ [n], Algorithm 4 satisfies  Pr[S ∩ Pt ̸= ∅] ≥ 1 −  �  B∈Bt  �  1 −  �  i∈B∩S  xit  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let Ft denote F at time t, and let Ct and Pt = Ct ∩ Ft be as in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Next, let  Ci,t = 1[i ∈ Ct] indicate whether i is a candidate at time t, and define Fi,t and Pi,t similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  By independence of Ci,t and Fi,t and since Pr[Ci,t] =  xi,t  1−si,t and Pr[Fi,t] = 1 − si,t (provable by a  simple induction on t ≥ 0), we have that Pr[Pi,t] = xi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Therefore, by linearity of expectation and  �  i∈B Ci,t ≤ 1, we have that Pr[Pt ∩ S ∩ B = ∅] = 1 − �  i∈B∩S xi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' It remains to show that the  events [Pt ∩S ∩B = ∅] for different B are negative cylinder dependent, giving the desired inequality  Pr[Pt ∩ S = ∅] = Pr  � �  B∈Bt  (Pt ∩ S ∩ B = ∅)  �  ≤  �  B∈Bt  Pr[Pt ∩ S ∩ B = ∅] =  �  B∈Bt  �  1 −  �  i∈B∩S  xi,t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  To this end, we will show that the events [Pt ∩ S ∩ B = ∅] are negatively associated (NA),  which by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3 yields the required above inequality, and hence yields the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' First, by  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4, for any bin B′ at time t′ < t, the binary variables Ci,t′ for all nodes i ∈ B′ ∩ S, whose  sum is at most one, are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover, the distributions over different bins and different time steps  are independent, and so by closure of NA under products (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5), all Ci,t′ are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Therefore,  the variables Fi,t = 1 − maxt′<t Ci,t′ are non-increasing functions of disjoint NA variables Ci,t′, and  so by closure of NA under such functions (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5), the variables Fi,t are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover, the  variables Ci,t are independent from the variables Fi,t (who are functions only of Ci,t′ for all t′ < t),  and consequently, by another application of closure of NA under products (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5), all variables  {Fi,t, Ci,t | i ∈ S} are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Finally, since 1[Pt ∩ S ∩ B = ∅] = 1 − �  i∈B∩S Ci,t · Fi,t are monotone  decreasing functions of disjoint NA variables (different bins B), then by closure of NA under such  functions (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5), the variables 1[Pt ∩ S ∩ B = ∅] are NA, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  The improved ODRS  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4 and Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3 allow us to argue that if much of t’s fractional degree is contributed by  neighbors i with low fractional degree, si,t ≤ θ, then Pr[S ∩ Pt ̸= ∅] ≥ 1 − 1  e + Ω(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1,  this allows to match edges of such nodes t with probability greater than (1 − 1  e) · xi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Indeed,  we can afford to decrease the fractions xi,t of low-degree nodes (that may be grouped effectively)  and still retain a similar 1 − 1  e + Ω(1) rounding ratio for such t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In return, we can afford to  increase the fractions xi,t of nodes of high degree (who might not be grouped).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' More precisely,  we consider the following transformation to x, capturing this aforementioned group discount and  11  individual markup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For θ =  δ  ǫ+δ, with ǫ, δ ∈ [0, 1] to be optimized shortly, we consider the following  assignment ˆx : E → R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  ˆxi,t = xi,t · (1 − ǫ) +  � max(θ, si,t+xi,t)  z=max(θ,si,t)  (ǫ + δ) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (3)  Intuitively, ˆx decreases/increases the x-value of (parts of) edges of i until/after i has fractional  degree θ, by a (1 − ǫ) and (1 + δ) factor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' So, for example, ˆxi,t ≥ xi,t · (1 − ǫ) if si,t ≤ θ  and ˆxi,t = xi,t · (1 + δ) if si,t > θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' While ˆx is not a fractional matching, since online nodes t may  have �  i ˆxi,t > 1, this assignment does satisfy the fractional degree constraints of offline nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For every offline node i ∈ [n] and time t, we have that ˆsi,t := �  t′<t ˆxi,t′ ≤ si,t ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If si,t ≤ θ, trivially ˆsi,t = si,t · (1 − ǫ) ≤ si,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Otherwise, by our choice of θ =  δ  ǫ+δ, we have  ˆsi,t = si,t · (1 − ǫ) + (si,t − θ)+ · (ǫ + δ) = si,t · (1 + δ) − δ ≤ si,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Our ODRS, given in Algorithm 5, first scales the fractions as in Equation (3) (easily computable  online, as these ˆxi,t only depend on information known by time t) and feeds the resultant vectors ˆx, x  (playing the roles of x and v, respectively) into our core ODRS subroutine, Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5  and Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2, we have that Algorithm 5 is a well-defined online algorithm, outputting a valid  matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' It remains to analyze this algorithm’s rounding ratio, starting with the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Algorithm 5 ODRS(ǫ, δ, x)  1: θ ←  δ  ǫ+δ  2: Init ODRS-core(θ · (1 − ǫ), ˆx, x)  3: for each time t do  4:  for each i ∈ [n] do  5:  si,t ← �  t′<t xi,t′  6:  Compute ˆxi,t from (xi,t, si,t, ǫ, δ) as in Equation (3)  7:  Run next step of ODRS-core(θ · (1 − ǫ), ˆx, x)  8: Output M computed by ODRS-core(θ · (1 − ǫ), ˆx, x)  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If ǫ, δ ∈ [0, 1] guarantee that for all z ≥ 1−θ  2  =  ǫ  2(ǫ+δ),  fǫ,δ(z) := exp(−z · (1 + δ)) − (1 − z · (1 − ǫ)) ≥ 0,  (4)  then Algorithm 5 with parameters ǫ and δ has rounding ratio at least  1 − exp  �  −1 − δ + ǫ + δ  1 − ǫ  �  1 − ǫ  1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By construction of ˆxi,t, we have that ˆsi,t ≤ θ · (1− ǫ) iff si,t ≤ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Define Lt := {i | si,t ≤ θ} =  {i | ˆsi,t ≤ θ · (1 − ǫ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4 and the Taylor expansion of 1 − exp(−z), for any offline node  set S ⊆ [n] and time t, the set of bidders Pt computed by ODRS-core(θ · (1 − ǫ), ˆx, x) satisfies  Pr[S ∩ Pt ̸= ∅] ≥ 1 −  �  B̸⊆Lt  �  1 −  �  i∈B∩S  ˆxit  � �  B⊆Lt  �  1 −  �  i∈B∩S  ˆxit  �  ≥ 1 − exp  \uf8eb  \uf8ed−  �  i∈S\\Lt  xit · (1 + δ)  \uf8f6  \uf8f8 ·  �  B⊆Lt  �  1 −  �  i∈B∩S  xit · (1 − ǫ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  12  We now lower bound the ratio Pr[S∩Pt̸=∅]  �  i∈S xit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' To do so, we note that our lower bound for Pr[S ∩Pt ̸= ∅]  is a concave function in �  i∈S xit in the range [0, 1], and so our lower bound for the above ratio  is minimized by virtually increasing �  i∈S xit until �  i∈S xit = 1, by adding all i /∈ S to the set  and possibly adding dummy nodes J with sj,t = θ for all j ∈ J and �  J xjt = 1 − �  i xit ≥ 0  (see Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5 On the other hand, by Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3, every bin B ⊆ Lt but at most one bin B∗ has  �  i∈B ˆxi,t ≥ 1−θ(1−ǫ)  2  ≥ (1−θ)(1−ǫ)  2  , and hence �  i∈B xi,t ≥ 1−θ  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Combining the above, we get  Pr[S ∩ Pt ̸= ∅]  �  i xit  ≥ 1 − exp  \uf8eb  \uf8ed−  \uf8eb  \uf8ed1 −  �  i̸∈Lt  xit  \uf8f6  \uf8f8 · (1 + δ)  \uf8f6  \uf8f8 ·  �  B⊆Lt  �  1 −  �  i∈B  xit · (1 − ǫ)  �  ≥ 1 − exp  �  −  �  1 −  �  i∈B∗  xit  �  (1 + δ)  � �  1 −  �  i∈B∗  xit · (1 − ǫ)  �  ≥ 1 − exp  �  −1 − δ + ǫ + δ  1 − ǫ  �  1 − ǫ  1 + δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (5)  Above, the second inequality follows from the lemma’s hypothesis, in Equation (4), and the last  inequality follows from the following claim, whose proof by inspection of the function g is deferred  to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If ǫ, δ ∈ [0, 1], the function g(y) := 1−exp (− (1 − y) · (1 + δ))·(1 − y · (1 − ǫ)) satisfies  ∀y ∈ R :  g(y) ≥ g  �  ǫ + δ  (1 − ǫ)(1 + δ)  �  = 1 − exp  �  −1 − δ + ǫ + δ  1 − ǫ  �  1 − ǫ  1 + δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Finally, the rounding ratio of the ODRS then follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1 and Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6 provides a sufficient condition for ǫ, δ to (possibly) allow for greater-than-(1 − 1/e)  rounding ratios, though this condition requires satisfying infinitely many non-linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The following lemma, which follows from convexity of fǫ,δ and is proven in Appendix C, provides a  simpler condition that is sufficient to guarantee all the above infinitely-many constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let fǫ,δ(z) be as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If fǫ,δ(a) ≥ 0 and f ′  ǫ,δ(a) ≥ 0, then fǫ,δ(b) ≥ 0 ∀b ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Combining the above, we are finally ready to optimize the rounding ratio of Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Algorithm 5 run with (ǫ, δ) ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='0480, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='0643) achieves a rounding ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='8, the rounding ratio of Algorithm 4 with parameters (ǫ, δ) is at least  as high as 1 − exp  �  −1 − δ + ǫ+δ  1−ǫ  �  1−ǫ  1+δ, provided that fǫ,δ  � 1−θ  2  �  ≥ 0 and f ′  ǫ,δ  �1−θ  2  �  ≥ 0, where  1−θ  2  =  ǫ  2(ǫ+δ) and fǫ,δ is as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Thus, the algorithm’s rounding ratio is at least as high as  the value of the following program, when run with parameters given by its optimal solution (ǫ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Off-the-shelf numerical solvers yield a maximum value of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='652 at (ǫ, δ) ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='0480, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='0643).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  max  α = 1 − exp  �  −1 − δ + ǫ + δ  1 − ǫ  �  1 − ǫ  1 + δ  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  exp  �  −  ǫ  2(ǫ + δ) · (1 + δ)  �  − 1 +  ǫ  2(ǫ + δ) · (1 − ǫ) ≥ 0  − (1 + δ) · exp  �  −  ǫ  2(ǫ + δ) · (1 + δ)  �  + 1 − ǫ ≥ 0  ǫ, δ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  5Concretely: adding any value to �  i∈S xi,t corresponding to adding (part of) another (dummy) element to S  decreases the value, so concavity of the resulting function together with Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5 completes the proof of this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  13  Computational considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The algorithm implied by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='9, as stated, takes expo-  nential time, due to its reliance on the CRS of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1 for non-independent bid distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  As we show in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1, a slight downscaling of the x-values allows us to achieve essentially  the same rounding ratio, while also guaranteeing that the number of bidders at each time step is  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This allows us to compute the support of the distribution of Pt in polynomial time, which  by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1 then gives us the following polytime counterpart to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For any γ > 0, there exists an nO(1/γ)-time bipartite matching ODRS with rounding  ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='652 − γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  The b-matching ODRS  In this section we extend our ODRS from matchings to b-matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' To avoid repetition, we only  outline the changes needed in the algorithm and analysis, and defer the proofs to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The change to the core algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' As we did (implicitly) in Algorithm 4, we let the online  level-set algorithm dictate the marginal probabilities with which an offline node i should bid to t,  based on the number of times i has bid by time t, which we denote by Si,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (This corresponds  to St in Algorithm 3 run from the perspective of the offline node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') We process offline nodes at  different times t as follows, with the choice of probabilities guided by Algorithm 3 to guarantee that  E[Si,t+1 − Si,t] = xi,t, by Property (P1), and Si,t ∈ [⌊si,t⌋, ⌈si,t⌉], by Property (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  For nodes with ⌈st+1⌉ = ⌈st⌉ — whose fractional degrees both before and after time t lie in the  range [⌊si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' ⌈si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t⌉] — we group these offline nodes and have them bid (at most one bid per bin) as  in Algorithm 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' with the following changes: si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t is replaced by its fractional part,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t − ⌊si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t⌋: this is  done both in the bidding probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' so these items have size  xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t  1−(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t−⌊si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t⌋) when bin packing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' and  in the condition si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t ≤ θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' which are now replaced by the condition si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t − ⌊si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t⌋ ≤ θ (and similarly  for the condition si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t > θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' with θ to be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover, we add a candidate offline node  i ∈ Ct to the list of potential matches Pt (the bidders) if Si,t < ⌊si,t⌋ (corresponding to i ∈ F ∩ Ct  in Algorithm 4 for simple matchings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  For all offline nodes with ⌊si,t+1⌋ = ⌊si,t⌋+1, we place these nodes in singleton bins, and we have  them bid with probability if Si,t = ⌊si,t⌋, or with probability si,t+1−⌊si,t+1⌋  si,t−⌊si,t⌋  if Si,t = ⌈si,t⌉ = ⌊si,t+1⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Change to core ODRS’ analysis: Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3 and its proof hold unchanged for our extension of  the core ODRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4, too, holds unchanged, although the proof, deferred to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2,  is now slightly more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Specifically, the key ingredient, proving that the events [Pt ∩ S ∩ B]  over different bins B are negatively associated (NA) requires a proof that the variables Si,t are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Change to the improved ODRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In our ODRS for b-matchings, since offline nodes i for which  ⌊si,t+1⌋ = ⌊si,t⌋ + 1 are grouped into singelton bins, we wish to apply a markup to these nodes’  xi,t-values, if xi,t is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We therefore consider the following assignment, with 0 ≤ θ1 ≤ θ2 ≤ 1  satisfying θ2 − θ1 =  δ  ǫ+δ, to be chosen shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  ˆxi,t =  � si,t+1  z=si,t  (1 − ǫ) · 1 [⌊z⌋ ∈ [θ1, θ2)] + (1 + δ) · 1 [⌊z⌋ ̸∈ [θ1, θ2)] dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (6)  In words: this assignment decreases the part of xi,t corresponding to the parts of i’s fractional  degree that are bounded away from 0 and 1 (specifically, those in the range [θ1, θ2), and increases  the part of the fractional degree that is somewhat close to 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Our new ODRS then runs the  modified core ODRS with inputs θ := θ2 · (1− ǫ) + θ1 · (δ + ǫ) and ˆx, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The choice of the parameter  θ is taken to have ˆsi,t − ⌊ˆsi,t⌋ ≤ θ if and only if si,t − ⌊si,t⌋ ≤ θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  14  Change to the improved ODRS’ analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' First, Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5 generalizes as follows, arguing that  modified fractional degrees ˆsi,t = �  t′<t ˆxi,t′ fall within the same integer range for all (i, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For every offline node i ∈ [n] and time t, we have that ⌊ˆsi,t⌋ = ⌊si,t⌋ and ⌈ˆsi,t⌉ = ⌈si,t⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='11 and Property (P2) of Algorithm 3 imply that offline node i makes some Si,t ∈  [⌊si,t⌋, ⌈si,t⌉] many bids by time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' As i is not matched more time than it bids, this implies that the  extended ODRS outputs a valid b-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Next, the key lemma in the analysis of Algorithm 5, namely Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6, needs to be extended  slightly, and in particular the condition of Equation (4) needs to hold for all z ≥ max  �  1−θ2  2(1+δ), θ1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  To make this constraint as less restrictive as possible, we take θ1 and θ2 to guarantee the needed  θ2 − θ1 =  δ  ǫ+δ, and also satisfy  1−θ2  2(1+δ) = θ1, resulting in θ1 =  ǫ  (3+2δ)(ǫ+δ) and θ2 =  ǫ+3δ+2δ2  (3+2δ)(ǫ+δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This  modification of the range for which we require Condition (4) to hold will again allow us to argue that  for all bins B except for at most one bin B∗ (to be characterized shortly), we have the following:  1 −  �  i∈B  ˆxi,t ≤ exp  �  −  �  i∈B  xi,t(1 + δ)  �  (7)  Specifically, for bins containing a single offline node i with ⌊si,t+1⌋ = ⌊si,t⌋+1, we have the following:  If si,t+1 − ⌊si,t+1⌋ ≥ θ1, then  xi,t = si,t+1 − si,t ≥ si,t+1 − ⌊si,t+1⌋ ≥ θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Similarly, if si,t − ⌊si,t+1⌋ ≤ θ2, then  xi,t = si,t+1 − si,t ≥ ⌊si,t+1⌋ − si,t = 1 + ⌊si,t⌋ − si,t ≥ 1 − θ2 ≥  1 − θ2  2(1 + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In both cases, Condition (4) holding for all z ≥ max  �  1−θ2  2(1+δ), θ1  �  implies Equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In the  alternative scenario where si,t+1 −⌊si,t+1⌋ ≤ θ1 and si,t−⌊si,t⌋ ≤ θ2, we have that ˆxi,t = (1+δ)·xi,t,  and Equation (7) follows from the basic inequality 1 − z ≤ exp(−z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Finally, by Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3, for all  bins B containing offline nodes i with ⌊si,t⌋ = ⌊si,t+1⌋ and ˆsi,t ≤ θ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', si,t ≤ θ2), except for at  most one bin B∗, we have that  �  i∈B  xi,t · (1 + δ) ≥  �  i∈B  ˆxi,t ≥ 1 − θ  2  ≥ 1 − θ2  2  ,  where the last inequality, whereby θ ≤ θ2, is proven similarly to Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  So, Condition (4)  holding for all z ≥  1−θ2  2(1+δ) implies Equation (7) for all remaining bins other than B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The analysis  then proceeds as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6, yielding the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If ǫ, δ ∈ [0, 1] guarantee that for all z ≥ max  �  1−θ2  2 , θ1  �  =  ǫ  (3+2δ)·(ǫ+δ),  fǫ,δ(z) := exp(−z · (1 + δ)) − (1 − z · (1 − ǫ)) ≥ 0,  (8)  then the extension of Algorithm 5 to b-matchings with parameters ǫ and δ has rounding ratio at least  1 − exp  �  −1 − δ + ǫ + δ  1 − ǫ  �  1 − ǫ  1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Combined with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='8, the above lemma then yields our b-matching ODRS’ rounding ratio,  after solving the appropriate modification of the mathematical program used to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The extension of Algorithm 5 to b-matchings, run with (ǫ, δ) ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='0347, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='0425),  achieves a rounding ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  15  5  Applications  In this section we provide applications and extensions of our online rounding algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Mirroring  the exposition in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1, we start with the application of our online (b-)matching algorithms,  omitting the edge-weighted matching application with ML predictions, which follows directly from  our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Applications and extensions of the matching ODRS  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Online edge coloring (multigraphs)  We show here how to obtain the first online algorithm that achieves a better-than-2 competitive  factor for online edge-coloring in bipartite multigraphs under adversarial arrivals, specifically under  one-sided node arrivals, as studied in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We assume that upon arrival of each node, its neighboring  edges are revealed and must be colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' As already mentioned, for online edge coloring, it is known  that the greedy algorithm’s competitive ratio of 2 is tight for graphs of low maximum degree  ∆ = O(log n) [8], while better competitive ratios are known for high-degree graphs under various  arrival models, all for simple graphs [6, 10, 27, 55, 65], with the exception of the random-order result  of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We obtain our improved bounds for multigraphs by extending a reduction from online edge  coloring to “fair” matching, suggested by [27] for simple graphs (see also Saberi and Wajc [65,  Appendix B]), given in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We say that a random matching M in a multigraph with maximum degree ∆ is  α-fair if for each (parallel) edge e we have that Pr[e ∈ M] ≥  1  α∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  An α-fair algorithm with α = 1 gives the highest uniform lower bound on edges’ matching  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' An α-fair algorithm with other α (α > 1) therefore α-approximates this maximally-fair  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  With the above terminology, we can now state the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' An online α-fair matching algorithm yields an online (α + o(1))∆-edge-coloring algo-  rithm for n-node bipartite multigraphs with maximum degree ∆ = ω(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We defer a brief proof sketch of the above to the end of the section, and turn to discussing its  consequences and converses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' An ODRS with rounding ratio 1/α yields an online (α + o(1))∆-edge-coloring  algorithm for n-node multigraphs with maximum degree ∆ = ω(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Consider the fractional matching x assigning value xe = κ(e)  ∆  to the simple edge e with κ(e)  parallel copies in the multigraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (Note that �  e∋v κ(e) ≤ ∆ by definition, so this is indeed a  fractional matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') Applying an ODRS with rounding ratio 1/α to this fractional matching x  yields a randomized matching M that matches every (parallel) edge in the graph with probability  1  α∆, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', it yields an α-fair matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We note that there is in fact an equivalence between α-fair matching algorithms and α-competitive  edge-coloring algorithms, as “the converse” also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' An online α∆-edge-coloring algorithm A for n-node multigraphs with maximum  degree ∆ yields an ODRS with rounding ratio 1/α for fractional matchings x with xe · ∆ integral  for all e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  16  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Given a fractional matching x ∈ QE in a simple graph G = (V, E), using that xe·∆ is integral  for each edge e, we provide to A a multigraph H = (V, F) with each edge e having multiplicity  κ(e) = xe · ∆ in H, and then randomly sample one of the α∆ edge colors (matchings) M computed  by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This yields an ODRS with rounding ratio of 1/α, since for each edge e ∈ E we have that  Pr[e ∈ M] = κ(e) ·  1  α∆ = xe  α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  By the equivalence between edge coloring multigraphs and ODRSes for matchings, together  with our (upper and lower) bounds on the rounding ratio of such ODRSes, we obtain the following  bounds on the number of colors needed to edge-color multigraphs online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In n-node bipartite multigraphs with maximum degree ∆ under adversarial one-sided  vertex arrivals, there exists an online 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='533∆-edge-coloring algorithm, assuming ∆ = ω(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In  contrast, no (1/(2  √  2 − 2 + ǫ))∆ ≈ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='207 − O(ǫ))∆-edge-coloring exists even for ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We note in passing that all known positive results for online edge coloring under adversarial  arrivals follow from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2, and indeed from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3 [27, 55, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The reason most of  these results do not extend to multigraphs is that their ODRSes required their input fractional  matchings x to be of the form xe = 1  ∆, or more generally for xe = o(1) for each edge e (including its  multiplicities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The work of Saberi and Wajc [65] is a lone exception to this rule, and combining its  ODRS with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2 also yields an algorithm for multigraphs, but using as many as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='9∆ colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We conclude the section with a brief proof sketch of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof sketch of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We describe the algorithm in an offline setting first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let C be such  that C = ω(log n) and C = o(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (Such C exist, by our necessary assumption that ∆ = ω(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=')  Initially, the entire multigraph is uncolored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For some ∆/C many rounds, compute α · C colors  as follows: In the uncolored subgraph U, compute α · C many α-fair matchings M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , MαC,  and let these occupy the next αC colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Since every vertex has at most αC = o(∆) many edges  colored during a round, any vertex with degree close to ∆ at the beginning of the round has degree  ∆ − o(∆) by the end of the round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Therefore, since these Mi are α-fair matchings, and so match  each (parallel) edge with probability at least  1  α·∆, by standard concentration bounds, all vertices  that have degree close to ∆ have some C · (1 − o(1)) many edges colored during a round (with  high probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This allows us to compute a high-probability upper bound of on the resultant  uncolored multigraph at the end of the round (possibly useful to compute α-fair matchings), and  moreover implies that after ∆/C many rounds (and so with (∆/C) · α · C = α · ∆ colors used), the  uncolored multigraph has maximum degree ∆ − (∆/C) · C · (1 − o(1)) = o(∆), and can be colored  greedily using a further 2 · o(∆) = o(∆) many colors, for (α + o(1))∆ colors overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  While the above algorithm was described in an offline setting, if one can compute an α-fair  algorithm online (possibly with information regarding the high-probability upper bound on ∆ in each  round), then the entire algorithm can be made to run online: for each (vertex/edge) arrival, perform  the next steps of the α∆ many α-fair matching algorithms outputting color classes M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Mα∆,  where for the i-th such algorithm, we simulate only the arrival of the parts of the subgraph that  are not contained M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Mi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The same approach is used to simulate the final greedy coloring  steps, using the fact that the greedy (2∆ − 1)-edge-coloring algorithm is an online algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  Stochastic extension  In this section, we generalize our matching ODRS and its analysis to the following stochastic online  bipartite matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' At each time t, an online node arrives with weight vector ⃗wt ∼ Dt,  17  where D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' are a priori known independent distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' A simple example which we focus  on for notational simplicity (though our approach generalizes to this problem in full generality)  is as follows: each online node t arrives according to a Bernoulli event At ∼ Ber(pt), each edge  (i, t) having some intrinsic value wit, but realized weight vit = wit · At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This problem (in its full  generality) is an online Bayesian selection problem, and a number of algorithms with competitive  ratio of 1  2 are known for it [35, 38, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By the classic lower bound of Krengel and Sucheston for the  single-item prophet inequality problem [54], the above ratio is best-possible when comparing with  the offline optimum algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In [62], Papadimitriou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' initiate the study of this problem in terms of the (polytime) approx-  imability of the optimum online algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Perhaps surprisingly, they show that while for simple  cases of the problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', the single-offline-node problem), the optimum online algorithm is com-  putable using a simple poly-sized dynamic program, in general it is pspace-hard to approximate the  optimum online algorithm within some absolute constant α < 1 (say, α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='999999), and showed  that a better-than-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5 approximation of the optimum online algorithm can be obtained by polytime  online algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This positive result was subsequently improved to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='526 implicitly [65] and to  1 − 1/e ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='632 explicitly [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' All these results are achieved by rounding the following LP, which  can be shown to upper bound the optimum online algorithm’s value [62, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  max  �  i,t  wi,t · xi,t  (LP OPTon)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  �  t,r  xi,t ≤ 1  ∀i ∈ [n]  (9)  �  i  xi,t ≤ pt  ∀t  (10)  xi,t ≤ pt ·  �  1 −  �  t′<t  xi,t′  �  ∀i, t  (11)  xi,t ≥ 0  ∀i, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (12)  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The optimum of the above LP upper bounds the expected gain of any online algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Most of these constraints are evident, and follow from (expected) matching constraints, which  apply also to offline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The one constraint which applies to online algorithms but not to  offline ones is Constraint (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This constraint follows from the fact that if offline node i is matched  to t, then t has already arrived, and that i is unmatched before time t — two events which are  independent for online algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We note that in their paper’s full version, Papadimitriou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' [62] (private communication) show  that the above LP has a “gap” of 1− 1  2e, in the sense that no online algorithm (even computationally-  unbounded ones) can get a better-than-(1 − 1  2e) fraction of this LP’s value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' So, an approximation  ratio greater than 1− 1  2e is unachievable using this LP, and a ratio of 1− 1  e is achievable [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Is the  latter bound tight?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In what follows, who show how the approach underlying our ODRSes, combined  with (and simplifying) the high-level analysis of [16], allows one to break this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  ODRS for the stochastic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' As in the non-stochastic setting of Section 4, specifically  Algorithm 4, we group offline nodes and have them coordinate a single bid per group (bin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Similarly,  our stochastic counterpart to Algorithm 5 applies group discounting and individual markup to the  target marginal bidding probability, and then calls the (soon-to-be) modified Algorithm 4 with  18  parameter θ · (1 − ǫ) and vector ˆxi,t given by Equation (3), restated below for ease of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  ˆxi,t = xi,t · (1 − ǫ) +  � max(θ, si,t+xi,t)  z=max(θ,si,t)  (ǫ + δ) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The three main differences compared to Section 4 are as follows: First, to account for the probability  pt of t to arrive, the conditional probability for i to bid if i is free and t arrives used in Line 6-Line 12  is now  ˆxi,t  pt·(1−ˆsi,t), for ˆsi,t := �  t′<t ˆxi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Second, we allow offline nodes to bid more than once, and  denote them as free (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', belonging to F) as long as they are not matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This avoids offline nodes  becoming positively correlated due to an unlikely online node’s arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The offline nodes’ matching  constraints are trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Finally, satisfying the online nodes’ matching constraints is even  easier than in Algorithm 4 for the non-stochastic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Rather than using a CRS, we have an  arriving online node t that receives bids simply greedily match to the bidding node i maximizing  the value wi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' See the pseudocode in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Algorithm 6 Stochastic Matching Algorithm  1: M ← ∅  2: x ← solution to (LP OPTon)  3: for each time t do  4:  for each i ∈ [n] do  5:  compute ˆxi,t as in Equation (3), and ˆsi,t = �  t′<t ˆxi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  6:  Bt ← FirstFit  ���  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  ˆxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t  pt·(1−ˆsi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t)  � ��� i ∈ [n],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t ≤ θ  ��  ⊲ Bucketing high-degree nodes  7:  Bt ← Bt ∪ {{i} | i ∈ [n],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t > θ}  ⊲ Bucketing low-degree nodes  8:  Ct ← ∅  ⊲ Candidates  9:  for each B ∈ Bt do  10:  U ∼ Uni[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 1]  11:  if U ≤ �  i∈B  ˆxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t  pt·(1−ˆsi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t) then  12:  Ct ← Ct ∪ mini  �  i ∈ [k]  ��� U ≤ �  j∈B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' j≤i  ˆxj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t  pt·(1−ˆsj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t)  �  13:  Pt ← Ct \\ V (M)  ⊲ Bidders = free candidates  14:  if Pt ̸= ∅ and t arrived then  15:  pick some i ∈ arg max{wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t | i ∈ Pt}  16:  M ← M ∪ {(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' t)}  17: Output M  Fact 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Algorithm 6 is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In particular,  ˆxi,t  pt·(1−ˆsi,t) ≤ 1 for every pair (i, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5, we have that ˆsi,t = si,t · (1 − ǫ) + (si,t − θ)+ · (ǫ + δ) ≤ si,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Therefore, by our  choice of θ =  δ  ǫ+δ and LP Constraint (11), we have that, indeed  ˆxi,t  pt · (1 − ˆsi,t) ≤  � xi,t·(1−ǫ)  pt·(1−si,t) ≤  xi,t  pt·(1−si,t) ≤ 1  si,t ≤ θ  xi,t·(1+δ)  pt·(1−si,t(1+δ)+δ) =  xi,t  pt·(1−si,t)) ≤ 1  si,t > θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Algorithm 6 is clearly a polytime algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' It remains to analyze its approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The following lemma makes explicit a fact implicit in the proof of the main theorem of [16]:  specifically, that a per-online-node “approximation ratio”, relating w(M(t), t)—the weight obtained  by matching t—to the LP solution’s value from t, implies a global approximation ratio of the same  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We provide a short self-contained proof of this fact for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  19  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If for each online node t and z ≥ 0 we have that Pr[w(M(t), t) ≥ z] ≥ �  i:wi,t≥z α·xi,t,  then the algorithm is an α-approximation of the optimum online algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Denote by wt := w(M(t), t) the weight of the matched edge of t, with wt = 0 if t is not  matched (possibly due to non-arrival).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, the weight of the output matching M satisfies  �  t  E[wt] =  �  t  � ∞  z=0  Pr[wt ≥ z] dz ≥  �  t  � ∞  z=0  α ·  �  i:wi,t≥z  xi,t dz = α ·  �  i,t  wi,t · xi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The proof is then completed by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6, implying that the RHS upper bounds α times the value  of the optimum online algorithm’s gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The objective of our analysis will therefore be to lower bound Pr[w(M(t)) ≥ z] for all z ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In our analysis, we let Ft = [n] \\ V (M) be the set of free offline nodes at time t, and let  Fi,t = 1[i ∈ Ft] = 1 − �  t′<t 1[(i, t′) ∈ M] indicate whether i is free by time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Similarly, we let  Ei,t = 1 − Fi,t indicate whether i was matched (“engaged”) before time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Our first observation is that our modification to the notion of freedom results in the following  lower bound on the probability of a node being free at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For each time t and offline i ∈ [n], we have that Pr[Fi,t] ≥ 1 − ˆsi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let B1  i,t indicate that i bid for the first before time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We prove that E[B1  i,t] = ˆsi,t, by  induction on t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The base case, is trivial, as Fi,1 ≡ 1 and ˆsi,1 ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Now, by induction, we have  E[B1  i,t+1 − B1  i,t] = pt ·  ˆxi,t  pt · (1 − ˆsi,t) · (1 − E[B1  i,t]) = ˆxi,t,  where the first equality relies on independence of t’s arrival, i becoming a candidate at time t, and  i not having yet bid, and the second equality follows from the inductive hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By linearity,  we then have that E[B1  i,t+1] = ˆsi,t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Since 1 − Fi,t = Ei,t ≤ B1  i,t, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Next, for any time t and offline node set S ⊆ [n], we let FS,t = �  i∈S Fi,t and ES,t = �  i∈S Ei,t  indicate whether all of S is free (respectively, engaged) by time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' So far, we have established that  Pr[Ei,t] ≤ ˆsi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The following lemma, whose statement and proof mirror that of [16, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3] (who  use xi,t and si,t, as opposed to ˆxi,t and ˆsi,t), asserts that this upper bound is “sub-multiplicative”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For each time t and set of offline nodes S ⊆ [n], we have that  Pr[ES,t] ≤  �  i∈S  ˆsi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  To prove the above, we need the following claim from [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For completeness, we provide a short  probabilistic proof of this lemma (see [16, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2] for a longer algebraic proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For all i ∈ [n], let qi ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then,  �  A⊆S  Pr[EA,t, FS\\A,t] ·  �  i∈S\\A  qi =  �  A⊆S  Pr[EA,t] ·  �  i∈A  (1 − qi)  �  i∈S\\A  qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Consider a probability space where at time t all offline nodes toss a coin with success prob-  ability qi, independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The LHS corresponds to all free nodes having their coin come up heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The RHS corresponds to the same event, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', all nodes whose coin came up tails were engaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  20  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The proof is by induction on t ≥ 1 for all sets S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The base case t = 1, for  which sit ≡ 0 for all i, is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For the inductive step, denote the marginal probability of i bidding  at time t (requiring t’s arrival) by  mi,t := Pr[i ∈ Pt] =  ˆxi,t  1 − ˆsi,t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We note that with this terminology, we have that  ˆsi,t+1 = ˆsi,t + ˆxi,t = ˆsi,t + mi,t · (1 − ˆsi,t) = ˆsi,t · (1 − mit) + mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (13)  The inductive step follows by noting that at most one of the nodes i ∈ S may be matched at time  t, and this requires that the single free node i ∈ S bid for t, which happens with probability mi,t  (conditioned on any event determined by randomness up to time t that implies that the node is  free), as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Pr[ES,t+1] ≤  �  A⊆S  |S\\A|≤1  Pr[EA,t, FS\\A,t] ·  �  i∈S\\A  mi,t  =  �  A⊆S  |S\\A|≤1  Pr[EA,t] ·  �  i∈A  (1 − mi,t)  �  i∈S\\A  mi,t  Lem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='11  ≤  �  A⊆S  Pr[EA,t] ·  �  i∈A  (1 − mi,t)  �  i∈S\\A  mi,t  ≤  �  A⊆S  �  i∈A  Ui,t ·  �  i∈A  (1 − mi,t)  �  i∈S\\A  mi,t  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  =  �  i∈S  (ˆsi,t · (1 − mit) + mit)  binomial expansion  ≤  �  i∈S  ˆsi,t+1  Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The preceding lemma allows us to prove the following counterpart to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4, key to our  improved approximation ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For any time t and set of offline nodes S ⊆ [n], Algorithm 6 satisfies  Pr[S ∩ Pt ̸= ∅] ≥ 1 −  �  B∈Bt  �  1 −  �  i∈B∩S  ˆxit/pt  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let Ct and Pt be as in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Since Pt = Ct \\{i | Ei,t = 1}, there are no bids from set  S (S ∩ Pt = ∅) iff all i ∈ A = Ct ∩ S are no longer free, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', iff EA,t holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' So,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' by total probability  21  over Ct ∩ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' we have that (⋆) := Pr[t arrives and S ∩ Pt = ∅] satisfies  (⋆) =  �  A⊆S  Pr[Ct ∩ S = A] · Pr[EA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t]  =  �  A⊆S  �  B:|B∩A|=0  Pr[Ct ∩ B ∩ A = ∅]  �  B:|B∩A|=1  �  i∈B∩A  Pr[Ct ∩ B ∩ A = {i}] · Pr[EA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t]  ≤  �  A⊆S  �  B:|B∩A|=0  Pr[Ct ∩ B ∩ A = ∅]  �  B:|B∩A|=1  �  i∈B∩A  Pr[Ct ∩ B ∩ A = {i}] · Pr[Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t]  =  �  B∈Bt  �  Pr[Ct ∩ B ∩ S = ∅] +  �  i∈B∩S  Pr[Ct ∩ B = {i}] · Pr[Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t]  �  =  �  B∈Bt  �  1 −  �  i∈B∩S  Pr[Ct ∩ B = {i}] · Pr[Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t]  �  ≤  �  B∈B  �  1 −  �  i∈B∩S  ˆxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t/pt  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Above, the second and third equality follow from independence of the sets {Ct∩B | B ∈ Bt} and from  |Ct ∩B| ≤ 1 always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The first inequality follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The fourth equality follows from  |Ct∩B| ≤ 1, and Fi,t = 1−Ei,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Finally, the last inequality follows from Pr[Ct∩B = {i}] =  ˆxi,t  pt·(1−ˆsi,t)  and Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='9, implying that Pr[Ct ∩ B = {i}] · Pr[Fi,t] ≥  ˆxi,t  pt·(1−ˆsi,t) · (1 − ˆsi,t) ≥ ˆxi,t/pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The following lemma, which is a counterpart to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6 for the non-stochastic setting, allows  us to obtain a concrete lower bound from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If ǫ, δ ∈ [0, 1] guarantee that for all z ≥ 1−θ  2  =  ǫ  2(ǫ+δ),  fǫ,δ(z) := exp(−z · (1 + δ)) − (1 − z · (1 − ǫ)) ≥ 0,  (14)  then Algorithm 6 with parameters ǫ and δ satisfies for each time t and set S ⊆ [n]  Pr[S ∩ Pt ̸= ∅] ≥  �  1 − exp  �  −1 − δ + ǫ + δ  1 − ǫ  �  1 − ǫ  1 + δ  � �  i∈S  xi,t/pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The proof is near identical to that of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6, and is therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6  Finally, the same optimization over ǫ, δ as in Section 4 yields our main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Algorithm 6 with parameters (ǫ, δ) ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='0480, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='0643) is a polytime online stochastic  weighted bipartite matching algorithm that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='652-approximates the optimum online algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let α = 1−exp  �  −1 − δ + ǫ+δ  1−ǫ  �  1−ǫ  1+δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By our algorithm’s greedy matching choice, t  gets matched to a node i with wi,t ≥ w iff t arrives and gets a bid from the set St,w := {i | wi,t ≥ w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  So, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='13, as (ǫ, δ) above satisfy Equation (14) for all z ≥ 0 (see the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='9),  Pr[w(M(t), t) ≥ w] = Pr[t arrives ∧ Pt ∩ St,w ̸= ∅] ≥ pt · α ·  �  i|wi,t≥w  xi,t/pt = α ·  �  i|wi,t≥w  xi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The approximation ratio of Algorithm 6 then follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='8, while the algorithm’s poly-  nomial running time is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  6The only changes to the proof are (1) the syntactic generalization of Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3 implies that all but one bin B ∈ Bt  computed in Line 6 has �  i ˆxi,t/pt ≥ 1−θ  2 , and (2) the convexity argument requires that we add dummy nodes J  with sj,t = θ for all j ∈ J and �  j∈J xj,t = pt − �  i xi,t, where this sum is non-negative, by LP Constraint 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  22  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  Application of the level-set ODRS  We now turn to discussing applications of our level-set algorithm, and its strong negative correlation  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Multi-stage stochastic optimization  We consider an application to multi-stage stochastic optimization, building on the framework of [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We recall briefly that in such multi-stage problems, various parameters of an optimization problem  materialize stochastically over some number k of stages, where each stage contains stochastic in-  formation about the following stage(s) and wherein we can take actions—with actions cheaper in  earlier stages, but possibly inaccurate since we only know the future stochastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Motivated by the results of [69], a broad class of multi-stage stochastic covering problems can  be cast as the following family of online problems: see Section 2 of [67], from which we quote the  framework almost verbatim next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' There is a hidden covering problem “minimize cT · x subject to  Ax ≥ b and with all variables in x being non-negative integers” that is revealed online as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='We  let m denote the number of rows of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' also, the variables in x are indexed as xj,ℓ, where 1 ≤ j ≤ n  and 1 ≤ ℓ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (We interchange m and n from [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') Such a covering problem, as well as a feasible  fractional solution x∗ for it (where each x∗  j,ℓ is allowed to be a non-negative real), are revealed to us  in k stages as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In each stage ℓ where 1 ≤ ℓ ≤ k, we are given the ℓth-stage fractional values  (x∗  j,ℓ :  1 ≤ j ≤ n) of the variables along with their respective columns in the coefficient matrix  A, and their respective coefficients in the objective-function vector c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Given this setup, we need to  round the variables (x∗  j,ℓ : 1 ≤ j ≤ n) immediately and irrevocably at stage ℓ, using randomization  if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (Note the direct connection to online computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') The goal is to develop such a  rounded vector (sj,ℓ :  1 ≤ j ≤ n, 1 ≤ ℓ ≤ k) that satisfies the constraints Ay ≥ b, and whose  (expected) approximation ratio E[cT · y]/cT · x∗ is small, where the expectation is only over any  random choices made by our online algorithm in this arrival model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This completes the description  of the framework from [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We make two contributions in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' A motivating family of problems to consider is  vertex-(multi-)cover on graphs and hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The multi-stage stochastic version of the classical  vertex-cover problem on a given graph G = (V, E) is basically as follows: the coverage constraint  corresponding to an edge e = (u, v) ∈ E is now  � k  �  ℓ=1  xu,ℓ  �  +  � k  �  ℓ=1  xv,ℓ  �  ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (15)  Improving on the 2k-approximation of Swamy and Shmoys [69] for this problem—in the “online  rounding” model mentioned in the previous paragraph—a 2-approximation is developed in [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Our first contribution is the following generalization of the 2-approximation for multi-stage  stochastic vertex cover from [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Consider the following more-general problem in d-regular hy-  pergraphs H = (V, E) (or hypergraphs with all edge-sizes lower bounded by d): for some integer  parameter t, we want the vertices in each edge e ∈ E to be covered in total to an extent of at  least t, where vertices can be multi-covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  That is, generalizing (15), the coverage constraint  corresponding to an edge e = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , vd} ∈ E is  d  �  i=1  k  �  ℓ=1  xvi,ℓ ≥ t,  (16)  where each xvi,ℓ is allowed to be any non-negative integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' note that the above-seen stochastic vertex  cover is the special case where d = 2 and t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Recalling our online arrival model, we need to  23  round the LP solution (x∗  v,ℓ : v ∈ V ) right away at stage ℓ, for each ℓ ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We proceed as follows  to obtain an α := (d + t − 1)/t–approximation in this model, generalizing the 2-approximation for  d = 2 and t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In particular, if t is large, this is essentially a 1-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Our algorithm works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For each vertex v ∈ V , consider the scaled vector z(v) :=  (α · x∗  v,ℓ : ℓ ∈ [k]) and independently for all v, run Algorithm 3 on z(v) to obtain the final rounded  values Xv,ℓ : ℓ ∈ [k]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (As usual, we imagine padding this vector with a dummy final element so  that the sum of the entries is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') We first verify that (16) is satisfied with probability one  for each e = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , vd} ∈ E:  d  �  i=1  k  �  ℓ=1  Xvi,ℓ ≥  d  �  i=1  � k  �  ℓ=1  α · x∗  vi,ℓ  �  (by Property (P2))  (17)  >  d  �  i=1  �� k  �  ℓ=1  α · x∗  vi,ℓ  �  − 1  �  (⌊β⌋ > β − 1 for all β)  (18)  ≥ α · t − d  (since x∗ satisfies (16))  = t − 1,  which implies that �d  i=1  �k  ℓ=1 Xvi,ℓ ≥ t, as required, since the LHS here is an integer and because  of the strict inequality connecting (17) and (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We thus obtain an online solution to the rounding  problem which satisfies all constraints with probability one, and whose expected objective function  value is at most α(cT · x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We observe that the work of [67] also runs an online rounding algo-  rithm, but that the problem considered there is much simpler since all we require there (in place of  Property (P2)) is that if the entries of the given input vector sum to at least one, then at least one  entry is rounded to one with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In particular, this does not address the multi-coverage  constraint we satisfy here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Our second contribution is that because of Property (P3), the objective function is strongly  concentrated around its mean (given by the Chernoff bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This is an especially useful property  in areas like stochastic optimization, where the algorithm can be run only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This is in comparison  to offline minimization problems with a non-negative objective where we can repeat the algorithm  O(1/ǫ) times and choose the best solution obtained, and apply Markov’s inequality to each iteration  to show that the best solution is at most (1 + ǫ) times the expected value with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Thus, the fact that Algorithm 3 yields sharp concentration as implied by Property (P3)—and does  not just guarantee Properties (P1) and (P2)—is very useful in the stochastic-optimization context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Summarizing the preceding discussion (and formalizing the preceding paragraph), we obtain the  following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For any k ≥ 1, there exists a k-stage stochastic hypergraph vertex cover algorithm  for d-regular hypergraphs where each hyper-edge must be covered t times, with approximation ratio  that is α = d+t−1  t  in expectation and is α(1 + o(1)) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' if costs are polynomially bounded and  OPT = ω(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Our algorithm satisfies the (multi-)coverage constraint, by the above, and has expected ap-  proximation ratio α by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The high-probability bound follows from standard Chernoff  bounds for negatively-associated variables, relying on Property (P3) and closure of NA under prod-  ucts (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5), relevant since we run independent copies of Algorithm 3 for each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  Negative association, fairness, diversity, and streaming  While Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1 only required the fact that Property (P3) yields sharp tail bounds for sums of  the Xi with non-negative weights, we go further now and use the negative association guaranteed by  24  Property (P3), which helps us correlate multiple such sums as well as handle submodular objectives  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The submodular-objective application further benefits from the fact that Algorithm 3 can  also be viewed as a data-stream algorithm: note that at when xt arrives, the algorithm need only  remember st−1 and St, and hence can be implemented with O(log n) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Intersectionality is a key notion in the study of fairness, where the intersection of multiple  attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', race, gender, age) can impact the outcomes of a person in a more fine-grained  manner than does standard group-fairness (where we typically take a single attribute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Its study  has naturally impacted AI/ML fairness as well—see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', [18] and its followups—and is a topic  of much debate in terms of precise definitions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', [53]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We make a contribution to this  nascent area, while being aware that many such formulations/contributions in the still-early stages  of research in AI/ML fairness are speculative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Consider people arriving online and requesting a resource whose availability expands over time:  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', vaccines for a new pathogen or a popular new model of car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' It is anticipated that by time t, at  most at units of this resource will be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' An ML system that adapts to the changing realities  on the ground (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', how the pathogen is spreading and which communities seem to be impacted the  most) decides online, the probability xt with which to allocate the resource to the request arriving  at time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' we naturally require �t  i=1 xi ≤ at for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We respond to this in real time by constructing  Xt ∈ {0, 1} using Algorithm 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Property (P2) ensures that we never exceed the supply-limits at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (In reality, a batch of requests will arrive at time t, which of course Algorithm 3 naturally extends  to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') The ML system outputting the vector x may be (required to be) unware of protected attributes  that characterize our underlying demographic groups and their intersections: given x, can we show  that while intersectional unfairness may be present in x (in which case the ML system needs to be  refined), our rounding algorithm tries to at least limit the scope of such intersectional unfairness in  some way?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We formulate this problem as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Suppose each person requesting the resource has three  attributes A1, A2, and A3: the “three" here is just for simplicity, and it is easy to see that the  framework below generalizes to any number of attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Suppose U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Uα is some given  partition of the population according to their values for A1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', if A1 is age, the Ui’s may be  disjoint age-intervals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Similarly, suppose we have a partition of the population according to A2  as V1, V2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Vβ and according to A3 as W1, W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Wγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In order to model intersectionality, let  us define the subgroup Gi,j,k := Ui ∩ Vj ∩ Wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let si,j,k = �  r∈Gi,j,k Xr be the random variable  denoting the amount of resource received in total by members of Gi,j,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We ask: “While intersectional  unfairness may be present, say inherently in the vector x, does our rounding at least limit the extent  of it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='" That this is indeed true follows from Property (P3) which guarantees negative association,  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For any sequence of thresholds (ti,j,k), the probabilities of different subgroups Gi,j,k  receiving intersectional unfairness in the sense of si,j,k ≤ ti,j,k, can be bounded as well as if the Xr’s  had been independent:  ∀S ⊆ ([α] × [β] × [γ]) , Pr[�  (i,j,k)∈S(si,j,k ≤ ti,j,k)] ≤ �  (i,j,k)∈S Pr[si,j,k ≤ ti,j,k];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' and, more  generally,  in disease-spread and meme-spread-like contexts, we are often interested in monotone non-  linear functions that model phase transitions: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', if a heavily-interacting subgroup receives  fewer than a threshold of vaccines, an explosive epidemic could happen within that subgroup—  with monotone behavior away from this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Given any sequence of monotone functions  (all increasing or all decreasing) (fi,j,k), we have  ∀S ⊆ ([α] × [β] × [γ]) , E  \uf8ee  \uf8f0  �  (i,j,k)∈S  fi,j,k(si,j,k)  \uf8f9  \uf8fb ≤  �  (i,j,k)∈S  E [fi,j,k(si,j,k)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  25  As is well-known, letting χ(·) be the indicator function, this implies that �  (i,j,k)∈S χ(si,j,k ≤  ti,j,k) has Chernoff-like concentration around its mean;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' in particular, while some subgroups in  S may face intersectional unfairness (that needs to be separately analyzed by inspecting x),  it is unlikely that many do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We next present an application to diversity in search results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In addition to the classical interest  in this problem in information retrieval [14], there has been much recent work in search-result  diversification, motivated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', by the fact that the document-set retrieved should account for the  interests of the user population [20, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By developing a model of knowledge about the topics the  query or the documents may refer to, the work of [4] presents an (1 − 1/e)–approximation for their  objective of maximizing average-user satisfaction with the search results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Given a search query q, a  corpus of N documents D, and a bound k on the number of documents to retrieve from D for q, a  monotone submodular function f : 2D → [0, 1] is developed in [4], where f(T) is the (approximate)  mix of relevance and diversity if the set T with |T| ≤ k is then returned to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The problem  is thus to approximately maximize f(T) subject to |T| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Consider the setting where D is very  large, we only get streaming access to it, and where, after encountering document d ∈ D, we get  ML advice on the probability xd with which to choose d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Algorithm 3 is tailor-made for this, as  it needs only O(log N) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Furthermore, by the already-known Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4, we obtain that  our final expected value F(X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , XN) is at least as large as if we had instead chosen the Xi’s  independently with marginals xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (Also see [5, 30] for applications of submodular functions toward  diversity in bipartite matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=')  6  Lower Bounds  In this section we present some impossibility results for ODRSes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  A simple example in [29] rules out a rounding ratio of one, or even just greater than 7  8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In this section we improve this bound to (2  √  2 − 2) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='828 with a similarly simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Before providing our lower bound, we provide an even simpler example ruling out a rounding  ratio of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  This example highlights the need of ODRSes with high rounding ratio to create  some form of negative correlation between all subsets of offline nodes, motivating our lower bound  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (We are also hopeful that this insight may inform future ODRSes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=')  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Suppose an ODRS A with rounding ratio of one exists, and apply it to the following  input on n = 3 offline nodes and 4 online nodes: For k = 1, 2, 3, online node k only neighbors offline  node k, and xkk = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' So, each offline node’s matched status so far is a Ber(1/2) random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Finally, online node t = 4 arrives, neighboring some two offline nodes, i, j, and xit = xjt = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The  above is clearly a feasible fractional matching, but since we assumed that A has a rounding ratio  of one, it must match t with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Since t cannot be matched if both i are j previously  matched, and both are matched with probability 1/2 before time t, this requires i’s and j’s matched  status before time t to be perfectly negatively correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' That is, i is unmatched iff j is matched,  and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Since i, j could be any two offline nodes, this requires perfect negative correlation  between any two of these three variables, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='7 Thus, we reach a contradiction, and  an ODRS with rounding ratio of one is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  To generalize the above and obtain a concrete numeric bound, we need the following fact,  whereby (near-)positive pairwise correlation is unavoidable in a large set of Bernoulli random vari-  ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (See Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3 for a potentially useful generalization to k-wise correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=')  7Perfect negative pairwise correlation between three binary variables X1, X2, X3 ∈ {0, 1} can be stated succinctly  as X1 + X2 = X1 + X3 = X2 + X3 = 1, but this linear system only has a fractional solution X1 = X2 = X3 = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  26  Fact 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let Yi ∼ Ber(p) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , n be (possibly dependent) Bernoulli variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then,  max  i,j Cov(Yi, Yj) ≥ −2p/(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This bound follows from non-negativity of variance and the following, after rearranging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  0 ≤ Var  ��  i  Yi  �  =  �  i  Var(Yi) +  �  i,j  Cov(Yi, Yj) ≤ n · p +  �n  2  �  max  ij  Cov(Yi, Yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Any online rounding algorithm for bipartite fractional matchings has some fractional  matching x = 1  2 · 1E on graph G = (V, E) such that for some edge e ∈ E, the output matching M  satisfies  Pr[e ∈ M] ≤ (2  √  2 − 2) · xe ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='828 · xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Consider an input with xe = 1/2 for each edge, with the first n online nodes neighboring  two distinct offline nodes, with one last online node neighboring two (judiciously chosen) offline  nodes, from two prior online nodes’ neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We wish to show that for some instantiation of  this family of x and G, some online node (and hence some edge) is matched with low probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Specifically, letting p = 2  √  2 − 2, we wish to show that for some such x and G,  ∃t ∈ [n + 1] such that Pr[t ∈ V (M)] ≤ p +  p  2(n − 1),  (19)  Since each online node t has �  i xit = 1, this implies that some edge e is matched with probability  at most (2  √  2 − 2 +  p  2(n−1)) · xe, by linearity of expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Taking n to be sufficiently large rules  out any (2  √  2 − 2 + ǫ)-approximate rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We turn to prove Equation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  If any of the first online nodes is matched with probability less than p, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Otherwise,  let Yt = 1[t ∈ V (M)] be indicators for online node t being matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We couple these with variables  Zt ≥ Ber(p) such that Yt ≥ Zt always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  By Fact 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1, there exist two online nodes t ̸= t′ with  Cov(Zt, Zt′) ≥ −2p/(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let N(t) = {i1, i2} and N(t′) = {i3, i4} be the neighborhoods of t and  t′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By a simple averaging argument, some pair of nodes (i, j) ∈ N(t) × N(t′) are both  matched before the online last arrival with probability at least  Pr[Yt, Yt′]  4  ≥ Pr[Zt, Zt′]  4  ≥ Pr[Zt] · Pr[Zt′] − 2p/(n − 1)  4  ≥ p2 − 2p/(n − 1)  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  So, if the last online node neighbors {i, j}, it is matched with probability at most 1 − p2−2p/(n−1)  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  However, as p = 2  √  2−2 is a root of 1−y− y2  4 , this probability is precisely 1− p2  4 +  p  2(n−1) = p+  p  2(n−1),  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Inevitable near-positive cylinder dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  A useful extension of Fact 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1 of possible  independent interest argues that for any (sufficiently large) subset of Bernoulli variables, some  subset of these variables must be (nearly) positively cylinder dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In Appendix D we prove  this Ramsey-theoretic lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let r ∈ N and 0 ≤ ǫ ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, there exists some n = nr(p, ǫ) such that any  Bernoulli variables Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Yn ∼ Ber(p) contains some subset I ⊆ [n] of size |I| = 2r such that  E  ��  i  Yi  �  ≥  �  i  E[Yi] − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  27  7  Summary and Open Questions  In this work we provided a methodical study of online dependent rounding schemes (ODRSes)  for bipartite b-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We obtained optimal rounding ratios with strong concentration for star  graphs (uniform-matroid rounding) and improved bounds beyond a ratio of 1− 1  e for (b-)matchings,  including the first results for b-matchings not obtained via OCRSes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (Consequently, ours is the only  ratio beyond 1/2 known for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') We furthermore provided a number of applications of our  ODRSes and their algorithmic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Beyond the natural question of improving our rounding  ratios for b-matchings (and the approximation/competitive ratios of their derived applications), our  work raises a number of directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Online applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We provided a number of online applications of our ODRSes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' What further  applications do these schemes have?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The bipartite setting seems particularly relevant for online  scheduling problems, with machines and jobs represented by offline and online nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  For minimization problems, it may prove useful to observe that our ODRSes for b-matching have  modest two-sided error: edges (i, t) are matched with probability at most the probability that i bids  for t, namely ˆxi,t ≤ (1 + δ)xi,t ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='064 · xi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover, these latter events satisfy strong upper-tail  bounds, by our ODRS’ strong negative correlation properties (see Property (P3) and Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Streaming applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We note that our ODRSes are not only online algorithms, but moreover  are streaming (for uniform matroids) or semi-streaming algorithms (for arbitrary b-matchings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Indeed, they require us to only remember O(log n)-bit partial sums and a single number per offline  node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Does this property have further applications in streaming or graph-analytics contexts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Richer ODRSes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Finally, we recall that online contention resolution schemes (OCRSes) yield  ODRSes, though this connection does not yield optimal rounding ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Recent years have seen an  explosion of work on OCRSes for increasingly-rich families of arrival models and combinatorial con-  straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' These have been fueled in large part by connections to various other online and economic  applications, most notably the prophet-inequality problem under more involved combinatorial con-  straints (see discussion in the influential work of Kleinberg and Weinberg [52]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Can a similarly  rich theory of online dependent rounding more broadly be developed, capturing other combinatorial  constraints beyond matchings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We thank Uri Feige for pointing out the connection of our work to offline CRSes, and thank Sahil  Singla for discussions about (lack of prior) examples of black-box application of offline CRSes to  online problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We thank Manish Purohit for asking about the two-sided error of our ODRSes,  which we anticipate may be of use in future applications for online minimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  APPENDIX  A  Additional Preliminaries  In this section we provide omitted proofs and additional preliminaries not covered in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Contention Resolution Schemes  We start with a brief, self-contained proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1, restated below for ease of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  28  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let D : 2[n] → R≥0 be a distribution over subsets of [n], with its support denoted by  supp(D) = {S ⊆ [n] | PrR∼D[R = S] ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, there exists a randomized algorithm CRS(R,⃗v)  which on input set R ∼ D and vector ⃗v ∈ Rn, outputs a subset O ⊆ R of size |O| ≤ 1 satisfying  Pr[i ∈ O] = vi · min  S⊆[n]  Pr[R ∩ S ̸= ∅]  �  i∈S vi  ∀i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The algorithm runs in time polynomial in n if D is a product distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Otherwise, it runs in  time poly(n, T), where T ≥ |supp(D)| is the time to compute and write down the support of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The lemma for product distributions follows from the work of Feige and Vondrák [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We  focus on the more general case, addressed by Bansal and Cohen [7], but proven here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Let α := minS⊆[n]  Pr[R∩S̸=∅]  �  i∈S vi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We prove the existence of conditional probabilities pi,S with  pi,S = 0 if i ̸∈ S and �  i pi,S = 1 for all S such that setting Pr[O = {i} | R = S] = pi,S yields  Pr[O = {i}] = �  S Pr[O = {i} | R = S] · Pr[R = S] ≥ α · vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We prove the existence of these  probabilities using the max-flow/min-cut theorem applied to the following directed flow network  that has a source vertex src, a sink vertex sink, and two other disjoint sets of vertices A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Vertices on side A correspond to subsets of elements in [n], with an edge of capacity Pr[R = S] from  src to the node in A representing set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Vertices on side B correspond to elements i ∈ [n], with an  edge of capacity α·vi leaving this vertex to sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Infinite-capacity directed edges go from each vertex  in A corresponding to some set S, to vertices in B corresponding to each i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Each vertex i ∈ S  in B, has a directed edge to sink with capacity α · vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This network is reminiscent of the network  used to prove Hall’s Theorem via the max-flow/min-cut theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Indeed, minS⊆[n]  Pr[R∩S̸=∅]  �  i∈S vi  ≥ α is  a (scaled) version of Hall’s condition, and implies by the same argument the existence of a “perfect”  flow f of value α·�  i vi, implying that each edge of the form (i, sink) has its capacity α·vi saturated  by this flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This in turn implies the existence of the probabilities pi,S as desired, where if the flows  through the edges (src, S) and (S, i) are fsrc,S and fS,i respectively, then pi,S = fS,i/ Pr[R = S] (if  Pr[R = S] = 0, we take arbitrary non-negative pi,S with pi,S = 0 if i ̸∈ S and �  i:i∈S pi,S = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Thus, the above algorithm outputs a set O ⊆ R of size at most one with each element belonging to  O with the requisite probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The running time of the algorithm for general distributions follows  by polytime solubility of the maximum flow problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  Negative correlation properties  Here we formalize the strong notion of negative correlation that our online level-set algorithm enjoys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' A distribution µ : 2[n] → R≥0 is strongly Rayleigh if its generating polynomial,  gµ(z) =  �  S⊆[n]  µ(S)  �  i∈S  zi,  is real stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', it has no root z ∈ Cn satisfying ℑ(zi) > 0 for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  As stated in Section 2 and proven in [13], the strong Rayleigh property implies NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Xn) are strongly Rayleigh binary r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='s, then they are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Moreover, since SRP is closed under conditioning [13], we find that SRP distributions are NA  even after conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The strong Rayleigh property, central to the area of geometry of polynomials, has been highly  influential in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  To the best of our knowledge, aside from the classic balls-and-bins  29  process, its variants and conditional counterparts (see [15]), our online level-set algorithm is the  only online algorithm known to satisfy such a strong negative-correlation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We believe that  this property of our algorithm will find further applications outside of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Submodular dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' As stated in Section 2, negative association implies stochastic dom-  inance in the submodular order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' To formally define the above, we briefly recall the definition of  submodular functions, which are set functions capturing the notion of diminishing returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' A set function f : 2[n] → R is submodular if for all sets A ⊆ B ⊆ [n] and element  x ∈ [n] \\ B, we have that  f(A ∪ {x}) − f(A) ≥ f(B ∪ {x}) − f(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The following submodular dominance result of Christofides and Vaggelatou [25] implies that  our online level-set algorithm not only preserves weighted objectives losslessly, but also preserves  submodular objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=')  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Xn be NA random variables and let X∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , X∗  n be independent random  variables that are place-wise equal to the X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Xn respectively in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, for any  monotone submodular function f,  E[f(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Xn)] ≥ E[f(X∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , X∗  n)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3  Odds and Ends  We will also make use of the following direct corollary of convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let f be a non-negative concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, for any 0 ≤ t ≤ α,  f(t)  t  ≥ f(α)  α  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Writing t as a convex combination of α and 0, the claim follows from convexity, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  f(t) = f  � t  α · α +  �  1 − t  α  �  0  �  ≥ t  α · f(α) +  �  1 − t  α  �  f(0) ≥ t  α · f(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  B  Deferred Proofs of Section 3  We start by briefly proving that Algorithm 1 terminates and preserves marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Fact B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Algorithm 1 terminates, and satisfies Pr[i ∈ S] = xi for all i ∈ [n] (Property (P1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' It is easy to see that STEP preserves the sum of its inputs/outputs while decreasing the  number of fractional such inputs/outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Consequently, �  i yi is integral during the execution  of Algorithm 1, while the number of fractional yi decreases and cannot be one, so this algorithm  terminates with all yi binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover, STEP is easily seen to preserve these values’ expectations,  and so by induction we have that E[yi] = xi for all i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', Algorithm 1 satisfies Property (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let sj := �  i≤j xi and Y t  j := �  i≤j yt  i, where yt  i is the value yi after t applications of  STEP in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, Y t  j ∈ [⌊sj⌋, ⌈sj⌉] for all j and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  30  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We prove this for all j by induction on t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The base case is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Fix a time t and let  i1 and i2 be as in Algorithm 1 at time t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If i1, i2 ̸∈ [j] or i1, i2 ∈ [j], then, since STEP(yi1, yi2)  does not affect Y t  j in the former case and preserves the sums yi1 + yi2 and Yj in the latter case,  the inductive hypothesis implies Y t+1  j  = Y t  j ∈ [⌊sj⌋, ⌈sj⌉].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Conversely, if i1 ∈ [j] and i2 ̸∈ [j] (and  therefore i1 = j), we have that yt+1  i  = yt  i ∈ {0, 1} for all i ∈ [j − 1], while yt+1  i1  ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Since by the  inductive hypothesis we have that Y t  j = Y t  j−1 + yt  j ∈ [⌊sj⌋, ⌈sj⌉] and Y t  j−1 = Y t+1  j−1 is the sum of 0-1  terms, while yt  j ∈ (0, 1), this implies that Y t+1  j  = Y t+1  j−1 + yt+1  j  = Y t  j−1 + yt+1  j  ∈ [⌊sj⌋, ⌈sj⌉].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Direct analysis of Algorithm 3  In this section we provide a simple, self-contained proof that Algorithm 3 satisfies the first two  properties we desire of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In the main paper body, we use this direct proof to prove that our  algorithm’s output distribution is the same as its offline counterpart, from which we also obtain the  third desideratum, namely Property (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Algorithm 3 is well-defined (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', pt ∈ [0, 1] for all times t and any realization of the  randomness) and it satisfies properties (P1) and (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We prove the above three properties (including pt ∈ [0, 1]) by strong induction on t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The base case, t = 0, is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' As our inductive hypothesis (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' ), we assume that these properties  hold for all t′ ≤ t − 1 and prove them for all t′ ≤ t, starting with Property (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By the I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=',  St−1 ≤ ⌈st−1⌉ ≤ ⌈st⌉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' so, since pt = 0 if St−1 = ⌈st⌉, we trivially have that St ≤ ⌈st⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover,  by the I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', St−1 ≥ ⌊st−1⌋ = ⌊st − xt⌋ ≥ ⌊st⌋ − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' so, since pt = 1 if St−1 < ⌊st⌋, in which case  St−1 = ⌊st⌋ − 1, we have that St ≥ ⌊st⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Thus, St ∈ {⌊st⌋, ⌈st⌉}, proving Property (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Next we prove that Algorithm 3 is well-defined, as pt ∈ [0, 1] (regardless of prior randomness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The first non-trivial case is when St−1 = ⌊st⌋ = ⌊st−1⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Non-negativity of pt—a ratio of non-negative  terms—is immediate, while  pt =  xt  ⌊st−1⌋ + 1 − st−1  =  st − st−1  ⌊st−1⌋ + 1 − st−1  ≤  ⌊st⌋ + 1 − st−1  ⌊st−1⌋ + 1 − st−1  = ⌊st−1⌋ + 1 − st−1  ⌊st−1⌋ + 1 − st−1  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The second non-trivial case is when St−1 = ⌊st⌋ > ⌊st−1⌋, in which case ⌊st⌋ = ⌊st−1⌋ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Again,  non-negativity of pt is immediate, while  pt =  st − ⌊st⌋  st−1 − ⌊st−1⌋ = st−1 + xt − ⌊st−1⌋ − 1  st−1 − ⌊st−1⌋  ≤ st−1 − ⌊st−1⌋  st−1 − ⌊st−1⌋ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  To prove Property (P1), we will need a closed form for qt := Pr[St = ⌊st⌋], useful when discussing  Pr[St−1 = ⌊st⌋], the probability that t may be added to S, and Pr[St−1 < ⌊st⌋], the probability that  t must be added to S (to satisfy Property (P2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Note that St ∼ ⌊st⌋ + Ber(1 − qt) by Property  (P2), while E[St] = �  t′≤t E[Xt′] = �  t′≤t xt′ = st, by Property (P1) (for all t′ ≤ t) and linearity of  expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Combining these observations, we obtain st = E[St] = ⌊st⌋ + 1 − qt, and therefore,  qt = ⌊st⌋ + 1 − st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (20)  We now use the closed form for qt−1 to prove Property (P1) for time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The easy case is when  ⌊st⌋ = ⌊st−1⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Here we have that E[Xt | St−1 = ⌊st⌋] =  xt  qt−1, while trivially E[Xt | St−1 = ⌈st⌉] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Consequently, since St−1 ∈ {⌊st−1⌋, ⌈st−1⌉} = {⌊st⌋, ⌈st⌉} in this case, then by total probability we  have that E[Xt] = qt−1 ·  xt  qt−1 + 0 = xt, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Finally, we consider the case where ⌊st⌋ > ⌊st−1⌋, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', ⌊st⌋ = ⌊st−1⌋ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If in addition st−1 =  ⌊st−1⌋ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', st−1 is integral, implying that st = st−1 + 1, and hence xt = st − st−1 = 1), we have by  31  Property (P2) that St−1 = st−1 < ⌊st⌋ (with probability one), and so we add t to S with probability  one, resulting in E[Xt] = 1 = xt, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If, conversely, we have that st−1 ̸= ⌊st−1⌋ < ⌊st⌋, then  by total probability, we obtain the desired equality as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  E[Xt] = Pr[St−1 = ⌊st−1⌋] · 1 + Pr[St−1 ̸= ⌊st−1⌋] ·  st − ⌊st⌋  st−1 − ⌊st−1⌋  = qt−1 + (1 − qt−1) · xt + st−1 − (⌊st−1⌋ + 1)  st−1 − ⌊st−1⌋  �  st = xt + st−1  ⌊st⌋ = ⌊st−1⌋ + 1  = qt−1 + (1 − qt−1) · xt − qt−1  1 − qt−1  Equation (20)  = xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  Coupling the offline and online algorithms  Here we prove that our online level-set algorithm, Algorithm 3, induces the same output distribution  as the offline algorithm of [66], Algorithm 1  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Fix a vector ⃗x ∈ [0, 1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let D and D′ denote the probability distributions on {0, 1}n  induced by the online and offline algorithms run on ⃗x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, D = D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We prove by induction on t ∈ [n] that the following holds:  ∀(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , bt−1) ∈ {0, 1}t−1, Pr  D [Xt = 1  �� (∀i < t, Xi = bi)] = Pr  D′[Xt = 1  �� (∀i < t, Xi = bi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (21)  The (strong) inductive hypothesis, whereby Equation (21) holds for all t′ < t clearly implies that for  all (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , bt) ∈ {0, 1}t−1 we have that PrD[(∀i < t, Xi = bi)] = PrD′[(∀i < t, Xi = bi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Similarly,  Equation (21) holds for all t ≤ n iff the two distributions are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The base case t = 1 follows from Property (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' So suppose t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Fix b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , bt−1 ∈ {0, 1},  and let E denote the event “∀i < t, Xi = bi" (measured w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' D or w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' D′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, by the strong  induction hypothesis, PrD[E] = PrD′[E], and in particular, PrD[E] ̸= 0 iff PrD′[E] ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' thus we  may assume that the events conditioned on in the LHS and in the RHS of (21) both have nonzero  (identical) probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This is the only place where we will need the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Recall that st−1 = �t−1  i=1 xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Furthermore, although St−1 technically depends on the rounding  algorithm, it is natural to take it to be �t−1  i=1 bi here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Note that since both the offline and online  algorithms satisfy Property (P2), we have that St−1 ∈ {⌊st−1⌋, ⌈st−1⌉}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Case I: st−1 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This is an easy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' It is immediate that PrD[Xt = 1  �� E] = xt here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This is  also not hard to see this for the offline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Recall that the offline algorithm starts with a  copy y of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Since st−1 ∈ Z, the offline algorithm would set Xi for exactly st−1 indices i ∈ [t − 1] to  one (and the rest in [t − 1] to zero), and would then recursively run on the suffix (yt, yt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , yn)  of y with no correlation with the rounding thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Furthermore, yj = xj for all j ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Thus, by  Property (P1) applied to this suffix, we have that PrD′[Xt = 1  �� E] = xt also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We may thus assume from now on that z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='= st−1 − ⌊st−1⌋ lies in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Case II: st−1 ̸∈ Z and z + xt ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' An easy sub-case here is that St−1 = ⌈st−1⌉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' here, since both  the offline and online algorithms satisfy Property (P2), it is immediate that in this sub-case  Pr  D [Xt = 1  �� E] = Pr  D′[Xt = 1  �� E] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We may thus assume that St−1 = ⌊st−1⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By the definition of the online algorithm 3 we have that  Pr  D [Xt = 1  �� E] =  xt  1 − z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (22)  32  We now analyze PrD′[Xt = 1  �� E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Consider the offline algorithm just before it involves xt in STEP:  at this point in time, it has:  set ⌊st−1⌋ of the variables (Xi : i < t) to one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  set t − 2 − ⌊st−1⌋ of the variables (Xi : i < t) to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' and, recalling again that the offline  algorithm starts with a copy y of x,  it has assigned the current value z to one variable yi∗ where i∗ ∈ [t − 1] is a random variable  from some distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  At this point, the offline algorithm is basically recursively run on the sequence (yi∗, yt, yt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , yn),  where yi∗ = z and yj = xj for all j ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Crucially, since St−1 = ⌊st−1⌋, the conditioning on E says  that the variable yi∗ is eventually rounded to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Thus, we can compute PrD′[Xt = 1  �� E] as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Suppose the offline algorithm is run on the sequence (u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , un−t+2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='= (yi∗, yt, yt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , yn), to  output a random bit-vector (U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Un−t+2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' then,  Pr  D′[Xt = 1  �� E] = Pr  D′[U2 = 1  �� U1 = 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (23)  But since u1 + u2 = z + xt ≤ 1, the first STEP applied to u1 and u2 ensures that at most one of u1  and u2 gets rounded to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This yields  Pr  D′[U2 = 1  �� U1 = 0] = PrD′[(U2 = 1) ∧ (U1 = 0)]  PrD′[U1 = 0]  = PrD′[U2 = 1]  PrD′[U1 = 0] =  xt  1 − z ,  where the second equality holds since at most one of U1 and U2 is one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' we are thus done by (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Case III: st−1 ̸∈ Z and z + xt > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  This is handled similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The easy sub-case here is  that St−1 = ⌊st−1⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' here, since both the offline and online algorithms satisfy Property (P2), it  is immediate that  Pr  D [Xt = 1  �� E] = Pr  D′[Xt = 1  �� E] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We may thus assume that St−1 = ⌈st−1⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By the definition of the online algorithm 3 we have that  Pr  D [Xt = 1  �� E] = xt + z − 1  z  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (24)  Now, the offline algorithm is, like in Case II, essentially run on the sequence (u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , un−t+2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='=  (yi∗, yt, yt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , yn)—where yi∗ is initially z—to output a random bit-vector (U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Un−t+2),  with the key difference from Case II being that U1 will eventually be set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' So,  Pr  D′[Xt = 1  �� E] = Pr  D′[U2 = 1  �� U1 = 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  (25)  Now, since u1 + u2 = z + xt > 1, the first STEP applied to u1 and u2 ensures that at least one of  u1 and u2 gets rounded to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' This implies  Pr  D′[U2 = 1  �� U1 = 1]  =  PrD′[U2 = 1] − Pr[(U2 = 1) ∧ (U1 = 0)]  PrD′[U1 = 1]  =  PrD′[U2 = 1] − PrD′[U1 = 0]  PrD′[U1 = 1]  =  xt + z − 1  z  ,  where the second equality holds since at least one of U1 and U2 is one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' we are thus done by (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  This concludes the proof of the inductive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  33  C  Deferred Proofs of Section 4  In this section we provide deferred proofs of claims from Section 4, restated below for ease of  reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If ǫ, δ ∈ [0, 1], the function g(y) := 1−exp (− (1 − y) · (1 + δ))·(1 − y · (1 − ǫ)) satisfies  ∀y ∈ R :  g(y) ≥ g  �  ǫ + δ  (1 − ǫ)(1 + δ)  �  = 1 − exp  �  −1 − δ + ǫ + δ  1 − ǫ  �  1 − ǫ  1 + δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof of Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Taking the derivative of g(y), we get  g′(y) = exp(−(1 − y) · (1 + δ)) · ((1 − ǫ) − (1 − y · (1 − ǫ)) · (1 + δ)) ,  which vanishes at y∗ =  ǫ+δ  (1−ǫ)(1+δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Next, we consider the second derivative of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  g′′(y) = g′(y) · (1 + δ) + exp(−(1 − y) · (1 + δ)) · (1 − ǫ) · (1 + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The first summand, g′(y) · (1 + δ), vanishes at y∗, by the above, while the second summand is  positive, since ǫ ≤ 1 and δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We conclude that y∗ is the global minimum of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let fǫ,δ(z) be as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' If fǫ,δ(a) ≥ 0 and f ′  ǫ,δ(a) ≥ 0, then fǫ,δ(b) ≥ 0 ∀b ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Fix ǫ, δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For notational simplicity, let f = fǫ,δ and let z∗ = z∗  ǫ,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Since f is the sum of twice-  differentiable convex functions (namely an exponential and a linear term), we have that f ′′(z) ≥ 0  for all z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Therefore, if f ′(z∗) ≥ 0, then for all z ≥ z∗ we have that  f ′(z) = f ′(z∗) +  � z  z∗ f ′′(x) dx ≥ f ′(z∗) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Thus, if moreover f(z∗) ≥ 0, then the desired statement follows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', for all x ≥ z∗ we have that  f(z) = f(z∗) +  � z  z∗ f ′(x) dx ≥ f(z∗) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1  Polytime implementation  So far, we ignored computational aspects of our ODRS of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In this section we show  how our grouping method used to increase our rounding ratio beyond 1 − 1  e moreover allows us to  compute Pr[Pt = S] for all sets S ⊆ [n] in polynomial time — and thus obtain a polytime ODRS  — while only decreasing our rounding ratio by an arbitrarily small constant γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For any γ > 0, there exists an nO(1/γ)-time bipartite matching ODRS with rounding  ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='652 − γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' First, we scale down all the fractions xi,t by a (1− γ) factor (this results in a valid fractional  matching).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' As this additional downscaling is linear, this results in each offline node having fractional  degree ˆsi,t ≤ 1−γ in the assignment ˆx, by Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Consequently, by the same argument as Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3,  each bin but at most one in B ∈ Bt has �  i∈B ˆxi,t ≤ γ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' But since �  i ˆxi,t ≤ �  i xi,t ·(1+δ) ≤ (1+δ),  this implies that each bin but one has total x-value at least �  i∈B xit ≥ γ/2(1+δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Consequently, by  the fractional matching constraint, whereby �  i xi,t ≤ 1, there are at most 2(1 + δ)/γ + 1 = O(1/γ)  such non-empty bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  34  The benefit of few bins is that now at each time t the set of bidders is a set of at most O(1/γ)  offline nodes, and so there are at most nO(1/ǫ) possible sets S ⊆ [n] in the support of the distribution  over Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover, for each such set S ⊆ [n], it is straightforward to compute the probability that  S is the set of first-time proposers at time t in time nO(1/γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (Briefly, for each subset S, we guess for  each of the O(1/γ) nodes in S at what time they made their first bid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The probability of each of these  nO(1/γ) guessed events is then easy to compute in polynomial time, by computing the probability  that these nodes did bid in these guessed time steps and no other times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=') All in all, this allows to  compute, exactly, the support of the distribution of Pt, in time nO(1/γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Therefore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1,  the CRS invocations take time poly(nO(1/γ) = nO(1/γ) per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' On the other hand, having  scaled down the values xi,t by a factor of (1 − γ) trivially incurs a multiplicative loss of (1 − γ) in  the rounding ratio, and so the obtained ODRS has rounding ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='652(1 − γ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='652 − γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2  Deferred proofs of the b-matching ODRS  In this section we prove that our extension of Algorithm 4 from matching to b-matchings results in  similar benefits when much grouping occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We start by introducing some useful notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Our online level-set rounding algorithm, Algorithm 3, makes at most one random choice when  deciding whether to allocate another online node to the center of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Our b-matching ODRS  follows the same logic (from the perspective of the offline node, playing the role of the star’s center).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Denote by Ci,t an indicator for the relevant coin for i coming up heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Our extended core ODRS,  Algorithm 4, correlates these coins for different offline nodes i at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let Si,t ∈ {⌊si,t⌋, ⌈si,t⌉} be  the number of times i bids by time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By our algorithm’s definition, we have the following expression  for [Si,t ̸= ⌈si,t⌉].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  [Si,t ̸= ⌈si,t⌉] =  �  [Si,t−1 ̸= ⌈si,t−1⌉] · (1 − Ci,t−1)  if ⌈si,t⌉ = ⌈si,t−1⌉  1 − [Si,t−1 = ⌈si,t−1⌉] · Ci,t−1  if ⌈si,t⌉ > ⌈si,t−1⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For any time t, the random variables {Si,t}i are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Proof by induction on t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The base case is trivial, as Si,1 = 0 deterministically for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We  turn to the inductive step for t, assuming that the variables Si,t−1 are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' First, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4, for  any bin B ∈ Bt−1 at time t−1, the binary variables Ci,t−1 for all nodes i ∈ B, whose sum is at most  one, are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover, the distributions over different bins are independent, and so by closure of  NA under products (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5), all Ci,t−1 are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Moreover, the variables Ci,t−1 are independent  from the variables Si,t−1 (who are functions only of Ci,t′ for all t′ < t − 1), and consequently, by  our inductive hypothesis and another application of closure of NA under products (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5),  all the variables {Si,t−1, Ci,t−1}i are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Therefore, since Si,t−1 ∈ [⌊si,t−1⌋, ⌈si,t−1⌉], we find that  [Si,t ̸= ⌈si,t⌉] is a decreasing function of Si,t−1 and Ci,t−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', they are decreasing functions of disjoint  NA variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Therefore, by yet another application of closure of NA under products (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5),  the events {[Si,t ̸= ⌈si,t⌉]}i are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Finally, since Si,t = ⌈si,t⌉−[Si,t ̸= ⌈si,t⌉] are monotone decreasing  functions of NA variables, the variables Si,t are themselves NA, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We are now ready to prove that our generalization of Algorithm 4 results in bins essentially  simulating offline nodes bidding with a probability equal to their constituent nodes’ bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' That is,  we are ready to prove the following generalization of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For any time t and set of offline nodes S ⊆ [n], the b-matching extension of Algo-  rithm 4 satisfies  Pr[S ∩ Pt ̸= ∅] ≥ 1 −  �  B∈Bt  �  1 −  �  i∈B∩S  xit  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  35  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We denote by B′  t ⊆ Bt the set of singleton bins B = {i} for which ⌈si,t+1⌉ > ⌈si,t⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For such  bins B ∈ B′  t, B = {i}, we have that Pr[Pt ∩ S ∩ B = ∅] = 1 − xi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' For other bins B ∈ Bt \\ B′  t, we  have that Pr[i ∈ Pt] = xi,t, by independence of Ci,t and Si,t and since Pr[Ci,t] =  xi,t  1−(si,t−⌊si,t⌋) and  Pr[Si,t = ⌊si,t⌋] = si,t − ⌊si,t⌋ (the latter following from our online level-set algorithm’s analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Therefore, by �  i∈B Ci,t ≤ 1 implying disjointness of the events [i ∈ Pt] for i ∈ B, we have that  Pr[Pt ∩ S ∩ B = ∅] = 1 − �  i∈B∩S xi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' It remains to show that the events [Pt ∩ S ∩ B = ∅] for  different B are negative cylinder dependent, giving the desired inequality,  Pr[Pt ∩ S = ∅] = Pr  � �  B∈Bt  (Pt ∩ S ∩ B = ∅)  �  ≤  �  B∈Bt  Pr[Pt ∩ S ∩ B = ∅] =  �  B∈Bt  �  1 −  �  i∈B∩S  xi,t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Now, for bins B = {i} ∈ B′  t, we have that 1[Pt ∩ S ∩ B = ∅] = 1 − [Si,t = ⌈si,t⌉] · Ci,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  For  non-singleton bins B ∈ Bt, we have that 1[Pt ∩S ∩B = ∅] = 1−�  i∈B∩S Ci,t ·[Si,t = ⌊si,t⌋].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In both  cases, the indicators 1[Pt ∩S ∩B = ∅] are NA, as they are monotone decreasing functions of disjoint  NA variables (here, we need to first apply decreasing functions that map Si,t to ⌈si,t⌉ − Si,t for all  i ∈ B ∈ Bt \\B′  t and all other variables to themselves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Hence, by closure of NA under such functions  (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='5), the variables 1[Pt ∩ S ∩ B = ∅] are NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Therefore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3, these variables are  negative cylinder dependent, yielding the required inequality above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  D  Deferred Proofs of Section 6  In this section we prove that any sufficiently many binary variables must contain some 2r with some  (near-)positive 2r-wise correlation, generalizing the special case of r = 1 given by Fact 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Let r ∈ N and 0 ≤ ǫ ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, there exists some n = nr(p, ǫ) such that any  Bernoulli variables Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Yn ∼ Ber(p) contains some subset I ⊆ [n] of size |I| = 2r such that  E  ��  i  Yi  �  ≥  �  i  E[Yi] − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We prove this claim for all 0 ≤ ǫ ≤ p ≤ 1 by induction on r ≥ 1, and note that the claim is  trivial if p2r ≤ ǫ, in which case the RHS is negative, so nr(p, ǫ) = 2r suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Fact 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1 proves the  base case, showing that n1(p, ǫ) ≤ ⌈2p  ǫ + 1⌉ suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' To prove the inductive step, consider a set of  k := n1(p,  ǫ  22r ) + 2m variables Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Yk ∼ Ber(p), for m to be determined shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, we find  a sequence of disjoint pairs I1, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Im ⊆ [n], where for each Is = {i, j}, s ∈ [m], we have that  Cov(Yi, Yj) ≥ − ǫ  22r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The existence of such pairs follows by repeatedly invoking Fact 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1 applied  to the variables {Yi | i ∈ [n] \\ �s  ℓ=1 Iℓ} to obtain the pair Is+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Now, for each pair Is = {i, j} as  above, define a new variable Zs := Yi · Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' By Cov(Yi, Yj) ≥ − ǫ  22r , we have that Pr[Zs] ≥ p2 −  ǫ  22r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Next, couple these combined variables with equiprobable variables As ∼ Ber(p2 −  ǫ  22r ) with Zs ≥ As  always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Then, if we take m = nr−1(p2 −  ǫ  22r , ǫ  2), the inductive hypothesis applied to the m variables  As implies the existence of a subset S ⊆ [m] of 2r−1 of these variables satisfing the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  E  \uf8ee  \uf8f0  �  i∈�  s∈S Is  Yi  \uf8f9  \uf8fb = E  ��  s∈S  Zs  �  ≥ E  ��  s∈S  As  �  ≥  �  p2 − ǫ  22r  �2r−1  − ǫ  2 ≥ p2r − 22r−1 · ǫ  22r − ǫ  2 ≥ p2r − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Above, the first inequality follows by the coupling Zs ≥ As, the second inequality follows from the  inductive hypothesis, the third inequality follows from (a − b)k ≥ ak − 2k · b for any a, b ∈ [0, 1] and  k ∈ N (as seen by expanding (a − b)k), and the final inequality follows from 22r−1 ≥ 22r−1 for r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  We conclude that {Yi | i ∈ �  s∈S Is} is the desired subset of 2r variables in Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' , Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  36  Remark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' We did not attempt to optimize the minimum possible nr(p, ǫ) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Nonetheless,  for concrete bounds, we note that if p2r ≥ ǫ, then our proof yields bounds satisfying the following  recurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  n1(p, ǫ) ≤  �2p  ǫ + 1  �  ≤ 4p  ǫ ,  nr(p, ǫ) ≤ n1  �  p, ǫ  22r  �  + 2nr−1  �  p2 − ǫ  22r , ǫ  2  �  ≤ 4 · 22rp  ǫ  + 2nr−1  �  p2 − ǫ  22r , ǫ  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Noting that this recurrence is monotone increasing in p, we have that nr(p, ǫ) = O  �  22r · p  ǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  References  [1] Algorithms with predictions (ALPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' https://algorithms-with-predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  Accessed: 2022-11-07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  Diverse weighted bipartite b-  matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' Online graph edge-coloring in the random-  order arrival model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Theory of Computing 8, 1, 567–595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Contention resolution, matrix scaling and fair allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In Proceedings of the 19th Workshop on Approximation and Online Algorithms (WAOA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  pages ↑4, ↑5, ↑29  [8] Bar-Noy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', Motwani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' The greedy algorithm is optimal for on-line  edge coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Information Processing Letters (IPL) 44, 5, 251–253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' Delegated Pandora’s box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' pages  ↑4  [10] Bhattacharya, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' Online edge coloring algorithms  via the nibble method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In Proceedings of the 32nd Annual ACM-SIAM Symposium on Discrete  Algorithms (SODA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' pages ↑2, ↑16  [11] Birnbaum, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' and Mathieu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  On-line bipartite matching made simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  ACM  SIGACT News 39, 1, 80–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑1  37  [12] Blanc, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' and Charikar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Multiway online correlated selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In Proceedings of  the 62nd Symposium on Foundations of Computer Science (FOCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' pages ↑1, ↑3  [13] Borcea, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', Brändén, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', and Liggett, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Negative dependence and the geometry  of polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Journal of the American Mathematical Society 22, 2, 521–567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑29  [14] Boyce, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Processing and Management 18, 105––109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  Negative dependence in sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Scandinavian  Journal of Statistics 39, 4, 830–838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑3, ↑7, ↑30  [16] Braverman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Max-weight online  stochastic matching: Improved approximations against the online benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In Proceedings of  the 23rd ACM Conference on Economics and Computation (EC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' pages ↑3, ↑18, ↑19,  ↑20  [17] Buchbinder, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Lossless online rounding for online  bipartite matching (despite its impossibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In Proceedings of the 34th Annual ACM-SIAM  Symposium on Discrete Algorithms (SODA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑1  [18] Buolamwini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' Gender shades: Intersectional accuracy disparities in  commercial gender classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' Friedler and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Wilson, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Proceedings of Machine Learning  Research (PMLR) Series, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  Maximizing a mono-  tone submodular function subject to a matroid constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  SIAM Journal on Computing  (SICOMP) 40, 6, 1740–1766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' SIGIR Forum 51, 2, 209–210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  In Proceedings of the 31st ACM SIGIR Conference on Research and Development in  Information Retrieval (SIGIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  BRICS Report Series 3, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' Matroid secretary is equivalent to contention resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms  (SODA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' Dependent  rounding and its applications to approximation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Journal of the ACM (JACM) 53, 3,  324–360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Improved online  correlated selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Class fairness in online  matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' In Proceedings of the 54th Annual ACM Symposium on Theory of Computing (STOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' Understanding zadimoghaddam’s edge-weighted online match-  ing algorithm: Unweighted case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' arXiv preprint arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  Are “intersectionally fair” ai algorithms really fair to women of color?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  a  philosophical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content='  Online  edge coloring via tree recurrences and correlation decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In Proceedings of the 54th Annual ACM  Symposium on Theory of Computing (STOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' Learnable and instance-robust  predictions for online matching, flows and load balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In Proceedings of the 29th Annual  European Symposium on Algorithms (ESA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' pages ↑3  [58] Lee, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Optimal online contention resolution schemes via ex-ante  prophet inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In Proceedings of the 26th Annual European Symposium on Algorithms  (ESA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+page_content=' Online unrelated machine load balancing with predictions revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 6523–6532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑3  [60] Mehta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', Saberi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', Vazirani, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', and Vazirani, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Adwords and generalized  online matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Journal of the ACM (JACM) 54, 5, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑1  [61] Mitzenmacher,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  and  Vassilvitskii,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Algorithms  with  predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  CoRR abs/2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='09123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑3  [62] Papadimitriou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', Pollner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', Saberi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', and Wajc, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Online stochastic  max-weight bipartite matching: Beyond prophet inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In Proceedings of the 22nd ACM  Conference on Economics and Computation (EC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 763–764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑1, ↑2, ↑3, ↑18  [63] Pemantle, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' and Peres, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Concentration of lipschitz functionals of determinantal  and other strong rayleigh measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Combinatorics, Probability and Computing 23, 1, 140–160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  pages ↑6  [64] Pollner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', Roghani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', Saberi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', and Wajc, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Improved online contention  resolution for matchings and applications to the gig economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In Proceedings of the 23rd ACM  Conference on Economics and Computation (EC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 321–322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑4  [65] Saberi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' and Wajc, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' The greedy algorithm is not optimal for on-line edge coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In Proceedings of the 48th International Colloquium on Automata, Languages and Programming  (ICALP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 109:1–109:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑1, ↑2, ↑3, ↑16, ↑17, ↑18  [66] Srinivasan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Distributions on level-sets with applications to approximation algo-  rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' In Proceedings of the 42nd Symposium on Foundations of Computer Science (FOCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  588–597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑1, ↑2, ↑3, ↑6, ↑7, ↑32  [67] Srinivasan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Approximation algorithms for stochastic and risk-averse optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 1305–  1313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑3, ↑23, ↑24  [68] Swamy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' and Shmoys, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Approximation algorithms for 2-stage stochastic opti-  mization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' ACM SIGACT News 37, 1, 33–46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑3  41  [69] Swamy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' and Shmoys, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Sampling-based approximation algorithms for multi-  stage stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' SIAM Journal on Computing (SICOMP) 41, 4, 975–1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages  ↑3, ↑23  [70] Tang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=', and Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' (fractional) online stochastic matching via fine-  grained offline statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  In Proceedings of the 54th Annual ACM Symposium on Theory of  Computing (STOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 77–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑5  [71] Torrico, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' and Toriello, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Dynamic relaxations for online bipartite matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  INFORMS Journal on Computing 34, 4, 1841–2382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑18  [72] Vazirani, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Approximation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑11  [73] Wajc, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Matching theory under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' thesis, Carnegie Mellon University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  pages ↑1  [74] Williamson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' and Shmoys, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  The design of approximation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content='  Cambridge university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
+page_content=' pages ↑11  42' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9FAT4oBgHgl3EQfwR5T/content/2301.08680v1.pdf'}
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+arXiv:2301.12942v1  [cs.LG]  30 Jan 2023
+Refined Regret for Adversarial MDPs with Linear Function
+Approximation
+Yan Dai ∗
+Haipeng Luo †
+Chen-Yu Wei ‡
+Julian Zimmert §
+Abstract
+We consider learning in an adversarial Markov Decision Process (MDP) where the loss functions can change
+arbitrarily over K episodes and the state space can be arbitrarily large. We assume that the Q-function of any
+policy is linear in some known features, that is, a linear function approximation exists. The best existing regret
+upper bound for this setting (Luo et al., 2021b) is of order �
+O(K2/3) (omitting all other dependencies), given access
+to a simulator. This paper provides two algorithms that improve the regret to �
+O(
+√
+K) in the same setting. Our
+first algorithm makes use of a refined analysis of the Follow-the-Regularized-Leader (FTRL) algorithm with the log-
+barrier regularizer. This analysis allows the loss estimators to be arbitrarily negative and might be of independent
+interest. Our second algorithm develops a magnitude-reduced loss estimator, further removing the polynomial
+dependency on the number of actions in the first algorithm and leading to the optimal regret bound (up to
+logarithmic terms and dependency on the horizon). Moreover, we also extend the first algorithm to simulator-free
+linear MDPs, which achieves �
+O(K8/9) regret and greatly improves over the best existing bound �
+O(K14/15). This
+algorithm relies on a better alternative to the Matrix Geometric Resampling procedure by Neu and Olkhovskaya
+(2020), which could again be of independent interest.
+1
+Introduction
+Markov Decision Processes (MDPs) have been widely used to model reinforcement learning problems, where an
+agent needs to make decisions sequentially and to learn from the feedback received from the environment. In this
+paper, we focus on adversarial MDPs where the loss functions can vary with time and the state space can also
+be arbitrarily large, capturing the fact that in real-world applications such as robotics, the environment can be
+non-stationary and the number of states can be prohibitively large.
+To handle large state spaces, one of the most common methods in the literature is to assume a linear-function
+approximation (Yang and Wang, 2020; Jin et al., 2020b; Wei et al., 2021; Zanette et al., 2021; Neu and Olkhovskaya,
+2021; Luo et al., 2021b), where the expected loss suffered by any policy from any state-action pair, commonly known
+as the Q-function, is linear in a set of known features.
+While such assumptions are commonly used in the literature, the minimax optimal regret attainable by the
+agent is poorly understood, especially in the adversarial setting we study in this paper.1 Specifically, for adversarial
+linear-Q MDPs, the best existing bound is of order K2/3 where K is the number of episodes (Luo et al., 2021b).2
+In that paper, the authors assume a transition simulator (i.e., the agent is allowed to draw a trajectory starting
+from any state-action pair, sampled from the actual transition and a given policy, without any cost) and achieve
+�O(d2/3H2K2/3) regret where H is the length of each episode and d is the dimension of the feature space. On the
+other hand, the best lower bound for this setting (induced from the special case of adversarial linear bandits) is of
+order Ω(
+√
+K) (Dani et al., 2008). Therefore, a natural question arises:
+Is it possible to design an algorithm in adversarial linear-Q MDPs that attains �O(
+√
+K) regret bound?
+∗IIIS, Tsinghua University. Email: yan-dai20@mails.tsinghua.edu.cn.
+†University of Southern California. Email: haipengl@usc.edu.
+‡Massachusetts Institute of Technology. Email: chenyuw@mit.edu.
+§Google. Email: zimmert@google.com.
+1To be more specific, we only consider bandit feedback in this paper, where the agent can only observe her experienced losses. For
+the easier full-information setting,
+√
+K-style near-optimal regret has already been achieved (He et al., 2022) (cf. Table 1).
+2Meanwhile, if a “good” exploratory policy π0 (formalized in Footnote 4) is granted,
+√
+K-style bounds are also achievable, though with
+some additional dependencies on the quality of π0 (Luo et al., 2021b; Neu and Olkhovskaya, 2021); see Table 1.
+1
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+Table 1: Overview of Our Results and Comparisons with the Most Related Works
+Algorithm
+Settinga
+Transition
+Assumptionb
+Regret
+Dilated Bonus
+(Luo et al., 2021b)
+Linear-Q MDP
+(Theorem 2.2)
+Simulator
+(Theorem 2.3)
+None
+�O(d2/3H2K2/3)
+Exploratory Policy
+�O
+�
+poly(d, H)(K/λ0)1/2�
+Algorithm 1 (This work)
+None
+�
+O(A1/2d1/2H3K1/2)
+Algorithm 2 (This work)
+�O(d1/2H3K1/2)
+POWERS (He et al., 2022)
+Linear Mixture MDP
+Unknown
+Full Information
+�
+O(dHK1/2)
+Online Q-REPS
+(Neu and Olkhovskaya, 2021)
+Linear MDP
+(Theorem 2.4)
+Known
+Exploratory Policy
+�O
+�
+poly(d, H)(K/λ0)1/2�c
+Dilated Bonus
+(Luo et al., 2021a;b)
+Unknown
+None
+�O(d2H4K14/15)
+Exploratory Policy
+�O
+�
+poly(d, H)
+�
+K/λ2/3
+0
+�6/7�
+Algorithm 6 (This work)
+None
+�
+O(H20/9A1/9d2/3K8/9)
+aLinear MDP is a special case of linear-Q MDP, while linear mixture MDP is generally incomparable to these two.
+bThe definitions of “exploratory policy” and λ0 are stated in Footnote 4, while “full information” means the entire loss function is
+revealed at the end of each episode (which is easier than our bandit-feedback setting).
+cThis is a refined version of the original O(√K log K) bound presented in their Theorem 1, which contains no explicit dependency on
+λ0 by assuming K = Ω(exp(λ−1
+0 )). See Review hYYK at https://openreview.net/forum?id=gviX23L1bqw.
+In this work, we answer this question in the affirmative by developing two algorithms that both attain �O(
+√
+K) re-
+gret in adversarial linear-Q MDPs when a simulator is granted, closing the gap with the lower bound. Both of our algo-
+rithms follow the same framework of the policy optimization algorithm with dilated exploration bonuses of (Luo et al.,
+2021b), but with important modifications. Specifically, our first algorithm applies Follow-the-Regularized-Leader
+(FTRL) with the log-barrier regularizer (instead of the negative entropy regularizer used by Luo et al. (2021b)) at
+each state, and we develop a new analysis inspired by Zimmert and Lattimore (2022) that allows the loss estimators
+to be arbitrarily negative (in contrast, in the usual analyses of FTRL, the loss estimator cannot be too negative). This
+new analysis is the key to improving the regret to O(
+√
+K) and might be of independent interest even for multi-armed
+bandits.
+Although our first algorithm is simple in design and analysis, the log-barrier regularizer causes an O(
+√
+A) factor
+in the regret bound �O(H3√
+AdK) where A is the number of actions. As a rescue, we develop another algorithm that
+uses the negative entropy regularizer with a magnitude-reduced loss estimator. This algorithm attains an �O(H3√
+dK)
+regret bound, which is optimal in d and K up to logarithmic factors and gets rid of the poly(A) dependency. We
+note that our algorithm is of interest even for the special case of adversarial linear bandits, as it removes the need of
+explicit John’s exploration introduced by Bubeck et al. (2012).
+At last, we also apply our method to simulator-free linear MDPs (formally defined in Theorem 2.4), yielding an
+efficient algorithm with �O(K8/9) regret and greatly outperforming the best existing bound �
+O(K14/15) (Luo et al.,
+2021a).3 Remarkably, in this application, we not only use our refined analysis for FTRL with the log-barrier reg-
+ularizer, but also develop a more sample-efficient alternative to the Matrix Geometric Resampling (MGR) method
+introduced by Neu and Olkhovskaya (2020) and later adopted by Neu and Olkhovskaya (2021) and Luo et al. (2021a),
+which could also be of independent interest.
+1.1
+Related Work
+MDPs with Linear-Function Approximation.
+Linear function approximation has been a standard technique
+for handling large state spaces in RL, but only recently have researchers provided strong regret guarantees for these
+algorithms under precise conditions.
+Yang and Wang (2020) introduced a linear function approximation scheme
+3(Luo et al., 2021a) is a refined version of (Luo et al., 2021b).
+2
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+called embedded linear transition MDPs where the transition kernels are bilinear, i.e., the probability of reaching
+state s′ in the h-th step of an episode after taking action a at state s is P(s′ | s, a) = φ(s, a)TMψ(s′) for some
+known feature mappings φ and ψ and an unknown M. Jin et al. (2020b) loosen the assumption to linear MDPs
+(Theorem 2.4), i.e., P(s′ | s, a) = φ(s, a)Tν(s′) where ν is unknown. Zhou et al. (2021) study the linear mixture
+MDP with P(s′ | s, a) = ψ(s′ | s, a)Tθ where θ is unknown. This generalizes embedded linear transition MDPs but is
+incomparable with linear MDPs. Another common model is linear-Q MDPs (Abbasi-Yadkori et al., 2019) where the
+Q-function with respect to any policy π can be written as Qπ
+h(s, a) = φT(s, a)θπ
+h for some unknown θπ
+h (Theorem 2.2).
+Linear MDPs are special cases of linear-Q MDPs.
+Adversarial MDPs.
+MDPs with adversarial losses were first studied in the tabular cases where the state space
+has a small size S ≪ ∞. Zimin and Neu (2013) first assume known transitions and achieve �
+O(H
+√
+K) regret when
+full information is available and �O(
+√
+HSAK) regret when only bandit feedback is available. Rosenberg and Mansour
+(2019) then study the unknown-transition case and get �O(HS
+√
+AK) regret with full information. Finally, Jin et al.
+(2020a) tackle the hardest case with unknown transitions and bandit feedback and achieve �O(HS
+√
+AK) regret as
+well.
+Results for adversarial MDPs with linear-function approximations are summarized in Table 1.
+Specifically,
+Cai et al. (2020) study unknown-transition, full-information linear mixture MDPs and get �
+O(dH3/2√
+K) regret;
+this result is further improved to �
+O(dH
+√
+K) by He et al. (2022). Neu and Olkhovskaya (2021) then study known-
+transition, bandit-feedback linear MDPs. Provided with a “good” exploratory policy,4 their algorithm achieves an
+�O(poly(d, H)
+�
+K/λ0) regret guarantee. Later, Luo et al. (2021b) study bandit-feedback linear-Q MDPs with a sim-
+ulator, providing an �O(d2/3H2K2/3) regret bound. A refined version (Luo et al., 2021a) considers simulator-free
+bandit-feedback linear MDPs, giving an �O(d2H4K14/15) bound. Meanwhile, provided with a good exploratory pol-
+icy (see Footnote 4), these bounds improve to �O(poly(d, H)
+�
+K/λ0) and �O(poly(d, H)λ−4/7
+0
+K6/7), respectively. As
+a final remark, we point out that exploration in MDPs with huge state spaces is challenging and assuming an ex-
+ploratory policy is unrealistic — as far as we know, there are no results ensuring even the existence of such a “good”
+exploratory policy, let alone finding it efficiently. We emphasize that our results do not require such unrealistic
+exploratory assumptions.
+Policy Optimization Algorithms.
+Policy optimization algorithms for RL directly optimize the learner’s policy.
+They are more resilient to model misspecification or even adversarial manipulation. But due to their local search
+nature, they suffer from the notorious distribution mismatch issue. Recent theoretical works address this issue by
+exploration bonuses; see (Agarwal et al., 2020; Shani et al., 2020; Zanette et al., 2021) for stochastic settings and
+(Luo et al., 2021b) for adversarial settings. While the latter work achieves near-optimal regret in tabular settings,
+there is a huge room for improvement when considering function approximation. Our work builds on top of their
+framework (especially the dilated bonus idea) and significantly improves their results in linear-function approximation
+settings.
+2
+Preliminaries
+Notations.
+For N ∈ N, [N] denotes the set {1, 2, . . . , N}. For a (possibly infinite) set X, we denote the probability
+simplex over X by △(X). For a random event E, denote its indicator by
+1[E]. For two square matrices A, B of the
+same size, ⟨A, B⟩ stands for Tr(ATB). For x ∈ R, define (x)− as min{x, 0}. We use �O to hide all logarithmic factors.
+No-Regret Learning in MDPs.
+An (episodic) adversarial MDP is specified by a tuple M = (S, A, P, ℓ) where S
+is the state space (possibly infinite), A is the action space (assumed to be finite with size A = |A|), P: S ×A → △(S)
+is the transition, and ℓ: [K] × S × A → [0, 1] is the loss function chosen arbitrarily by an adversary. The state space
+is assumed to be layered, i.e., S = S1 ∪ S2 ∪ · · · ∪ SH where Sh ∩ Sh′ = ∅ for any 1 ≤ h < h′ ≤ H, and transition is
+only possible from one layer to the next one, that is, P(s′ | s, a) ̸= 0 only when s ∈ Sh and s′ ∈ Sh+1 for some h < H.
+We also assume that there is an initial state s1 such that S1 = {s1}.
+4Formally, a “good” exploratory policy π0 ensures that λmin(Σπ0
+h ) ≥ λ0 for all h ∈ [H] and a positive constant λ0, where Σπ0
+h
+means
+the covariance of π0 in layer h (see Eq. (2)).
+3
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+The game lasts for K episodes, each with length H. For each episode k, the agent is initialized at state s1. For
+each step h ∈ [H], she chooses an action ah ∈ A, suffers and observes the loss ℓk(sh, ah), and transits to a new state
+sh+1 independently sampled from the transition P(· | sh, ah).
+A policy π of the agent is a mapping from S to △(A). Let Π be the set of all policies. For each episode k ∈ [K],
+let πk ∈ Π be the policy deployed by the agent. Its expected loss is then indicated by V πk
+k (s1), where the state-value
+function (or V-function in short) V π
+k (s1) is defined as follows for any episode k and policy π:
+V π
+k (s1) ≜ E
+� H
+�
+h=1
+ℓk(sh, ah)
+�����(sh, ah) ∼ π, ∀h ∈ [H]
+�
+,
+with (sh, ah) ∼ π, ∀h ∈ [H] denoting a trajectory sampled from π. The agent aims to minimize the cumulative total
+loss collected in all episodes, or equivalently, the regret:
+Definition 2.1 (Regret). The regret of the agent is
+RK ≜ E
+� K
+�
+k=1
+V π1
+k (s1)
+�
+−
+K
+�
+k=1
+V π∗
+k (s1),
+where the expectation is taken over the randomness of both the agent and the transition, and π∗ is the optimal policy
+in hindsight (i.e., π∗ ∈ argminπ∈Π
+�K
+k=1 V π
+k (s1)).
+2.1
+Linear-Q MDP and Linear MDP
+A concept closely related to the V-function is the action-value function (a.k.a.
+Q-function), which denotes
+the expected loss suffered by a policy π starting from a given state-action pair (s, a). Formally, we define for all
+(s, a) ∈ S × A:
+Qπ
+k(s, a) = ℓk(s, a) +
+1[s /∈ SH]
+E
+s′∼P(·|s,a)
+a′∼π(·|s′)
+[Qπ
+k(s′, a′)] .
+(1)
+We can then define linear-Q MDPs as follows.
+Definition 2.2 ((Luo et al., 2021b, Assumption 1)). In a linear-Q MDP, each state-action pair (s, a) is associated
+with a known feature φ(s, a) ∈ Rd with ∥φ(s, a)∥2 ≤ 1. Moreover, for any policy π ∈ Π, episode k ∈ [K], and layer
+h ∈ [H], there exists a (hidden) vector θπ
+k,h ∈ Rd such that
+Qπ
+k(sh, ah) = φ(sh, ah)Tθπ
+k,h,
+∀sh ∈ Sh, ah ∈ A.
+We assume that ∥θπ
+k,h∥2 ≤
+√
+dH for all k, h, π.
+Theorem 2.2 does not specify any specific structure on the transition, making learning extremely difficult. Con-
+sequently, prior works all assume the availability of a simulator that allows us to sample from the hidden transition:
+Definition 2.3 (Simulator). A simulator accepts a state-action pair (s, a) ∈ S ×A and generates a next-state output
+sampled from the true transition, i.e., s′ ∼ P(· | s, a).
+When a simulator is unavailable, we consider a special case called linear MDPs which further impose a linear
+structure on the transition, enabling the agent to learn:
+Definition 2.4 ((Luo et al., 2021b, Assumption 3)). A linear MDP is a linear-Q MDP that additionally satisfies
+the following property: for any h ∈ [H], the transition from layer h to layer h + 1 can be written as follows:
+P(sh+1 | sh, ah) = ⟨φ(sh, ah), ν(sh+1)⟩,
+∀sh+1 ∈ Sh+1, sh ∈ Sh, ah ∈ A.
+Here, the mapping ν : S → Rd is also unrevealed to the agent. We also assume that ∥ν(s)∥2 ≤
+√
+d for all s ∈ S.
+In MDPs with linear function approximation, another important quantity associated with each policy π is its
+covariance matrix at each layer h ∈ [H], defined as follows:
+Σπ
+h =
+E
+(sh,ah)∼π[φ(sh, ah)φ(sh, ah)T],
+∀h ∈ [H].
+(2)
+4
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+2.2
+Dilated Bonuses for Policy Optimization
+We now briefly introduce the policy optimization method and the key dilated bonus idea of Luo et al. (2021b),
+upon which our algorithms are built. The foundation of policy optimization is the performance difference lemma
+(Kakade and Langford, 2002), which asserts that the regret of the agent can be viewed as the weighted average of
+the regret of some local bandit problem over each state. Formally, we have
+RK =
+H
+�
+h=1
+E
+sh∼π∗
+� K
+�
+k=1
+�
+a
+(πk(a|sh) − π∗(a|sh)) Qπk
+k (sh, a)
+�
+,
+where the part inside the expectation is exactly the regret of a multi-armed bandit (MAB) problem at state sh
+with “loss” Qπk
+k (sh, a) (instead of ℓk(sh, a)) for action a. Policy optimization algorithms then naturally run a bandit
+algorithm at each state with an appropriate Q-function estimator to learn the best policy directly. For example, in
+linear-Q MDPs, since the “loss" Qπk
+k (sh, a) is linear in some feature, it suggests running an adversarial linear bandit
+algorithm such as Exp2 (Bubeck et al., 2012) at each state.
+However, as discussed in detail by Luo et al. (2021b), the bias/variance of the Q-function estimator often leads to a
+regret term of the form �
+k,h E(sh,ah)∼π∗[bk(sh, ah)] for some non-negative functions b1, . . . , bK : S ×A → R≥0, where
+bk(s, a) is often prohibitively large if (s, a) is rarely visited by the agent. Hence, the expectation over (sh, ah) ∼ π∗
+could be potentially large as well, while the expectation over (sh, ah) ∼ πk is relatively small. This is the well-known
+distribution mismatch issue: for example, when applying a linear bandit algorithm at each state, bk(sh, ah) is roughly
+β∥φ(sh, ah)∥2
+(Σ
+πk
+h )−1 for some β > 0. Thus, E(sh,ah)∼πk[bk(sh, ah)] = β
+�
+(Σπk
+h )−1, E(sh,ah)∼πk[φ(sh, ah)φ(sh, ah)T]
+�
+=
+βd; on the other hand, its counterpart with (sh, ah) drawn from π∗ (i.e., E(sh,ah)∼π∗[bk(sh, ah)]) could be arbitrarily
+large.
+To address this distribution mismatch issue and “convert” the measure from πk to π∗, Luo et al. (2021b) consider
+treating these functions as exploration bonuses and further propose the so-called “dilated bonus” functions Bk(s, a):
+Bk(s, a) = bk(s, a) +
+1[s /∈ SH]
+�
+1 + 1
+H
+�
+E
+s′∼P(·|s,a)
+a′∼πk(·|s′)
+[Bk(s′, a′)] .
+(3)
+Compared to Eq. (1), Bk can be viewed the Q-function of bk, except that it assigns slightly more weight to deeper
+layers via the extra weighting 1 +
+1
+H to encourage more exploration to those layers. Their algorithms then try
+to minimize the regret with respect to the “optimistic" loss Qπk
+k
+− Bk, instead of just Qπk
+k , at each state. Their
+analysis relies on the following key lemma, which shows that if the regret w.r.t.
+Qπk
+k
+− Bk at each state is in
+some particular form, then the distribution mismatch issue can be resolved (i.e., the final regret RK is in terms of
+�
+k,h E(sh,ah)∼πk[bk(sh, ah)] instead of �
+k,h E(sh,ah)∼π∗[bk(sh, ah)]).
+Lemma 2.5 (Lemma 3.1 by Luo et al. (2021a)). If {bk}K
+k=1 are non-negative, Bk(s, a) is defined as in Eq. (3), and
+the following holds for any state s:
+E
+� K
+�
+k=1
+�
+a
+(πk(a|s) − π∗(a|s)) (Qπk
+k (s, a) − Bk(s, a))
+�
+≤ X(s) + E
+� K
+�
+k=1
+�
+a∈A
+�
+π∗(a|s)bk(s, a) + 1
+H πk(a|s)Bk(s, a)
+��
+,
+where X(s) is an arbitrary function, then
+RK ≤
+H
+�
+h=1
+E
+sh∼π∗[X(sh)] + 3
+K
+�
+k=1
+H
+�
+h=1
+E
+(sh,ah)∼πk
+[bk(sh, ah)].
+Luo et al. (2021b) show that the per-state regret bound in the condition of Theorem 2.5 indeed holds when
+deploying Follow-the-Regularized-Leader (FTRL) with the negative entropy regularizer at each state. However, for
+technical issues (which we will discuss and resolve in Sections 3 and 4), they only achieve �O(K2/3) regret in linear-Q
+MDPs.
+5
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+3
+�
+O(
+√
+K) Regret for Linear-Q MDPs via Refined Log-Barrier Analysis
+As mentioned, our first algorithm replaces the negative entropy regularizer in FTRL used by Luo et al. (2021b)
+with another regularizer called the log-barrier. Specifically, FTRL applied to the action space A maintains a sequence
+of distributions x1, x2, . . . , xT ∈ △([A]) via the FTRL update rule xt = argminx∈△([A]){η⟨x, �
+τ<t cτ⟩+Ψ(x)}, where
+c1, . . . , cT ∈ RA is the loss sequence, Ψ : △([A]) → R is the regularizer, and η > 0 is the learning rate. The classical
+MAB algorithm Exp3 (Auer et al., 2002) and its linear-bandit variant Exp2 (Bubeck et al., 2012) both use the
+negative entropy regularizer Ψ(x) = �A
+i=1 pi ln pi.
+Starting from (Foster et al., 2016), a sequence of works discover many nice properties of a different regularizer
+called log-barrier, defined as Ψ(p) = �A
+i=1 ln 1
+pi .
+Here, we provide yet another new and useful property of log-
+barrier, summarized in the following lemma. Our proof is inspired by the analysis of the log-determinant regularizer
+by Zimmert and Lattimore (2022) and is deferred to Appendix B.1.
+Lemma 3.1. Let x1, . . . , xT ∈ △([A]) be defined as
+xt = argmin
+x∈△([A])
+�
+η
+�
+x,
+�
+τ<t
+cτ
+�
++ Ψ(x)
+�
+,
+∀t = 1, . . . , T,
+where ct ∈ RA is an arbitrary loss vector corresponding to the t-th iteration and Ψ(p) = �A
+i=1 ln 1
+pi is the log-barrier
+regularizer. Then the regret against any distribution y ∈ △([A]) with respect to {ct}T
+t=1 is bounded as:
+T
+�
+t=1
+⟨xt − y, ct⟩ ≤ Ψ(y) − Ψ(x1)
+η
++ η
+T
+�
+t=1
+A
+�
+i=1
+xt,ic2
+t,i.
+Readers familiar with the Exp2/Exp3 analysis would immediately recognize this regret bound since it is also the
+same bound that FTRL with negative entropy (also known as Hedge) enjoys. However, the key distinction is that
+for log-barrier, this holds without any requirement on the magnitude of ct,i, while for negative entropy, one must
+require ηct,i ≥ −1 for all t ∈ [T ] and i ∈ [A] (i.e., losses cannot be too negative; see Theorem C.1 in the appendix for
+more details). This turns out to be critical for improving the regret when applying it to linear-Q MDPs, as discussed
+later.
+Remark 3.2. Our Theorem 3.1 also answers the open question raised by Zheng et al. (2019) (see their remark after
+Lemma 14): it is indeed possible for FTRL with log-barrier to attain a �
+i xt,ic2
+t,i-style bound without any restrictions
+on the losses. As log-barrier regularizers are widely used in the literature for its better data-adaptivity (Wei and Luo,
+2018; Ito, 2021), we expect this result to be of independent interest.
+Linear-Q Algorithm.
+Our final algorithm for linear-Q MDPs is shown in Algorithm 1, which is nearly the same
+as (Luo et al., 2021b, Algorithm 2) except for the part marked in blue where we use the log-barrier regularizer, as
+mentioned. Specifically, based on the discussions in Section 2.2, the loss fed to FTRL is �Qk−Bk. Here, �Qk (defined in
+Eq. (5)) is a standard estimator for Qπk
+k involving an estimate �Σ†
+k,h for the inverse of Σπk
+h , which is constructed via the
+Matrix Geometric Resampling procedure (Neu and Olkhovskaya, 2020) with the help of the simulator. Meanwhile,
+Bk is the dilated bonus following the idea of Eq. (3) with bk(s, a) = β
+�
+∥φ(s, a)∥2
+�Σ†
+k,h + E�a∼πk(·|s)
+�
+∥φ(s, �a)∥2
+�Σ†
+k,h
+��
+for
+s ∈ Sh (which again uses the simulator; see Algorithm 4 as well as detailed discussions in (Luo et al., 2021a)).
+With the help of our new log-barrier analysis, we show that our algorithm achieves �
+O(
+√
+K) regret, as follows.
+Theorem 3.3. Algorithm 1 when applied to a linear-Q MDP (Theorem 2.2) with a simulator ensures the following
+when 12ηβH2 ≤ γ, 8ηH2 ≤ β, and ǫ ≤ (H2K)−1:
+RK = �
+O
+�
+βdHK + γ
+β dH3K + AH
+η
++ β
+γ AH2 + A
+√
+dH2 + ηH3
+γ2K2
+�
+.
+Picking η =
+�
+A
+dH4K , β = 8
+�
+A
+dK , γ = 96A
+dK , and ǫ =
+1
+H2K , we have RK = �
+O(H3√
+AdK).
+We defer the full proof to Appendix B and provide a sketch below highlighting why removing any constraints on
+the losses fed to FTRL is critical to improving the regret.
+6
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+Algorithm 1 Improved Linear-Q Algorithm (Using Log-Barrier Regularizers)
+Input: Learning rate η, bonus parameter β, MGR parameters γ and ǫ, FTRL regularizer Ψ(p) = �|A|
+i=1 ln 1
+pi .
+1: for k = 1, 2, . . ., K do
+2:
+Let πk ∈ Π be defined as follows for all s ∈ S:
+πk(s) = argmin
+p∈△(A)
+�
+Ψ(p) + η
+�
+k′<k
+⟨p(·), �Qk′(s, ·) − Bk′(s, ·)⟩
+�
+,
+(4)
+where Bk(s, a) is calculated in Algorithm 4.
+3:
+Execute πk, observing the trajectory (sk,h, ak,h) and losses ℓk(sk,h, ak,h) for all h ∈ [H].
+4:
+Construct covariance matrix inverse estimate �Σ†
+k,h using Matrix Geometric Resampling (Algorithm 3 in the
+appendix) s.t. ∥�Σ†
+k,h∥2 ≤ 1
+γ ,
+���E[�ӆ
+k,h] − (γI + Σπk
+h )−1���
+2 ≤ ǫ,
+and the following holds w.p. 1 − K−3:
+���(�ӆ
+k,h)1/2Σπk
+h (�Σ†
+k,h)1/2���
+2 ≤ 2.
+5:
+Let Lk,h = �
+h′≥h ℓk(sk,h′, ak,h′). Estimate the Q-function Qπk
+k (s, a) as follows (for s ∈ Sh):
+�Qk(s, a) = φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h)Lk,h.
+(5)
+6: end for
+Proof Sketch. To bound RK by the dilated bonus lemma (Theorem 2.5), we focus on a single (k, s)-pair and bound
+�
+a(πk(a | s) − π∗(a | s))(Qπk
+k (s, a) − Bk(s, a)).
+(6)
+As the FTRL lemma only applies to the losses fed into Eq. (4) (namely �Qk(s, a) − Bk(s, a)), we add and substract
+�Qk(s, a) in Eq. (6). Hence, after summing over k ∈ [K] and s ∼ π∗, we need to consider the following three terms
+to figure out the term X(s) in Theorem 2.5:
+• Bias-1 = �
+k,h
+E
+sh∼π∗
+�
+E
+ah∼πk
+�
+Qπk
+k (sh, ah) − �Qk(sh, ah)
+��
+, measuring the under-estimation of �Qk w.r.t. πk.
+• Bias-2 = �
+k,h
+E
+sh∼π∗
+�
+E
+ah∼π∗
+�
+�Qk(sh, ah) − Qπk
+k (sh, ah)
+��
+, measuring the over-estimation of �Qk w.r.t. π∗.
+• Reg-Term = �
+k,h
+E
+sh∼π∗
+� �
+a
+(πk(a|s) − π∗(a|s))( �Qk(s, a) − Bk(s, a))
+�
+, which can be tackled by Theorem 3.1.
+As Bias-1 and Bias-2 are independent of the regularizer, they are handled similarly to the original analysis
+and both contribute �
+O( γ
+β dH3K) to �
+h Esh∼π∗[X(sh)] (see Theorem B.1 in the appendix). To handle Reg-Term,
+we apply Theorem 3.1. As in standard log-barrier analyses, since Ψ(π∗(· | s)) − Ψ(π1(· | s)) is potentially infinity,
+we introduce a smooth version �π∗ defined via �π∗(a | s) = (1 − AK−1)π∗(a | s) + K−1. Given an s ∈ S, we have
+Ψ(�π∗(· | s)) − Ψ(π1(· | s)) ≤ A log K. Hence, fixing the state s, we can write
+K
+�
+k=1
+�
+a∈A
+(πk(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))
+≤
+K
+�
+k=1
+�
+a∈A
+(�π∗(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))+
+A log K
+η
++ η
+K
+�
+k=1
+�
+a∈A
+πk(a | s)2( �Qk(s, a) − Bk(s, a))2.
+7
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+After summing over h and taking expectation over s, we show that the first term is of order �
+O(AH2(
+√
+d + β/γ)),
+while the last term’s contribution to �
+h Esh∼π∗[X(sh)] is of order �O( ηH3
+γ2K2 ) by the same analysis of (Luo et al.,
+2021b). Finally, we combine everything, apply Theorem 2.5, and show that the term �
+k,h E(sh,ah)∼πk[bk(sh, ah)]
+in this case is �O(βdHK), finishing the proof for the first claimed regret bound. This bound is almost identical
+to that of (Luo et al., 2021b). However, because of the use of negative-entropy, their analysis requires 2Hη ≤ γ,
+which prevents them from picking a very small γ and eventually leads to �
+O(K2/3) regret. On the other hand, our
+log-barrier analysis allows us to drop this requirement and pick a γ as small as K−1, though bearing some extra
+factors polynomial in A, i.e., the number of actions. Indeed, optimizing the bound over the parameters (see the
+second claim), we achieve RK = �O(H3√
+AdK), which is of order
+√
+K.
+4
+Removing Polynomial Dependency on A via a New Magnitude-Reduced
+Estimator
+One drawback of using log-barrier is that the term Ψ(y) − Ψ(x1) from Theorem 3.1 leads to poly(A) dependency
+(while for negative entropy, this is only log A). To get around this issue, we go back to using negative entropy. Recall
+that the issue of this regularizer is that to ensure the same bound as Theorem 3.1, we require ηct,i ≥ −1, which
+translates to the requirement η( �Qk(s, a) − Bk(s, a)) ≥ −1 in the context of linear-Q MDPs. The challenging part
+here is that �Qk(s, a), as defined in Eq. (5), could be as negative as −H/γ, which then restricts η to be of order
+O(γ/H), as mentioned.
+To resolve this, we propose a new Q-function estimator that has a smaller magnitude in the negative direction.
+Our high-level idea is the following: for a possibly negative random variable Z, defined another magnitude-reduced
+random variable �Z = Z − (Z)− + E[(Z)−] (recall (Z)− = min{Z, 0}). �Z then has the same expectation as Z and
+also the same order of second moment: E[ �Z2] ≤ 2 E[Z2] + 2(E[(Z)−])2 = O(E[Z2]). More importantly, we have
+�Z ≥ E[(Z)−], and thus the smallest possible value of �Z is E[(Z)−], no less than (often much larger than) the smallest
+possible value of Z. Thus, it becomes much easier to ensure ηct,i ≥ −1.
+Applying this idea to our context, we propose a new Q-function estimator as in Eq. (9), where mk(s, a) exactly
+takes the role of E[(Z)−], except that we have no access to the real expectation but have to approximate it with
+samples; see Line 7 of Algorithm 2. To make sure that these samples are “consistent" with those used in constructing
+�Σ†
+k,h, we perform a check in Line 6 and repeat the sampling until the check passes.
+Since both Eq.
+(8) and
+∥�Σk,h − Σk,h∥2 ≤ γ hold with probability at least 1 − K−3, the expected number of trials is only 1 + o(1). Our
+complete algorithm in shown in Algorithm 2, where the parts different from (Luo et al., 2021b) are again highlighted
+in blue.
+Our analysis shows that the new Q-function estimator indeed has a significantly smaller magnitude: it can only
+be as negative as −H/√γ (as opposed to the previous −H/γ bound). This only restricts η to be of order O(√γ/H),
+enabling us to pick γ ≈ 1/K as in Theorem 3.3 and achieve �
+O(
+√
+K) regret again, as formalized in the next theorem.
+Note that our final regret bound not only has no poly(A) dependency, but is also optimal in both d and K as one
+cannot do better even in the special case of linear bandits.
+Theorem 4.1. When applied to a linear-Q MDP (Theorem 2.2) with a simulator, Algorithm 2 ensures the following
+when ηβγ−1 ≤
+1
+12H2 , η2γ−1 ≤
+1
+12H2 , 8ηH2 ≤ β, ǫ ≤ (H2K)−1, and M = 32γ−2 log K:
+RK = �
+O
+�
+βdHK + γ
+β dH3K + H
+η + ηdH3K +
+η
+γ2K2 H3
+�
+.
+Picking η =
+1
+√
+dH4K , β =
+8
+√
+dK , γ = 96
+dK , and ǫ =
+1
+H2K , we have RK = �
+O(H3√
+dK).
+Proof Sketch. Due to space limitations, we only sketch why �Qk(s, a) is now at least − O(H/√γ): since Lk,h ∈ [0, H],
+we have �Qk(s, a) ≥ Hmk(s, a). By Jensen’s inequality, we can bound mk(s, a)2 for all (s, a) as:
+mk(s, a)2 ≤ 1
+M
+M
+�
+m=1
+�
+φ(s, a)T�Σ†
+k,hφ(sm,h, am,h)
+�2
+8
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+Algorithm 2 Improved Linear-Q Algorithm (Using Magnitude-Reduced Loss Estimators)
+Input: Learning rate η, bonus parameter β, MGR parameters γ and ǫ, covariance estimation parameter M.
+1: for k = 1, 2, . . ., K do
+2:
+Let πk ∈ Π be defined as follows for all s ∈ S:
+πk(a | s) ∝ exp
+�
+− η
+�
+k′<k
+( �Qk′(s, a) − Bk′(s, a))
+�
+,
+where Bk(s, a) is calculated in Algorithm 4.
+3:
+Execute πk, observing the trajectory (sk,h, ak,h) and losses ℓk(sk,h, ak,h) for all h ∈ [H].
+4:
+Construct �Σ†
+k,h using Matrix Geometric Resampling (Algorithm 3 in the appendix) s.t. ∥�Σ†
+k,h∥2 ≤ 1
+γ ,
+���E[�ӆ
+k,h] − (γI + Σπk
+h )−1���
+2 ≤ ǫ,
+(7)
+and the following holds w.p. 1 − K−3:
+���(�ӆ
+k,h)1/2Σπk
+h (�Σ†
+k,h)1/2���
+2 ≤ 2.
+(8)
+5:
+Simulate πk for M times, giving {(sm,h, am,h)}m,h. Estimate the covariance matrix Σπk
+h
+as
+�Σk,h = 1
+M
+M
+�
+m=1
+φ(sm,h, am,h)φ(sm,h, am,h)T.
+6:
+If ∥(�Σ†
+k,h)1/2�Σk,h(�Σ†
+k,h)1/2∥2 ≥ 3, goto Line 4.
+7:
+For each s ∈ Sh and a ∈ A, define mk(s, a) as
+mk(s, a) = 1
+M
+M
+�
+m=1
+�
+φ(s, a)T�Σ†
+k,hφ(sm,h, am,h)
+�
+− .
+8:
+Let Lk,h = �
+h′≥h ℓk(sk,h′, ak,h′). Estimate the Q-function Qπk
+k (s, a) as follows (for s ∈ Sh):
+�Qk(s, a) = φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h)Lk,h−
+H
+�
+φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h)
+�
+− + Hmk(s, a).
+(9)
+9: end for
+= φ(s, a)T�Σ†
+k,h�Σπk
+h �Σ†
+k,hφ(s, a) ≤ 3∥φ(s, a)∥2
+�Σ†
+k,h ≤ 3
+γ ,
+where the second inequality is by Line 6. Therefore, we have �Qk(s, a) ≥ −
+√
+3H
+√γ , a reduced magnitude compared to
+that of a standard estimator (defined in Eq. (5)).
+5
+Generalizing to Linear MDPs
+When a simulator is unavailable, we consider a special case of linear-Q MDPs: linear MDPs (Theorem 2.4), where
+the transition is also linear in the features. As Luo et al. (2021a) show, this makes the dilated bonuses linear as
+well, and thus we can efficiently estimate them without using a simulator. Due to various technical obstacles, they
+only achieve �O(K14/15) regret. We improve it to �O(K8/9) via two key modifications to their algorithm. As the
+algorithm is fairly lengthy due to the lack of simulators and the estimation of the dilated bonus function, we defer it
+9
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+(Algorithm 6) to the appendix. We refer the reader to (Luo et al., 2021a) for a detail description of their algorithm,
+and only focus on our two modifications (highlighted in blue) described below.
+The first obvious modification is to apply one of the techniques introduced in the last two sections. However, since
+the magnitude-reduced estimator we developed in Algorithm 2 requires extra samples (see Line 5), which, without a
+simulator, can only be done via real episodes and in turn introduces more regret, we go with the log-barrier approach
+instead, even though it leads to extra poly(A) factors.
+But it turns out that this only leads to a mild improvement in the regret, since the constraint on η is not the
+bottleneck of their analysis. Instead, one important bottleneck comes from using the Matrix Geometric Resampling
+(MGR) procedure (Neu and Olkhovskaya, 2020) to estimate the covariance matrix inverse, which again, without a
+simulator, can only be done via running the same policy for multiple real episodes (called an epoch) to collect samples.
+It is thus critical to make this step as sample efficient as possible.
+To resolve this issue, our key observation is that MGR uses too many samples to produce an estimate that is
+in a sense more accurate than required: specifically, it needs O(ǫ−2γ−3) samples to ensure a bound like Eq. (7).
+Instead, we find that a weaker multiplicative approximation guarantee is enough, which only requires O(γ−2) samples.
+Moreover, this is achieved by simply taking the (regularized) inverse of the empirical average of O(γ−2) samples.
+More concretely, for a policy �πj (which mixes πj with some exploration policy; see Algorithm 6 for more details)
+in epoch j of the algorithm, we collect roughly O(γ−2) samples using this policy and construct an empirical average
+covariance matrix �Σj,h for layer h. Then we let �Σ†
+j,h = (γI + �Σj,h)−1 be the estimation for (γI + Σ�πj
+h )−1 (see Line 17
+of Algorithm 6). We then prove the following:
+Lemma 5.1. If the epoch length W in Algorithm 6 is set to (4d log d
+δ )γ−2, then
+(1 − √γ)(γI + Σ�πj
+h ) ⪯ (γI + �Σj,h) ⪯ (1 + √γ)(γI + Σ�πj
+h )
+holds with probability at least 1 − 2δ.
+The proof relies on a new matrix concentration bound (Theorem A.4 in the appendix), which can be of independent
+interest. A direct corollary of this lemma is:
+Corollary 5.2. If W = (4d log d
+δ )γ−2 and γ ≤ 1
+4, then with probability 1 − 2δ, we have the following:
+(1 − 2√γ)I ⪯ (�Σ†
+j,h)1/2(γI + Σ�πj
+h )(�Σ†
+j,h)1/2 ⪯ (1 + 2√γ)I.
+Proof. Simply left and right multiply (�Σ†
+j,h)1/2 = (γI + �Σj,h)−1/2 in the bound of Theorem 5.1 and use the fact
+[
+1
+1+√γ ,
+1
+1−√γ ] ⊆ [1 − 2√γ, 1 + 2√γ] when γ ≤ 1
+4.
+Together with a new analysis to bound bias terms in the regret, such approximations turn out to be enough: the
+bias terms enjoy almost the same bound (up to constants) as we had in previous sections. Hence, we finally get the
+following theorem; see Appendix D for a full proof.
+Theorem 5.3 (Informal version of Theorem D.1). When applied to linear MDPs, Algorithm 6 with proper tuning
+ensures RK = �O(K8/9) (omitting all other dependencies).
+Proof Sketch. Ignoring less important parts, at a high level we still decompose the regret into several bias terms and
+a Reg-Term. In this sketch, we only provide key ideas in bounding the bias terms using Theorem 5.2. Specifically,
+we illustrate how to bound E[Bias-1], defined as follows:
+Bias-1 ≜ W
+�
+j,h
+E
+sh∼π∗
+�
+E
+ah∼πj
+�
+Q
+πj
+j (sh, ah) − �Qj(sh, ah)
+��
+,
+where Q
+πj
+j (s, a) = φ(s, a)Tθ
+πj
+j,h is the average Q-functions in epoch j induced by policy πj, and �Qj(s, a) is its estimator.
+By direct calculation, one may check that
+E
+�
+Q
+πj
+j (s, a) − �Qj(s, a)
+�
+= φ(s, a)T(I − �Σ†
+j,hΣ�πj
+h )θ
+πj
+j,h,
+= φ(s, a)T�Σ†
+j,h(γI + �Σj,h − Σ�πj
+h )θ
+πj
+j,h
+10
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+≤ ∥φ(s, a)∥�Σ†
+j,h
+���(γI + �Σj,h − Σ�πj
+h )θ
+πj
+j,h
+����ӆ
+j,h
+≤ ∥φ(s, a)∥�Σ†
+j,h
+����γθ
+πj
+j,h(s, a)
+����ӆ
+j,h
++
+���(�Σj,h − Σ�πj
+h )θ
+πj
+j,h
+����ӆ
+j,h
+�
+≤ β
+4 ∥φ(s, a)∥2
+�Σ†
+j,h + 2
+β
+���γθ
+πj
+j,h(s, a)
+���
+2
+�Σ†
+j,h
++ 2
+β
+���(�Σj,h − Σ�πj
+h )θ
+πj
+j,h
+���
+2
+�Σ†
+j,h
+,
+where we apply Cauchy-Schwarz inequality, triangle inequality, and AM-GM inequality (twice). The first term above
+contributes to O(βdHK) after summing over k and h and applying Theorem 2.5, while the second term is bounded
+by O( γ
+β dH2) for any j (as ∥�Σ†
+j,h∥2 ≤ γ−1). By some algebraic manipulations, the last term translates to
+2
+β
+����(�ӆ
+j,h)−1/2�
+I − (�Σ†
+j,h)1/2(γI + Σ�πj
+h )(�Σ†
+j,h)1/2�2(�Σ†
+j,h)−1/2
+����
+2
+dH2.
+Thanks to Theorem 5.2, we know that
+−2√γI ⪯ I − (�Σ†
+j,h)1/2(γI + Σ�πj
+h )(�Σ†
+j,h)1/2 ⪯ 2√γI,
+and thus the last term is again of order O( γ
+βdH2). All other bias terms are bounded analogously. At last, it only
+remains to bound the Reg-Term, which is directly calculated and bounded like we did in the proof of Theorem 3.3.
+6
+Conclusion
+In this paper, we study policy optimization algorithms in adversarial MDPs with linear function approximation.
+Building on top of the dilated bonus approach introduced by Luo et al. (2021b), we derive two algorithms which both
+achieve the first �
+O(
+√
+K)-style regret bounds in linear-Q MDPs with the help of a simulator. Technically speaking, the
+first algorithm use a novel refined analysis for FTRL with the log-barrier regularizer, while the second one develops a
+new magnitude-reduced loss estimator. Moreover, we also generalize the first approach to simulator-free linear MDPs
+and get �
+O(K8/9) regret, greatly improving over the best-known �
+O(T 14/15) regret bound (Luo et al., 2021a). This
+generalization also contains an alternative to the Matrix Geometric Resampling procedure (Neu and Olkhovskaya,
+2020) using a new matrix concentration bound (Theorem A.4). We expect all these techniques to be of independent
+interest and potentially useful for other problems. Further improving the result for linear MDPs is the key future
+direction.
+References
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+Peter Auer, Nicolo Cesa-Bianchi, Yoav Freund, and Robert E Schapire.
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+Refined Regret for Adversarial MDPs with Linear Function Approximation
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+in Neural Information Processing Systems, 20, 2008.
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+on Learning Theory, pages 2552–2583. PMLR, 2021.
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+with bandit feedback and unknown transition. In International Conference on Machine Learning, pages 4860–4869.
+PMLR, 2020a.
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+Tor Lattimore and Csaba Szepesvári. Bandit algorithms. Cambridge University Press, 2020.
+Haipeng Luo, Chen-Yu Wei, and Chung-Wei Lee. Policy optimization in adversarial mdps: Improved exploration
+via dilated bonuses. arXiv preprint arXiv:2107.08346, 2021a.
+Haipeng Luo, Chen-Yu Wei, and Chung-Wei Lee. Policy optimization in adversarial mdps: Improved exploration
+via dilated bonuses. Advances in Neural Information Processing Systems, 34:22931–22942, 2021b.
+Lingsheng Meng and Bing Zheng. The optimal perturbation bounds of the moore–penrose inverse under the frobenius
+norm. Linear algebra and its applications, 432(4):956–963, 2010.
+Gergely Neu and Julia Olkhovskaya. Efficient and robust algorithms for adversarial linear contextual bandits. In
+Conference on Learning Theory, pages 3049–3068. PMLR, 2020.
+Gergely Neu and Julia Olkhovskaya. Online learning in mdps with linear function approximation and bandit feedback.
+Advances in Neural Information Processing Systems, 34:10407–10417, 2021.
+Aviv Rosenberg and Yishay Mansour. Online convex optimization in adversarial markov decision processes. In
+International Conference on Machine Learning, pages 5478–5486. PMLR, 2019.
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+In International Conference on Machine Learning, pages 8604–8613. PMLR, 2020.
+Joel A Tropp. User-friendly tail bounds for sums of random matrices. Foundations of computational mathematics,
+12(4):389–434, 2012.
+Ruosong Wang, Simon S Du, Lin Yang, and Russ R Salakhutdinov. On reward-free reinforcement learning with
+linear function approximation. Advances in neural information processing systems, 33:17816–17826, 2020.
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+Theory, pages 1263–1291. PMLR, 2018.
+Chen-Yu Wei, Mehdi Jafarnia Jahromi, Haipeng Luo, and Rahul Jain. Learning infinite-horizon average-reward
+mdps with linear function approximation. In International Conference on Artificial Intelligence and Statistics,
+pages 3007–3015. PMLR, 2021.
+Lin Yang and Mengdi Wang. Reinforcement learning in feature space: Matrix bandit, kernels, and regret bound. In
+International Conference on Machine Learning, pages 10746–10756. PMLR, 2020.
+12
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+Refined Regret for Adversarial MDPs with Linear Function Approximation
+Andrea Zanette, Ching-An Cheng, and Alekh Agarwal. Cautiously optimistic policy optimization and exploration
+with linear function approximation. In Conference on Learning Theory, pages 4473–4525. PMLR, 2021.
+Kai Zheng, Haipeng Luo, Ilias Diakonikolas, and Liwei Wang. Equipping experts/bandits with long-term memory.
+Advances in Neural Information Processing Systems, 32, 2019.
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+feature mapping. In International Conference on Machine Learning, pages 12793–12802. PMLR, 2021.
+Alexander Zimin and Gergely Neu. Online learning in episodic markovian decision processes by relative entropy
+policy search. Advances in neural information processing systems, 26, 2013.
+Julian Zimmert and Tor Lattimore.
+Return of the bias: Almost minimax optimal high probability bounds for
+adversarial linear bandits. In Conference on Learning Theory, pages 3285–3312. PMLR, 2022.
+13
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+Supplementary Materials
+A Auxiliary Lemmas and Omitted Algorithms
+15
+A.1 Matrix Geometric Resampling Procedure
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+A.2 Dilated Bonus Calculation in Linear-Q MDP Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+A.3 PolicyCover Algorithm in Linear MDP Algorithms
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+A.4 Stochastic Matrix Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+B Omitted Proofs in Section 3 (Linear-Q Algorithm Using Log-Barrier Regularizers)
+19
+B.1
+Property of FTRL with Log-Barrier Regularizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+B.2
+Proof of Main Theorem
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+B.3
+Bounding Bias-1 and Bias-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+B.4
+Bounding Reg-Term
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+22
+C Omitted Proofs in Section 4 (Linear-Q Algorithm Using Magnitude-Reduced Estimators)
+24
+C.1 Property of FTRL with Negative-Entropy Regularizers (a.k.a. Hedge) . . . . . . . . . . . . . . . . . .
+24
+C.2 Proof of Main Theorem
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+C.3 Bounding Bias-1 and Bias-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+26
+C.4 Bounding Reg-Term
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+26
+D Omitted Proofs in Section 5 (Linear MDP Algorithm)
+30
+D.1 Pseudocode of the Improved Linear MDP Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+30
+D.2 Alternative to the Matrix Geometric Resampling Procedure . . . . . . . . . . . . . . . . . . . . . . . .
+30
+D.3 Proof of Main Theorem
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+31
+D.4 Bounding the Bias Terms
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+33
+D.5 Bounding Reg-Term
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+34
+D.6 Bounding the Magnitudes of Bonuses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+35
+14
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+A
+Auxiliary Lemmas and Omitted Algorithms
+A.1
+Matrix Geometric Resampling Procedure
+Matrix Geometric Resampling algorithm, introduced by Neu and Olkhovskaya (2020) and improved by Luo et al.
+(2021b), generates a close estimation of (γI + Σπ
+h)−1 for given π ∈ Π and h ∈ [H].
+Formally, we present it in
+Algorithm 3 and state its performance guarantees in Theorem A.1.
+Algorithm 3 Matrix Geometric Resampling Algorithm
+Input: Policy π, regularization parameter γ, desired precision ǫ.
+Output: A set of matrices {�Σ†
+h}H
+h=1.
+1: Set M ≥ 24 ln(dHT )
+ǫ2γ2
+and N ≥ 2
+γ ln 1
+ǫγ . Set c = 1
+2.
+2: Draw MN trajectories of π using the simulator; denote them by {(sm,n,h, am,n,h)}m∈[M],n∈[N],h∈[H].
+3: for m = 1, 2, . . ., M do
+4:
+for n = 1, 2, . . ., N do
+5:
+Set Yn,h = γI + φ(sm,n,h, am,n,h)φ(sm,n,h, am,n,h)T and Zn,h = �n
+n′=1(I − cYn′,h) for all h ∈ [H].
+6:
+end for
+7:
+Calculate �Σ†
+m,h = cI + c �N
+n=1 Zn,h for all h ∈ [H].
+8: end for
+9: Set �Σ†
+h =
+1
+M
+�M
+m=1 �Σ†
+m,h for all h ∈ [H].
+Lemma A.1 (Lemma D.1 of Luo et al. (2021b)). With the configurations of M and N stated in Algorithm 3, for
+any policy π, we have ∥�Σ†
+h∥2 ≤ γ−1, ∥E[�Σ†
+h] − (γI + Σπ
+h)−1∥2 ≤ ǫ. Moreover, there exists a good event that happens
+with probability 1 −
+1
+K3 , under which the following two properties hold:
+∥E[�Σ†
+h] − (γI + Σπ
+h)−1∥2 ≤ 2ǫ,
+∥�Σ†
+hΣπ
+h∥2 ≤ 1 + 2ǫ.
+A.2
+Dilated Bonus Calculation in Linear-Q MDP Algorithms
+The following algorithm (which is Algorithm 3 of Luo et al. (2021b)) indicates how we calculate the dilated bonus
+function Bk(s, a) with the help of the simulator in Algorithms 1 and 2. In both algorithms, we define the bonus as
+bk(s, a) = β
+�
+∥φ(s, a)∥2
+�Σ†
+j,h +
+E
+�a∼πk(·|s)
+�
+∥φ(s, �a)∥2
+�Σ†
+j,h
+��
+.
+Algorithm 4 Dilated Bonus Calculation in Linear-Q Algorithms
+Input: Episode k ∈ [K], state s ∈ S, action a ∈ A.
+Output: The dilated bonus function Bk(s, a).
+1: if Bonus(k, s, a) is called before then return the value calculated at that time.
+2: Let h be the layer that s lies in, i.e., s ∈ Sh. if h = H + 1 then return 0.
+3: Call the simulator for a next state s′ ∼ P(· | s, a). Calculate πk(· | s) and πk(· | s′) according to the algorithm.
+4: return β
+�
+∥φ(s, a)∥2
+�Σ†
+k,h + E�a∼πk(·|s)
+�
+∥φ(s, �a)∥2
+�Σ†
+k,h
+��
++ (1 + 1
+H )Bk(s′, a′) where a′ is a sample from πk(· | s′).
+Note that, during the calculation of πk and Bk(s′, a′), we are essentially recursively calling this algorithm.
+A.3
+PolicyCover Algorithm in Linear MDP Algorithms
+The following algorithm is Algorithm 6 of Luo et al. (2021a) (which itself builds upon Algorithm 1 of Wang et al.
+(2020); we refer the readers to the original paper for more details). This algorithm generates a mixture of policies,
+which we call the policy cover πcov, together with an estimate of the (regularized) inverse of its covariance matrix,
+namely {�Σcov
+h }H
+h=1. It ensures the following property:
+15
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+Lemma A.2 (Lemma D.4 by Luo et al. (2021a)). With probability 1 − δ, we have the following for all policies π
+and any h ∈ [H]:
+Pr
+sh∼π [sh ̸∈ K] = �
+O
+�dH
+α
+�
+,
+where K is set of known states defined as follows:
+K =
+�
+s ∈ S
+���∀a ∈ A, ∥φ(s, a)∥2
+(�Σcov
+h
+)−1 ≤ α
+�
+,
+where h is the layer that s lies in.
+Algorithm 5 PolicyCover for Linear MDPs
+Input: Configurations M0, N0, α, failure probability δ.
+1: Set Γ1,h ← I for all h ∈ [H]. Set �β = 60dH
+�
+ln K
+δ .
+2: for m = 1, 2, . . ., M0 do
+3:
+Set �Vm(xh+1) ← 0.
+4:
+for h = H, H − 1, . . . , 1 do
+5:
+For all (s, a) ∈ Sh × A, let
+�Qm(s, a) = min{rm(s, a) + �β∥φ(s, a)∥Γ−1
+m,h + φ(s, a)T�θm,h, H},
+�Vm(s) = max
+a∈A
+�Qm(s, a),
+πm(a | s) =
+1[a = argmax
+a′∈A
+�Qm(s, a′)],
+where
+rm(s, a) = ramp 1
+K (∥φ(s, a)∥Γ−1
+m,h − α
+M0
+),
+�θm,h = Γ−1
+m,h( 1
+N0
+m−1
+�
+m′=1
+N0
+�
+n′=1
+φ(sm′,n′,h, am′,n′,h)�Vm(sm′,n′,h+1)).
+Here, rampz(y) is 0 if y ≤ −z, 1 if y ≥ 0, and y
+z + 1 otherwise.
+6:
+end for
+7:
+for n = 1, 2, . . ., N0 do
+8:
+Execute πm and get trajectory {(sm,n,h, am,n,h)}H
+h=1.
+9:
+end for
+10:
+Compute Γm+1,h as
+Γm+1,h ← Γm,h + 1
+N0
+N0
+�
+n=1
+φ(sm,n,h, am,n,h)φ(sm,n,h, am,n,h)T.
+11:
+return πcov defined as the uniform mixture of π1, π2, . . . , πM0 and �Σcov
+h
+defined as
+1
+M0 ΓM0+1,h (for all h).
+12: end for
+A.4
+Stochastic Matrix Concentration
+We first state the well-known Matrix Azuma inequality.
+Lemma A.3 (Matrix Azuma; see (Tropp, 2012, Theorem 7.1)). Let {Xk}n
+k=1 be a adapted sequence of d × d self-
+adjoint matrices. Let {Ak}n
+k=1 be a fixed sequence of self-adjoint matrices such that
+Ek[Xk] = 0,
+X2
+k ⪯ A2
+k almost surely.
+16
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+Let σ2 = ∥ 1
+n
+�n
+k=1 A2
+k∥2. Then for all ǫ > 0:
+Pr
+������
+1
+n
+n
+�
+k=1
+Xk
+�����
+2
+≥ ǫ
+�
+≤ d exp
+�
+− nǫ2
+8σ2
+�
+.
+Then, we state the following lemma, which we use to replace the MGR procedure in Linear MDPs.
+Lemma A.4. Let H1, H2, . . . , Hn be i.i.d. PSD matrices s.t. E[Hi] = H, Hi ⪯ I a.s., and H ⪰
+1
+dn log d
+δ I, then with
+probability 1 − 2δ,
+−
+�
+d
+n log d
+δ H1/2 ⪯ 1
+n
+n
+�
+i=1
+Hi − H ⪯
+�
+d
+n log d
+δ H1/2.
+Proof. Let G =
+�
+1
+dn log d
+δ H−1/2. We first show that
+−2 log d
+δ
+n
+G−1 ⪯ 1
+n
+n
+�
+i=1
+Hi − H
+holds with probability 1 − δ. We have
+Pr
+�
+−2 log( d
+δ )
+n
+G−1 ̸⪯ 1
+n
+n
+�
+i=1
+Hi − H
+�
+= Pr
+� n
+�
+i=1
+G1/2 (H − Hi) G1/2 − log(d
+δ )I ̸⪯ log(d
+δ )I
+�
+.
+Since log is operator monotone, we have for PSD matrices A, B: exp(A) ⪯ exp(B) ⇒ log(exp(A)) ⪯ log(exp(B))
+(note that the reverse does not generally hold). We have Pr{exp(A) ⪯ exp(B)} ≤ Pr{A ⪯ B} and hence Pr{A ̸⪯
+B} ≤ Pr{exp(A) ̸⪯ exp(B)}. Hence
+Pr
+� n
+�
+i=1
+G1/2 (H − Hi) G1/2 − log(d
+δ )I ̸⪯ log(d
+δ )I
+�
+≤ Pr
+�
+exp
+� n
+�
+i=1
+G1/2 (H − Hi) G1/2 − log(d
+δ )I
+�
+̸⪯ d
+δ I
+�
+≤ Pr
+�
+Tr
+�
+exp
+� n
+�
+i=1
+G1/2 (H − Hi) G1/2 − log(d
+δ )I
+��
+> d
+δ
+�
+≤ E
+�
+Tr
+�
+exp
+��n
+i=1 G1/2 (H − Hi) G1/2 − log( d
+δ )I
+���
+d
+δ
+(Markov’s inequality)
+Using the Golden-Thompson inequality, we have
+E
+�
+Tr
+�
+exp
+� n
+�
+i=1
+G1/2 (H − Hi) G1/2 − log(d
+δ )I
+���
+≤ E
+�
+Tr
+�
+exp
+�n−1
+�
+i=1
+G1/2 (H − Hi) G1/2 − log( d
+δ )
+n
+I
+�
+exp(G1/2 (H − Hn) G1/2 − log( d
+δ )
+n
+I)
+��
+= Tr
+�
+E
+�
+exp
+�n−1
+�
+i=1
+G1/2 (H − Hi) G1/2 − log( d
+δ )
+n
+I
+��
+E
+�
+exp(G1/2 (H − Hn) G1/2�
+exp
+�
+−log( d
+δ )
+n
+I)
+��
+We have due to G1/2(H − Hn)G1/2 ⪯ I:
+E
+�
+exp(G1/2 (H − Hn) G1/2)
+�
+⪯ E[I + (G1/2 (H − Hn) G1/2) + (G1/2 (H − Hn) G1/2)2] = I + E
+�
+(G1/2HnG1/2)2�
+.
+By the Araki–Lieb–Thirring inequality, we have Tr((ABA)2) ≤ Tr(A2B2A2), hence
+Tr
+�
+E[(G1/2HnG1/2)2]
+�
+≤ E[Tr(GH2
+nG)] ≤ E[Tr(GHnG)]
+(Hn ⪯ I)
+17
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+= Tr(GHG) ≤ log( d
+δ )
+n
+.
+Combining this with the previous result yields
+E
+�
+exp(G1/2 (H − Hn) G1/2)
+�
+exp
+�
+−log( d
+δ )
+n
+I)
+�
+⪯
+�
+1 + log( d
+δ )
+n
+�
+I exp
+�
+−log( d
+δ )
+n
+I)
+�
+⪯ I .
+Applying this recursively yields
+E
+�
+Tr
+�
+exp
+� n
+�
+i=1
+G1/2 (H − Hi) G1/2 − log(d
+δ )I
+���
+≤ Tr (I) = d ,
+which concludes the proof of the claim. By symmetry and union bound, we have the following with probability 1−2δ:
+−2 log d
+δ
+n
+G−1 ⪯ 1
+n
+n
+�
+i=1
+Hi − H ⪯ 2 log d
+δ
+n
+G−1.
+This then directly simplifies to our conclusion given H ⪰
+1
+dn log d
+δ I.
+18
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+B
+Omitted Proofs in Section 3 (Linear-Q Algorithm Using Log-Barrier
+Regularizers)
+B.1
+Property of FTRL with Log-Barrier Regularizers
+Proof of Theorem 3.1. We first introduce the notation of Bregman divergences DΨ(y, x) = Ψ(y)−Ψ(x)−⟨∇Ψ(x), y−
+x⟩, which is heavily used in FTRL analyses. By standard FTRL analysis (see, e.g., Theorem 28.5 by Lattimore and Szepesvári
+(2020)), we know
+T
+�
+t=1
+⟨xt − y, ct⟩ ≤ Ψ(y) − Ψ(x1)
+η
++
+T
+�
+t=1
+�
+⟨xt − xt+1, ct⟩ − η−1DΨ(xt+1, xt)
+�
+.
+(10)
+Consider DΨ(xt+1, xt). By definition of DΨ and our choice that Ψ(p) = �A
+i=1 ln 1
+pi , we have
+DΨ(xt+1, xt) = Ψ(xt+1) − Ψ(xt) − ⟨∇Ψ(xt), xt+1 − xt⟩
+=
+A
+�
+i=1
+�
+ln xt
+xt+1
++ xt+1 − xt
+xt
+�
+.
+Observe that the following holds for all x ∈ (0, 1) and x + ∆ ∈ (0, 1):
+ln
+x
+x + ∆ + ∆
+x =
+� ∆
+0
+α
+x(x + α)dα ≥
+� ∆
+0
+α
+x dα = ∆2
+2x ,
+For all i, we apply the inequality about with x = xt,i and ∆ = xt+1,i − xt,i (as xt and xt+1 both belong to the
+simplex, the conditions x ∈ (0, 1) and (x + ∆) ∈ (0, 1) indeed hold). We then get the following lower bound on
+DΨ(xt+1, xt):
+DΨ(xt+1, xt) ≥
+A
+�
+i=1
+(xt+1 − xt)2
+2xt
+.
+We further have
+⟨xt − xt+1, ct⟩ − η−1DΨ(xt+1, xt) ≤
+A
+�
+i=1
+�
+(xt,i − xt+1,i)ct,i − (xt+1 − xt)2
+2xt
+�
+≤
+A
+�
+i=1
+1
+2xt,iηc2
+t,i,
+where the last step uses AM-GM inequality −a2 + 2ab ≤ b2. Plugging this back to Eq. (10) gives our conclusion.
+B.2
+Proof of Main Theorem
+Proof of Theorem 3.3. As sketched in the main text, we consider the following expression in order to apply the
+dilated bonus lemma (Theorem 2.5):
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+� �
+ah∈A
+(πk(ah | sh) − π∗(ah | sh))(Qπk
+k (sh, ah) − Bk(sh, ah))
+�
+=
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)
+�
+Qπk
+k (sh, ah) − �Qk(sh, ah)
+��
+�
+��
+�
+Bias-1
++
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼π∗(·|sh)
+�
+�Qk(sh, ah) − Qπk
+k (sh, ah)
+��
+�
+��
+�
+Bias-2
++
+19
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+��
+πk(· | sh) − π∗(· | sh), �Qk(sh, ·) − Bk(sh, ·)
+��
+�
+��
+�
+Reg-Term
+.
+(11)
+According to Theorem B.1, we know that
+E[Bias-1] + E[Bias-2] ≤ β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++
+β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼π∗(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++ O
+�γ
+β dH3K + ǫ(H + β)HK
+�
+.
+Moreover, by Theorem B.2, we can see that
+E[Reg-Term] ≤ Hη−1A ln K + O
+�
+AH2
+�
+ǫ +
+√
+d + β
+γ
+��
++
+2ηH2
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)
+�
+∥φ(sh, ah)∥�Σ†
+k,h
+��
++ O
+� ηH3
+γ2K2
+�
++
+1
+H
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh) [Bk(sh, ah)]
+�
+.
+Plugging back into Eq. (11), we know that
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+� �
+ah∈A
+(πk(ah | sh) − π∗(ah | sh))(Qπk
+k (sh, ah) − Bk(sh, ah))
+�
+≤ β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++
+β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼π∗(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++
+2ηH2
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)
+�
+∥φ(sh, ah)∥2
+�Σ†
+k,h
+��
++
+1
+H
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh) [Bk(sh, ah)]
+�
++
+�O
+�γ
+β dH3K + ǫ(H + β)HK + H
+η A + AH2
+�
+ǫ +
+√
+d + β
+γ
+�
++ ηH3
+γ2K2
+�
+.
+Using (6ηH2 + β
+4 ) ≤ β, we apply Theorem 2.5 with bk(s, a) = ∥φ(s, a)∥2
+�Σ†
+k,h + Ea′∼πk(·|s)[∥φ(s, a′)∥2
+�Σ†
+k,h], giving
+RK ≤ β E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼πk
+�
+E
+ah∼πk(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++
+�
+O
+�γ
+β dH3K + ǫ(H + β)HK + H
+η A + AH2
+�
+ǫ +
+√
+d + β
+γ
+�
++ ηH3
+γ2K2
+�
+.
+Fixing an episode k ∈ [K] and h ∈ [H], we have
+E
+sh∼πk
+��
+a
+πk(a | s)∥φ(s, a)∥2
+�Σ†
+k,h
+�
+20
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+≤
+E
+sh∼πk
+��
+a
+πk(a | s)∥φ(s, a)∥2
+(γI+Σ
+πk
+h )−1
+�
++ O(ǫ)
+≤
+E
+sh∼πk
+��
+a
+πk(a | s)∥φ(s, a)∥2
+(Σ
+πk
+h )−1
+�
++ O(ǫ)
+=
+�
+(Σπk
+h )−1,
+E
+(sh,ah)∼πk
+[φ(sh, ah)φ(sh, ah)T]
+�
+= d.
+(12)
+Therefore, we can conclude the following if 12ηβH2 ≤ γ and 8ηH2 ≤ β:
+RK = �
+O
+�
+βdHK + γ
+β dH3K + ǫ(H + β)HK + H
+η A + AH2
+�
+ǫ +
+√
+d + β
+γ
+�
++ ηH3
+γ2K2
+�
+.
+It remains to tune the parameters. We first pick ǫ =
+1
+H2K , which makes all terms related to ǫ constantly-bounded.
+Setting η ≈ K−1/2 and γ ≈ K−1, the last term is also o(1). Hence, removing all constantly-bounded terms give
+RK = �O
+�
+AH 1
+η + βdHK + γ
+β dH3K + AH2 β
+γ
+�
+.
+We then get RK = �O(
+√
+AdH6K) = �O(A1/2d1/2H3K1/2) by picking:
+η =
+�
+A
+dH4K , β = 8
+�
+A
+dK , γ = 96 A
+dK .
+It’s straightforward to verify that they satisfy 12ηβH2 ≤ γ and 8ηH2 ≤ β.
+B.3
+Bounding Bias-1 and Bias-2
+Lemma B.1. In Algorithm 1, we have
+E[Bias-1] + E[Bias-2] ≤ β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++
+β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼π∗(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++ O
+�γ
+β dH3K + ǫ(H + β)HK
+�
+.
+Proof. As Bias-1 has nothing to do with the choice of the regularizer, it can be bounded the same as the original
+algorithm (Luo et al., 2021b, Lemma D.2), which we also include below for completeness: fixing a specific (k, s, a)
+and suppose that s ∈ Sh. Then we have the following, where every expectation is taken to the randomness in the
+k-th episode:
+E
+�
+Qπk
+k (s, a) − �Qk(s, a)
+�
+= φ(s, a)Tθπk
+k,h − φ(s, a)T E
+�
+�Σ†
+k,hφ(sk,h, ak,h)Lk
+�
+= φ(s, a)Tθπk
+k,h − φ(s, a)T E
+�
+�Σ†
+k,h
+�
+E
+�
+φ(sk,h, ak,h)φ(sk,h, ak,h)Tθπk
+k,h
+�
+(a)
+≤ φ(s, a)T �
+I − (γI + Σπk
+h )−1Σπk
+h
+�
+θπk
+k,h + ǫH
+= γφ(s, a)T(γI + Σπk
+h )−1θπk
+k,h + ǫH
+(b)
+≤ γ∥φ(s, a)∥(γI+Σ
+πk
+h )−1∥θπk
+k,h∥(γI+Σ
+πk
+h )−1 + ǫH
+(c)
+≤ β
+4 ∥φ(s, a)∥2
+(γI+Σ
+πk
+h )−1 + γ2
+β ∥θπk
+k,h∥2
+(γI+Σ
+πk
+h )−1 + ǫH
+(d)
+≤ β
+4 E
+�
+∥φ(s, a)∥2
+�Σ†
+k,h
+�
++ γ
+β dH2 + ǫH + ǫβ
+4 .
+21
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+where (a) used Eq. (7) (which follows from Theorem A.1) and the assumption that ∥φ(s, a)∥ ≤ 1 and Lk ≤ H, (b)
+used Cauchy-Schwartz inequality, (c) used AM-GM inequality, and (d) used the assumption that ∥θπk
+k,h∥ ≤
+√
+dH (see
+Theorem 2.2) and again Eq. (7). Hence, we have
+E[Bias-1] ≤ β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++ O
+�γ
+β dH3K + ǫ(H + β)HK
+�
+.
+Similarly, we have
+E[Bias-2] ≤ β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼π∗(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++ O
+�γ
+β dH3K + ǫ(H + β)HK
+�
+.
+Combining these two parts together gives our conclusion.
+B.4
+Bounding Reg-Term
+Lemma B.2. Under the assumption that 12ηβH2 ≤ γ, we have the following in Algorithm 1:
+E[Reg-Term] ≤ Hη−1A ln K + O
+�
+AH2
+�
+ǫ +
+√
+d + β
+γ
+��
++
+2ηH2
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)
+�
+∥φ(sh, ah)∥2
+�Σ†
+k,h
+��
++ O
+� ηH3
+γ2K2
+�
++
+1
+H
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh) [Bk(sh, ah)]
+�
+.
+Proof. Using Theorem 3.1, we get the following for all k ∈ [K], s ∈ Sh (where h ∈ [H]), and �π∗ ∈ Π:
+K
+�
+k=1
+�
+a∈A
+(πk(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))
+≤ Ψ(�π∗(· | s)) − Ψ(π1(· | s))
+η
++
+K
+�
+k=1
+�
+a∈A
+(�π∗(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))+
+η
+K
+�
+k=1
+�
+a∈A
+πk(a | s)( �Qk(s, a) − Bk(s, a))2.
+By picking �π∗(a | s) = (1 − AK−1)π∗(a | s) + K−1, the first term is bounded by
+Ψ(�π∗(· | s)) − Ψ(π1(· | s))
+η
+= 1
+η
+�
+a∈A
+ln π1(a | s)
+�π∗(a | s) ≤ 1
+η
+�
+a∈A
+ln π1(a | s)
+K−1
+≤ η−1A ln K.
+Meanwhile, the second term is bounded by
+K
+�
+k=1
+�
+a∈A
+(�π∗(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))
+=
+K
+�
+k=1
+�
+a∈A
+(−AK−1π∗(a | s) + K−1)( �Qk(s, a) − Bk(s, a)).
+22
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+Firstly, we have the following as ∥�Σ†
+k,h∥2 ≤ γ−1:
+Bk(s, a) ≤ H
+�
+1 + 1
+H
+�H
+× 2β sup
+s,a,h
+∥φ(s, a)∥2
+�Σ†
+k,h ≤ 6βHγ−1.
+(13)
+Moreover, according to the calculation in Theorem B.1, we know that
+E[ �Qk(s, a) − Qπk
+k (s, a)] ≤ γφ(s, a)T(γI + Σπk
+h )−1θπk
+k,h + ǫH ≤ O((ǫ +
+√
+d)H),
+while, at the same time, Qπk
+k (s, a) ∈ [0, H].
+Hence, after taking expectations on both sides, we know that
+E
+� K
+�
+k=1
+�
+a∈A
+(�π∗(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))
+�
+≤
+K
+�
+k=1
+�
+a∈A
+(|−AK−1π∗(a | s)| + |K−1|)|E[ �Qk(s, a) − Bk(s, a)]|
+≤ K × 2AK−1 × O
+�
+(ǫ +
+√
+d)H + H + β
+γ H
+�
+= O
+�
+AH
+�
+ǫ +
+√
+d + β
+γ
+��
+.
+Then consider the last term, which is directly bounded by
+η
+K
+�
+k=1
+�
+a∈A
+πk(a | s)2( �Qk(s, a) − Bk(s, a))2
+≤ 2η
+K
+�
+k=1
+�
+a∈A
+πk(a | s) �Qk(s, a)2 + 2η
+K
+�
+k=1
+�
+a∈A
+πk(a | s)Bk(s, a)2.
+The first term can be calculated as follows, following the original proof (Luo et al., 2021b, Lemma D.3):
+E[ �Qk(s, a)2] ≤ H2 E[φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h)φ(sk,h, ak,h)T�Σ†
+k,hφ(s, a)]
+= H2 E[φ(s, a)T�Σ†
+k,hΣπk
+h �Σ†
+k,hφ(s, a)]
+(a)
+≤ 2H2 E[φ(s, a)T�Σ†
+k,hφ(s, a)] + O
+� H2
+γ2K3
+�
+= 2H2 E
+�
+∥φ(s, a)∥2
+�Σ†
+k,h
+�
++ O
+� H2
+γ2K3
+�
+,
+where (a) used Eq. (8), which happens with probability 1 − K−3 for each k (when it does not hold, we simply use
+the bound ∥�Σ†
+k,h∥2 ≤ γ−1). Then we can conclude the following by adding back the summation over h ∈ [H] and
+sh ∼ π∗:
+E[Reg-Term] ≤ Hη−1A ln K + O
+�
+AH2
+�
+ǫ +
+√
+d + β
+γ
+��
++
+2ηH2
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)
+�
+∥φ(sh, ah)∥2
+�Σ†
+k,h
+��
++ O
+� ηH3
+γ2K2
+�
++
+1
+H
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh) [Bk(sh, ah)]
+�
+,
+where the last term comes from the magnitude of Bk (Eq. (13)) and the assumption that 12ηβH2 ≤ γ.
+23
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+C
+Omitted Proofs in Section 4 (Linear-Q Algorithm Using Magnitude-
+Reduced Estimators)
+C.1
+Property of FTRL with Negative-Entropy Regularizers (a.k.a. Hedge)
+The following result is a classic result for the Hedge algorithm. For the sake of completeness, we also include a
+proof here.
+Lemma C.1. Let x0, x1, x2, . . . , xT ∈ RA be defined as
+xt+1,i = (xt,i exp(−ηct,i))
+�� A
+�
+i′=1
+xt,i′ exp(−ηct,i′)
+�
+,
+∀0 ≤ t < T,
+where ct ∈ RA is the loss corresponding to the t-th iteration. Suppose that ηct,i ≥ −1 for all t ∈ [T ] and i ∈ [A].
+Then
+T
+�
+t=1
+⟨xt − y, ct⟩ ≤ log A
+η
++ η
+T
+�
+t=1
+A
+�
+i=1
+xt,ic2
+t,i
+holds for any distribution y ∈ △([A]) when x0 = ( 1
+A, 1
+A, . . . , 1
+A).
+Proof. By linearity, it suffices to prove the inequality for all one-hot y’s. Without loss of generality, let y = 1i∗ where
+i∗ ∈ [A]. Define Ct,i = �t
+t′=1 ct′,i as the prefix sum of ct,i. Let
+Φt = 1
+η ln
+� A
+�
+i=1
+exp (−ηCt,i)
+�
+,
+then by definition of xt, we have
+Φt − Φt−1 = 1
+η ln
+� �A
+i=1 exp(−ηCt,i)
+�A
+i=1 exp(−ηCt−1,i)
+�
+= 1
+η ln
+� A
+�
+i=1
+xt,i exp(−ηct,i)
+�
+(a)
+≤ 1
+η ln
+� A
+�
+i=1
+xt,i(1 − ηct,i + η2c2
+t,i)
+�
+= 1
+η ln
+�
+1 − η⟨xt, ct⟩ + η2
+A
+�
+i=1
+xt,ic2
+t,i
+�
+(b)
+≤ −⟨xt, ct⟩ + η
+A
+�
+i=1
+xt,ic2
+t,i,
+where (a) used exp(−x) ≤ 1 − x + x2 for all x ≥ −1 and (b) used ln(1 + x) ≤ x (again for all x ≥ −1). Therefore,
+summing over t = 1, 2, . . ., T gives
+T
+�
+t=1
+⟨xt, ct⟩ ≤ Φ0 − ΦT + η
+T
+�
+t=1
+A
+�
+i=1
+xt,ic2
+t,i
+≤ ln N
+η
+− 1
+η ln (exp(−ηCT,i∗)) + η
+T
+�
+t=1
+N
+�
+i=1
+pt(i)ℓ2
+t(i)
+≤ ln A
+η
++ LT(i∗) + η
+T
+�
+t=1
+A
+�
+i=1
+xt,ic2
+t,i.
+Moving Ct,i∗ to the LHS then shows the inequality for y = 1i∗. The result then extends to all y ∈ △([A]) by
+linearity.
+24
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+C.2
+Proof of Main Theorem
+Proof of Theorem 4.1. We first consider Line 6 in the algorithm. As sketched in the main text, we shall expect such
+an operation to be repeated for 1 + o(1) times because Eq. (8) and ∥�Σk,h − Σπk
+h ∥2 ≤ γ both happens with probability
+1 − K−3 — the first claim follows from Theorem A.1 and the second one comes from Theorem A.3 (where we set
+Xm = φ(sm,h, am,h)φ(sm,h, am,h)T − Σπk
+h
+and Am,h = I ⪰ Xm,h). With these two conditions and the fact that
+∥�Σ†
+k,h∥2 ≤ γ−1, the desired condition trivially holds. Therefore, such an operation brings neither extra regret nor
+extra computational complexity, and we focus on the regret analysis from now on.
+We define Bias-1, Bias-2, and Reg-Term exactly the same as Theorem 3.3 (i.e., Eq. (11)). The Bias-1 and
+Bias-2 terms are bounded by Theorem C.2, as follows:
+E[Bias-1] + E[Bias-2] ≤ β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++
+β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼π∗(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++ O
+�γ
+β dH3K + ǫ(H + β)HK
+�
+.
+Assuming 12η2H2 ≤ γ and 12ηβH2 ≤ γ, Reg-Term is bounded by Theorem C.3 as
+E[Reg-Term] ≤ Hη−1 ln A + 6ηH2
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)
+�
+∥φ(sh, ah)∥2
+�Σ†
+k,h
+��
++ 1
+H
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh) [Bk(sh, ah)]
+�
+.
+Plugging into the regret decomposition, we get
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+� �
+ah∈A
+(πk(ah | sh) − π∗(ah | sh))(Qπk
+k (sh, ah) − Bk(sh, ah))
+�
+≤ β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++
+β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼π∗(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++
+6ηH2 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)
+�
+∥φ(sh, ah)∥2
+�Σ†
+k,h
+���
++
+1
+H E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh) [Bk(sh, ah)]
+��
++ �O
+�H
+η + γ
+β dH3K + ǫ(H + β)HK
+�
+.
+Using the condition that (6ηH2 + β
+4 ) ≤ β, we can apply Theorem 2.5 to conclude that
+RK = �
+O
+�
+E
+�
+β
+H
+�
+h=1
+E
+sh∼πk
+� K
+�
+k=1
+�
+a
+πk(a | s)∥φ(s, a)∥2
+�Σ†
+k,h
+��
++ H
+η + γ
+β dH3K + ǫ(H + β)HK
+�
+.
+By Eq. (12), we can conclude the following when assuming 8ηH2 ≤ β:
+RK = �O
+�H
+η + γ
+β dH3K + ǫ(H + β)HK + βdHK
+�
+.
+Plugging in the configurations that (again, one can see that this configuration satisfies all the conditions)
+η =
+1
+√
+dKH2 , β =
+8
+√
+dK
+, γ = 96
+dK , ǫ =
+1
+H2K ,
+we then conclude that RK = �
+O(
+√
+dH6K).
+25
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+C.3
+Bounding Bias-1 and Bias-2
+Lemma C.2. In Algorithm 2, we have
+E[Bias-1] + E[Bias-2] ≤ β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++
+β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼π∗(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++ O
+�γ
+β dH3K + ǫ(H + β)HK
+�
+.
+Proof. Fixing a specific (k, s, a) and assume s ∈ Sh. Then we have (again, all expectations are taken w.r.t. random-
+ness in the k-th episode)
+E
+�
+�Qk(s, a)
+�
+= E
+�
+φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h)Lk,h
+�
+− H E
+��
+φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h)
+�
+−
+�
++
+H E
+�
+1
+M
+M
+�
+m=1
+�
+φ(s, a)T�Σ†
+k,hφ(sm,h, am,h)
+�
+−
+�
+= E
+�
+φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h)Lk,h
+�
+,
+where the last equality is because (sk,h, ak,h) and (sm,h, am,h) are both sampled from πk. The rest of the proof
+is then identical to Theorem B.1.
+C.4
+Bounding Reg-Term
+Lemma C.3. Suppose that 12H2η2 ≤ γ, 12ηβH2 ≤ γ, 8ηH2 ≤ γ, and M = 32γ−2 log K. Then in Algorithm 2, we
+have
+E[Reg-Term] ≤ Hη−1 ln A + 6ηH2 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)
+�
+∥φ(sh, ah)∥2
+�Σ†
+k,h
+���
++ 1
+H E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh) [Bk(sh, ah)]
+��
+.
+Proof. To apply the Hedge lemma (Theorem C.1), we need to ensure that
+η( �Qk(s, a) − Bk(s, a)) ≥ −1,
+∀(s, a) ∈ S × A, k ∈ [K].
+Fix an (s, a, k) tuple and assume that s ∈ Sh. We have
+�Qk(s, a) − Hmk(s, a) = φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h)Lk,h − H(φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h))−
+=
+�
+φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h)Lk,h,
+φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h) ≥ 0
+−φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h)(H − Lk,h),
+φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h) < 0
+≥ 0,
+as Lk,h ∈ [0, H].
+Hence, �Qk(s, a) ≥ Hmk(s, a) always holds.
+We consider �Qk(s, a) ≥ Hmk(s, a) and Bk(s, a)
+separately.
+We first claim that ηHmk(s, a) ≥ − 1
+2, which ensures η �Qk(s, a) ≥ − 1
+2.
+To see this, we only need to show
+η2H2mk(s, a)2 ≤ 1
+4. By definition, mk(s, a)2 can be written as the following:
+mk(s, a)2 =
+�
+1
+M
+M
+�
+m=1
+�
+φ(s, a)T�Σ†
+k,hφ(sm,h, am,h)
+�
+−
+�2
+26
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+(a)
+≤
+1
+M
+M
+�
+m=1
+�
+φ(s, a)T�Σ†
+k,hφ(sm,h, am,h)
+�2
+−
+= φ(s, a)T�Σ†
+k,h
+�
+1
+M
+M
+�
+m=1
+φ(sm,h, am,h)φ(sm,h, am,h)T
+�
+�Σ†
+k,hφ(s, a)
+= φ(s, a)T�Σ†
+k,h�Σk,h�Σ†
+k,hφ(s, a)
+(b)
+≤ 3φ(s, a)T�Σ†
+k,hφ(s, a) ≤ 3γ−1.
+where (a) used Jensen inequality, (b) used Line 6 of Algorithm 2, and the last step uses the fact that ∥�Σ†
+k,h∥2 ≤ γ−1.
+Hence, it only remains to ensure that 12H2η2γ−1 ≤ 1, which is guaranteed by the assumption.
+Meanwhile, we claim that ηBk(s, a) ≤ 1
+2. By definition of Bk(s, a) and the fact that ∥�Σ†
+k,h∥2 ≤ γ−1, we have
+ηBk(s, a) ≤ ηH
+�
+1 + 1
+H
+�H
+× 2β sup
+s,a,h
+∥φ(s, a)∥2
+�Σ†
+k,h ≤ 6ηβHγ−1,
+(14)
+which is bounded by
+1
+2H according to the condition that 12ηβH2 ≤ γ.
+Therefore, fixing h ∈ [H] and s ∈ Sh, we can apply the Hedge lemma (Theorem C.1):
+E
+� K
+�
+k=1
+�
+a
+(πk(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))
+�
+≤ ln A
+η
++ 2η E
+� K
+�
+k=1
+�
+a
+πk(a | s) �Qk(s, a)2
+�
++ 2η E
+� K
+�
+k=1
+�
+a
+πk(a | s)Bk(s, a)2
+�
+.
+For the second term, we can write
+�Qk(s, a)2 = (φ(s, a)T �Σ†
+k,hφ(sk,h, ak,h)Lk,h − H(φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h))− + Hmk(s, a))2
+≤ 2H2(φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h))2 + 2H2(φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h))2
+− + 2H2m2
+k(s, a).
+After taking expectations on both sides, we have
+E[ �Q2
+k(s, a)] ≤ 2H2 E[(φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h))2] + 2H2 E[(φ(s, a)T �Σ†
+k,hφ(sk,h, ak,h))2
+−]+
+2H2 E
+�
+1
+M
+M
+�
+m=1
+(φ(s, a)T�Σ†
+k,hφ(sm,h, am,h))−
+�2
+≤ 6H2 E[(φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h))2],
+where we used (X)2
+− ≤ X2, Jensen’s inequality, and the fact that (sk,h, ak,h) and (sm,h, am,h) are both sampled from
+πk. Meanwhile, we can also calculate that
+E[(φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h))2]
+= E
+�
+φ(s, a)T�Σ†
+k,hφ(sk,h, ak,h)φ(sk,h, ak,h)T�Σ†
+k,hφ(s, a)
+�
+= E
+�
+φ(s, a)T�Σ†
+k,hΣπk
+k �Σ†
+k,hφ(s, a)
+�
+≤ E
+�
+∥φ(s, a)∥2
+�Σ†
+k,h
+�
+,
+where the last inequality again uses Line 6 of Algorithm 2. Hence, after summing up over the expectations when
+sh ∼ π∗ for all h ∈ [H], we can conclude that
+E[Reg-Term] ≤ Hη−1 ln A + 6ηH2
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)
+�
+∥φ(sh, ah)∥2
+�Σ†
+k,h
+��
+27
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
++ 1
+H
+K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh) [Bk(sh, ah)]
+�
+,
+where the last term comes from Eq. (14) and the condition that 12ηβH2 ≤ γ.
+28
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+Algorithm 6 Improved Linear MDP Algorithm
+Input: Learning rate η, PolicyCover parameters α and T0, bonus parameter β, covariance estimation parameter
+γ ∈ (0, 1
+4), epoch length W, FTRL regularizer Ψ(p) = �A
+i=1 ln 1
+pi , exploration probability δe
+1: Let πcov and �Σcov
+h
+be the outputs of the PolicyCover algorithm (see Algorithm 5).
+2: Let K = {s ∈ S | ∥φ(s, a)∥2
+(�Σcov
+h
+)−1 ≤ α, ∀a ∈ A} be all the “known” states (defined in Theorem A.2).
+3: for j = 1, 2, . . ., J = (T − T0)/W do
+4:
+Calculate πj ∈ Π as follows for all s ∈ Sh:
+πj(s) = argmin
+p∈△(A)
+�
+Ψ(p) + η
+�
+τ<j
+�
+a∈A
+p(a)
+� �Qτ(s, a) − �Bτ(s, a)
+��
+,
+where �Qτ(s, a) = φ(s, a)T�θτ,h, �Bτ(s, a) = bτ(s, a) + φ(s, a)T�Λτ,h, and the bonus function is defined as
+bτ(s, a) =
+1[s ∈ K] × β
+�
+∥φ(s, a)∥2
+�Σ†
+τ,h +
+E
+a′∼πτ (·|s)
+�
+∥φ(s, a′)∥2
+�Σ†
+τ,h
+��
+.
+5:
+Randomly partition the episodes in the current epoch, i.e., {K0 + (j − 1)W + 1, . . . , K0 + jW}, into two halves
+Tj and T ′
+j , such that where |Tj| = |T ′
+j | = W
+2 .
+6:
+for k = K0 + (j − 1)W + 1, . . . , K0 + jW do
+7:
+Let Yk be a sample from a Bernoulli distribution Ber(δe).
+8:
+if Yk = 0 then
+9:
+Execute πj for this episode and observe {(sk,h, ak,h)}H
+h=1 (together with ℓk(sk,h, ak,h)).
+10:
+else if Yt = 1 and k ∈ Tj then
+11:
+Execute πcov and observe {(sk,h, ak,h)}H
+h=1 together with ℓk(sk,h, ak,h).
+12:
+else
+13:
+Let hk be uniformly sampled from [H]. Execute πcov for steps 1, 2, . . ., h − 1 and πk for the remaining
+ones. Again, observe the trajectory {(sk,h, ak,h)}H
+h=1 and the losses ℓk(sk,h, ak,h).
+14:
+end if
+15:
+end for
+16:
+Define �πj as the mixture of (1 − δe) times πj and δe times πcov (i.e., expected policy played in epoch j).
+17:
+Estimate the covariance matrix Σ�πj
+h as follows, and estimate (γI + Σ�πj
+h )−1 by �Σ†
+j,h = (γI + �Σj,h)−1.
+�Σj,h =
+1
+|Tj|
+�
+k∈Tj
+φ(sk,h, ak,h)φ(sk,h, ak,h)T,
+18:
+Estimate the average Q-function kernel θ
+πj
+j,h ≜
+1
+|T ′
+j |
+�
+k∈T ′
+j θπj
+k,h as follows:
+�θj,h = �Σ†
+j,h
+
+ 1
+|T ′
+j |
+�
+k∈T ′
+j
+((1 − Yk) + YkH
+1[h = hk]) φ(sk,h, ak,h)Lk,h
+
+ ,
+where Lk,h = �H
+h′=h ℓk(sk,h′, ak,h′).
+19:
+Estimate the average dilated bonus kernel Λ
+πj
+j,h ≜
+1
+|T ′
+j |
+�
+k∈T ′
+j Λπj
+k,h as follows:
+�Λj,h = �Σ†
+j,h
+
+ 1
+|T ′
+j |
+�
+k∈T ′
+j
+((1 − Yk) + YkH
+1[h = hk]) φ(sk,h, ak,h)Dk,h
+
+ ,
+where Dk,h = �H
+h′=h+1(1 + 1
+H )i−hbj(sk,h, ak,h).
+20: end for
+29
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+D
+Omitted Proofs in Section 5 (Linear MDP Algorithm)
+D.1
+Pseudocode of the Improved Linear MDP Algorithm
+This section briefly discusses the linear MDP algorithm (presented in Algorithm 6). Apart from the new covariance
+estimation technique introduced in the main text, it is also different from Algorithm 1 in some other aspects due to
+the distinct nature of linear-Q MDPs and (simulator-free) linear MDPs, listed as follows:
+Firstly, as there are no simulators, we cannot calculate the dilated bonus function recursively like Algorithm 4.
+Fortunately, in linear MDPs, as observed by Luo et al. (2021a), for any k associated with some policy πk and bonus
+function bk, we can write Bk(s, a) defined in Eq. (3) as Bk(s, a) = bk(s, a) + φ(s, a)TΛπk
+k,h, where (ν is defined in
+Theorem 2.4)
+Λπ
+k,h ≜
+�
+1 + 1
+H
+� �
+sh+1
+E
+ah+1∼π(·|sh+1) [Bk(sh+1, ah+1)] ν(sh+1)dsh+1.
+Hence, Bk(s, a) is also linear in φ(s, a). This allows us to estimate the Bk just like Qπk
+k , as we see in Line 19.
+Notice that, as there are no more simulators, we cannot directly “assume” a good covariance estimation like in
+Algorithm 1 (which ensures Eqs. (7) and (8)). Instead, we should divide the time horizon into several epochs and
+execute (nearly) the same policy during each epoch to ensure a good estimation. See Algorithm 6 for more details.
+D.2
+Alternative to the Matrix Geometric Resampling Procedure
+Proof of Theorem 5.1. By definition, we know the following holds for all k ∈ Tj:
+E[φ(sk,h, ak,h)φ(sk,h, ak,h)T] = Σ�πj
+h ,
+φ(sk,h, ak,h)φ(sk,h, ak,h)T ⪯ I
+Moreover, each (sk,h, ak,h) is i.i.d. Thus, we can apply Theorem A.4 with
+Hi = 1
+2
+�
+γI + φ(ski,h, aki,h)φ(ski,h, aki,h)T�
+, H = 1
+2
+�
+γI + Σ�πj
+h
+�
+,
+i = 1, 2, . . . , |Tj|,
+where ki stands for the i-th element in Tj. We then have the following according to Theorem A.4:
+−
+�
+d
+|Tj| log d
+δ H1/2 ⪯
+1
+|Tj|
+|Tj|
+�
+i=1
+Hi − H ⪯
+�
+d
+|Tj| log d
+δ H1/2,
+if γ ≥ 2 d
+|Tj| log d
+δ .
+(15)
+Let the empirical average of all Hi’s be �H, i.e.,
+�H =
+1
+|Tj|
+|Tj|
+�
+i=1
+Hi = 1
+2(γI + �Σj,h).
+Then we can arrive at the following under the same condition as Eq. (15):
+I −
+�
+d
+|Tj| log d
+δ H−1/2 ⪯ �HH−1 ⪯ I +
+�
+d
+|Tj| log d
+δ H−1/2.
+Moreover, by the definition of H, we know H−1/2 ⪯
+√
+2(γI)−1/2.
+Setting W = 4d log d
+δ γ−2 (which ensures
+γ ≥ 2
+d
+|Tj| log d
+δ as |Tj| = W
+2 ), the LHS and RHS become (1 − √γ)I and (1 + √γ)I, respectively. Hence,
+(1 − √γ) 1
+2(γI + Σ�πj
+h ) ⪯ 1
+2(γI + �Σj,h) ⪯ (1 + √γ) 1
+2(γI + Σ�πj
+h ),
+which gives our conclusion after multiplying 2 on both sides.
+30
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+D.3
+Proof of Main Theorem
+We first state the formal version of Theorem 5.3:
+Theorem D.1. Suppose that α =
+δe
+6β, M0 ≥ α2dH2, N0 ≥ 100 M3
+0
+α2 log K
+δ , W = 4d log d
+δ γ−2, γ ≥ 36 β2
+δ2e , and
+100ηH4 ≤ β. Further pick δ = K−3. Then Algorithm 6 applied to linear MDPs (Theorem 2.4) ensures
+RK = �
+O
+� δ6
+e
+β6 d4H8 + δeK + βdHK + γ
+β dH3K + H
+η Adγ−2 + Ad3/2H2γ−2 + H3
+γ2 K
+�
+.
+With some proper tuning, we can ensure RK = �
+O((H20Ad6)1/9K8/9).
+Proof of Theorem 5.3. The regret decomposition is the same as Theorem 6.1 by Luo et al. (2021a), which we include
+below. As sketched in the main text, we decompose the episodes into three parts: those executing PolicyCover,
+those using exploratory policies (i.e., the Ber(δe) gives 1), and the ones executing πj.
+For the first part, it’s trivially bounded by O(K0H). For the second part, as we explore with probability δe for
+each episode, the total regret gets bounded by O(δeKH). For the last part, it suffices to bound the following to apply
+Theorem 2.5 (where we still consider those exploratory episodes as they only bring extra regret), where J = K−K0
+W
+denotes the number of epochs for simplicity and Q
+π
+j (s, a) = φ(s, a)Tθ
+π
+j,h denotes the average Q-function in T ′
+j (h is
+such that s ∈ Sh):
+J
+�
+j=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+W
+�
+ah∈A
+(πj(ah | sh) − π∗(ah | sh))(Q
+πj
+j (sh, ah) − Bj(sh, ah))
+�
+.
+As Bk(s, a) is only bounded for the known states (by definition of K), we first consider the unknown states:
+J
+�
+j=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+W
+1[x ̸∈ K]
+�
+ah∈A
+(πj(ah | sh) − π∗(ah | sh))(Q
+πj
+j (s, a) − Bj(sh, ah))
+�
+≤ J
+E
+�
+h=1
+E
+sh∼π∗[W
+1[sh ̸∈ K]3H] = �O
+�
+HK dH3
+α
+�
+according to Theorem A.2. For the remaining, we decompose (πj(ah | sh) − π∗(ah | sh))(Q
+πj
+j (sh, ah) − Bj(sh, ah))
+into the biases of �Qj(sh, ah) and �Bj(s, ah) plus the FTRL regret (πj(ah | sh) − π∗(ah | sh))( �Qj(sh, ah) − �Bj(sh, ah)),
+i.e.,
+J
+�
+j=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+W
+1[sh ∈ K]
+�
+ah∈A
+(πj(ah | sh) − π∗(ah | sh))(Q
+πj
+j (sh, ah) − Bj(sh, ah))
+�
+= W
+J
+�
+j=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+1[sh ∈ K]
+E
+ah∼πj(·|sh)
+�
+Q
+πj
+j (sh, ah) − �Qj(sh, ah))
+��
+�
+��
+�
+Bias-1
++
+W
+J
+�
+j=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+1[sh ∈ K]
+E
+ah∼π∗(·|sh)
+�
+�Qj(sh, ah)) − Q
+πj
+j (sh, ah)
+��
+�
+��
+�
+Bias-2
++
+W
+J
+�
+j=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+1[sh ∈ K]
+E
+ah∼πj(·|sh)
+�
+�Bj(sh, ah) − Bj(sh, ah)
+��
+�
+��
+�
+Bias-3
++
+W
+J
+�
+j=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+1[sh ∈ K]
+E
+ah∼π∗(·|sh)
+�
+Bj(sh, ah) − �Bj(sh, ah)
+��
+�
+��
+�
+Bias-4
++
+31
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+W
+J
+�
+j=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+1[sh ∈ K]
+�
+ah∈A
+(πj(ah | sh) − π∗(ah | sh))( �Qj(sh, ah) − �Bj(sh, ah))
+�
+�
+��
+�
+Reg-Term
+.
+All the bias terms can be bounded similarly, as we will show in Theorems D.2 and D.3, we can bound them as
+E[Bias-1] + E[Bias-2] + E[Bias-3] + E[Bias-4]
+≤ β
+2 E
+
+
+J
+�
+j=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πj(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+j,h]
+�
+ +
+β
+2 E
+
+
+J
+�
+j=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼π∗(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+j,h]
+�
+ + O
+�γ
+β dH3J
+�
+.
+Different from Theorems B.1 and C.2, in that proof, we need to handle the estimation error β−1∥(�Σj,h−Σπj
+h )θ
+πj
+j,h(s, a)∥2
+�Σ†
+j,h
+by the multiplicative bound Theorem 5.2 instead of the additive one (e.g., Eq. (7)). See Eq. (16) for more details.
+For the Reg-Term, we again apply the new FTRL lemma Theorem 3.1, giving the following expression:
+E[Reg-Term] ≤ �
+O
+�H
+η A + A
+√
+dH2 + H3
+γ2 δJ
+�
++ 2ηH3
+H
+�
+h=1
+E
+sh∼π∗
+
+
+J
+�
+j=1
+E
+ah∼πj(·|sh)
+�
+∥φ(sh, ah)∥2
+�Σ†
+j,h
+�
+
++
+H
+�
+h=1
+E
+sh∼π∗
+
+
+J
+�
+j=1
+E
+ah∼πj(·|sh) [bj(s, a)]
+
+ ,
+whose formal proof is in Theorem D.4. In that proof, we need to bound the magnitudes of the bonuses to write
+�Bj(sh, ah)2 ≲ 1
+H �Bj(sh, ah). This is done by applying Theorem D.5 later in this section.
+By summing up all terms and multiplying W, we can apply Theorem 2.5 by again using 100ηH4 ≤ β. Hence, we
+get the following by using Eq. (12):
+RK ≤ O(K0) + O(δeK) + �O
+�
+βdHK + γ
+β dH3K + H
+η AW + A
+√
+dH2W + H3
+γ2 δK
+�
+,
+where we conditioned on some good event with probability 1 − O(K−2 + Kδ). Picking δ = K−3, the total regret
+when the good event does not happen is of order O(K−2HK) = o(1).
+Moreover, by the conditions α =
+δe
+6β, M0 ≥ α2dH2, and N0 ≥ 100 M3
+0
+α2 log K
+δ , we know that K0 = M0N0 =
+�O( δ6
+e
+β6 d4H8). Meanwhile, by W = 4d log d
+δ γ−2, we know that W = �
+O(dγ−2). Thus, we have
+RK = �O
+� δ6
+e
+β6 d4H8 + δeK + βdHK + γ
+β dH3K + H
+η Adγ−2 + Ad3/2H2γ−2
+�
+.
+The only conditions are then γ ≥ 36 β2
+δe , 100ηH4 ≤ β, which allows us to set
+δe = C(H20Ad6)1/9K−1/9, β = C2
+36 (H13A2d3K−2)1/9K−2/9, γ = C3
+36 (H2A1)1/3K−1/3, η = C2
+3600(H−23A2d3K−2)1/9K−2/9,
+where C is a constant. This ensures RK = �O((H20Ad6)1/9K8/9) (note that only the 2nd, 4th, and 5th term have a
+K8/9 dependency, which means all other terms can be ignored when stating the bound).
+32
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+D.4
+Bounding the Bias Terms
+Lemma D.2. When W = 4d log d
+δ γ−2 and γ < 1
+4, the Bias-1 and Bias-2 terms in Algorithm 6 is bounded by
+E[Bias-1] + E[Bias-2] ≤ β
+4 E
+
+
+J
+�
+j=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πj(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+j,h]
+�
+ +
+β
+4 E
+
+
+J
+�
+j=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼π∗(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+j,h]
+�
+ + O
+�γ
+β dH3K
+�
+.
+Proof. By direct calculation like Theorem B.1, we get the following for any j ∈ [J], h ∈ [H], s ∈ Sh and a ∈ A (recall
+that �Σ†
+j,h only depends on the episodes in Tj; again, all expectations are only taken to the randomness in epoch j)
+E
+�
+Q
+πj
+j (s, a) − �Qj(s, a)
+�
+= φ(s, a)Tθ
+πj
+j,h − φ(s, a)T E
+
+�Σ†
+j,h
+
+ 1
+|T ′
+j |
+�
+k∈T ′
+j
+((1 − Yk) + YkH
+1[h = hk])φ(sk,h, ak,h)Lk,h
+
+
+
+
+= φ(s, a)Tθ
+πj
+j,h − φ(s, a)T E
+�
+�Σ†
+j,h
+�
+E
+
+
+
+ 1
+|T ′
+j |
+�
+k∈T ′
+j
+((1 − Yk) + YkH
+1[h = hk])φ(sk,h, ak,h)φ(sk,h, ak,h)Tθπj
+k,h
+
+
+
+
+= φ(s, a)Tθ
+πj
+j,h − φ(s, a)T E
+�
+�Σ†
+j,h
+�
+Σ�πj
+h θ
+πj
+j,h
+= φ(s, a)T(I − E[�Σ†
+j,h]Σ�πj
+h )θ
+πj
+j,h.
+By using Cauchy-Schwartz inequality, triangle inequality, and the AM-GM inequality, we get the following:
+φ(s, a)T(I − �Σ†
+j,hΣ�πj
+h )θ
+πj
+j,h = φ(s, a)T�Σ†
+j,h(γI + �Σj,h − Σ�πj
+h )θ
+πj
+j,h
+≤ ∥φ(s, a)∥�Σ†
+j,h
+����(γI)θ
+πj
+j,h
+����ӆ
+j,h
++
+���(�Σj,h − Σ�πj
+h )θ
+πj
+j,h
+����ӆ
+j,h
+�
+≤ β
+4 ∥φ(s, a)∥2
+�Σ†
+j,h + 2
+β
+���γθ
+πj
+j,h(s, a)
+���
+2
+�Σ†
+j,h
++ 2
+β
+���(�Σj,h − Σ�πj
+h )θ
+πj
+j,h
+���
+2
+�Σ†
+j,h
+.
+The first term is the usual bonus term in Theorem 2.5, while the second term easily translates to the following
+using the assumption that ∥θπj
+k,h∥ ≤
+√
+dH for all k ∈ T ′
+j :
+2
+β
+���γθ
+πj
+j,h
+���
+2
+�Σ†
+j,h
+≤ 2γ2
+β
+���(γI + �Σj,h)−1���
+2 dH2 ≤ 2 γ
+β dH2,
+while the last term translates to the following by algebraic manipulations:
+2
+β
+���(�Σj,h − Σ�πj
+h )θ
+πj
+j,h
+���
+2
+�Σ†
+j,h
+= 2
+β
+���(�Σj,h − Σ�πj
+h )�Σ†
+j,h(�Σj,h − Σ�πj
+h )
+���
+2 dH2
+= 2
+β
+����(�ӆ
+j,h)−1/2 �
+I − (�Σ†
+j,h)1/2(γI + Σ�πj
+h )(�Σ†
+j,h)1/2�2
+(�Σ†
+j,h)−1/2
+����
+2
+dH2,
+(16)
+where we wrote (�Σj,h − Σ�πj
+h )�Σ†
+j,h(�Σj,h − Σ�πj
+h ) = (�Σ†
+j,h)−1/2(�Σ†
+j,h)1/2(�Σj,h − Σ�πj
+h )�Σ†
+j,h(�Σj,h − Σ�πj
+h )(�Σ†
+j,h)1/2(�Σ†
+j,h)−1/2
+and made a square in the middle.
+Using Theorem 5.2, the squared-matrix in the middle has its operator norm bounded by 4γ with high probability
+(if the good event does not hold, then one can directly bound the last term by matrix Azuma and the operator norm
+of �Σ†
+k,h, giving γ−3; as this only happens with probability K−3, this part contributes o(K−2) to the total regret and
+we thus omit it). Meanwhile, the first and last term both has their operator norms bounded by
+√
+2. Thus,
+2
+β
+���(�Σj,h − Σπj
+h )θ
+πj
+j,h
+���
+2
+�Σ†
+j,h
+dH2 ≤ 16 γ
+β dH2.
+33
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+In other words, the last term gets absorbed by the second term up to constants. Hence, the conclusion follows
+by the same argument as Theorem B.1.
+Lemma D.3. When W = 4d log d
+δ γ−2 and γ < 1
+4, the Bias-3 and Bias-4 terms in Algorithm 6 is bounded by the
+following when the good event in Theorem D.5 holds:
+E[Bias-3] + E[Bias-4] ≤ β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼πk(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++
+β
+4 E
+� K
+�
+k=1
+H
+�
+h=1
+E
+sh∼π∗
+�
+E
+ah∼π∗(·|sh)[∥φ(sh, ah)∥2
+�Σ†
+k,h]
+��
++ O
+�γ
+β dH3K
+�
+.
+Proof. The proof is identical to Theorem D.2 except for Lk,h is replaced by Dk,h and θ
+πj
+j,h is replaced by Λπj
+j,h. As we
+also have bj(s, a) ∈ [0, 1] by Theorem D.5, the same bound holds.
+D.5
+Bounding Reg-Term
+Lemma D.4. Assuming 100HηH4 ≤ β and the good events in Theorem D.5 hold. Then the Reg-Term has its
+expectation bounded by
+E[Reg-Term] ≤ �
+O
+�H
+η A + A
+√
+dH2 + H3
+γ2 δJ
+�
++ 2ηH3
+H
+�
+h=1
+E
+sh∼π∗
+
+
+J
+�
+j=1
+E
+ah∼πj(·|sh)
+�
+∥φ(sh, ah)∥2
+�Σ†
+j,h
+�
+
++
+H
+�
+h=1
+E
+sh∼π∗
+
+
+J
+�
+j=1
+E
+ah∼πj(·|sh) [bj(s, a)]
+
+ .
+Proof. The proof generally follows from Theorem B.2, except for some tiny differences due to epoching.
+Using Theorem 3.1, we get the following for all j ∈ [J], s ∈ Sh ∩ K (where h ∈ [H]), and �π∗ ∈ Π:
+J
+�
+j=1
+�
+a∈A
+(πj(a | s) − π∗(a | s))( �Qj(s, a) − �Bj(s, a))
+≤ Ψ(�π∗(· | s)) − Ψ(π1(· | s))
+η
++
+J
+�
+j=1
+�
+a∈A
+(�π∗(a | s) − π∗(a | s))( �Qj(s, a) − �Bj(s, a))+
+η
+J
+�
+j=1
+�
+a∈A
+πj(a | s)( �Qj(s, a) − �Bj(s, a))2.
+By picking �π∗(a | s) = (1 − AJ−1)π∗(a | s) + J−1 as Theorem B.2, the first term is bounded by η−1A ln J.
+Meanwhile, the second term is bounded by
+J
+�
+j=1
+�
+a∈A
+(�π∗(a | s) − π∗(a | s))( �Qj(s, a) − �Bj(s, a))
+=
+J
+�
+j=1
+�
+a∈A
+(−AJ−1π∗(a | s) + J−1)( �Qj(s, a) − �Bj(s, a)).
+Like what we did in Theorem B.2, we consider the expected difference between �Qj and Q
+πj
+j :
+E[ �Qj(s, a) − Q
+πj
+j (s, a)] = φ(s, a)T(I − E[�Σ†
+j,hΣ�πj
+h ])θ
+πj
+j,h ≤ 2
+√
+dH,
+34
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+where the first inequality is due to Theorem D.2 and the second one uses Theorem 5.1. Moreover, according to
+Theorem D.5, the same bound also holds for E[ �Bj(s, a) − B
+πj
+j (s, a)].
+Hence, after taking expectations on both sides, we know that
+E
+
+
+J
+�
+j=1
+�
+a∈A
+(�π∗(a | s) − π∗(a | s))( �Qj(s, a) − �Bj(s, a))
+
+
+≤
+J
+�
+j=1
+�
+a∈A
+(|−AJ−1π∗(a | s)| + |J−1|)|E[ �Qj(s, a) − �Bj(s, a)]|
+≤ J × 2AJ−1 × 4
+√
+dH = O
+�
+A
+√
+dH
+�
+.
+Then consider the last term. We still write ( �Qj(s, a)− �Bj(s, a))2 ≤ 2 �Qj(s, a)2+2 �Bj(s, a)2, which can be calculated
+as follows:
+E[ �Qj(s, a)2] ≤ H2 E
+
+ 1
+|T ′
+j |
+�
+k∈T ′
+j
+φ(s, a)T�Σ†
+j,h
+�
+((1 − Yk) + YkH
+1[h = hk])2φ(sk,h, ak,h)φ(sk,h, ak,h)T� �Σ†
+j,hφ(s, a)
+
+
+= H2 E
+�
+φ(s, a)T�Σ†
+j,h
+�
+(1 − δe)Σπj
+h + HδeΣcov
+h
+� �Σ†
+j,hφ(s, a)
+�
+≤ H3 E
+�
+φ(s, a)T�Σ†
+j,hΣ�πj
+h �Σ†
+j,hφ(s, a)
+�
+.
+Then we use Theorem 5.2.
+If the good event does not happen, then this term is bounded by O(H3γ−2δ).
+Otherwise, it can be written as 2H3 E
+�
+∥φ(s, a)∥2
+�Σ†
+j,h
+�
+. Then we consider the estimated dilated bonus term:
+E[ �Bj(s, a)2] ≤ 2 E[bj(s, a)2] + 2 E[(φ(s, a)T�Λj,h)2].
+According to Theorem D.5 (whose failure only contributes in total o(1) regret as we pick δ = K−3), we know
+bj(s, a) ≤ 1, which means Dk,h ≤ 3H. Thus, the second term is also bounded by O(H3 E[∥φ(s, a)∥2
+�Σ†
+j,h] + H3γ−2δ).
+Moreover, as bj(s, a) ≤ 1, the first term is bounded by E[bj(s, a)]. Thus, by definition of bj(s, a), we get
+E[ �Bj(s, a)2] ≤ O
+�H3
+β
+�
+bj(s, a) + O(H3γ−2δ)).
+Putting everything together gives
+E[Reg-Term] ≤ �
+O
+�H
+η A + A
+√
+dH2 + H3
+γ2 δJ
+�
++ 2ηH3
+H
+�
+h=1
+E
+sh∼π∗
+
+
+J
+�
+j=1
+E
+ah∼πj(·|sh)
+�
+∥φ(sh, ah)∥2
+�Σ†
+j,h
+�
+
++ 100ηH3
+β
+H
+�
+h=1
+E
+sh∼π∗
+
+
+J
+�
+j=1
+E
+ah∼πj(·|sh) [bj(s, a)]
+
+ .
+This then translates to our conclusion using the condition that 100H4η ≤ β.
+D.6
+Bounding the Magnitudes of Bonuses
+Lemma D.5. Let α = δe
+6β, M0 ≥ α2dH2, N0 ≥ 100 M3
+0
+α2 log K
+δ , and γ ≥ 36 β2
+δe . Then with probability 1 − (K−2 + Kδ),
+bj(s, a) =
+1[(s, a) ∈ K] × β(∥φ(s, a)∥2
+�Σ†
+j,h + Ea′∼πj(·|s)[∥φ(s, a′)∥�Σ†
+j,h]) ≤ 1 for all j ∈ [J], h ∈ [H], s ∈ Sh and a ∈ A.
+35
+
+Refined Regret for Adversarial MDPs with Linear Function Approximation
+The proof is similar to, but different from Lemma F.1 of Luo et al. (2021a). Here, we use the alternative for the
+MGR procedure. We also refine the analysis on M −1
+0
++ Σcov
+h
+≈ �Σcov
+h
+for a smaller N0, which is critical for our new
+regret bound.
+Proof. It suffices to show that β∥φ(s, a′)∥2
+�Σ†
+k,h ≤ 1
+2 for any s ∈ K and a′ ∈ A. Firstly, we have the following with
+high probability because �Σ†
+j,h is a multiplicative approximation (i.e., Theorem 5.2):
+β∥φ(s, a′)∥�Σ†
+j,h ≤ 2β∥φ(s, a′)∥2
+(γI+Σ
+�πj
+h )−1 ≤ 2 β
+δe
+∥φ(s, a′)∥2
+( γ
+δe I+Σcov
+h
+)−1 ≤ 2 β
+δe
+∥φ(s, a′)∥2
+(M−1
+0
+I+Σcov
+h
+)−1,
+where Σcov
+h
+is the short-hand notation of Σπcov
+h
+and the last step uses the fact that
+γ
+δe M0 ≥
+γ
+δe
+δ2
+e
+36β2 ≥ 1. Moreover,
+we argue that M −1
+0
++ Σcov
+h
+≈ �Σcov
+h . By definition of �Σcov
+h
+(see Algorithm 5):
+�Σcov
+h
+=
+1
+M0
+I +
+1
+M0N0
+M0
+�
+m=1
+N0
+�
+n=1
+φ(sm,n,h, am,n,h)φ(sm,n,h, am,n,h)T,
+we can apply the Matrix Azuma inequality (Theorem A.3) to ensure that (where we simply pick XmN0+n =
+φ(sm,n,h, am,n,h)φ(sm,n,a, am,n,h)T − Σπm
+h
+and AmN0+n = I ⪰ XmN0+n; there are in total M0N0 matrices):
+�����Σcov
+h
+−
+1
+M0
+I − Σcov
+h
+����
+2
+≤
+�
+8
+M0N0
+log d
+δ .
+The rest follows the proof of the original lemma (Luo et al., 2021a, Lemma F.1).
+By following the proof of
+Theorem 2.1 of Meng and Zheng (2010), we can conclude that
+�����
+�
+�Σcov
+h
+�−1
+−
+� 1
+M0
+I − Σcov
+h
+�−1�����
+2
+≤ M 2
+0
+�����Σcov
+h
+−
+1
+M0
+I − Σcov
+h
+����
+2
+≤ M 2
+0
+�
+8
+M0N0
+log d
+δ =
+�
+8M 3
+0
+N0
+log d
+δ .
+Therefore, we only need M0 = O(N 3
+0 ) to make it bounded by α
+2 . Consequently, as ∥φ(s, a′)∥2 ≤ 1,
+β∥φ(s, a′)∥�Σ†
+j,h ≤ 2 β
+δe
+∥φ(s, a′)∥(M−1
+0
+I+Σcov
+h
+)−1
+≤ 2 β
+δe
+�
+∥φ(s, a′)∥2
+(�Σcov
+h
+)−1 +
+�����
+�
+�Σcov
+h
+�−1
+−
+� 1
+M0
+I − Σcov
+h
+�−1�����
+2
+�
+≤ 2 β
+δe
+�
+α + α
+2
+�
+≤ 1
+2,
+by the condition that s ∈ K and our choice of α.
+36
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf,len=1952
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='12942v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='LG]  30 Jan 2023  Refined Regret for Adversarial MDPs with Linear Function  Approximation  Yan Dai ∗  Haipeng Luo †  Chen-Yu Wei ‡  Julian Zimmert §  Abstract  We consider learning in an adversarial Markov Decision Process (MDP) where the loss functions can change  arbitrarily over K episodes and the state space can be arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We assume that the Q-function of any  policy is linear in some known features, that is, a linear function approximation exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The best existing regret  upper bound for this setting (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b) is of order �  O(K2/3) (omitting all other dependencies), given access  to a simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This paper provides two algorithms that improve the regret to �  O(  √  K) in the same setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Our  first algorithm makes use of a refined analysis of the Follow-the-Regularized-Leader (FTRL) algorithm with the log-  barrier regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This analysis allows the loss estimators to be arbitrarily negative and might be of independent  interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Our second algorithm develops a magnitude-reduced loss estimator, further removing the polynomial  dependency on the number of actions in the first algorithm and leading to the optimal regret bound (up to  logarithmic terms and dependency on the horizon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Moreover, we also extend the first algorithm to simulator-free  linear MDPs, which achieves �  O(K8/9) regret and greatly improves over the best existing bound �  O(K14/15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This  algorithm relies on a better alternative to the Matrix Geometric Resampling procedure by Neu and Olkhovskaya  (2020), which could again be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  1  Introduction  Markov Decision Processes (MDPs) have been widely used to model reinforcement learning problems, where an  agent needs to make decisions sequentially and to learn from the feedback received from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In this  paper, we focus on adversarial MDPs where the loss functions can vary with time and the state space can also  be arbitrarily large, capturing the fact that in real-world applications such as robotics, the environment can be  non-stationary and the number of states can be prohibitively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  To handle large state spaces, one of the most common methods in the literature is to assume a linear-function  approximation (Yang and Wang, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Zanette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Neu and Olkhovskaya,  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b), where the expected loss suffered by any policy from any state-action pair, commonly known  as the Q-function, is linear in a set of known features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  While such assumptions are commonly used in the literature, the minimax optimal regret attainable by the  agent is poorly understood, especially in the adversarial setting we study in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 Specifically, for adversarial  linear-Q MDPs, the best existing bound is of order K2/3 where K is the number of episodes (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2  In that paper, the authors assume a transition simulator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', the agent is allowed to draw a trajectory starting  from any state-action pair, sampled from the actual transition and a given policy, without any cost) and achieve  �O(d2/3H2K2/3) regret where H is the length of each episode and d is the dimension of the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' On the  other hand, the best lower bound for this setting (induced from the special case of adversarial linear bandits) is of  order Ω(  √  K) (Dani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Therefore, a natural question arises:  Is it possible to design an algorithm in adversarial linear-Q MDPs that attains �O(  √  K) regret bound?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  ∗IIIS, Tsinghua University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Email: yan-dai20@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  †University of Southern California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Email: haipengl@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  ‡Massachusetts Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Email: chenyuw@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  §Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Email: zimmert@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  1To be more specific, we only consider bandit feedback in this paper, where the agent can only observe her experienced losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For  the easier full-information setting,  √  K-style near-optimal regret has already been achieved (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2022) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  2Meanwhile, if a “good” exploratory policy π0 (formalized in Footnote 4) is granted,  √  K-style bounds are also achievable, though with  some additional dependencies on the quality of π0 (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Neu and Olkhovskaya, 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  1  Refined Regret for Adversarial MDPs with Linear Function Approximation  Table 1: Overview of Our Results and Comparisons with the Most Related Works  Algorithm  Settinga  Transition  Assumptionb  Regret  Dilated Bonus  (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b)  Linear-Q MDP  (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2)  Simulator  (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3)  None  �O(d2/3H2K2/3)  Exploratory Policy  �O  �  poly(d, H)(K/λ0)1/2�  Algorithm 1 (This work)  None  �  O(A1/2d1/2H3K1/2)  Algorithm 2 (This work)  �O(d1/2H3K1/2)  POWERS (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2022)  Linear Mixture MDP  Unknown  Full Information  �  O(dHK1/2)  Online Q-REPS  (Neu and Olkhovskaya, 2021)  Linear MDP  (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4)  Known  Exploratory Policy  �O  �  poly(d, H)(K/λ0)1/2�c  Dilated Bonus  (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='b)  Unknown  None  �O(d2H4K14/15)  Exploratory Policy  �O  �  poly(d, H)  �  K/λ2/3  0  �6/7�  Algorithm 6 (This work)  None  �  O(H20/9A1/9d2/3K8/9)  aLinear MDP is a special case of linear-Q MDP, while linear mixture MDP is generally incomparable to these two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  bThe definitions of “exploratory policy” and λ0 are stated in Footnote 4, while “full information” means the entire loss function is  revealed at the end of each episode (which is easier than our bandit-feedback setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  cThis is a refined version of the original O(√K log K) bound presented in their Theorem 1, which contains no explicit dependency on  λ0 by assuming K = Ω(exp(λ−1  0 )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' See Review hYYK at https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='net/forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='id=gviX23L1bqw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  In this work, we answer this question in the affirmative by developing two algorithms that both attain �O(  √  K) re-  gret in adversarial linear-Q MDPs when a simulator is granted, closing the gap with the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Both of our algo-  rithms follow the same framework of the policy optimization algorithm with dilated exploration bonuses of (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=',  2021b), but with important modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Specifically, our first algorithm applies Follow-the-Regularized-Leader  (FTRL) with the log-barrier regularizer (instead of the negative entropy regularizer used by Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021b)) at  each state, and we develop a new analysis inspired by Zimmert and Lattimore (2022) that allows the loss estimators  to be arbitrarily negative (in contrast, in the usual analyses of FTRL, the loss estimator cannot be too negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This  new analysis is the key to improving the regret to O(  √  K) and might be of independent interest even for multi-armed  bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Although our first algorithm is simple in design and analysis, the log-barrier regularizer causes an O(  √  A) factor  in the regret bound �O(H3√  AdK) where A is the number of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' As a rescue, we develop another algorithm that  uses the negative entropy regularizer with a magnitude-reduced loss estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This algorithm attains an �O(H3√  dK)  regret bound, which is optimal in d and K up to logarithmic factors and gets rid of the poly(A) dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We  note that our algorithm is of interest even for the special case of adversarial linear bandits, as it removes the need of  explicit John’s exploration introduced by Bubeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  At last, we also apply our method to simulator-free linear MDPs (formally defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4), yielding an  efficient algorithm with �O(K8/9) regret and greatly outperforming the best existing bound �  O(K14/15) (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=',  2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3 Remarkably, in this application, we not only use our refined analysis for FTRL with the log-barrier reg-  ularizer, but also develop a more sample-efficient alternative to the Matrix Geometric Resampling (MGR) method  introduced by Neu and Olkhovskaya (2020) and later adopted by Neu and Olkhovskaya (2021) and Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021a),  which could also be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1  Related Work  MDPs with Linear-Function Approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Linear function approximation has been a standard technique  for handling large state spaces in RL, but only recently have researchers provided strong regret guarantees for these  algorithms under precise conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Yang and Wang (2020) introduced a linear function approximation scheme  3(Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021a) is a refined version of (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  2  Refined Regret for Adversarial MDPs with Linear Function Approximation  called embedded linear transition MDPs where the transition kernels are bilinear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', the probability of reaching  state s′ in the h-th step of an episode after taking action a at state s is P(s′ | s, a) = φ(s, a)TMψ(s′) for some  known feature mappings φ and ψ and an unknown M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2020b) loosen the assumption to linear MDPs  (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', P(s′ | s, a) = φ(s, a)Tν(s′) where ν is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021) study the linear mixture  MDP with P(s′ | s, a) = ψ(s′ | s, a)Tθ where θ is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This generalizes embedded linear transition MDPs but is  incomparable with linear MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Another common model is linear-Q MDPs (Abbasi-Yadkori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2019) where the  Q-function with respect to any policy π can be written as Qπ  h(s, a) = φT(s, a)θπ  h for some unknown θπ  h (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Linear MDPs are special cases of linear-Q MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Adversarial MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  MDPs with adversarial losses were first studied in the tabular cases where the state space  has a small size S ≪ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Zimin and Neu (2013) first assume known transitions and achieve �  O(H  √  K) regret when  full information is available and �O(  √  HSAK) regret when only bandit feedback is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Rosenberg and Mansour  (2019) then study the unknown-transition case and get �O(HS  √  AK) regret with full information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Finally, Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (2020a) tackle the hardest case with unknown transitions and bandit feedback and achieve �O(HS  √  AK) regret as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Results for adversarial MDPs with linear-function approximations are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Specifically,  Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2020) study unknown-transition, full-information linear mixture MDPs and get �  O(dH3/2√  K) regret;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  this result is further improved to �  O(dH  √  K) by He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Neu and Olkhovskaya (2021) then study known-  transition, bandit-feedback linear MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Provided with a “good” exploratory policy,4 their algorithm achieves an  �O(poly(d, H)  �  K/λ0) regret guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Later, Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021b) study bandit-feedback linear-Q MDPs with a sim-  ulator, providing an �O(d2/3H2K2/3) regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' A refined version (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021a) considers simulator-free  bandit-feedback linear MDPs, giving an �O(d2H4K14/15) bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Meanwhile, provided with a good exploratory pol-  icy (see Footnote 4), these bounds improve to �O(poly(d, H)  �  K/λ0) and �O(poly(d, H)λ−4/7  0  K6/7), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' As  a final remark, we point out that exploration in MDPs with huge state spaces is challenging and assuming an ex-  ploratory policy is unrealistic — as far as we know, there are no results ensuring even the existence of such a “good”  exploratory policy, let alone finding it efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We emphasize that our results do not require such unrealistic  exploratory assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Policy Optimization Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Policy optimization algorithms for RL directly optimize the learner’s policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  They are more resilient to model misspecification or even adversarial manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' But due to their local search  nature, they suffer from the notorious distribution mismatch issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Recent theoretical works address this issue by  exploration bonuses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' see (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Shani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Zanette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021) for stochastic settings and  (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b) for adversarial settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' While the latter work achieves near-optimal regret in tabular settings,  there is a huge room for improvement when considering function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Our work builds on top of their  framework (especially the dilated bonus idea) and significantly improves their results in linear-function approximation  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  2  Preliminaries  Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  For N ∈ N, [N] denotes the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For a (possibly infinite) set X, we denote the probability  simplex over X by △(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For a random event E, denote its indicator by  1[E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For two square matrices A, B of the  same size, ⟨A, B⟩ stands for Tr(ATB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For x ∈ R, define (x)− as min{x, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We use �O to hide all logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  No-Regret Learning in MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  An (episodic) adversarial MDP is specified by a tuple M = (S, A, P, ℓ) where S  is the state space (possibly infinite), A is the action space (assumed to be finite with size A = |A|), P: S ×A → △(S)  is the transition, and ℓ: [K] × S × A → [0, 1] is the loss function chosen arbitrarily by an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The state space  is assumed to be layered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', S = S1 ∪ S2 ∪ · · · ∪ SH where Sh ∩ Sh′ = ∅ for any 1 ≤ h < h′ ≤ H, and transition is  only possible from one layer to the next one, that is, P(s′ | s, a) ̸= 0 only when s ∈ Sh and s′ ∈ Sh+1 for some h < H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  We also assume that there is an initial state s1 such that S1 = {s1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  4Formally, a “good” exploratory policy π0 ensures that λmin(Σπ0  h ) ≥ λ0 for all h ∈ [H] and a positive constant λ0, where Σπ0  h  means  the covariance of π0 in layer h (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  3  Refined Regret for Adversarial MDPs with Linear Function Approximation  The game lasts for K episodes, each with length H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For each episode k, the agent is initialized at state s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For  each step h ∈ [H], she chooses an action ah ∈ A, suffers and observes the loss ℓk(sh, ah), and transits to a new state  sh+1 independently sampled from the transition P(· | sh, ah).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  A policy π of the agent is a mapping from S to △(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Let Π be the set of all policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For each episode k ∈ [K],  let πk ∈ Π be the policy deployed by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Its expected loss is then indicated by V πk  k (s1), where the state-value  function (or V-function in short) V π  k (s1) is defined as follows for any episode k and policy π:  V π  k (s1) ≜ E  � H  �  h=1  ℓk(sh, ah)  �����(sh, ah) ∼ π, ∀h ∈ [H]  �  ,  with (sh, ah) ∼ π, ∀h ∈ [H] denoting a trajectory sampled from π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The agent aims to minimize the cumulative total  loss collected in all episodes, or equivalently, the regret:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 (Regret).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The regret of the agent is  RK ≜ E  � K  �  k=1  V π1  k (s1)  �  −  K  �  k=1  V π∗  k (s1),  where the expectation is taken over the randomness of both the agent and the transition, and π∗ is the optimal policy  in hindsight (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', π∗ ∈ argminπ∈Π  �K  k=1 V π  k (s1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1  Linear-Q MDP and Linear MDP  A concept closely related to the V-function is the action-value function (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Q-function), which denotes  the expected loss suffered by a policy π starting from a given state-action pair (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Formally, we define for all  (s, a) ∈ S × A:  Qπ  k(s, a) = ℓk(s, a) +  1[s /∈ SH]  E  s′∼P(·|s,a)  a′∼π(·|s′)  [Qπ  k(s′, a′)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (1)  We can then define linear-Q MDPs as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2 ((Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b, Assumption 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In a linear-Q MDP, each state-action pair (s, a) is associated  with a known feature φ(s, a) ∈ Rd with ∥φ(s, a)∥2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Moreover, for any policy π ∈ Π, episode k ∈ [K], and layer  h ∈ [H], there exists a (hidden) vector θπ  k,h ∈ Rd such that  Qπ  k(sh, ah) = φ(sh, ah)Tθπ  k,h,  ∀sh ∈ Sh, ah ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  We assume that ∥θπ  k,h∥2 ≤  √  dH for all k, h, π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2 does not specify any specific structure on the transition, making learning extremely difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Con-  sequently, prior works all assume the availability of a simulator that allows us to sample from the hidden transition:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3 (Simulator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' A simulator accepts a state-action pair (s, a) ∈ S ×A and generates a next-state output  sampled from the true transition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', s′ ∼ P(· | s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  When a simulator is unavailable, we consider a special case called linear MDPs which further impose a linear  structure on the transition, enabling the agent to learn:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4 ((Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b, Assumption 3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' A linear MDP is a linear-Q MDP that additionally satisfies  the following property: for any h ∈ [H], the transition from layer h to layer h + 1 can be written as follows:  P(sh+1 | sh, ah) = ⟨φ(sh, ah), ν(sh+1)⟩,  ∀sh+1 ∈ Sh+1, sh ∈ Sh, ah ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Here, the mapping ν : S → Rd is also unrevealed to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We also assume that ∥ν(s)∥2 ≤  √  d for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  In MDPs with linear function approximation, another important quantity associated with each policy π is its  covariance matrix at each layer h ∈ [H], defined as follows:  Σπ  h =  E  (sh,ah)∼π[φ(sh, ah)φ(sh, ah)T],  ∀h ∈ [H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (2)  4  Refined Regret for Adversarial MDPs with Linear Function Approximation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2  Dilated Bonuses for Policy Optimization  We now briefly introduce the policy optimization method and the key dilated bonus idea of Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021b),  upon which our algorithms are built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The foundation of policy optimization is the performance difference lemma  (Kakade and Langford, 2002), which asserts that the regret of the agent can be viewed as the weighted average of  the regret of some local bandit problem over each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Formally, we have  RK =  H  �  h=1  E  sh∼π∗  � K  �  k=1  �  a  (πk(a|sh) − π∗(a|sh)) Qπk  k (sh, a)  �  ,  where the part inside the expectation is exactly the regret of a multi-armed bandit (MAB) problem at state sh  with “loss” Qπk  k (sh, a) (instead of ℓk(sh, a)) for action a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Policy optimization algorithms then naturally run a bandit  algorithm at each state with an appropriate Q-function estimator to learn the best policy directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For example, in  linear-Q MDPs, since the “loss" Qπk  k (sh, a) is linear in some feature, it suggests running an adversarial linear bandit  algorithm such as Exp2 (Bubeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2012) at each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  However, as discussed in detail by Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021b), the bias/variance of the Q-function estimator often leads to a  regret term of the form �  k,h E(sh,ah)∼π∗[bk(sh, ah)] for some non-negative functions b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , bK : S ×A → R≥0, where  bk(s, a) is often prohibitively large if (s, a) is rarely visited by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Hence, the expectation over (sh, ah) ∼ π∗  could be potentially large as well, while the expectation over (sh, ah) ∼ πk is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This is the well-known  distribution mismatch issue: for example, when applying a linear bandit algorithm at each state, bk(sh, ah) is roughly  β∥φ(sh, ah)∥2  (Σ  πk  h )−1 for some β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Thus, E(sh,ah)∼πk[bk(sh, ah)] = β  �  (Σπk  h )−1, E(sh,ah)∼πk[φ(sh, ah)φ(sh, ah)T]  �  =  βd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' on the other hand, its counterpart with (sh, ah) drawn from π∗ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', E(sh,ah)∼π∗[bk(sh, ah)]) could be arbitrarily  large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  To address this distribution mismatch issue and “convert” the measure from πk to π∗, Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021b) consider  treating these functions as exploration bonuses and further propose the so-called “dilated bonus” functions Bk(s, a):  Bk(s, a) = bk(s, a) +  1[s /∈ SH]  �  1 + 1  H  �  E  s′∼P(·|s,a)  a′∼πk(·|s′)  [Bk(s′, a′)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (3)  Compared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (1), Bk can be viewed the Q-function of bk, except that it assigns slightly more weight to deeper  layers via the extra weighting 1 +  1  H to encourage more exploration to those layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Their algorithms then try  to minimize the regret with respect to the “optimistic" loss Qπk  k  − Bk, instead of just Qπk  k , at each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Their  analysis relies on the following key lemma, which shows that if the regret w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Qπk  k  − Bk at each state is in  some particular form, then the distribution mismatch issue can be resolved (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', the final regret RK is in terms of  �  k,h E(sh,ah)∼πk[bk(sh, ah)] instead of �  k,h E(sh,ah)∼π∗[bk(sh, ah)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 by Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' If {bk}K  k=1 are non-negative, Bk(s, a) is defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (3), and  the following holds for any state s:  E  � K  �  k=1  �  a  (πk(a|s) − π∗(a|s)) (Qπk  k (s, a) − Bk(s, a))  �  ≤ X(s) + E  � K  �  k=1  �  a∈A  �  π∗(a|s)bk(s, a) + 1  H πk(a|s)Bk(s, a)  ��  ,  where X(s) is an arbitrary function, then  RK ≤  H  �  h=1  E  sh∼π∗[X(sh)] + 3  K  �  k=1  H  �  h=1  E  (sh,ah)∼πk  [bk(sh, ah)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021b) show that the per-state regret bound in the condition of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5 indeed holds when  deploying Follow-the-Regularized-Leader (FTRL) with the negative entropy regularizer at each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' However, for  technical issues (which we will discuss and resolve in Sections 3 and 4), they only achieve �O(K2/3) regret in linear-Q  MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  5  Refined Regret for Adversarial MDPs with Linear Function Approximation  3  �  O(  √  K) Regret for Linear-Q MDPs via Refined Log-Barrier Analysis  As mentioned, our first algorithm replaces the negative entropy regularizer in FTRL used by Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021b)  with another regularizer called the log-barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Specifically, FTRL applied to the action space A maintains a sequence  of distributions x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , xT ∈ △([A]) via the FTRL update rule xt = argminx∈△([A]){η⟨x, �  τ<t cτ⟩+Ψ(x)}, where  c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , cT ∈ RA is the loss sequence, Ψ : △([A]) → R is the regularizer, and η > 0 is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The classical  MAB algorithm Exp3 (Auer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2002) and its linear-bandit variant Exp2 (Bubeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2012) both use the  negative entropy regularizer Ψ(x) = �A  i=1 pi ln pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Starting from (Foster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2016), a sequence of works discover many nice properties of a different regularizer  called log-barrier, defined as Ψ(p) = �A  i=1 ln 1  pi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Here, we provide yet another new and useful property of log-  barrier, summarized in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Our proof is inspired by the analysis of the log-determinant regularizer  by Zimmert and Lattimore (2022) and is deferred to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , xT ∈ △([A]) be defined as  xt = argmin  x∈△([A])  �  η  �  x,  �  τ<t  cτ  �  + Ψ(x)  �  ,  ∀t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , T,  where ct ∈ RA is an arbitrary loss vector corresponding to the t-th iteration and Ψ(p) = �A  i=1 ln 1  pi is the log-barrier  regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Then the regret against any distribution y ∈ △([A]) with respect to {ct}T  t=1 is bounded as:  T  �  t=1  ⟨xt − y, ct⟩ ≤ Ψ(y) − Ψ(x1)  η  + η  T  �  t=1  A  �  i=1  xt,ic2  t,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Readers familiar with the Exp2/Exp3 analysis would immediately recognize this regret bound since it is also the  same bound that FTRL with negative entropy (also known as Hedge) enjoys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' However, the key distinction is that  for log-barrier, this holds without any requirement on the magnitude of ct,i, while for negative entropy, one must  require ηct,i ≥ −1 for all t ∈ [T ] and i ∈ [A] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', losses cannot be too negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' see Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 in the appendix for  more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This turns out to be critical for improving the regret when applying it to linear-Q MDPs, as discussed  later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Our Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 also answers the open question raised by Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2019) (see their remark after  Lemma 14): it is indeed possible for FTRL with log-barrier to attain a �  i xt,ic2  t,i-style bound without any restrictions  on the losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' As log-barrier regularizers are widely used in the literature for its better data-adaptivity (Wei and Luo,  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Ito, 2021), we expect this result to be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Linear-Q Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Our final algorithm for linear-Q MDPs is shown in Algorithm 1, which is nearly the same  as (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b, Algorithm 2) except for the part marked in blue where we use the log-barrier regularizer, as  mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Specifically, based on the discussions in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2, the loss fed to FTRL is �Qk−Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Here, �Qk (defined in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (5)) is a standard estimator for Qπk  k involving an estimate �Σ†  k,h for the inverse of Σπk  h , which is constructed via the  Matrix Geometric Resampling procedure (Neu and Olkhovskaya, 2020) with the help of the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Meanwhile,  Bk is the dilated bonus following the idea of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (3) with bk(s, a) = β  �  ∥φ(s, a)∥2  �Σ†  k,h + E�a∼πk(·|s)  �  ∥φ(s, �a)∥2  �Σ†  k,h  ��  for  s ∈ Sh (which again uses the simulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' see Algorithm 4 as well as detailed discussions in (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  With the help of our new log-barrier analysis, we show that our algorithm achieves �  O(  √  K) regret, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Algorithm 1 when applied to a linear-Q MDP (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2) with a simulator ensures the following  when 12ηβH2 ≤ γ, 8ηH2 ≤ β, and ǫ ≤ (H2K)−1:  RK = �  O  �  βdHK + γ  β dH3K + AH  η  + β  γ AH2 + A  √  dH2 + ηH3  γ2K2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Picking η =  �  A  dH4K , β = 8  �  A  dK , γ = 96A  dK , and ǫ =  1  H2K , we have RK = �  O(H3√  AdK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  We defer the full proof to Appendix B and provide a sketch below highlighting why removing any constraints on  the losses fed to FTRL is critical to improving the regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  6  Refined Regret for Adversarial MDPs with Linear Function Approximation  Algorithm 1 Improved Linear-Q Algorithm (Using Log-Barrier Regularizers)  Input: Learning rate η, bonus parameter β, MGR parameters γ and ǫ, FTRL regularizer Ψ(p) = �|A|  i=1 ln 1  pi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  1: for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', K do  2:  Let πk ∈ Π be defined as follows for all s ∈ S:  πk(s) = argmin  p∈△(A)  �  Ψ(p) + η  �  k′<k  ⟨p(·), �Qk′(s, ·) − Bk′(s, ·)⟩  �  ,  (4)  where Bk(s, a) is calculated in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  3:  Execute πk, observing the trajectory (sk,h, ak,h) and losses ℓk(sk,h, ak,h) for all h ∈ [H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  4:  Construct covariance matrix inverse estimate �Σ†  k,h using Matrix Geometric Resampling (Algorithm 3 in the  appendix) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ∥�Σ†  k,h∥2 ≤ 1  γ ,  ���E[�Σ†  k,h] − (γI + Σπk  h )−1���  2 ≤ ǫ,  and the following holds w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' 1 − K−3:  ���(�Σ†  k,h)1/2Σπk  h (�Σ†  k,h)1/2���  2 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  5:  Let Lk,h = �  h′≥h ℓk(sk,h′, ak,h′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Estimate the Q-function Qπk  k (s, a) as follows (for s ∈ Sh):  �Qk(s, a) = φ(s, a)T�Σ†  k,hφ(sk,h, ak,h)Lk,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (5)  6: end for  Proof Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' To bound RK by the dilated bonus lemma (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5), we focus on a single (k, s)-pair and bound  �  a(πk(a | s) − π∗(a | s))(Qπk  k (s, a) − Bk(s, a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (6)  As the FTRL lemma only applies to the losses fed into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (4) (namely �Qk(s, a) − Bk(s, a)), we add and substract  �Qk(s, a) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Hence, after summing over k ∈ [K] and s ∼ π∗, we need to consider the following three terms  to figure out the term X(s) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5:  Bias-1 = �  k,h  E  sh∼π∗  �  E  ah∼πk  �  Qπk  k (sh, ah) − �Qk(sh, ah)  ��  , measuring the under-estimation of �Qk w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' πk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Bias-2 = �  k,h  E  sh∼π∗  �  E  ah∼π∗  �  �Qk(sh, ah) − Qπk  k (sh, ah)  ��  , measuring the over-estimation of �Qk w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Reg-Term = �  k,h  E  sh∼π∗  � �  a  (πk(a|s) − π∗(a|s))( �Qk(s, a) − Bk(s, a))  �  , which can be tackled by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  As Bias-1 and Bias-2 are independent of the regularizer, they are handled similarly to the original analysis  and both contribute �  O( γ  β dH3K) to �  h Esh∼π∗[X(sh)] (see Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' To handle Reg-Term,  we apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' As in standard log-barrier analyses, since Ψ(π∗(· | s)) − Ψ(π1(· | s)) is potentially infinity,  we introduce a smooth version �π∗ defined via �π∗(a | s) = (1 − AK−1)π∗(a | s) + K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Given an s ∈ S, we have  Ψ(�π∗(· | s)) − Ψ(π1(· | s)) ≤ A log K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Hence, fixing the state s, we can write  K  �  k=1  �  a∈A  (πk(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))  ≤  K  �  k=1  �  a∈A  (�π∗(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))+  A log K  η  + η  K  �  k=1  �  a∈A  πk(a | s)2( �Qk(s, a) − Bk(s, a))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  7  Refined Regret for Adversarial MDPs with Linear Function Approximation  After summing over h and taking expectation over s, we show that the first term is of order �  O(AH2(  √  d + β/γ)),  while the last term’s contribution to �  h Esh∼π∗[X(sh)] is of order �O( ηH3  γ2K2 ) by the same analysis of (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=',  2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Finally, we combine everything, apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5, and show that the term �  k,h E(sh,ah)∼πk[bk(sh, ah)]  in this case is �O(βdHK), finishing the proof for the first claimed regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This bound is almost identical  to that of (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' However, because of the use of negative-entropy, their analysis requires 2Hη ≤ γ,  which prevents them from picking a very small γ and eventually leads to �  O(K2/3) regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' On the other hand, our  log-barrier analysis allows us to drop this requirement and pick a γ as small as K−1, though bearing some extra  factors polynomial in A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', the number of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Indeed, optimizing the bound over the parameters (see the  second claim), we achieve RK = �O(H3√  AdK), which is of order  √  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  4  Removing Polynomial Dependency on A via a New Magnitude-Reduced  Estimator  One drawback of using log-barrier is that the term Ψ(y) − Ψ(x1) from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 leads to poly(A) dependency  (while for negative entropy, this is only log A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' To get around this issue, we go back to using negative entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Recall  that the issue of this regularizer is that to ensure the same bound as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1, we require ηct,i ≥ −1, which  translates to the requirement η( �Qk(s, a) − Bk(s, a)) ≥ −1 in the context of linear-Q MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The challenging part  here is that �Qk(s, a), as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (5), could be as negative as −H/γ, which then restricts η to be of order  O(γ/H), as mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  To resolve this, we propose a new Q-function estimator that has a smaller magnitude in the negative direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Our high-level idea is the following: for a possibly negative random variable Z, defined another magnitude-reduced  random variable �Z = Z − (Z)− + E[(Z)−] (recall (Z)− = min{Z, 0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' �Z then has the same expectation as Z and  also the same order of second moment: E[ �Z2] ≤ 2 E[Z2] + 2(E[(Z)−])2 = O(E[Z2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' More importantly, we have  �Z ≥ E[(Z)−], and thus the smallest possible value of �Z is E[(Z)−], no less than (often much larger than) the smallest  possible value of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Thus, it becomes much easier to ensure ηct,i ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Applying this idea to our context, we propose a new Q-function estimator as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (9), where mk(s, a) exactly  takes the role of E[(Z)−], except that we have no access to the real expectation but have to approximate it with  samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' see Line 7 of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' To make sure that these samples are “consistent" with those used in constructing  �Σ†  k,h, we perform a check in Line 6 and repeat the sampling until the check passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Since both Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (8) and  ∥�Σk,h − Σk,h∥2 ≤ γ hold with probability at least 1 − K−3, the expected number of trials is only 1 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Our  complete algorithm in shown in Algorithm 2, where the parts different from (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b) are again highlighted  in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Our analysis shows that the new Q-function estimator indeed has a significantly smaller magnitude: it can only  be as negative as −H/√γ (as opposed to the previous −H/γ bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This only restricts η to be of order O(√γ/H),  enabling us to pick γ ≈ 1/K as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3 and achieve �  O(  √  K) regret again, as formalized in the next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Note that our final regret bound not only has no poly(A) dependency, but is also optimal in both d and K as one  cannot do better even in the special case of linear bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' When applied to a linear-Q MDP (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2) with a simulator, Algorithm 2 ensures the following  when ηβγ−1 ≤  1  12H2 , η2γ−1 ≤  1  12H2 , 8ηH2 ≤ β, ǫ ≤ (H2K)−1, and M = 32γ−2 log K:  RK = �  O  �  βdHK + γ  β dH3K + H  η + ηdH3K +  η  γ2K2 H3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Picking η =  1  √  dH4K , β =  8  √  dK , γ = 96  dK , and ǫ =  1  H2K , we have RK = �  O(H3√  dK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Due to space limitations, we only sketch why �Qk(s, a) is now at least − O(H/√γ): since Lk,h ∈ [0, H],  we have �Qk(s, a) ≥ Hmk(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' By Jensen’s inequality, we can bound mk(s, a)2 for all (s, a) as:  mk(s, a)2 ≤ 1  M  M  �  m=1  �  φ(s, a)T�Σ†  k,hφ(sm,h, am,h)  �2  8  Refined Regret for Adversarial MDPs with Linear Function Approximation  Algorithm 2 Improved Linear-Q Algorithm (Using Magnitude-Reduced Loss Estimators)  Input: Learning rate η, bonus parameter β, MGR parameters γ and ǫ, covariance estimation parameter M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  1: for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', K do  2:  Let πk ∈ Π be defined as follows for all s ∈ S:  πk(a | s) ∝ exp  �  − η  �  k′<k  ( �Qk′(s, a) − Bk′(s, a))  �  ,  where Bk(s, a) is calculated in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  3:  Execute πk, observing the trajectory (sk,h, ak,h) and losses ℓk(sk,h, ak,h) for all h ∈ [H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  4:  Construct �Σ†  k,h using Matrix Geometric Resampling (Algorithm 3 in the appendix) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ∥�Σ†  k,h∥2 ≤ 1  γ ,  ���E[�Σ†  k,h] − (γI + Σπk  h )−1���  2 ≤ ǫ,  (7)  and the following holds w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' 1 − K−3:  ���(�Σ†  k,h)1/2Σπk  h (�Σ†  k,h)1/2���  2 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (8)  5:  Simulate πk for M times, giving {(sm,h, am,h)}m,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Estimate the covariance matrix Σπk  h  as  �Σk,h = 1  M  M  �  m=1  φ(sm,h, am,h)φ(sm,h, am,h)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  6:  If ∥(�Σ†  k,h)1/2�Σk,h(�Σ†  k,h)1/2∥2 ≥ 3, goto Line 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  7:  For each s ∈ Sh and a ∈ A, define mk(s, a) as  mk(s, a) = 1  M  M  �  m=1  �  φ(s, a)T�Σ†  k,hφ(sm,h, am,h)  �  − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  8:  Let Lk,h = �  h′≥h ℓk(sk,h′, ak,h′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Estimate the Q-function Qπk  k (s, a) as follows (for s ∈ Sh):  �Qk(s, a) = φ(s, a)T�Σ†  k,hφ(sk,h, ak,h)Lk,h−  H  �  φ(s, a)T�Σ†  k,hφ(sk,h, ak,h)  �  − + Hmk(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (9)  9: end for  = φ(s, a)T�Σ†  k,h�Σπk  h �Σ†  k,hφ(s, a) ≤ 3∥φ(s, a)∥2  �Σ†  k,h ≤ 3  γ ,  where the second inequality is by Line 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Therefore, we have �Qk(s, a) ≥ −  √  3H  √γ , a reduced magnitude compared to  that of a standard estimator (defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  5  Generalizing to Linear MDPs  When a simulator is unavailable, we consider a special case of linear-Q MDPs: linear MDPs (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4), where  the transition is also linear in the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' As Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021a) show, this makes the dilated bonuses linear as  well, and thus we can efficiently estimate them without using a simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Due to various technical obstacles, they  only achieve �O(K14/15) regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We improve it to �O(K8/9) via two key modifications to their algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' As the  algorithm is fairly lengthy due to the lack of simulators and the estimation of the dilated bonus function, we defer it  9  Refined Regret for Adversarial MDPs with Linear Function Approximation  (Algorithm 6) to the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We refer the reader to (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021a) for a detail description of their algorithm,  and only focus on our two modifications (highlighted in blue) described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  The first obvious modification is to apply one of the techniques introduced in the last two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' However, since  the magnitude-reduced estimator we developed in Algorithm 2 requires extra samples (see Line 5), which, without a  simulator, can only be done via real episodes and in turn introduces more regret, we go with the log-barrier approach  instead, even though it leads to extra poly(A) factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  But it turns out that this only leads to a mild improvement in the regret, since the constraint on η is not the  bottleneck of their analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Instead, one important bottleneck comes from using the Matrix Geometric Resampling  (MGR) procedure (Neu and Olkhovskaya, 2020) to estimate the covariance matrix inverse, which again, without a  simulator, can only be done via running the same policy for multiple real episodes (called an epoch) to collect samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  It is thus critical to make this step as sample efficient as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  To resolve this issue, our key observation is that MGR uses too many samples to produce an estimate that is  in a sense more accurate than required: specifically, it needs O(ǫ−2γ−3) samples to ensure a bound like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Instead, we find that a weaker multiplicative approximation guarantee is enough, which only requires O(γ−2) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Moreover, this is achieved by simply taking the (regularized) inverse of the empirical average of O(γ−2) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  More concretely, for a policy �πj (which mixes πj with some exploration policy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' see Algorithm 6 for more details)  in epoch j of the algorithm, we collect roughly O(γ−2) samples using this policy and construct an empirical average  covariance matrix �Σj,h for layer h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Then we let �Σ†  j,h = (γI + �Σj,h)−1 be the estimation for (γI + Σ�πj  h )−1 (see Line 17  of Algorithm 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We then prove the following:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' If the epoch length W in Algorithm 6 is set to (4d log d  δ )γ−2, then  (1 − √γ)(γI + Σ�πj  h ) ⪯ (γI + �Σj,h) ⪯ (1 + √γ)(γI + Σ�πj  h )  holds with probability at least 1 − 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  The proof relies on a new matrix concentration bound (Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4 in the appendix), which can be of independent  interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' A direct corollary of this lemma is:  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' If W = (4d log d  δ )γ−2 and γ ≤ 1  4, then with probability 1 − 2δ, we have the following:  (1 − 2√γ)I ⪯ (�Σ†  j,h)1/2(γI + Σ�πj  h )(�Σ†  j,h)1/2 ⪯ (1 + 2√γ)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Simply left and right multiply (�Σ†  j,h)1/2 = (γI + �Σj,h)−1/2 in the bound of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 and use the fact  [  1  1+√γ ,  1  1−√γ ] ⊆ [1 − 2√γ, 1 + 2√γ] when γ ≤ 1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Together with a new analysis to bound bias terms in the regret, such approximations turn out to be enough: the  bias terms enjoy almost the same bound (up to constants) as we had in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Hence, we finally get the  following theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' see Appendix D for a full proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3 (Informal version of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' When applied to linear MDPs, Algorithm 6 with proper tuning  ensures RK = �O(K8/9) (omitting all other dependencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Ignoring less important parts, at a high level we still decompose the regret into several bias terms and  a Reg-Term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In this sketch, we only provide key ideas in bounding the bias terms using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Specifically,  we illustrate how to bound E[Bias-1], defined as follows:  Bias-1 ≜ W  �  j,h  E  sh∼π∗  �  E  ah∼πj  �  Q  πj  j (sh, ah) − �Qj(sh, ah)  ��  ,  where Q  πj  j (s, a) = φ(s, a)Tθ  πj  j,h is the average Q-functions in epoch j induced by policy πj, and �Qj(s, a) is its estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  By direct calculation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' one may check that  E  �  Q  πj  j (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a) − �Qj(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)  �  = φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)T(I − �Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='hΣ�πj  h )θ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  = φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)T�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h(γI + �Σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − Σ�πj  h )θ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  10  Refined Regret for Adversarial MDPs with Linear Function Approximation  ≤ ∥φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)∥�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  ���(γI + �Σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − Σ�πj  h )θ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  ����ӆ  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  ≤ ∥φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)∥�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  ����γθ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)  ����ӆ  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  +  ���(�Σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − Σ�πj  h )θ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  ����ӆ  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  �  ≤ β  4 ∥φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)∥2  �Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h + 2  β  ���γθ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)  ���  2  �ӆ  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  + 2  β  ���(�Σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − Σ�πj  h )θ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  ���  2  �ӆ  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  where we apply Cauchy-Schwarz inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' triangle inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' and AM-GM inequality (twice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The first term above  contributes to O(βdHK) after summing over k and h and applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5, while the second term is bounded  by O( γ  β dH2) for any j (as ∥�Σ†  j,h∥2 ≤ γ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' By some algebraic manipulations, the last term translates to  2  β  ����(�Σ†  j,h)−1/2�  I − (�Σ†  j,h)1/2(γI + Σ�πj  h )(�Σ†  j,h)1/2�2(�Σ†  j,h)−1/2  ����  2  dH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Thanks to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2, we know that  −2√γI ⪯ I − (�Σ†  j,h)1/2(γI + Σ�πj  h )(�Σ†  j,h)1/2 ⪯ 2√γI,  and thus the last term is again of order O( γ  βdH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' All other bias terms are bounded analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' At last, it only  remains to bound the Reg-Term, which is directly calculated and bounded like we did in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  6  Conclusion  In this paper, we study policy optimization algorithms in adversarial MDPs with linear function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Building on top of the dilated bonus approach introduced by Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021b), we derive two algorithms which both  achieve the first �  O(  √  K)-style regret bounds in linear-Q MDPs with the help of a simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Technically speaking, the  first algorithm use a novel refined analysis for FTRL with the log-barrier regularizer, while the second one develops a  new magnitude-reduced loss estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Moreover, we also generalize the first approach to simulator-free linear MDPs  and get �  O(K8/9) regret, greatly improving over the best-known �  O(T 14/15) regret bound (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This  generalization also contains an alternative to the Matrix Geometric Resampling procedure (Neu and Olkhovskaya,  2020) using a new matrix concentration bound (Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We expect all these techniques to be of independent  interest and potentially useful for other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Further improving the result for linear MDPs is the key future  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  References  Yasin Abbasi-Yadkori, Peter Bartlett, Kush Bhatia, Nevena Lazic, Csaba Szepesvari, and Gellért Weisz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Politex:  Regret bounds for policy iteration using expert prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In International Conference on Machine Learning,  pages 3692–3702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content='  The nonstochastic multiarmed bandit  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content='  Sébastien Bubeck, Nicolo Cesa-Bianchi, and Sham M Kakade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content=' JMLR Workshop and Conference Proceedings,  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content='  Provably efficient exploration in policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  In  International Conference on Machine Learning, pages 1283–1294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content=' Learning in games: Robustness  of fast convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content=' PMLR,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content='  PMLR, 2020a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content=' PMLR, 2020b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content=' 19th  International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content=' Bandit algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Cambridge University Press, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Haipeng Luo, Chen-Yu Wei, and Chung-Wei Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Policy optimization in adversarial mdps: Improved exploration  via dilated bonuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content='  Haipeng Luo, Chen-Yu Wei, and Chung-Wei Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Policy optimization in adversarial mdps: Improved exploration  via dilated bonuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content='  Lingsheng Meng and Bing Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The optimal perturbation bounds of the moore–penrose inverse under the frobenius  norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Linear algebra and its applications, 432(4):956–963, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Gergely Neu and Julia Olkhovskaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Online learning in mdps with linear function approximation and bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Advances in Neural Information Processing Systems, 34:10407–10417, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Aviv Rosenberg and Yishay Mansour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Online convex optimization in adversarial markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In  International Conference on Machine Learning, pages 5478–5486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Lior Shani, Yonathan Efroni, Aviv Rosenberg, and Shie Mannor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Optimistic policy optimization with bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  In International Conference on Machine Learning, pages 8604–8613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Joel A Tropp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' User-friendly tail bounds for sums of random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Foundations of computational mathematics,  12(4):389–434, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Ruosong Wang, Simon S Du, Lin Yang, and Russ R Salakhutdinov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' On reward-free reinforcement learning with  linear function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Advances in neural information processing systems, 33:17816–17826, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Chen-Yu Wei and Haipeng Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' More adaptive algorithms for adversarial bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In Conference On Learning  Theory, pages 1263–1291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Chen-Yu Wei, Mehdi Jafarnia Jahromi, Haipeng Luo, and Rahul Jain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Learning infinite-horizon average-reward  mdps with linear function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In International Conference on Artificial Intelligence and Statistics,  pages 3007–3015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Lin Yang and Mengdi Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Reinforcement learning in feature space: Matrix bandit, kernels, and regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In  International Conference on Machine Learning, pages 10746–10756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  12  Refined Regret for Adversarial MDPs with Linear Function Approximation  Andrea Zanette, Ching-An Cheng, and Alekh Agarwal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content=' In Conference on Learning Theory, pages 4473–4525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content='  Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Dongruo Zhou, Jiafan He, and Quanquan Gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Provably efficient reinforcement learning for discounted mdps with  feature mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In International Conference on Machine Learning, pages 12793–12802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Alexander Zimin and Gergely Neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Online learning in episodic markovian decision processes by relative entropy  policy search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Advances in neural information processing systems, 26, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Julian Zimmert and Tor Lattimore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Return of the bias: Almost minimax optimal high probability bounds for  adversarial linear bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In Conference on Learning Theory, pages 3285–3312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  13  Refined Regret for Adversarial MDPs with Linear Function Approximation  Supplementary Materials  A Auxiliary Lemmas and Omitted Algorithms  15  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 Matrix Geometric Resampling Procedure  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content='  34  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='6 Bounding the Magnitudes of Bonuses .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  35  14  Refined Regret for Adversarial MDPs with Linear Function Approximation  A  Auxiliary Lemmas and Omitted Algorithms  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1  Matrix Geometric Resampling Procedure  Matrix Geometric Resampling algorithm, introduced by Neu and Olkhovskaya (2020) and improved by Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (2021b), generates a close estimation of (γI + Σπ  h)−1 for given π ∈ Π and h ∈ [H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Formally, we present it in  Algorithm 3 and state its performance guarantees in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Algorithm 3 Matrix Geometric Resampling Algorithm  Input: Policy π, regularization parameter γ, desired precision ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Output: A set of matrices {�Σ†  h}H  h=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  1: Set M ≥ 24 ln(dHT )  ǫ2γ2  and N ≥ 2  γ ln 1  ǫγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Set c = 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  2: Draw MN trajectories of π using the simulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' denote them by {(sm,n,h, am,n,h)}m∈[M],n∈[N],h∈[H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  3: for m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', M do  4:  for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', N do  5:  Set Yn,h = γI + φ(sm,n,h, am,n,h)φ(sm,n,h, am,n,h)T and Zn,h = �n  n′=1(I − cYn′,h) for all h ∈ [H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  6:  end for  7:  Calculate �Σ†  m,h = cI + c �N  n=1 Zn,h for all h ∈ [H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  8: end for  9: Set �Σ†  h =  1  M  �M  m=1 �Σ†  m,h for all h ∈ [H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 (Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 of Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' With the configurations of M and N stated in Algorithm 3, for  any policy π, we have ∥�Σ†  h∥2 ≤ γ−1, ∥E[�Σ†  h] − (γI + Σπ  h)−1∥2 ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Moreover, there exists a good event that happens  with probability 1 −  1  K3 , under which the following two properties hold:  ∥E[�Σ†  h] − (γI + Σπ  h)−1∥2 ≤ 2ǫ,  ∥�Σ†  hΣπ  h∥2 ≤ 1 + 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2  Dilated Bonus Calculation in Linear-Q MDP Algorithms  The following algorithm (which is Algorithm 3 of Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021b)) indicates how we calculate the dilated bonus  function Bk(s, a) with the help of the simulator in Algorithms 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In both algorithms, we define the bonus as  bk(s, a) = β  �  ∥φ(s, a)∥2  �Σ†  j,h +  E  �a∼πk(·|s)  �  ∥φ(s, �a)∥2  �Σ†  j,h  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Algorithm 4 Dilated Bonus Calculation in Linear-Q Algorithms  Input: Episode k ∈ [K], state s ∈ S, action a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Output: The dilated bonus function Bk(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  1: if Bonus(k, s, a) is called before then return the value calculated at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  2: Let h be the layer that s lies in, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', s ∈ Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' if h = H + 1 then return 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  3: Call the simulator for a next state s′ ∼ P(· | s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Calculate πk(· | s) and πk(· | s′) according to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  4: return β  �  ∥φ(s, a)∥2  �Σ†  k,h + E�a∼πk(·|s)  �  ∥φ(s, �a)∥2  �Σ†  k,h  ��  + (1 + 1  H )Bk(s′, a′) where a′ is a sample from πk(· | s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Note that, during the calculation of πk and Bk(s′, a′), we are essentially recursively calling this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3  PolicyCover Algorithm in Linear MDP Algorithms  The following algorithm is Algorithm 6 of Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021a) (which itself builds upon Algorithm 1 of Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' we refer the readers to the original paper for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This algorithm generates a mixture of policies,  which we call the policy cover πcov, together with an estimate of the (regularized) inverse of its covariance matrix,  namely {�Σcov  h }H  h=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' It ensures the following property:  15  Refined Regret for Adversarial MDPs with Linear Function Approximation  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2 (Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4 by Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' With probability 1 − δ, we have the following for all policies π  and any h ∈ [H]:  Pr  sh∼π [sh ̸∈ K] = �  O  �dH  α  �  ,  where K is set of known states defined as follows:  K =  �  s ∈ S  ���∀a ∈ A, ∥φ(s, a)∥2  (�Σcov  h  )−1 ≤ α  �  ,  where h is the layer that s lies in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Algorithm 5 PolicyCover for Linear MDPs  Input: Configurations M0, N0, α, failure probability δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  1: Set Γ1,h ← I for all h ∈ [H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Set �β = 60dH  �  ln K  δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  2: for m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', M0 do  3:  Set �Vm(xh+1) ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  4:  for h = H, H − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , 1 do  5:  For all (s, a) ∈ Sh × A, let  �Qm(s, a) = min{rm(s, a) + �β∥φ(s, a)∥Γ−1  m,h + φ(s, a)T�θm,h, H},  �Vm(s) = max  a∈A  �Qm(s, a),  πm(a | s) =  1[a = argmax  a′∈A  �Qm(s, a′)],  where  rm(s, a) = ramp 1  K (∥φ(s, a)∥Γ−1  m,h − α  M0  ),  �θm,h = Γ−1  m,h( 1  N0  m−1  �  m′=1  N0  �  n′=1  φ(sm′,n′,h, am′,n′,h)�Vm(sm′,n′,h+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Here, rampz(y) is 0 if y ≤ −z, 1 if y ≥ 0, and y  z + 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  6:  end for  7:  for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', N0 do  8:  Execute πm and get trajectory {(sm,n,h, am,n,h)}H  h=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  9:  end for  10:  Compute Γm+1,h as  Γm+1,h ← Γm,h + 1  N0  N0  �  n=1  φ(sm,n,h, am,n,h)φ(sm,n,h, am,n,h)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  11:  return πcov defined as the uniform mixture of π1, π2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , πM0 and �Σcov  h  defined as  1  M0 ΓM0+1,h (for all h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  12: end for  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4  Stochastic Matrix Concentration  We first state the well-known Matrix Azuma inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3 (Matrix Azuma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' see (Tropp, 2012, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Let {Xk}n  k=1 be a adapted sequence of d × d self-  adjoint matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Let {Ak}n  k=1 be a fixed sequence of self-adjoint matrices such that  Ek[Xk] = 0,  X2  k ⪯ A2  k almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  16  Refined Regret for Adversarial MDPs with Linear Function Approximation  Let σ2 = ∥ 1  n  �n  k=1 A2  k∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Then for all ǫ > 0:  Pr  ������  1  n  n  �  k=1  Xk  �����  2  ≥ ǫ  �  ≤ d exp  �  − nǫ2  8σ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Then, we state the following lemma, which we use to replace the MGR procedure in Linear MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Let H1, H2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , Hn be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' PSD matrices s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' E[Hi] = H, Hi ⪯ I a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', and H ⪰  1  dn log d  δ I, then with  probability 1 − 2δ,  −  �  d  n log d  δ H1/2 ⪯ 1  n  n  �  i=1  Hi − H ⪯  �  d  n log d  δ H1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Let G =  �  1  dn log d  δ H−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We first show that  −2 log d  δ  n  G−1 ⪯ 1  n  n  �  i=1  Hi − H  holds with probability 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We have  Pr  �  −2 log( d  δ )  n  G−1 ̸⪯ 1  n  n  �  i=1  Hi − H  �  = Pr  � n  �  i=1  G1/2 (H − Hi) G1/2 − log(d  δ )I ̸⪯ log(d  δ )I  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Since log is operator monotone, we have for PSD matrices A, B: exp(A) ⪯ exp(B) ⇒ log(exp(A)) ⪯ log(exp(B))  (note that the reverse does not generally hold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We have Pr{exp(A) ⪯ exp(B)} ≤ Pr{A ⪯ B} and hence Pr{A ̸⪯  B} ≤ Pr{exp(A) ̸⪯ exp(B)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Hence  Pr  � n  �  i=1  G1/2 (H − Hi) G1/2 − log(d  δ )I ̸⪯ log(d  δ )I  �  ≤ Pr  �  exp  � n  �  i=1  G1/2 (H − Hi) G1/2 − log(d  δ )I  �  ̸⪯ d  δ I  �  ≤ Pr  �  Tr  �  exp  � n  �  i=1  G1/2 (H − Hi) G1/2 − log(d  δ )I  ��  > d  δ  �  ≤ E  �  Tr  �  exp  ��n  i=1 G1/2 (H − Hi) G1/2 − log( d  δ )I  ���  d  δ  (Markov’s inequality)  Using the Golden-Thompson inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' we have  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='exp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='� n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='i=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='G1/2 (H − Hi) G1/2 − log(d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='δ )I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='≤ E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='exp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�n−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='i=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='G1/2 (H − Hi) G1/2 − log( d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='δ )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='exp(G1/2 (H − Hn) G1/2 − log( d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='δ )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='I)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='= Tr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='exp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�n−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='i=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='G1/2 (H − Hi) G1/2 − log( d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='δ )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='exp(G1/2 (H − Hn) G1/2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='exp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='−log( d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='δ )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='I)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='We have due to G1/2(H − Hn)G1/2 ⪯ I:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='exp(G1/2 (H − Hn) G1/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='⪯ E[I + (G1/2 (H − Hn) G1/2) + (G1/2 (H − Hn) G1/2)2] = I + E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='(G1/2HnG1/2)2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  By the Araki–Lieb–Thirring inequality, we have Tr((ABA)2) ≤ Tr(A2B2A2), hence  Tr  �  E[(G1/2HnG1/2)2]  �  ≤ E[Tr(GH2  nG)] ≤ E[Tr(GHnG)]  (Hn ⪯ I)  17  Refined Regret for Adversarial MDPs with Linear Function Approximation  = Tr(GHG) ≤ log( d  δ )  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Combining this with the previous result yields  E  �  exp(G1/2 (H − Hn) G1/2)  �  exp  �  −log( d  δ )  n  I)  �  ⪯  �  1 + log( d  δ )  n  �  I exp  �  −log( d  δ )  n  I)  �  ⪯ I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Applying this recursively yields  E  �  Tr  �  exp  � n  �  i=1  G1/2 (H − Hi) G1/2 − log(d  δ )I  ���  ≤ Tr (I) = d ,  which concludes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' By symmetry and union bound, we have the following with probability 1−2δ:  −2 log d  δ  n  G−1 ⪯ 1  n  n  �  i=1  Hi − H ⪯ 2 log d  δ  n  G−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  This then directly simplifies to our conclusion given H ⪰  1  dn log d  δ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  18  Refined Regret for Adversarial MDPs with Linear Function Approximation  B  Omitted Proofs in Section 3 (Linear-Q Algorithm Using Log-Barrier  Regularizers)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1  Property of FTRL with Log-Barrier Regularizers  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We first introduce the notation of Bregman divergences DΨ(y, x) = Ψ(y)−Ψ(x)−⟨∇Ψ(x), y−  x⟩, which is heavily used in FTRL analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' By standard FTRL analysis (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', Theorem 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5 by Lattimore and Szepesvári  (2020)), we know  T  �  t=1  ⟨xt − y, ct⟩ ≤ Ψ(y) − Ψ(x1)  η  +  T  �  t=1  �  ⟨xt − xt+1, ct⟩ − η−1DΨ(xt+1, xt)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (10)  Consider DΨ(xt+1, xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' By definition of DΨ and our choice that Ψ(p) = �A  i=1 ln 1  pi , we have  DΨ(xt+1, xt) = Ψ(xt+1) − Ψ(xt) − ⟨∇Ψ(xt), xt+1 − xt⟩  =  A  �  i=1  �  ln xt  xt+1  + xt+1 − xt  xt  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Observe that the following holds for all x ∈ (0, 1) and x + ∆ ∈ (0, 1):  ln  x  x + ∆ + ∆  x =  � ∆  0  α  x(x + α)dα ≥  � ∆  0  α  x dα = ∆2  2x ,  For all i, we apply the inequality about with x = xt,i and ∆ = xt+1,i − xt,i (as xt and xt+1 both belong to the  simplex, the conditions x ∈ (0, 1) and (x + ∆) ∈ (0, 1) indeed hold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We then get the following lower bound on  DΨ(xt+1, xt):  DΨ(xt+1, xt) ≥  A  �  i=1  (xt+1 − xt)2  2xt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  We further have  ⟨xt − xt+1, ct⟩ − η−1DΨ(xt+1, xt) ≤  A  �  i=1  �  (xt,i − xt+1,i)ct,i − (xt+1 − xt)2  2xt  �  ≤  A  �  i=1  1  2xt,iηc2  t,i,  where the last step uses AM-GM inequality −a2 + 2ab ≤ b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Plugging this back to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (10) gives our conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2  Proof of Main Theorem  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' As sketched in the main text, we consider the following expression in order to apply the  dilated bonus lemma (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5):  K  �  k=1  H  �  h=1  E  sh∼π∗  � �  ah∈A  (πk(ah | sh) − π∗(ah | sh))(Qπk  k (sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah) − Bk(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah))  �  =  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)  �  Qπk  k (sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah) − �Qk(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)  ��  �  ��  �  Bias-1  +  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼π∗(·|sh)  �  �Qk(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah) − Qπk  k (sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)  ��  �  ��  �  Bias-2  +  19  Refined Regret for Adversarial MDPs with Linear Function Approximation  K  �  k=1  H  �  h=1  E  sh∼π∗  ��  πk(· | sh) − π∗(· | sh),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' �Qk(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ·) − Bk(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ·)  ��  �  ��  �  Reg-Term  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (11)  According to Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1, we know that  E[Bias-1] + E[Bias-2] ≤ β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  +  β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼π∗(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  + O  �γ  β dH3K + ǫ(H + β)HK  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Moreover, by Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2, we can see that  E[Reg-Term] ≤ Hη−1A ln K + O  �  AH2  �  ǫ +  √  d + β  γ  ��  +  2ηH2  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)  �  ∥φ(sh, ah)∥�Σ†  k,h  ��  + O  � ηH3  γ2K2  �  +  1  H  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh) [Bk(sh, ah)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Plugging back into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' we know that  K  �  k=1  H  �  h=1  E  sh∼π∗  � �  ah∈A  (πk(ah | sh) − π∗(ah | sh))(Qπk  k (sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah) − Bk(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah))  �  ≤ β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)[∥φ(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)∥2  �Σ†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h]  ��  +  β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼π∗(·|sh)[∥φ(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)∥2  �Σ†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h]  ��  +  2ηH2  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)  �  ∥φ(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)∥2  �Σ†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  ��  +  1  H  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh) [Bk(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)]  �  +  �O  �γ  β dH3K + ǫ(H + β)HK + H  η A + AH2  �  ǫ +  √  d + β  γ  �  + ηH3  γ2K2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Using (6ηH2 + β  4 ) ≤ β, we apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5 with bk(s, a) = ∥φ(s, a)∥2  �Σ†  k,h + Ea′∼πk(·|s)[∥φ(s, a′)∥2  �Σ†  k,h], giving  RK ≤ β E  � K  �  k=1  H  �  h=1  E  sh∼πk  �  E  ah∼πk(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  +  �  O  �γ  β dH3K + ǫ(H + β)HK + H  η A + AH2  �  ǫ +  √  d + β  γ  �  + ηH3  γ2K2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Fixing an episode k ∈ [K] and h ∈ [H], we have  E  sh∼πk  ��  a  πk(a | s)∥φ(s, a)∥2  �Σ†  k,h  �  20  Refined Regret for Adversarial MDPs with Linear Function Approximation  ≤  E  sh∼πk  ��  a  πk(a | s)∥φ(s, a)∥2  (γI+Σ  πk  h )−1  �  + O(ǫ)  ≤  E  sh∼πk  ��  a  πk(a | s)∥φ(s, a)∥2  (Σ  πk  h )−1  �  + O(ǫ)  =  �  (Σπk  h )−1,  E  (sh,ah)∼πk  [φ(sh, ah)φ(sh, ah)T]  �  = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (12)  Therefore, we can conclude the following if 12ηβH2 ≤ γ and 8ηH2 ≤ β:  RK = �  O  �  βdHK + γ  β dH3K + ǫ(H + β)HK + H  η A + AH2  �  ǫ +  √  d + β  γ  �  + ηH3  γ2K2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  It remains to tune the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We first pick ǫ =  1  H2K , which makes all terms related to ǫ constantly-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Setting η ≈ K−1/2 and γ ≈ K−1, the last term is also o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Hence, removing all constantly-bounded terms give  RK = �O  �  AH 1  η + βdHK + γ  β dH3K + AH2 β  γ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  We then get RK = �O(  √  AdH6K) = �O(A1/2d1/2H3K1/2) by picking:  η =  �  A  dH4K , β = 8  �  A  dK , γ = 96 A  dK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  It’s straightforward to verify that they satisfy 12ηβH2 ≤ γ and 8ηH2 ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3  Bounding Bias-1 and Bias-2  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In Algorithm 1, we have  E[Bias-1] + E[Bias-2] ≤ β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  +  β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼π∗(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  + O  �γ  β dH3K + ǫ(H + β)HK  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' As Bias-1 has nothing to do with the choice of the regularizer, it can be bounded the same as the original  algorithm (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b, Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2), which we also include below for completeness: fixing a specific (k, s, a)  and suppose that s ∈ Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Then we have the following,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' where every expectation is taken to the randomness in the  k-th episode:  E  �  Qπk  k (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a) − �Qk(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)  �  = φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)Tθπk  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)T E  �  �Σ†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='hφ(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)Lk  �  = φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)Tθπk  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)T E  �  �Σ†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  �  E  �  φ(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)φ(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)Tθπk  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  �  (a)  ≤ φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)T �  I − (γI + Σπk  h )−1Σπk  h  �  θπk  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h + ǫH  = γφ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)T(γI + Σπk  h )−1θπk  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h + ǫH  (b)  ≤ γ∥φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)∥(γI+Σ  πk  h )−1∥θπk  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h∥(γI+Σ  πk  h )−1 + ǫH  (c)  ≤ β  4 ∥φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)∥2  (γI+Σ  πk  h )−1 + γ2  β ∥θπk  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h∥2  (γI+Σ  πk  h )−1 + ǫH  (d)  ≤ β  4 E  �  ∥φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)∥2  �Σ†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  �  + γ  β dH2 + ǫH + ǫβ  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  21  Refined Regret for Adversarial MDPs with Linear Function Approximation  where (a) used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (7) (which follows from Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1) and the assumption that ∥φ(s, a)∥ ≤ 1 and Lk ≤ H, (b)  used Cauchy-Schwartz inequality, (c) used AM-GM inequality, and (d) used the assumption that ∥θπk  k,h∥ ≤  √  dH (see  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2) and again Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Hence, we have  E[Bias-1] ≤ β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  + O  �γ  β dH3K + ǫ(H + β)HK  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Similarly, we have  E[Bias-2] ≤ β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼π∗(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  + O  �γ  β dH3K + ǫ(H + β)HK  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Combining these two parts together gives our conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4  Bounding Reg-Term  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Under the assumption that 12ηβH2 ≤ γ, we have the following in Algorithm 1:  E[Reg-Term] ≤ Hη−1A ln K + O  �  AH2  �  ǫ +  √  d + β  γ  ��  +  2ηH2  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)  �  ∥φ(sh, ah)∥2  �Σ†  k,h  ��  + O  � ηH3  γ2K2  �  +  1  H  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh) [Bk(sh, ah)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1, we get the following for all k ∈ [K], s ∈ Sh (where h ∈ [H]), and �π∗ ∈ Π:  K  �  k=1  �  a∈A  (πk(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))  ≤ Ψ(�π∗(· | s)) − Ψ(π1(· | s))  η  +  K  �  k=1  �  a∈A  (�π∗(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))+  η  K  �  k=1  �  a∈A  πk(a | s)( �Qk(s, a) − Bk(s, a))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  By picking �π∗(a | s) = (1 − AK−1)π∗(a | s) + K−1, the first term is bounded by  Ψ(�π∗(· | s)) − Ψ(π1(· | s))  η  = 1  η  �  a∈A  ln π1(a | s)  �π∗(a | s) ≤ 1  η  �  a∈A  ln π1(a | s)  K−1  ≤ η−1A ln K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Meanwhile, the second term is bounded by  K  �  k=1  �  a∈A  (�π∗(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))  =  K  �  k=1  �  a∈A  (−AK−1π∗(a | s) + K−1)( �Qk(s, a) − Bk(s, a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  22  Refined Regret for Adversarial MDPs with Linear Function Approximation  Firstly, we have the following as ∥�Σ†  k,h∥2 ≤ γ−1:  Bk(s, a) ≤ H  �  1 + 1  H  �H  × 2β sup  s,a,h  ∥φ(s, a)∥2  �Σ†  k,h ≤ 6βHγ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (13)  Moreover, according to the calculation in Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1, we know that  E[ �Qk(s, a) − Qπk  k (s, a)] ≤ γφ(s, a)T(γI + Σπk  h )−1θπk  k,h + ǫH ≤ O((ǫ +  √  d)H),  while, at the same time, Qπk  k (s, a) ∈ [0, H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Hence, after taking expectations on both sides, we know that  E  � K  �  k=1  �  a∈A  (�π∗(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))  �  ≤  K  �  k=1  �  a∈A  (|−AK−1π∗(a | s)| + |K−1|)|E[ �Qk(s, a) − Bk(s, a)]|  ≤ K × 2AK−1 × O  �  (ǫ +  √  d)H + H + β  γ H  �  = O  �  AH  �  ǫ +  √  d + β  γ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Then consider the last term, which is directly bounded by  η  K  �  k=1  �  a∈A  πk(a | s)2( �Qk(s, a) − Bk(s, a))2  ≤ 2η  K  �  k=1  �  a∈A  πk(a | s) �Qk(s, a)2 + 2η  K  �  k=1  �  a∈A  πk(a | s)Bk(s, a)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  The first term can be calculated as follows, following the original proof (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021b, Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3):  E[ �Qk(s, a)2] ≤ H2 E[φ(s, a)T�Σ†  k,hφ(sk,h, ak,h)φ(sk,h, ak,h)T�Σ†  k,hφ(s, a)]  = H2 E[φ(s, a)T�Σ†  k,hΣπk  h �Σ†  k,hφ(s, a)]  (a)  ≤ 2H2 E[φ(s, a)T�Σ†  k,hφ(s, a)] + O  � H2  γ2K3  �  = 2H2 E  �  ∥φ(s, a)∥2  �Σ†  k,h  �  + O  � H2  γ2K3  �  ,  where (a) used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (8), which happens with probability 1 − K−3 for each k (when it does not hold, we simply use  the bound ∥�Σ†  k,h∥2 ≤ γ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Then we can conclude the following by adding back the summation over h ∈ [H] and  sh ∼ π∗:  E[Reg-Term] ≤ Hη−1A ln K + O  �  AH2  �  ǫ +  √  d + β  γ  ��  +  2ηH2  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)  �  ∥φ(sh, ah)∥2  �Σ†  k,h  ��  + O  � ηH3  γ2K2  �  +  1  H  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh) [Bk(sh, ah)]  �  ,  where the last term comes from the magnitude of Bk (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (13)) and the assumption that 12ηβH2 ≤ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  23  Refined Regret for Adversarial MDPs with Linear Function Approximation  C  Omitted Proofs in Section 4 (Linear-Q Algorithm Using Magnitude-  Reduced Estimators)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1  Property of FTRL with Negative-Entropy Regularizers (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Hedge)  The following result is a classic result for the Hedge algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For the sake of completeness, we also include a  proof here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Let x0, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , xT ∈ RA be defined as  xt+1,i = (xt,i exp(−ηct,i))  �� A  �  i′=1  xt,i′ exp(−ηct,i′)  �  ,  ∀0 ≤ t < T,  where ct ∈ RA is the loss corresponding to the t-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Suppose that ηct,i ≥ −1 for all t ∈ [T ] and i ∈ [A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Then  T  �  t=1  ⟨xt − y, ct⟩ ≤ log A  η  + η  T  �  t=1  A  �  i=1  xt,ic2  t,i  holds for any distribution y ∈ △([A]) when x0 = ( 1  A, 1  A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , 1  A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' By linearity, it suffices to prove the inequality for all one-hot y’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Without loss of generality, let y = 1i∗ where  i∗ ∈ [A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Define Ct,i = �t  t′=1 ct′,i as the prefix sum of ct,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Let  Φt = 1  η ln  � A  �  i=1  exp (−ηCt,i)  �  ,  then by definition of xt, we have  Φt − Φt−1 = 1  η ln  � �A  i=1 exp(−ηCt,i)  �A  i=1 exp(−ηCt−1,i)  �  = 1  η ln  � A  �  i=1  xt,i exp(−ηct,i)  �  (a)  ≤ 1  η ln  � A  �  i=1  xt,i(1 − ηct,i + η2c2  t,i)  �  = 1  η ln  �  1 − η⟨xt, ct⟩ + η2  A  �  i=1  xt,ic2  t,i  �  (b)  ≤ −⟨xt, ct⟩ + η  A  �  i=1  xt,ic2  t,i,  where (a) used exp(−x) ≤ 1 − x + x2 for all x ≥ −1 and (b) used ln(1 + x) ≤ x (again for all x ≥ −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Therefore,  summing over t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', T gives  T  �  t=1  ⟨xt, ct⟩ ≤ Φ0 − ΦT + η  T  �  t=1  A  �  i=1  xt,ic2  t,i  ≤ ln N  η  − 1  η ln (exp(−ηCT,i∗)) + η  T  �  t=1  N  �  i=1  pt(i)ℓ2  t(i)  ≤ ln A  η  + LT(i∗) + η  T  �  t=1  A  �  i=1  xt,ic2  t,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Moving Ct,i∗ to the LHS then shows the inequality for y = 1i∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The result then extends to all y ∈ △([A]) by  linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  24  Refined Regret for Adversarial MDPs with Linear Function Approximation  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2  Proof of Main Theorem  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We first consider Line 6 in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' As sketched in the main text, we shall expect such  an operation to be repeated for 1 + o(1) times because Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (8) and ∥�Σk,h − Σπk  h ∥2 ≤ γ both happens with probability  1 − K−3 — the first claim follows from Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 and the second one comes from Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3 (where we set  Xm = φ(sm,h, am,h)φ(sm,h, am,h)T − Σπk  h  and Am,h = I ⪰ Xm,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' With these two conditions and the fact that  ∥�Σ†  k,h∥2 ≤ γ−1, the desired condition trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Therefore, such an operation brings neither extra regret nor  extra computational complexity, and we focus on the regret analysis from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  We define Bias-1, Bias-2, and Reg-Term exactly the same as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The Bias-1 and  Bias-2 terms are bounded by Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2, as follows:  E[Bias-1] + E[Bias-2] ≤ β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  +  β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼π∗(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  + O  �γ  β dH3K + ǫ(H + β)HK  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Assuming 12η2H2 ≤ γ and 12ηβH2 ≤ γ, Reg-Term is bounded by Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3 as  E[Reg-Term] ≤ Hη−1 ln A + 6ηH2  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)  �  ∥φ(sh, ah)∥2  �Σ†  k,h  ��  + 1  H  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh) [Bk(sh, ah)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Plugging into the regret decomposition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' we get  K  �  k=1  H  �  h=1  E  sh∼π∗  � �  ah∈A  (πk(ah | sh) − π∗(ah | sh))(Qπk  k (sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah) − Bk(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah))  �  ≤ β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)[∥φ(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)∥2  �Σ†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h]  ��  +  β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼π∗(·|sh)[∥φ(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)∥2  �Σ†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h]  ��  +  6ηH2 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)  �  ∥φ(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)∥2  �Σ†  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  ���  +  1  H E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh) [Bk(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)]  ��  + �O  �H  η + γ  β dH3K + ǫ(H + β)HK  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Using the condition that (6ηH2 + β  4 ) ≤ β, we can apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5 to conclude that  RK = �  O  �  E  �  β  H  �  h=1  E  sh∼πk  � K  �  k=1  �  a  πk(a | s)∥φ(s, a)∥2  �Σ†  k,h  ��  + H  η + γ  β dH3K + ǫ(H + β)HK  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (12), we can conclude the following when assuming 8ηH2 ≤ β:  RK = �O  �H  η + γ  β dH3K + ǫ(H + β)HK + βdHK  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Plugging in the configurations that (again, one can see that this configuration satisfies all the conditions)  η =  1  √  dKH2 , β =  8  √  dK  , γ = 96  dK , ǫ =  1  H2K ,  we then conclude that RK = �  O(  √  dH6K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  25  Refined Regret for Adversarial MDPs with Linear Function Approximation  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3  Bounding Bias-1 and Bias-2  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In Algorithm 2, we have  E[Bias-1] + E[Bias-2] ≤ β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  +  β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼π∗(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  + O  �γ  β dH3K + ǫ(H + β)HK  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Fixing a specific (k, s, a) and assume s ∈ Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Then we have (again, all expectations are taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' random-  ness in the k-th episode)  E  �  �Qk(s, a)  �  = E  �  φ(s, a)T�Σ†  k,hφ(sk,h, ak,h)Lk,h  �  − H E  ��  φ(s, a)T�Σ†  k,hφ(sk,h, ak,h)  �  −  �  +  H E  �  1  M  M  �  m=1  �  φ(s, a)T�Σ†  k,hφ(sm,h, am,h)  �  −  �  = E  �  φ(s, a)T�Σ†  k,hφ(sk,h, ak,h)Lk,h  �  ,  where the last equality is because (sk,h, ak,h) and (sm,h, am,h) are both sampled from πk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The rest of the proof  is then identical to Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4  Bounding Reg-Term  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Suppose that 12H2η2 ≤ γ, 12ηβH2 ≤ γ, 8ηH2 ≤ γ, and M = 32γ−2 log K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Then in Algorithm 2, we  have  E[Reg-Term] ≤ Hη−1 ln A + 6ηH2 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)  �  ∥φ(sh, ah)∥2  �Σ†  k,h  ���  + 1  H E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh) [Bk(sh, ah)]  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' To apply the Hedge lemma (Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1), we need to ensure that  η( �Qk(s, a) − Bk(s, a)) ≥ −1,  ∀(s, a) ∈ S × A, k ∈ [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Fix an (s, a, k) tuple and assume that s ∈ Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We have  �Qk(s, a) − Hmk(s, a) = φ(s, a)T�Σ†  k,hφ(sk,h, ak,h)Lk,h − H(φ(s, a)T�Σ†  k,hφ(sk,h, ak,h))−  =  �  φ(s, a)T�Σ†  k,hφ(sk,h, ak,h)Lk,h,  φ(s, a)T�Σ†  k,hφ(sk,h, ak,h) ≥ 0  −φ(s, a)T�Σ†  k,hφ(sk,h, ak,h)(H − Lk,h),  φ(s, a)T�Σ†  k,hφ(sk,h, ak,h) < 0  ≥ 0,  as Lk,h ∈ [0, H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Hence, �Qk(s, a) ≥ Hmk(s, a) always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  We consider �Qk(s, a) ≥ Hmk(s, a) and Bk(s, a)  separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  We first claim that ηHmk(s, a) ≥ − 1  2, which ensures η �Qk(s, a) ≥ − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  To see this, we only need to show  η2H2mk(s, a)2 ≤ 1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' By definition, mk(s, a)2 can be written as the following:  mk(s, a)2 =  �  1  M  M  �  m=1  �  φ(s, a)T�Σ†  k,hφ(sm,h, am,h)  �  −  �2  26  Refined Regret for Adversarial MDPs with Linear Function Approximation  (a)  ≤  1  M  M  �  m=1  �  φ(s, a)T�Σ†  k,hφ(sm,h, am,h)  �2  −  = φ(s, a)T�Σ†  k,h  �  1  M  M  �  m=1  φ(sm,h, am,h)φ(sm,h, am,h)T  �  �Σ†  k,hφ(s, a)  = φ(s, a)T�Σ†  k,h�Σk,h�Σ†  k,hφ(s, a)  (b)  ≤ 3φ(s, a)T�Σ†  k,hφ(s, a) ≤ 3γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  where (a) used Jensen inequality, (b) used Line 6 of Algorithm 2, and the last step uses the fact that ∥�Σ†  k,h∥2 ≤ γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Hence, it only remains to ensure that 12H2η2γ−1 ≤ 1, which is guaranteed by the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Meanwhile, we claim that ηBk(s, a) ≤ 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' By definition of Bk(s, a) and the fact that ∥�Σ†  k,h∥2 ≤ γ−1, we have  ηBk(s, a) ≤ ηH  �  1 + 1  H  �H  × 2β sup  s,a,h  ∥φ(s, a)∥2  �Σ†  k,h ≤ 6ηβHγ−1,  (14)  which is bounded by  1  2H according to the condition that 12ηβH2 ≤ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Therefore, fixing h ∈ [H] and s ∈ Sh, we can apply the Hedge lemma (Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1):  E  � K  �  k=1  �  a  (πk(a | s) − π∗(a | s))( �Qk(s, a) − Bk(s, a))  �  ≤ ln A  η  + 2η E  � K  �  k=1  �  a  πk(a | s) �Qk(s, a)2  �  + 2η E  � K  �  k=1  �  a  πk(a | s)Bk(s, a)2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  For the second term, we can write  �Qk(s, a)2 = (φ(s, a)T �Σ†  k,hφ(sk,h, ak,h)Lk,h − H(φ(s, a)T�Σ†  k,hφ(sk,h, ak,h))− + Hmk(s, a))2  ≤ 2H2(φ(s, a)T�Σ†  k,hφ(sk,h, ak,h))2 + 2H2(φ(s, a)T�Σ†  k,hφ(sk,h, ak,h))2  − + 2H2m2  k(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  After taking expectations on both sides, we have  E[ �Q2  k(s, a)] ≤ 2H2 E[(φ(s, a)T�Σ†  k,hφ(sk,h, ak,h))2] + 2H2 E[(φ(s, a)T �Σ†  k,hφ(sk,h, ak,h))2  −]+  2H2 E  �  1  M  M  �  m=1  (φ(s, a)T�Σ†  k,hφ(sm,h, am,h))−  �2  ≤ 6H2 E[(φ(s, a)T�Σ†  k,hφ(sk,h, ak,h))2],  where we used (X)2  − ≤ X2, Jensen’s inequality, and the fact that (sk,h, ak,h) and (sm,h, am,h) are both sampled from  πk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Meanwhile, we can also calculate that  E[(φ(s, a)T�Σ†  k,hφ(sk,h, ak,h))2]  = E  �  φ(s, a)T�Σ†  k,hφ(sk,h, ak,h)φ(sk,h, ak,h)T�Σ†  k,hφ(s, a)  �  = E  �  φ(s, a)T�Σ†  k,hΣπk  k �Σ†  k,hφ(s, a)  �  ≤ E  �  ∥φ(s, a)∥2  �Σ†  k,h  �  ,  where the last inequality again uses Line 6 of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Hence, after summing up over the expectations when  sh ∼ π∗ for all h ∈ [H], we can conclude that  E[Reg-Term] ≤ Hη−1 ln A + 6ηH2  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)  �  ∥φ(sh, ah)∥2  �Σ†  k,h  ��  27  Refined Regret for Adversarial MDPs with Linear Function Approximation  + 1  H  K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh) [Bk(sh, ah)]  �  ,  where the last term comes from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (14) and the condition that 12ηβH2 ≤ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  28  Refined Regret for Adversarial MDPs with Linear Function Approximation  Algorithm 6 Improved Linear MDP Algorithm  Input: Learning rate η, PolicyCover parameters α and T0, bonus parameter β, covariance estimation parameter  γ ∈ (0, 1  4), epoch length W, FTRL regularizer Ψ(p) = �A  i=1 ln 1  pi , exploration probability δe  1: Let πcov and �Σcov  h  be the outputs of the PolicyCover algorithm (see Algorithm 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  2: Let K = {s ∈ S | ∥φ(s, a)∥2  (�Σcov  h  )−1 ≤ α, ∀a ∈ A} be all the “known” states (defined in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  3: for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', J = (T − T0)/W do  4:  Calculate πj ∈ Π as follows for all s ∈ Sh:  πj(s) = argmin  p∈△(A)  �  Ψ(p) + η  �  τ<j  �  a∈A  p(a)  � �Qτ(s, a) − �Bτ(s, a)  ��  ,  where �Qτ(s, a) = φ(s, a)T�θτ,h, �Bτ(s, a) = bτ(s, a) + φ(s, a)T�Λτ,h, and the bonus function is defined as  bτ(s, a) =  1[s ∈ K] × β  �  ∥φ(s, a)∥2  �Σ†  τ,h +  E  a′∼πτ (·|s)  �  ∥φ(s, a′)∥2  �Σ†  τ,h  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  5:  Randomly partition the episodes in the current epoch, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', {K0 + (j − 1)W + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , K0 + jW}, into two halves  Tj and T ′  j , such that where |Tj| = |T ′  j | = W  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  6:  for k = K0 + (j − 1)W + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , K0 + jW do  7:  Let Yk be a sample from a Bernoulli distribution Ber(δe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  8:  if Yk = 0 then  9:  Execute πj for this episode and observe {(sk,h, ak,h)}H  h=1 (together with ℓk(sk,h, ak,h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  10:  else if Yt = 1 and k ∈ Tj then  11:  Execute πcov and observe {(sk,h, ak,h)}H  h=1 together with ℓk(sk,h, ak,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  12:  else  13:  Let hk be uniformly sampled from [H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Execute πcov for steps 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', h − 1 and πk for the remaining  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Again, observe the trajectory {(sk,h, ak,h)}H  h=1 and the losses ℓk(sk,h, ak,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  14:  end if  15:  end for  16:  Define �πj as the mixture of (1 − δe) times πj and δe times πcov (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', expected policy played in epoch j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  17:  Estimate the covariance matrix Σ�πj  h as follows, and estimate (γI + Σ�πj  h )−1 by �Σ†  j,h = (γI + �Σj,h)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  �Σj,h =  1  |Tj|  �  k∈Tj  φ(sk,h, ak,h)φ(sk,h, ak,h)T,  18:  Estimate the average Q-function kernel θ  πj  j,h ≜  1  |T ′  j |  �  k∈T ′  j θπj  k,h as follows:  �θj,h = �Σ†  j,h  \uf8eb  \uf8ed 1  |T ′  j |  �  k∈T ′  j  ((1 − Yk) + YkH  1[h = hk]) φ(sk,h, ak,h)Lk,h  \uf8f6  \uf8f8 ,  where Lk,h = �H  h′=h ℓk(sk,h′, ak,h′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  19:  Estimate the average dilated bonus kernel Λ  πj  j,h ≜  1  |T ′  j |  �  k∈T ′  j Λπj  k,h as follows:  �Λj,h = �Σ†  j,h  \uf8eb  \uf8ed 1  |T ′  j |  �  k∈T ′  j  ((1 − Yk) + YkH  1[h = hk]) φ(sk,h, ak,h)Dk,h  \uf8f6  \uf8f8 ,  where Dk,h = �H  h′=h+1(1 + 1  H )i−hbj(sk,h, ak,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  20: end for  29  Refined Regret for Adversarial MDPs with Linear Function Approximation  D  Omitted Proofs in Section 5 (Linear MDP Algorithm)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1  Pseudocode of the Improved Linear MDP Algorithm  This section briefly discusses the linear MDP algorithm (presented in Algorithm 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Apart from the new covariance  estimation technique introduced in the main text, it is also different from Algorithm 1 in some other aspects due to  the distinct nature of linear-Q MDPs and (simulator-free) linear MDPs, listed as follows:  Firstly, as there are no simulators, we cannot calculate the dilated bonus function recursively like Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Fortunately, in linear MDPs, as observed by Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021a), for any k associated with some policy πk and bonus  function bk, we can write Bk(s, a) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (3) as Bk(s, a) = bk(s, a) + φ(s, a)TΛπk  k,h, where (ν is defined in  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4)  Λπ  k,h ≜  �  1 + 1  H  � �  sh+1  E  ah+1∼π(·|sh+1) [Bk(sh+1, ah+1)] ν(sh+1)dsh+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Hence, Bk(s, a) is also linear in φ(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This allows us to estimate the Bk just like Qπk  k , as we see in Line 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Notice that, as there are no more simulators, we cannot directly “assume” a good covariance estimation like in  Algorithm 1 (which ensures Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (7) and (8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Instead, we should divide the time horizon into several epochs and  execute (nearly) the same policy during each epoch to ensure a good estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' See Algorithm 6 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2  Alternative to the Matrix Geometric Resampling Procedure  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' By definition, we know the following holds for all k ∈ Tj:  E[φ(sk,h, ak,h)φ(sk,h, ak,h)T] = Σ�πj  h ,  φ(sk,h, ak,h)φ(sk,h, ak,h)T ⪯ I  Moreover, each (sk,h, ak,h) is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Thus, we can apply Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4 with  Hi = 1  2  �  γI + φ(ski,h, aki,h)φ(ski,h, aki,h)T�  , H = 1  2  �  γI + Σ�πj  h  �  ,  i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' , |Tj|,  where ki stands for the i-th element in Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We then have the following according to Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4:  −  �  d  |Tj| log d  δ H1/2 ⪯  1  |Tj|  |Tj|  �  i=1  Hi − H ⪯  �  d  |Tj| log d  δ H1/2,  if γ ≥ 2 d  |Tj| log d  δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (15)  Let the empirical average of all Hi’s be �H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=',  �H =  1  |Tj|  |Tj|  �  i=1  Hi = 1  2(γI + �Σj,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Then we can arrive at the following under the same condition as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (15):  I −  �  d  |Tj| log d  δ H−1/2 ⪯ �HH−1 ⪯ I +  �  d  |Tj| log d  δ H−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Moreover, by the definition of H, we know H−1/2 ⪯  √  2(γI)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Setting W = 4d log d  δ γ−2 (which ensures  γ ≥ 2  d  |Tj| log d  δ as |Tj| = W  2 ), the LHS and RHS become (1 − √γ)I and (1 + √γ)I, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Hence,  (1 − √γ) 1  2(γI + Σ�πj  h ) ⪯ 1  2(γI + �Σj,h) ⪯ (1 + √γ) 1  2(γI + Σ�πj  h ),  which gives our conclusion after multiplying 2 on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  30  Refined Regret for Adversarial MDPs with Linear Function Approximation  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3  Proof of Main Theorem  We first state the formal version of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3:  Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Suppose that α =  δe  6β, M0 ≥ α2dH2, N0 ≥ 100 M3  0  α2 log K  δ , W = 4d log d  δ γ−2, γ ≥ 36 β2  δ2e , and  100ηH4 ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Further pick δ = K−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Then Algorithm 6 applied to linear MDPs (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4) ensures  RK = �  O  � δ6  e  β6 d4H8 + δeK + βdHK + γ  β dH3K + H  η Adγ−2 + Ad3/2H2γ−2 + H3  γ2 K  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  With some proper tuning, we can ensure RK = �  O((H20Ad6)1/9K8/9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The regret decomposition is the same as Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 by Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021a), which we include  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' As sketched in the main text, we decompose the episodes into three parts: those executing PolicyCover,  those using exploratory policies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', the Ber(δe) gives 1), and the ones executing πj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  For the first part, it’s trivially bounded by O(K0H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For the second part, as we explore with probability δe for  each episode, the total regret gets bounded by O(δeKH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For the last part, it suffices to bound the following to apply  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5 (where we still consider those exploratory episodes as they only bring extra regret), where J = K−K0  W  denotes the number of epochs for simplicity and Q  π  j (s, a) = φ(s, a)Tθ  π  j,h denotes the average Q-function in T ′  j (h is  such that s ∈ Sh):  J  �  j=1  H  �  h=1  E  sh∼π∗  �  W  �  ah∈A  (πj(ah | sh) − π∗(ah | sh))(Q  πj  j (sh, ah) − Bj(sh, ah))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  As Bk(s, a) is only bounded for the known states (by definition of K), we first consider the unknown states:  J  �  j=1  H  �  h=1  E  sh∼π∗  �  W  1[x ̸∈ K]  �  ah∈A  (πj(ah | sh) − π∗(ah | sh))(Q  πj  j (s, a) − Bj(sh, ah))  �  ≤ J  E  �  h=1  E  sh∼π∗[W  1[sh ̸∈ K]3H] = �O  �  HK dH3  α  �  according to Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' For the remaining, we decompose (πj(ah | sh) − π∗(ah | sh))(Q  πj  j (sh, ah) − Bj(sh, ah))  into the biases of �Qj(sh, ah) and �Bj(s, ah) plus the FTRL regret (πj(ah | sh) − π∗(ah | sh))( �Qj(sh, ah) − �Bj(sh, ah)),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  J  �  j=1  H  �  h=1  E  sh∼π∗  �  W  1[sh ∈ K]  �  ah∈A  (πj(ah | sh) − π∗(ah | sh))(Q  πj  j (sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah) − Bj(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah))  �  = W  J  �  j=1  H  �  h=1  E  sh∼π∗  �  1[sh ∈ K]  E  ah∼πj(·|sh)  �  Q  πj  j (sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah) − �Qj(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah))  ��  �  ��  �  Bias-1  +  W  J  �  j=1  H  �  h=1  E  sh∼π∗  �  1[sh ∈ K]  E  ah∼π∗(·|sh)  �  �Qj(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)) − Q  πj  j (sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)  ��  �  ��  �  Bias-2  +  W  J  �  j=1  H  �  h=1  E  sh∼π∗  �  1[sh ∈ K]  E  ah∼πj(·|sh)  �  �Bj(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah) − Bj(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)  ��  �  ��  �  Bias-3  +  W  J  �  j=1  H  �  h=1  E  sh∼π∗  �  1[sh ∈ K]  E  ah∼π∗(·|sh)  �  Bj(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah) − �Bj(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah)  ��  �  ��  �  Bias-4  +  31  Refined Regret for Adversarial MDPs with Linear Function Approximation  W  J  �  j=1  H  �  h=1  E  sh∼π∗  �  1[sh ∈ K]  �  ah∈A  (πj(ah | sh) − π∗(ah | sh))( �Qj(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah) − �Bj(sh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ah))  �  �  ��  �  Reg-Term  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  All the bias terms can be bounded similarly, as we will show in Theorems D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3, we can bound them as  E[Bias-1] + E[Bias-2] + E[Bias-3] + E[Bias-4]  ≤ β  2 E  \uf8ee  \uf8f0  J  �  j=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πj(·|sh)[∥φ(sh, ah)∥2  �Σ†  j,h]  �\uf8f9  \uf8fb +  β  2 E  \uf8ee  \uf8f0  J  �  j=1  H  �  h=1  E  sh∼π∗  �  E  ah∼π∗(·|sh)[∥φ(sh, ah)∥2  �Σ†  j,h]  �\uf8f9  \uf8fb + O  �γ  β dH3J  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Different from Theorems B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2, in that proof, we need to handle the estimation error β−1∥(�Σj,h−Σπj  h )θ  πj  j,h(s, a)∥2  �Σ†  j,h  by the multiplicative bound Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2 instead of the additive one (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (16) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  For the Reg-Term, we again apply the new FTRL lemma Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1, giving the following expression:  E[Reg-Term] ≤ �  O  �H  η A + A  √  dH2 + H3  γ2 δJ  �  + 2ηH3  H  �  h=1  E  sh∼π∗  \uf8ee  \uf8f0  J  �  j=1  E  ah∼πj(·|sh)  �  ∥φ(sh, ah)∥2  �Σ†  j,h  �\uf8f9  \uf8fb  +  H  �  h=1  E  sh∼π∗  \uf8ee  \uf8f0  J  �  j=1  E  ah∼πj(·|sh) [bj(s, a)]  \uf8f9  \uf8fb ,  whose formal proof is in Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' In that proof, we need to bound the magnitudes of the bonuses to write  �Bj(sh, ah)2 ≲ 1  H �Bj(sh, ah).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This is done by applying Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5 later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  By summing up all terms and multiplying W, we can apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5 by again using 100ηH4 ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Hence, we  get the following by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (12):  RK ≤ O(K0) + O(δeK) + �O  �  βdHK + γ  β dH3K + H  η AW + A  √  dH2W + H3  γ2 δK  �  ,  where we conditioned on some good event with probability 1 − O(K−2 + Kδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Picking δ = K−3, the total regret  when the good event does not happen is of order O(K−2HK) = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Moreover, by the conditions α =  δe  6β, M0 ≥ α2dH2, and N0 ≥ 100 M3  0  α2 log K  δ , we know that K0 = M0N0 =  �O( δ6  e  β6 d4H8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Meanwhile, by W = 4d log d  δ γ−2, we know that W = �  O(dγ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Thus, we have  RK = �O  � δ6  e  β6 d4H8 + δeK + βdHK + γ  β dH3K + H  η Adγ−2 + Ad3/2H2γ−2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  The only conditions are then γ ≥ 36 β2  δe , 100ηH4 ≤ β, which allows us to set  δe = C(H20Ad6)1/9K−1/9, β = C2  36 (H13A2d3K−2)1/9K−2/9, γ = C3  36 (H2A1)1/3K−1/3, η = C2  3600(H−23A2d3K−2)1/9K−2/9,  where C is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' This ensures RK = �O((H20Ad6)1/9K8/9) (note that only the 2nd, 4th, and 5th term have a  K8/9 dependency, which means all other terms can be ignored when stating the bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  32  Refined Regret for Adversarial MDPs with Linear Function Approximation  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4  Bounding the Bias Terms  Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' When W = 4d log d  δ γ−2 and γ < 1  4, the Bias-1 and Bias-2 terms in Algorithm 6 is bounded by  E[Bias-1] + E[Bias-2] ≤ β  4 E  \uf8ee  \uf8f0  J  �  j=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πj(·|sh)[∥φ(sh, ah)∥2  �Σ†  j,h]  �\uf8f9  \uf8fb +  β  4 E  \uf8ee  \uf8f0  J  �  j=1  H  �  h=1  E  sh∼π∗  �  E  ah∼π∗(·|sh)[∥φ(sh, ah)∥2  �Σ†  j,h]  �\uf8f9  \uf8fb + O  �γ  β dH3K  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' By direct calculation like Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1, we get the following for any j ∈ [J], h ∈ [H], s ∈ Sh and a ∈ A (recall  that �Σ†  j,h only depends on the episodes in Tj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' all expectations are only taken to the randomness in epoch j)  E  �  Q  πj  j (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a) − �Qj(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)  �  = φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)Tθ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)T E  \uf8ee  \uf8f0�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  \uf8eb  \uf8ed 1  |T ′  j |  �  k∈T ′  j  ((1 − Yk) + YkH  1[h = hk])φ(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)Lk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  \uf8f6  \uf8f8  \uf8f9  \uf8fb  = φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)Tθ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)T E  �  �Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  �  E  \uf8ee  \uf8f0  \uf8eb  \uf8ed 1  |T ′  j |  �  k∈T ′  j  ((1 − Yk) + YkH  1[h = hk])φ(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)φ(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)Tθπj  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  \uf8f6  \uf8f8  \uf8f9  \uf8fb  = φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)Tθ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)T E  �  �Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  �  Σ�πj  h θ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  = φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' a)T(I − E[�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h]Σ�πj  h )θ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  By using Cauchy-Schwartz inequality, triangle inequality, and the AM-GM inequality, we get the following:  φ(s, a)T(I − �Σ†  j,hΣ�πj  h )θ  πj  j,h = φ(s, a)T�Σ†  j,h(γI + �Σj,h − Σ�πj  h )θ  πj  j,h  ≤ ∥φ(s, a)∥�Σ†  j,h  ����(γI)θ  πj  j,h  ����Σ†  j,h  +  ���(�Σj,h − Σ�πj  h )θ  πj  j,h  ����Σ†  j,h  �  ≤ β  4 ∥φ(s, a)∥2  �Σ†  j,h + 2  β  ���γθ  πj  j,h(s, a)  ���  2  �Σ†  j,h  + 2  β  ���(�Σj,h − Σ�πj  h )θ  πj  j,h  ���  2  �Σ†  j,h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  The first term is the usual bonus term in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' while the second term easily translates to the following  using the assumption that ∥θπj  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h∥ ≤  √  dH for all k ∈ T ′  j :  2  β  ���γθ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  ���  2  �ӆ  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  ≤ 2γ2  β  ���(γI + �Σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)−1���  2 dH2 ≤ 2 γ  β dH2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  while the last term translates to the following by algebraic manipulations:  2  β  ���(�Σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − Σ�πj  h )θ  πj  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  ���  2  �ӆ  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h  = 2  β  ���(�Σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − Σ�πj  h )�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h(�Σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − Σ�πj  h )  ���  2 dH2  = 2  β  ����(�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)−1/2 �  I − (�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)1/2(γI + Σ�πj  h )(�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)1/2�2  (�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)−1/2  ����  2  dH2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  (16)  where we wrote (�Σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − Σ�πj  h )�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h(�Σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − Σ�πj  h ) = (�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)−1/2(�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)1/2(�Σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − Σ�πj  h )�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h(�Σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h − Σ�πj  h )(�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)1/2(�Σ†  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='h)−1/2  and made a square in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2, the squared-matrix in the middle has its operator norm bounded by 4γ with high probability  (if the good event does not hold, then one can directly bound the last term by matrix Azuma and the operator norm  of �Σ†  k,h, giving γ−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' as this only happens with probability K−3, this part contributes o(K−2) to the total regret and  we thus omit it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Meanwhile, the first and last term both has their operator norms bounded by  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Thus,  2  β  ���(�Σj,h − Σπj  h )θ  πj  j,h  ���  2  �Σ†  j,h  dH2 ≤ 16 γ  β dH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  33  Refined Regret for Adversarial MDPs with Linear Function Approximation  In other words, the last term gets absorbed by the second term up to constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Hence, the conclusion follows  by the same argument as Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' When W = 4d log d  δ γ−2 and γ < 1  4, the Bias-3 and Bias-4 terms in Algorithm 6 is bounded by the  following when the good event in Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5 holds:  E[Bias-3] + E[Bias-4] ≤ β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼πk(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  +  β  4 E  � K  �  k=1  H  �  h=1  E  sh∼π∗  �  E  ah∼π∗(·|sh)[∥φ(sh, ah)∥2  �Σ†  k,h]  ��  + O  �γ  β dH3K  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The proof is identical to Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2 except for Lk,h is replaced by Dk,h and θ  πj  j,h is replaced by Λπj  j,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' As we  also have bj(s, a) ∈ [0, 1] by Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5, the same bound holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5  Bounding Reg-Term  Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Assuming 100HηH4 ≤ β and the good events in Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Then the Reg-Term has its  expectation bounded by  E[Reg-Term] ≤ �  O  �H  η A + A  √  dH2 + H3  γ2 δJ  �  + 2ηH3  H  �  h=1  E  sh∼π∗  \uf8ee  \uf8f0  J  �  j=1  E  ah∼πj(·|sh)  �  ∥φ(sh, ah)∥2  �Σ†  j,h  �\uf8f9  \uf8fb  +  H  �  h=1  E  sh∼π∗  \uf8ee  \uf8f0  J  �  j=1  E  ah∼πj(·|sh) [bj(s, a)]  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' The proof generally follows from Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2, except for some tiny differences due to epoching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1, we get the following for all j ∈ [J], s ∈ Sh ∩ K (where h ∈ [H]), and �π∗ ∈ Π:  J  �  j=1  �  a∈A  (πj(a | s) − π∗(a | s))( �Qj(s, a) − �Bj(s, a))  ≤ Ψ(�π∗(· | s)) − Ψ(π1(· | s))  η  +  J  �  j=1  �  a∈A  (�π∗(a | s) − π∗(a | s))( �Qj(s, a) − �Bj(s, a))+  η  J  �  j=1  �  a∈A  πj(a | s)( �Qj(s, a) − �Bj(s, a))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  By picking �π∗(a | s) = (1 − AJ−1)π∗(a | s) + J−1 as Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2, the first term is bounded by η−1A ln J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Meanwhile, the second term is bounded by  J  �  j=1  �  a∈A  (�π∗(a | s) − π∗(a | s))( �Qj(s, a) − �Bj(s, a))  =  J  �  j=1  �  a∈A  (−AJ−1π∗(a | s) + J−1)( �Qj(s, a) − �Bj(s, a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Like what we did in Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2, we consider the expected difference between �Qj and Q  πj  j :  E[ �Qj(s, a) − Q  πj  j (s, a)] = φ(s, a)T(I − E[�Σ†  j,hΣ�πj  h ])θ  πj  j,h ≤ 2  √  dH,  34  Refined Regret for Adversarial MDPs with Linear Function Approximation  where the first inequality is due to Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2 and the second one uses Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Moreover, according to  Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5, the same bound also holds for E[ �Bj(s, a) − B  πj  j (s, a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Hence, after taking expectations on both sides, we know that  E  \uf8ee  \uf8f0  J  �  j=1  �  a∈A  (�π∗(a | s) − π∗(a | s))( �Qj(s, a) − �Bj(s, a))  \uf8f9  \uf8fb  ≤  J  �  j=1  �  a∈A  (|−AJ−1π∗(a | s)| + |J−1|)|E[ �Qj(s, a) − �Bj(s, a)]|  ≤ J × 2AJ−1 × 4  √  dH = O  �  A  √  dH  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Then consider the last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We still write ( �Qj(s, a)− �Bj(s, a))2 ≤ 2 �Qj(s, a)2+2 �Bj(s, a)2, which can be calculated  as follows:  E[ �Qj(s, a)2] ≤ H2 E  \uf8ee  \uf8f0 1  |T ′  j |  �  k∈T ′  j  φ(s, a)T�Σ†  j,h  �  ((1 − Yk) + YkH  1[h = hk])2φ(sk,h, ak,h)φ(sk,h, ak,h)T� �Σ†  j,hφ(s, a)  \uf8f9  \uf8fb  = H2 E  �  φ(s, a)T�Σ†  j,h  �  (1 − δe)Σπj  h + HδeΣcov  h  � �Σ†  j,hφ(s, a)  �  ≤ H3 E  �  φ(s, a)T�Σ†  j,hΣ�πj  h �Σ†  j,hφ(s, a)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Then we use Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  If the good event does not happen, then this term is bounded by O(H3γ−2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Otherwise, it can be written as 2H3 E  �  ∥φ(s, a)∥2  �Σ†  j,h  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Then we consider the estimated dilated bonus term:  E[ �Bj(s, a)2] ≤ 2 E[bj(s, a)2] + 2 E[(φ(s, a)T�Λj,h)2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  According to Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5 (whose failure only contributes in total o(1) regret as we pick δ = K−3), we know  bj(s, a) ≤ 1, which means Dk,h ≤ 3H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Thus, the second term is also bounded by O(H3 E[∥φ(s, a)∥2  �Σ†  j,h] + H3γ−2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Moreover, as bj(s, a) ≤ 1, the first term is bounded by E[bj(s, a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Thus, by definition of bj(s, a), we get  E[ �Bj(s, a)2] ≤ O  �H3  β  �  bj(s, a) + O(H3γ−2δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Putting everything together gives  E[Reg-Term] ≤ �  O  �H  η A + A  √  dH2 + H3  γ2 δJ  �  + 2ηH3  H  �  h=1  E  sh∼π∗  \uf8ee  \uf8f0  J  �  j=1  E  ah∼πj(·|sh)  �  ∥φ(sh, ah)∥2  �Σ†  j,h  �\uf8f9  \uf8fb  + 100ηH3  β  H  �  h=1  E  sh∼π∗  \uf8ee  \uf8f0  J  �  j=1  E  ah∼πj(·|sh) [bj(s, a)]  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  This then translates to our conclusion using the condition that 100H4η ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='6  Bounding the Magnitudes of Bonuses  Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Let α = δe  6β, M0 ≥ α2dH2, N0 ≥ 100 M3  0  α2 log K  δ , and γ ≥ 36 β2  δe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Then with probability 1 − (K−2 + Kδ),  bj(s, a) =  1[(s, a) ∈ K] × β(∥φ(s, a)∥2  �Σ†  j,h + Ea′∼πj(·|s)[∥φ(s, a′)∥�Σ†  j,h]) ≤ 1 for all j ∈ [J], h ∈ [H], s ∈ Sh and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  35  Refined Regret for Adversarial MDPs with Linear Function Approximation  The proof is similar to, but different from Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 of Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Here, we use the alternative for the  MGR procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' We also refine the analysis on M −1  0  + Σcov  h  ≈ �Σcov  h  for a smaller N0, which is critical for our new  regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' It suffices to show that β∥φ(s, a′)∥2  �Σ†  k,h ≤ 1  2 for any s ∈ K and a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Firstly, we have the following with  high probability because �Σ†  j,h is a multiplicative approximation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='2):  β∥φ(s, a′)∥�Σ†  j,h ≤ 2β∥φ(s, a′)∥2  (γI+Σ  �πj  h )−1 ≤ 2 β  δe  ∥φ(s, a′)∥2  ( γ  δe I+Σcov  h  )−1 ≤ 2 β  δe  ∥φ(s, a′)∥2  (M−1  0  I+Σcov  h  )−1,  where Σcov  h  is the short-hand notation of Σπcov  h  and the last step uses the fact that  γ  δe M0 ≥  γ  δe  δ2  e  36β2 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Moreover,  we argue that M −1  0  + Σcov  h  ≈ �Σcov  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' By definition of �Σcov  h  (see Algorithm 5):  �Σcov  h  =  1  M0  I +  1  M0N0  M0  �  m=1  N0  �  n=1  φ(sm,n,h, am,n,h)φ(sm,n,h, am,n,h)T,  we can apply the Matrix Azuma inequality (Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='3) to ensure that (where we simply pick XmN0+n =  φ(sm,n,h, am,n,h)φ(sm,n,a, am,n,h)T − Σπm  h  and AmN0+n = I ⪰ XmN0+n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' there are in total M0N0 matrices):  �����Σcov  h  −  1  M0  I − Σcov  h  ����  2  ≤  �  8  M0N0  log d  δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  The rest follows the proof of the original lemma (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=', 2021a, Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  By following the proof of  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='1 of Meng and Zheng (2010), we can conclude that  �����  �  �Σcov  h  �−1  −  � 1  M0  I − Σcov  h  �−1�����  2  ≤ M 2  0  �����Σcov  h  −  1  M0  I − Σcov  h  ����  2  ≤ M 2  0  �  8  M0N0  log d  δ =  �  8M 3  0  N0  log d  δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  Therefore, we only need M0 = O(N 3  0 ) to make it bounded by α  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content=' Consequently, as ∥φ(s, a′)∥2 ≤ 1,  β∥φ(s, a′)∥�Σ†  j,h ≤ 2 β  δe  ∥φ(s, a′)∥(M−1  0  I+Σcov  h  )−1  ≤ 2 β  δe  �  ∥φ(s, a′)∥2  (�Σcov  h  )−1 +  �����  �  �Σcov  h  �−1  −  � 1  M0  I − Σcov  h  �−1�����  2  �  ≤ 2 β  δe  �  α + α  2  �  ≤ 1  2,  by the condition that s ∈ K and our choice of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
+page_content='  36' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFOT4oBgHgl3EQf2jRR/content/2301.12942v1.pdf'}
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+Evidencing the squeezed dark nuclear spin state in lead halide perovskites
+E. Kirstein,1 D. S. Smirnov,2, ∗ E. A. Zhukov,1, 2 D. R. Yakovlev,1, 2 N. E.
+Kopteva,1 D. N. Dirin,3 O. Hordiichuk,3 M. V. Kovalenko,3, 4 and M. Bayer1, 2
+1Experimentelle Physik 2, Technische Universit¨at Dortmund, 44221 Dortmund, Germany
+2Ioffe Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia
+3Department of Chemistry and Applied Biosciences,
+Laboratory of Inorganic Chemistry, ETH Z¨urich, 8093 Z¨urich, Switzerland
+4Department of Advanced Materials and Surfaces, Laboratory for Thin Films and Photovoltaics,
+Empa - Swiss Federal Laboratories for Materials Science and Technology, 8600 D¨ubendorf, Switzerland
+Coherent many-body states are highly promising for robust and scalable quantum information
+processing. While far-reaching theoretical predictions have been made for various implementations,
+direct experimental evidence of their appealing properties can be challenging. Here, we demon-
+strate coherent optical manipulation of the nuclear spin ensemble in the lead halide perovskite
+semiconductor FAPbBr3 (FA=formamidinium), targeting a long-postulated collective dark state
+that is insensitive to optical pumping. Via optical orientation of localized hole spins we drive the
+nuclear many-body system into an entangled state, requiring a weak magnetic field of only a few
+Millitesla strength at cryogenic temperatures. During its fast build-up, the nuclear polarization
+along the optical axis remains small, while the transverse nuclear spin fluctuations are strongly re-
+duced, corresponding to spin squeezing as evidenced by a strong violation of the generalized nuclear
+squeezing-inequality with ξs < 0.3. The dark state evidenced in this process corresponds to an ap-
+proximately 750-body entanglement between the nuclei. Dark nuclear spin states can be exploited
+to store quantum information benefiting from their long-lived many-body coherence and to perform
+quantum measurements with a precision beyond the standard limit.
+INTRODUCTION
+The range of many-body states in condensed matter is
+formidably rich: the fractional quantum Hall [1, 2] and
+the Wigner crystal [3–5] states are just two examples [6].
+Quantum coherence is essential for the formation of many
+of them.
+The many-body state of localized spins in solid state
+is arguably one of the best controlled and potentially
+scalable hardware to that end. For example, an electron
+spin can be used to create entanglement between multiple
+photons for one-way quantum computing [7–9]. However,
+electron and photon spins suffer from fast decoherence
+or short lifetime, limiting their applicability for quantum
+information storage and many-body unitary operations.
+Nuclear spins, on the other hand, are largely isolated
+from their environment resulting in extended coherence
+times.
+Thus, they complement electron spins [10–12],
+particularly due to possible electron-nuclear spin inter-
+facing [13–17].
+Spin-based concepts for solid state quantum comput-
+ing and quantum metrology rely on many-body entan-
+glement in combination with nuclear spin squeezing. The
+latter of which is the ultimate goal of nuclear spin manip-
+ulation [18, 19]. Almost 20 years ago, a coherent many-
+body dark nuclear spin state (DNSS) was predicted that
+can be formed by orienting optically the spins of resident
+charge carriers interacting with the nuclear spin bath in
+the host lattice [10, 20]. Mathematically, the DNSS is
+the state which vanishes upon action of the rising oper-
+ator of the total nuclear spin. That is implemented by
+the optical pumping in the experiment: DNSS formation
+blocks the optical nuclear spin pumping in analogy with
+optically dark states in ensembles of atoms, which are
+immune to the light-matter interaction [21]. The DNSS
+belongs to the class of maximally entangled singlet states,
+so that it can be exploited not only to suppress the de-
+phasing of localized electron and hole spin qubits [22, 23],
+but also for storage of quantum information [10, 11] or
+for quantum metrology applications [24].
+The DNSS is characterized by destructive interference
+of the nuclear spin amplitudes in the collective trans-
+verse components, which leads to their strong suppres-
+sion without large nuclear spin polarization [Fig. 1(a)].
+The destructive interference arises from quantum correla-
+tions and entanglement between the nuclear spins, some-
+what analogous to the spin structure in antiferromagnetic
+materials. Realization and demonstration of the DNSS
+have turned out to be challenging due to threats from
+the dipole-dipole and quadrupole nuclear interactions. In
+prior works on III-V semiconductor quantum dots [25–
+27], where the nuclear spin polarization was measured,
+the limited polarization value was attributed to DNSS
+formation, but without proof of suppression of the trans-
+verse spin fluctuations [20]. Later, detailed investigations
+of GaAs quantum dots revealed 80% nuclear polariza-
+tion without any signatures of DNSS formation, i.e. the
+nuclear spin state in this system could be described by
+an effective nuclear spin temperature and the absence of
+correlations [28]. Correlations between nuclear spins was
+evidenced for non-thermal nuclear spin states [29], nar-
+rowed nuclear spin states [30–32], and nuclear frequency
+focusing [33]. Thus none of these studies evidenced the
+DNSS. In a recent study [34], the authors reconstructed
+arXiv:2301.11460v1  [cond-mat.mtrl-sci]  26 Jan 2023
+
+2
+FIG. 1. Measurement of the nuclear spin polarization in FAPbBr3. (a) Formation of the DNSS: after optical orientation
+(green light pulse) of a localized hole (magenta arrow), individual nuclear spins Ik (small coral arrows) rotate through the
+hyperfine interaction, each by the same angle, so that the total nuclear spin polarization I (large red arrow) is directed along
+the optical axis.
+(b) Time-resolved Kerr ellipticity dynamics measured for σ+ and σ− pump pulse polarization measured
+at 2.191 eV photon energy, in which oscillations associated with electron and hole spin precession about the magnetic field
+are superimposed. Arrows mark the second oscillation minimum in the hole signal after the pump pulse at time zero. (c)
+Illustration of nuclear spin polarization in tilted magnetic field for σ+ (left) and σ− (right) pump polarization. (d) Time-
+resolved, measurement of the Overhauser field rise and decay, showing changes on time scales below the 2 ms low pass of the
+lock-in amplifier. Experimental parameters: T = 1.6 K, P = 2 mW, B = 0.1 T, θ = 60◦, ∆t = 3.6 ns. (e) Time-resolved Kerr
+ellipticity dynamics measured at different pump powers for fixed pump polarization. Dashed line indicates the temporal shift
+of the hole oscillation for increased pump power. (f,g) Overhauser field for holes and electrons, respectively, extracted from the
+Kerr ellipticity dynamics, using the hole and electron g-factors of gh = +0.4 and ge = +2.4.
+the nuclear spin statistics and suggested destructive in-
+terference between different nuclear spins.
+However, a
+measurement of the transverse spin components, manda-
+tory for DNSS demonstration, is missing.
+Here we study the intertwined spin dynamics of nu-
+clei and charge carriers in a FAPbBr3 lead halide
+perovskite crystal, using tailored optical pump-probe
+schemes. More specifically, we develop a method for as-
+sessing the nuclear spin inertia, which allows us to access
+the regime of weak magnetic fields in the milliTesla range.
+We find that excitation with circularly polarized light
+leads to the disappearance of the transverse nuclear spin
+fluctuations with simultaneous absence of a significant
+longitudinal nuclear spin polarization. The combination
+of these two factors directly evidences the destructive in-
+terference of nuclear spins and thus the DNSS formation.
+This many-body correlated state builds up fast on short
+time scales in the ms-range and shows significant spin
+squeezing evaluated through the Kitagawa-Ueda param-
+eter [35] ξs = 0.29, corresponding to entanglement of
+about 750 nuclear spins.
+RESULTS
+The exciton resonance in the hybrid organic-inorganic
+lead halide perovskite FAPbBr3 crystals is located at
+2.191 eV. We have measured the time-resolved Kerr el-
+lipticity (TRKE) at this energy. We give basic informa-
+tion on the charge carrier spin dynamics and the pho-
+toluminescence spectrum in the Supplementary Infor-
+mation I.A. The nuclear spin polarization has been ad-
+dressed by the Overhauser field experienced by the res-
+ident hole spins, which we observe in TRKE, recorded
+at the temperature of T = 1.6 K in tilted magnetic field,
+Figure 1(b). The spin dynamics consists of slow and fast
+oscillating components related to the hole (h) and elec-
+tron (e) spin precession about the total magnetic field.
+The corresponding carrier spin g-factors are gh = +0.4
+and ge = +2.4 (Supplementary Information I.A).
+
+3
+−150
+−100
+−50
+0
+50
+100
+150
+20
+30
+40
+50
+BF (mT)
+Kerr Ellipticity (arb. u.)
+0
+1
+2
+3
+4
+5
+0
+20
+40
+fmod (kHz)
+|KE| (arb. u.)
+0
+1
+2
+3
+4
+5
+0
+20
+40
+fmod (kHz)
+HWHM (mT)
+−50
+0
+50
+20
+30
+40
+50
+BF (mT)
+Kerr Ellipticity (arb. u.)
+(a)
+(b)
+(c)
+(d)
+fmod = 1 kHz
+5 kHz
+0.1 kHz
+2 kHz
+1 kHz
+fmod = 
+x10
+FIG. 2. Nuclear spin inertia measurement. (a) Representative PRCs each consisting of a broad (orange filling) and a
+narrow (green filling) component. (b) PRCs at different polarization modulation frequencies. (c,d) Polarization modulation
+frequency dependencies of widths and amplitudes of the broad (red symbols) and narrow PRC components (green symbols).
+Lines are guides to the eye. T = 5 K.
+The strong spin-orbit interaction in perovskites pro-
+vides a selectivity for light absorption of different circular
+polarizations [37]. As a result, optical spin orientation
+is feasible for resident electrons and holes, localized at
+different sites in the sample at low temperatures, which
+represents a situation similar to that provided by quan-
+tum confinement. The angular momentum of a σ+ or
+σ− photon is transferred to the charge carrier spin po-
+larization, and further to the host lattice nuclei through
+the hyperfine interaction, which leads to build-up of an
+Overhauser field BN parallel or antiparallel to the exter-
+nal magnetic field, see Fig. 1(c). This field can be di-
+rectly determined through measurement of the electron
+and hole Larmor precession frequencies about the total
+magnetic field, to which the external field and the Over-
+hauser field contribute.
+With increasing pump power,
+the Overhauser field rises [Fig. 1(e)], but saturates at
+relatively small values of BN,h = 15 mT for the holes
+[Fig. 1(f)] and BN,e = 0.4 mT for the electrons [Fig. 1(g)].
+Similar to other perovskites [38, 39], the hyperfine inter-
+action is much stronger for the holes due to the dominant
+s-type orbital contribution to the Bloch wave functions
+at the top of the valence band. Thus, we focus on the
+holes in our study, which dominate the Kerr ellipticity
+dynamics. The measured dependence of the Overhauser
+field on the direction and strength of the external field is
+presented in the Supplementary Information [Fig. S2].
+The small measured Overhauser field, as compared to
+the maximum possible strength of 1.3 T evaluated from
+the hyperfine coupling constants [40], indicates an un-
+usual nuclear spin statistics which can be highlighted fur-
+ther by measurement of its rise and decay times. There-
+for, we first close the pump beam to erase the nuclear
+spin polarization, and then open the shutter after which
+we observe almost immediately a nuclear spin polariza-
+tion, see Fig. 1(d). After closing the shutter again, the
+spin polarization decays on the same time scale of about
+100 ms. This nuclear spin response time is much shorter
+than the previously reported longitudinal nuclear spin re-
+laxation times of T1,N = 5 s in FA0.9Cs0.1PbI2.8Br0.2 [40]
+and 960 s in MAPbI3 [41]. In fact, the variation is limited
+by the shutter mechanical opening/closing times in ex-
+periment, so that the nuclear spin dynamics occur even
+faster.
+To access this surprisingly fast nuclear spin response
+time, we develop a technique for measuring the nuclear
+spin inertia. Similarly to the electron spin inertia tech-
+nique [42, 43], it monitors the Kerr ellipticity signal
+as function of the longitudinal magnetic field (Faraday
+geometry) with simultaneous modulation of the pump
+helicity at different frequencies fmod.
+An example for
+fmod = 1 kHz is shown in Fig. 2(a).
+The application
+of a longitudinal magnetic field leads to a spin polariza-
+tion recovery due to the suppression of nuclei-induced
+spin relaxation of the holes [44]. Strikingly, the shape
+of this polarization recovery curve (PRC) changes with
+
+4
+−20
+−10
+0
+10
+20
+BV (mT)
+Kerr Ellipticity (arb. u.)
+−40 −20 0
+20 40
+BV (mT)
+Kerr Ellipticity (arb. u.)
+25 kHz
+50 kHz
+75 kHz
+100 kHz
+0 20 40 60 80 100
+0
+5
+10
+15
+fmod (kHz)
+BNMR (mT)
+8.56 MHz/T
+fmod = 
+RSA
+hole
+electron 
+ 
+ZP
+NMR
+BNMR
+(b)
+(a)
+(c)
+FIG. 3. ODNMR measurement. (a) Resonant spin ampli-
+fication signals at different modulation frequencies. (b) De-
+composition of the RSA signal at fmod = 50 kHz into four
+components by fitting.
+From top to bottom:
+RSA signal
+(blue) together with the sum of the four components (red),
+contributions of the hole (green), the electron (dark red), the
+zero peak [ZP] (light blue), the nuclear magnetic resonances
+[NMR] (orange). (c) Dependence of the resonance field on the
+modulation frequency, in agreement with the 207Pb gyromag-
+netic ratio. T = 5 K.
+the modulation frequency [Fig. 2(b)]: at the high fre-
+quency of fmod = 5 kHz, the PRC resembles a wide dip
+of Lorentzian shape with the half width at half maximum
+(HWHM) of 40 mT. With decreasing frequency down to
+0.1 kHz, this dip gradually disappears and is replaced by
+a much narrower dip, having the HWHM of about 1 mT.
+Generally, a PRC consists of the broad and the narrow
+component, each of which can be phenomenologically de-
+scribed by a Lorentzian. Their widths depend weakly on
+fmod [Fig. 2(c)], and their amplitudes change in opposite
+ways [Fig. 2(d)].
+The spin relaxation time of a hole is of the order of
+1 µs (as measured using the same spin inertia method
+at higher modulation frequencies), so that we attribute
+the PRC changes with modulation frequency to fast hole
+spin-induced nuclear spin dynamics occurring on time
+scales of 1 ms, which is shorter than the typical quantum
+decoherence time of nuclear spins. Further details will be
+given in the Theory section, additional pump-power de-
+pendencies and measurements for fmod = 0 are presented
+in the Supplementary Information [Figs. S3 and S4].
+Further, we identify the specific nuclear isotope dom-
+inating the hyperfine interaction. For this, we use the
+Voigt geometry with the magnetic field normal to the op-
+tical axis, where a nuclear magnetic resonance appears if
+the carrier polarization modulation matches the nuclear
+Zeeman splitting. The carrier polarization modulation
+equals the pump modulation fmod [45]. The Kerr ellip-
+ticity signal [Fig. 3(a)] can be separated into an electron-
+related component and a hole-related component, both
+representing damped oscillations, in addition a nuclei-
+related zero field peak and nuclear magnetic resonance
+(NMR) appears [Fig. 3(b)]. The latter dominate the sig-
+nal, their position is given by BNMR = fmod/γ with
+γ = 8.56 MHz/T [Fig. 3(a,c)]. This value closely cor-
+responds to the gyromagnetic ratio of 207Pb given by
+8.88 MHz [see Supplementary Information, Tab. I]. As
+in previous measurements of optically detected nuclear
+magnetic resonance in perovskites, the small chemical
+shift may occur due to the presence of a spin-polarized
+hole [40].
+When the light polarization modulation frequency is
+high, the nuclear spin distribution remains in equilib-
+rium.
+The polarization recovery effect in Fig. 2(b) at
+fmod = 5 kHz then demonstrates nuclei-dominated hole
+spin relaxation [44]: In the absence of the external mag-
+netic field, the spin precession in the randomly oriented
+Overhauser field BN leads to spin dephasing [Fig. 4(a)],
+while application of a longitudinal field B rotates the
+total magnetic field experienced by the holes, Btot =
+BN + B, towards the optical axis and suppresses the
+spin dephasing. Thus, the HWHM of the PRC in this
+case gives the typical fluctuation of the Overhauser field,
+∆B = 40 mT (Supplementary Information II.A).
+The PRC directly measures the average angle θ be-
+tween the optical z axis and the hole spin precession fre-
+quency vector, which includes the Overhauser and exter-
+nal field contributions [inset in Fig. 4(f)]. The nuclear
+spin inertia measurements reveal the recovery of the hole
+spin polarization at magnetic fields above 1 mT, when
+decreasing the polarization modulation frequency from
+5 kHz down to 0.1 kHz. This appears to be a paradox,
+since the typical nuclear spin polarization rate is much
+smaller than these rates. Moreover, the largest nuclear
+spin polarization of BN,h = 15 mT shown in Fig. 1(e)
+is considerably smaller than the typical fluctuation of
+the Overhauser field ∆B = 40 mT. However, the sup-
+pression of the transverse Overhauser field fluctuations
+in absence of a significant nuclear spin polarization un-
+ambiguously evidences the destructive interference be-
+tween nuclear spins due to their quantum correlations,
+and exactly matches the predictions for the DNSS for-
+mation [20].
+THEORY
+The established concept of dynamic nuclear spin po-
+larization relies on the assumption of an effective nu-
+clear spin temperature and the absence of correlations
+between the nuclei. In the DNSS, by contrast, quantum
+
+5
+correlations and destructive interference in the collective
+transverse nuclear spin components necessitate a purely
+quantum mechanical description. Thus, to model the co-
+herent nuclear spin dynamics, we apply the central spin
+box model described by the following Hamiltonian:
+H = AIS + ℏΩL,hSz.
+(1)
+Here I is the total nuclear spin in the hole localization
+volume, S is the hole spin, A is the hyperfine coupling
+constant of the holes, ΩL is the hole Larmor precession
+frequency in the external magnetic field applied along the
+optical z axis, and ℏ is the reduced Planck constant. The
+spin precession about the nuclear field is included in the
+renormalization of ΩL.
+The total nuclear spin is formed by N individual
+207Pb 1/2 spins Ik: I = �N
+k=1 Ik. The number of nu-
+clear spins can be estimated from the PRC width us-
+ing the hyperfine interaction constant A = 33 µeV [40]:
+N = [A/(ghµB∆B)]2 = 1300. Using the lattice constant
+of a0 = 0.6 nm, this gives the hole localization length
+l = a0(N/0.22)1/3 = 11 nm (with 0.22 being the natu-
+ral lead spin abundance), which is similar to other per-
+ovskites [38, 40]. Since N ≫ 1, Eq. (1) implies hole spin
+precession between two pump pulses with the constant
+frequency AI/ℏ + ΩLez (ez is the unit vector along the
+z axis), which leads to hole spin dephasing in weak fields
+for a randomly oriented total nuclear spin I, close to
+equilibrium.
+Continuous hole spin pumping and its dephasing lead
+to the transfer of angular momentum to the nuclei, as can
+be seen from the conservation of the total angular mo-
+mentum Iz +Sz by the Hamiltonian (1). Simultaneously,
+the conservation of the absolute value of the total nuclear
+spin I prevents the build-up of a nuclear spin polarization
+larger than ∼ 1/
+√
+N. Therefore, the total nuclear spin is
+rotated towards the z axis, while its transverse compo-
+nents decrease, but its absolute value does not increase.
+The collective nuclear spin dynamics driven by the inter-
+action with a single hole spin produces quantum correla-
+tions and leads to entanglement between the nuclei. As a
+result, the DNSS is formed [Fig. 4(b)], in agreement with
+the original theoretical predictions [10, 20].
+The
+quantum
+mechanical
+solution
+of
+the
+box
+model [46–49] allows us to fit the experimentally mea-
+sured amplitude of the broad PRC dip as function of the
+polarization modulation frequency, see Fig. 4(d).
+The
+modification of the nuclear spin distribution with de-
+creasing modulation frequency is shown in the corre-
+sponding insets. The alignment of all nuclear spin fluc-
+tuations along the optical axis at low modulation fre-
+quencies cancels the hole spin precession [Fig. 4(c)] and
+restores the hole spin polarization to the same value
+as in large magnetic fields, where the hole and nuclear
+spins become in effect decoupled (the hole spin dephas-
+ing by the nuclei is suppressed). Thereby also the whole
+PRC amplitude rises, as shown in Fig. 4(f). From fit-
+ting the nuclear spin inertia we find the DNSS forma-
+tion rate (typical frequency of nuclear spin rotation) of
+ν0 = 0.9 ms−1 (Supplementary Information II.C).
+ANALYSIS AND DISCUSSION
+The ratio of the hole spin polarization at a given mag-
+netic field and in saturation (at B ≳ 100 mT) represents
+a collective measurement of the total nuclear spin com-
+ponents ⟨(I2
+x + I2
+y)/I2⟩ [44]. This allows us to quantify
+the suppression of the transverse nuclear spin fluctua-
+tions by the Kitagawa and Ueda spin squeezing parame-
+ter ξ2
+s = 4 ⟨I2
+x⟩ /N [19, 35] (⟨I2
+x⟩ = ⟨I2
+y⟩), which for uncor-
+related spins equals to unity. This parameter extracted
+from the measured PRC amplitude is plotted in Fig. 4(e)
+by the dots, and its simulated frequency dependence with
+the same DNSS formation rate is shown by the solid line
+(Supplementary Information II.D). For almost complete
+suppression of the broad PRC dip to values below 0.05
+at fmod = 100 Hz, the spin squeezing parameter is 0.29,
+which is limited by nuclear spin diffusion due to incom-
+plete hole spin polarization and quantum fluctuations of
+the transverse total nuclear spin components.
+The quantum correlations between the nuclear spins
+suggest their entanglement. This can be shown using the
+generalized spin squeezing inequality [19, 50]
+⟨I2
+x⟩ + ⟨I2
+y⟩ + ⟨(Iz − ⟨Iz⟩)2⟩ ≥ M/2.
+(2)
+Its violation requires M-body entanglement between N
+nuclear spins, meaning that there are at least M spins,
+which are entangled with the rest of the ensemble [18, 51].
+To show its violation, we use the upper boundary for the
+longitudinal nuclear spin fluctuations, ⟨(Iz − ⟨Iz⟩)2⟩ ≤
+N/4, which yields the lower limit of ξ(0)
+s
+= 0.71 for the
+entangled states. States with ξs < ξ(0)
+s
+violate Eq. (2)
+with M = N and are entangled. The maximum achieved
+DNSS with ξs = 0.29 is at least M = 750-body entan-
+gled. However, our theoretical simulations of the nuclear
+spin fluctuations in the DNSS suggest much deeper en-
+tanglement.
+There are good reasons for FAPbBr3 perovskites to be
+the material system for experimental observation of the
+DNSS: (i) Lead has either nuclear spin 0(206Pb, 208Pb
+with abundance of 77.9%) or 1/2 (207Pb with abundance
+of 22.1%), which excludes quadrupole splitting of the
+nuclear spin levels due to strain or electric field gradi-
+ents. (ii) The abundance of nonzero lead spins is rela-
+tively low and the elementary cell is relatively large. This
+suppresses the nuclear dipole-dipole interactions, which
+threaten the DNSS. (iii) The magnetic fields that have
+to be applied are relatively weak of the order of the fluc-
+tuations of the Overhauser field, which accelerates the
+DNSS formation ∝ 1/B2.
+
+6
+FIG. 4.
+Modeling the nuclear spin inertia for DNSS formation. (a) The hole spin precession in the randomly oriented
+Overhauser field leads to partial spin dephasing. (b) During the DNSS formation, the Overhauser field rotates towards the
+optical axis. (c) This leads to the suppression of hole spin dephasing. (d) Comparison of the measured (dots) and calculated
+(solid line) frequency dependencies of the amplitude of the broad PRC dip. The insets show the nuclear spin distribution
+functions at the corresponding modulation frequencies normalized to their maxima, evidencing the spin squeezing mostly in
+transverse direction. (e) Nuclear spin squeezing parameter ξs calculated from the experimental data (dots) and calculated
+numerically (solid line). States below the blue dashed line are entangled. (f) Calculated frequency dependence of the PRC.
+Our findings establish lead halide perovskites as a
+promising platform for exploration and exploitation of
+intertwined hole and nuclear spin dynamics to excite
+non-classical collective spin states with quantum corre-
+lations. The nuclear spin inertia method is powerful for
+clearly demonstrating DNSS formation, especially for ap-
+plication to other systems with 1/2 nuclear spins includ-
+ing other perovskites or to unstrained quantum dots. In
+combination with existing demonstrations of nuclear spin
+based quantum registers [17, 52] and collective spin mea-
+surements [34, 53], the DNSS may be the most reliable
+platform for quantum metrology, quantum information
+storage and processing with solid state spins beyond the
+standard quantum limit [54], due to the low decoherence
+of the DNSS, caused by destructive interference in the
+transverse spin fluctuations.
+∗ Electronic address: smirnov@mail.ioffe.ru
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+METHODS
+Growth of FAPbBr3 crystals
+The FAPbBr3, with FA being formamidinium, per-
+ovskite crystal was grown using the inverse temperature
+crystallization technique (sample code: OH0071a).
+In
+essence, the reactant salts (FABr and PbBr2) were dis-
+solved in a mixture of DMF:GBL (1:1 v/v), forming the
+precursor solution. By rising the temperature of the solu-
+tion, the sample crystallized due to retrograde solubility
+of the perovskite crystals in the chosen solvent mixture,
+see Ref. [55]. The studied crystal has a reddish color, see
+the inset in Fig. S1(a), with a size of 5 × 5 × 2 mm3.
+Magneto-optical measurements
+The sample was placed in a cryostat at a temperature
+variable from 1.6 K up to 300 K. For T = 1.6 K the sam-
+ple was immersed in superfluid helium, while for temper-
+atures in the range from 4.2 K to 300 K the sample was
+held in cooling helium gas. The cryostat is equipped with
+a vector magnet composed of three superconducting split
+coils orthogonal to each other. This allows us to apply
+magnetic fields up to 3 T in any direction. All magnetic
+fields, the light vector, and sample surface normal are set
+to the horizontal plane. Note that the 3D vector magnet
+allows precise compensation of the residual fields. Mag-
+netic fields parallel to the light wave vector k are denoted
+as BF (Faraday geometry) and magnetic fields perpen-
+dicular to k as BV (Voigt geometry). The angle θ defines
+the tilt of the magnetic field from the Faraday geometry.
+Photoluminescence
+The
+photoluminescence
+(PL)
+was
+excited
+by
+a
+continuous-wave laser with the photon energy of 3.06 eV
+(405 nm).
+The emitted light was coupled into a 0.5 m
+monochromator equipped with a Peltier cooled charge
+coupled device (CCD) via a fiber.
+Time-resolved Kerr ellipticity (TRKE)
+The coherent spin dynamics of electrons and holes in-
+teracting with the nuclear spins were measured by a
+degenerate pump-probe setup, where pump and probe
+have the same photon energy [56]. A titanium-sapphire
+(Ti:Sa) laser generates 1.5 ps long pulses with aspectral
+width of about 1 nm (about 1.5 meV) and pulse repe-
+tition rate of 76 MHz (repetition period TR = 13.2 ns).
+The Ti:Sa laser beam was fed into an optical paramet-
+ric oscillator with an internal frequency doubling and the
+output photon energy was adjusted to values around the
+exciton resonance to meet the maximum of the Kerr ro-
+tation signal at 2.191 eV (566 nm). The laser output was
+split into the pump and probe beams. The probe pulses
+were delayed relative to the pump pulses by a double-
+pass mechanical delay line with one meter length. The
+pump and probe beams were modulated using a photoe-
+lastic modulator (PEM) for the probe and an electroop-
+tical modulator (EOM) for the pump. The probe beam
+was always linearly polarized and amplitude modulated
+at a frequency of 84 kHz. The pump beam was either
+
+9
+helicity modulated between the σ+/σ− circular polariza-
+tions, or amplitude modulated with fixed helicity, either
+σ+ or σ−, in the frequency range from 0 to 5 MHz. In
+all cases fmod refers to the helicity modulation frequency.
+Amplitude modulation can be in effect considered as 0 Hz
+helicity modulation, as the signal is independent from the
+bare amplitude modulation frequency. In the experiment
+typically 20 Hz to 100 kHz helicity and amplitude mod-
+ulation frequencies were used. The polarization of the
+reflected probe beam was analyzed in respect of the ro-
+tation of its elliptical polarization (Kerr ellipticity) with
+a balanced photodiode, using a lock-in technique.
+ACKNOWLEDGMENTS
+We thankful to M. M. Glazov for fruitful discussions.
+This work was supported by the Deutsche Forschungsge-
+meinschaft through the International Collaborative Re-
+search Centre TRR 160 (Project A1). The work at ETH
+Z¨urich (O.H. and M.V.K.) was financially supported by
+the Swiss National Science Foundation (grant agreement
+186406, funded in conjunction with SPP2196 through
+DFG-SNSF bilateral program) and by the ETH Z¨urich
+through the ETH+ Project SynMatLab. D.S.S. acknowl-
+edges the RF President Grant No.
+MK-5158.2021.1.2,
+and the Foundation for the Advancement of Theoreti-
+cal Physics and Mathematics “BASIS”. The theoretical
+modeling of DNSS formation by D.S.S. was supported by
+the Russian Science Foundation (grant No. 21-72-10035).
+
+S1
+Supplementary Information
+“Evidencing the squeezed dark nuclear spin state in lead halide perovskites”
+I. EXPERIMENTAL DETAILS
+A. Photoluminescence and time-resolved Kerr
+ellipticity
+A photo of the hybrid organic-inorganic FAPbBr3 per-
+ovskite crystal under study is shown in the inset of
+Fig. S1(a). It is a crystal with a size of 5 × 5 × 2 mm3
+and a glossy semi-transparent reddish color. Its low tem-
+perature photoluminescence measured at T = 5 K has
+maximum at 2.1776 eV, see Fig. S1(a).
+In the time-resolved Kerr ellipticity (TRKE) measure-
+ments the laser was tuned to the energy of 2.191 eV, reso-
+nant with the free exciton. The pump beam was helicity
+modulated between σ+/σ− circular polarization at the
+frequency of 50 kHz to avoid dynamical nuclear polar-
+ization.
+A typical TRKE dynamics trace measured in
+BV = 0.5 T magnetic field applied in the Voigt geome-
+try is shown in Fig. S1(b). It is contributed by the co-
+herent spin dynamics of charge carriers, which spins are
+initially oriented by the circularly polarized pump and
+then precess about the magnetic field. One can clearly
+see two Larmor precession frequencies, corresponding to
+electrons (e) and holes (h), which is the typical phe-
+nomenology for lead halide perovskite crystals [S1]. We
+have shown that the coherent spin dynamics in perovskite
+semiconductors are provided by resident electrons and
+holes, which are localized in spatially different sites [S2–
+S4].
+The TRKE dynamics were fitted using the function:
+TRKE = Ae cos (ΩL,et) exp (−t/T ∗
+2,e)
++ Ah cos (ΩLt) exp (−t/T ∗
+2,h),
+(S1)
+as shown by the yellow dashed line in Fig. S1(b). Ae(h)
+are the Kerr ellipticity amplitudes, reflecting the de-
+grees of carrier spin polarization, ΩL = ghµBB/ℏ and
+ΩL,e = geµBB/ℏ are the Larmor frequencies with the
+carrier g-factors ge(h), and T ∗
+2,e(h) are the ensemble spin
+dephasing times. There we show the individual compo-
+nents of the hole (blue line) and the electron (red line)
+spin dynamics obtained form the fit with the parameters
+gh = +0.4 and T ∗
+2,h = 780 ps for the holes and ge = +2.4
+and T ∗
+2,e = 1080 ps for the electrons. The assignment of
+the components is made based on the universal depen-
+dence of their g-factors on the band gap energy that we
+have published recently [S1].
+The time delay is fixed at the negative value of −75 ps,
+marked by the arrow in Fig. S1(b), for the measurements
+of the resonant spin amplification (RSA) and polarization
+recovery curves (PRCs).
+2.15
+2.2
+2.25
+0
+0.2
+0.4
+0.6
+0.8
+1
+Energy (eV)
+Photoluminesence (arb. u.)
+FAPbBr3
+0
+500
+1000
+1500
+0
+Time (ps)
+Kerr Ellipticity (arb. u.)
+gh = +0.4 T2
+* = 780 ps
+ge = +2.4 T2
+* = 1076 ps
+BV = 500 mT, T = 6 K
+T = 5 K
+(a)
+(b)
+5 mm
+FIG. S1. (a) Photoluminescence spectrum of the FAPbBr3
+crystal measured under continuous-wave excitation with the
+photon energy of 3.06 eV at T = 5 K. Inset shows a photo
+of the studied crystal. (b) Example of the TRKE dynamics
+(black line) at the pump-probe energy of 2.191 eV. The pump
+beam was helicity modulated between σ+/σ− circular polar-
+ization at the frequency of 50 kHz. Its fit with Eq. (S1) is
+shown by the yellow dashed line. The electron and hole spin
+precession components evaluated from the fit are shown below
+by the red and blue lines, respectively.
+B. Nuclear spin polarization in tilted magnetic field
+Using constant pump helicity, additional measure-
+ments of the nuclear spin polarization were performed
+using the same scheme as for Fig. 1.
+Figures S2(a,b)
+show that the Overhauser field for the electrons does not
+depend on the strength of the external magnetic field,
+as expected for dynamic nuclear spin polarization [S5].
+The Overhauser field for the holes decreases from 22 mT
+at B = 150 mT to 12 mT at B = 250 mT and then
+stays about constant at 10 mT up to the magnetic field
+of 750 mT. Nevertheless, it remains small in compari-
+son with its typical stochastic fluctuation in equilibrium,
+∆B = 40 mT.
+In the angular dependence of the Overhauser field for
+both electron and hole spins, despite a certain scattering
+of the achieved field value, a dependence according to a
+
+S2
+0
+10
+20
+30
+40
+50
+BN,h (mT)
+0
+200
+400
+600
+800
+0
+0.2
+0.4
+0.6
+0.8
+1
+B (mT)
+BN,e (mT)
+θ = 30°
+0
+5
+10
+15
+20
+25
+BN,h (mT)
+0
+30
+60
+90
+0
+0.2
+0.4
+0.6
+0.8
+1
+Angle θ (°)
+BN,e (mT)
+B = 125 mT
+0
+30
+60
+90
+0
+5
+10
+15
+20
+25
+Angle θ (°)
+BN,h (mT)
+B=
+125 mT
+250 mT
+500 mT
+750 mT
+(a)
+(b)
+(c)
+(d)
+(e)
+hole
+hole
+hole
+electron
+electron
+ΔB
+θ = 30°
+B = 125 mT
+FIG. S2. (a,b) Overhauser field (BN,e(h)) experienced by the hole and electron spins as function of the magnetic field tilted by
+θ = 30◦. Lines are guides to the eye. (c,d) Overhauser fields as function of the tilting angle of the external magnetic field with
+strength B = 125 mT. Lines are fits according to BN ∝ cos (θ). (e) Same as (c) for different magnetic field strengths. For all
+panels T = 1.6 K and the pump power is 8 mW.
+TABLE I. Isotopes with nonzero nuclear spin in FAPbBr3
+with FA = CH2(NH)2.
+Nuclear abundance Spin NMR γN
+isotope
+[%]
+I
+[MHz/T]
+207Pb
+22.10
+1/2
+8.882
+79Br
+50.69
+3/2
+10.704
+81Br
+49.31
+3/2
+11.538
+1H
+99.98
+1/2
+42.577
+13C
+1.07
+1/2
+10.708
+14N
+99.63
+1
+3.078
+cosine-function is found, see Figs. S2(c,d). The angular
+dependence of the Overhauser field experienced by the
+holes for several magnetic fields is given in Fig. S2(e).
+C. PRC at constant helicity pumping
+In order to analyze the limit of fmod → 0, we performed
+additional PRC measurements for constant pump helic-
+ity, σ+ or σ−. Figure S3(a) shows the two corresponding
+Kerr ellipticity dynamics. They merge with each other
+when reversing amplitude and magnetic field.
+In par-
+ticular, when their amplitudes are summed, the narrow
+peak around BF = 0 becomes fully compensated and
+cannot be seen [Fig. S3(b)]. We plot also the difference
+of the signals, which corresponds to the nuclear spin in-
+ertia signal at fmod = 0 and compare it with the case of
+fmod = 20 Hz in Fig. S3(c). Generally, the signals are
+similar, however the latter shows two weak maxima at
+BF ≈ ±10 mT around the zero-field dip in agreement
+with the theoretical modeling [Fig. 4(f), green and blue
+curves].
+D. PRC power dependence
+Here we present data for the pump power dependence
+of the PRC signals. In Figures S4(a,b) we compare the
+PRC curves for two different regimes: when the DNSS
+is formed (fmod = 20 Hz) and in absence of the DNSS
+(fmod = 5 kHz).
+All curves were evaluated by fits with respect to
+their amplitude, the ratio of broad dip (Ab) to the
+high field (Ahf) amplitude and the dip width, shown in
+Figs. S4(b,c,d), respectively. To start with 5 kHz mod-
+ulation frequency (full symbols), with increasing pump
+power the amplitude of the PRC dip and the high field
+amplitude rise linearly before showing a saturation trend.
+The narrow dip shows up as a minor peak with magni-
+tude and width near the noise level so that is cannot
+be evaluated reliably for this frequency.
+The ratio of
+the PRC dip to the amplitude at high magnetic fields
+(≈ 150 mT) [Fig. S4(c)] decreases slightly with increas-
+ing pump power.
+The width stays about constant in
+Fig. S4(d) for 5 kHz modulation.
+In contrast, for the low modulation frequency of 20 Hz,
+the PRC shape changes drastically with varying pump
+power. Again, the amplitudes at high field (Ahf), of the
+broad dip (Ab) and of the narrow dip (An) rise nonlin-
+early with a tendency to saturate with increasing pump
+
+S3
+σ+
+σ-
+−100
+0
+100
+−200
+0
+200
+BF (mT)
+Kerr Ellipticity (arb. u.)
+−100
+0
+100
+−200
+0
+200
+BF (mT)
+Kerr Ellipticity (arb. u.)
+σ++ σ-
+−100
+0
+100
+200
+300
+400
+500
+BF (mT)
+Kerr Ellipticity (arb. u.)
+σ+- σ-
+σ+/σ-
+fmod=20 Hz
+σ+/σ- - alternating
+σ+-only
+σ--only
+(a)
+(b)
+(c)
+(d)
+FIG. S3. (a) PRC with constant pump helicity σ+ (blue) or σ− (red). T = 1.6 K, pump power 6 mW. (b) Sum of the two
+signals from panel (a). (c) Difference of the two signals from panel (a) (green), corresponding to the nuclear spin inertia signal
+at fmod = 0, and nuclear spin inertia signal at fmod = 20 Hz (magenta). (d) Scheme of the applied excitation.
+−100
+−50
+0
+50
+100
+150
+0
+100
+200
+300
+BF (mT)
+Kerr Ellipticity (arb. u.)
+0.2 mW
+0.6 mW
+1 mW
+2 mW
+P=6 mW
+−100
+−50
+0
+50
+100
+150
+0
+100
+200
+300
+BF (mT)
+Kerr Ellipticity (arb. u.)
+0.5 mW
+1 mW
+2 mW
+P=6 mW
+(a)
+(c)
+20 Hz
+5 kHz
+0
+1
+2
+3
+4
+5
+6
+7
+0
+0.5
+1
+Pump Power (mW)
+|Ab/Ahf| (arb. u.)
+0
+1
+2
+3
+4
+5
+6
+7
+0
+20
+40
+60
+80
+100
+120
+Pump Power (mW)
+Width (mT)
+An
+0
+1
+2
+3
+4
+5
+6
+7
+−200
+0
+200
+400
+Pump Power (mW)
+Kerr Ellipticity (arb. u.)
+Ahf
+Ab
+(e)
+(b)
+(d)
+x10
+20 Hz
+5 kHz
+20 Hz
+5 kHz
+20 Hz
+5 kHz
+FIG. S4. PRC at different pump powers P, for (a) fmod = 20 Hz and (b) fmod = 5 kHz. The curves are not shifted. (c,d,e)
+Parameters evaluated from curves (a,b), full symbols for fmod = 5 kHz and open symbols for fmod = 20 Hz. (c) Pump power
+dependence of the amplitudes of the broad Ab and narrow An dips (red & dark red diamonds and green triangles, respectively);
+high field (150 mT) values of the amplitudes Ahf (blue & light blue squares). (d) Pump power dependence of the ratio of
+amplitude of the broad dip and its high field value (Ab/Ahf) for fmod = 20 Hz (red) and fmod = 5 kHz (blue). (e) Pump power
+dependencies of the HWHM of the broad (open orange diamonds) and narrow (open green triangles) PRC dips at fmod = 20 Hz,
+and of the broad dip at fmod = 5 kHz (filled orange diamonds). For all panels T = 1.6 K.
+power. However, by contrast to the previous case, the
+ratio Ab/Ahf here drops in a nonlinear fashion much
+stronger. Further, the width of the broad peak broad-
+ens significantly with excitation power.
+The width of
+the narrow peak remains constant. Note, in both cases
+of 5 kHz and 20 Hz, the power dependencies of the spin
+signals match each other at the high magnetic field of
+BF = ±150 mT, when the hyperfine interaction plays a
+minor role, see the blue symbols.
+Overall the PRC pump power dependence for the low
+
+time
+00
+timetime
+0S4
+modulation frequency [Fig. S4(a)] is similar to the fre-
+quency dependence, shown in Fig. 2(b). As the DNSS
+formation rate is determined by the flux of angular mo-
+mentum to the nuclear spin system, this shows that it is
+also proportional to the flux of angular momentum from
+exciting photons to a hole in agreement with the theo-
+retical prediction, Eq. (S19).
+E. DNSS at zero magnetic field
+Surprisingly, we do not observe the hole spin polariza-
+tion recovery with decreasing modulation frequency at
+zero magnetic field, which evidences the fragility of the
+DNSS in the field range ≲ 1 mT. The exact reason for the
+DNSS not to form at zero field is unclear so far, but we
+note that the ODNMR in Fig. 3 has the same linewidth
+of about 1 mT as the narrow PRC dip.
+This suggests
+that the DNSS is destroyed by the non-secular part of
+the nuclear dipole-dipole interaction. Application of the
+magnetic field with strengths exceeding that of the local
+fields suppresses this interaction and stabilizes the DNSS.
+II. SIMULATION OF DNSS FORMATION
+We describe the nuclear spin dynamics in the pump-
+probe experiments using the central spin box model. The
+Hamiltonian of the system reads
+H = AIS + ℏΩLSz,
+(S2)
+where S is the hole spin, A is the joint hyperfine inter-
+action constant for all N nuclei in the hole localization
+volume, I = �N
+n=1 In is the total nuclear spin composed
+of the individual spins In. The hyperfine interaction is
+assumed here to be isotropic because of the dominant S-
+type Bloch wave functions at the top of the valence band
+in perovskites. The same description is valid for electrons
+in conventional GaAs-like semiconductors. However, in
+this work on perovskites the hyperfine interaction of the
+holes plays the main role.
+A. PRC without DNSS
+In equilibrium or at high enough polarization modu-
+lation frequencies, the nuclear spin distribution is Gaus-
+sian:
+F(BN) = exp(−2B2
+N/∆2
+B)
+(
+�
+π/2∆B)3
+(S3)
+with the parameter
+∆B =
+√
+NA
+ghµB
+,
+(S4)
+determining the dispersion of the distribution. In these
+conditions, the PRC is described by [S10]
+TRKE = 1 − 2
+3
+∆2
+B
+∆2
+B + B2 .
+(S5)
+One can see from this expression that the HWHM of the
+PRC is given by ∆B, which in combination with Eq. (S4)
+yields
+N = (ghµB∆B/A)2,
+(S6)
+as used in the main text to determine the number of
+nuclei N.
+B. Nuclear spin dynamics
+To describe the nuclear spin dynamics, we note that
+in Eq. (S2) the total nuclear spin I is conserved. For N
+nuclear spins 1/2, the number of possible realizations of
+spin I is CN/2−I
+N
+− CN/2−I−1
+N
+, where Ck
+n is the binomial
+coefficient, and the probability to find the spin I is
+PI = (2I + 1)(CN/2−I
+N
+− CN/2−I−1
+N
+)
+2N
+.
+(S7)
+The eigenstates of the system can be labeled by the total
+angular momentum Fz = Iz + Sz, and have the form
+Ψ±(Fz) = A±(Fz) |Fz + 1/2, ↓⟩ + B±(Fz) |Fz − 1/2, ↑⟩ ,
+(S8)
+where ↑ (↓) corresponds to a hole with spin Sz = +1/2
+(−1/2) and
+A+(Fz) = −B−(Fz) =
+Ωx
+�
+2Ω(Ω + Ωz)
+,
+(S9a)
+A−(Fz) = B+(Fz) =
+�
+Ω + Ωz
+2Ω
+,
+(S9b)
+with
+Ωx = A
+ℏ
+�
+I(I + 1) − F 2z + 1/4,
+(S10a)
+Ωy = 0,
+(S10b)
+Ωz = ΩL + A
+ℏ Fz
+(S10c)
+and
+Ω = |Ω| = 1
+ℏ
+�
+A2I(I + 1) + ℏ2Ω2
+L + 2AFzℏΩL + A2/4.
+(S11)
+
+S5
+For a given Iz after initialization of the hole in the
+spin-up state, the wave function is |Iz, ↑⟩, which can be
+represented as
+|Iz, ↑⟩ = αΨ+(Iz + 1/2) + βΨ−(Iz + 1/2),
+(S12)
+where
+α2 =
+A2
+−(Iz + 1/2)
+A2
++(Iz + 1/2) + A2
+−(Iz + 1/2),
+(S13a)
+β2 =
+A2
++(Iz + 1/2)
+A2
++(Iz + 1/2) + A2
+−(Iz + 1/2).
+(S13b)
+Then the coherence between the states Ψ+(Iz + 1/2)
+and Ψ−(Iz + 1/2) is lost [S11] on the timescale T ∗
+2,h ∼
+ℏ/(AI) ≪ TR, where TR is the repetition period of the
+pump pulses. As a result, the nuclear spin component Iz
+increases by unity with the probability
+v(Iz + 1/2) = |αA+(Iz + 1/2)|2 + |βA−(Iz + 1/2)|2
+= 2A2
+−(Iz + 1/2)A2
++(Iz + 1/2)
+A2
+−(Iz + 1/2) + A2
++(Iz + 1/2) = 1
+2
+�Ωx
+Ω
+�2
+.
+(S14)
+Here the argument Iz + 1/2 is used to denote the total
+angular momentum component of the nuclei and the hole.
+In the limit of classical I, this reduces to v(Iz + 1/2) =
+A2(I2
+x + I2
+y)/(2|AI + ℏΩL|2). For a hole with spin down,
+the nuclear spin projection reduces by unity with the
+probability v(Iz − 1/2).
+In the limit of many nuclear spins, N ≫ 1, it is conve-
+nient to introduce the distribution function g(x), where
+x = Iz/I, which is normalized such that
+1
+�
+−1
+g(x)dx = 1.
+(S15)
+It satisfies the Fokker-Planck equation
+I2TR
+∂g
+∂t = Pe(t)I ∂
+∂x (vg) + ∂
+∂x
+�v
+2
+∂g
+∂x
+�
+,
+(S16)
+where v = (1 − x2)/2. Introducing τ = t/(I2TR) and
+Π(t) = Pe(t)I, we obtain
+∂g
+∂τ = Π(t) ∂
+∂x (vg) + ∂
+∂x
+�v
+2
+∂g
+∂x
+�
+.
+(S17)
+As boundary condition, g(x) should be finite at x = ±1.
+C. Nuclear spin inertia
+To describe the nuclear spin inertia, we assume that
+each pump pulse excites a trion with the probability Γ0
+and after its recombination, the hole spin polarization
+increases by Ph. According to the optical selection rules,
+the transverse hole spin components subsequently are
+erased [S12]. This allows us to neglect the off-diagonal
+components of the hole and nuclei density matrix. In this
+case, the nuclear spin state is described by the probabil-
+ities Pm for finding the spin I in the state with Iz = m.
+We note that after the hole spin decoherence, the hole
+spin reduces by a factor of v(Fz), which provides an in-
+crease of Iz. Thus, if the polarization modulation period
+consists of Nmod pulses, the signal measured by the nu-
+clear spin inertia method is given by [S13]
+L =
+����Nmod/2
+k=1
+�I
+m=−I Pm(k)v(m + 1/2)
+���
+����Nmod/2
+k=1
+exp (2πik/Nmod)
+���
+,
+(S18)
+where Pm(k) describes the distribution of Iz = m after
+k pulses, starting from the beginning of the period, and
+we assume that it is sufficient to consider only the first
+half of the period.
+This formalism allows us to simulate the evolution of
+the nuclear spin distribution function under pulsed exci-
+tation with modulation of the light helicity and to cal-
+culate the nuclear spin inertia signal as function of the
+magnetic field and the modulation frequency.
+Due to
+the numerical difficulties with the boundary conditions
+for Eq. (S17), we have used a finite number of nuclei,
+namely N = 1300, as estimated in the main text.
+To fit the frequency dependence of the PRC amplitude
+in Fig. 4(d), we have used Γ0 = 0.03 and Ph = 0.5. The
+latter parameter determines the saturation of the PRC
+amplitude at low modulation frequencies, and the total
+DNSS formation rate
+ν0 = Γ0Ph/(TRN) = 0.9 ms−1
+(S19)
+determines the characteristic time scale of the nuclear
+spin dynamics. Thus, all parameters can be reliably de-
+termined from the experimental results.
+D. Nuclear spin squeezing and entanglement
+We characterize the shrinking of the nuclear spin dis-
+tribution function along the transverse directions by the
+Kitagawa and Ueda spin squeezing parameter ξs [S14,
+S15], which is defined by
+ξ2
+s = 4
+N ⟨I2
+x⟩ .
+(S20)
+During the simulation of the nuclear spin inertia, it can
+be extracted as
+ξ2
+s = 4
+N
+�
+I
+�
+m=−I
+Pm
+I(I + 1) − m2
+2
+�
+,
+(S21)
+
+S6
+where the angular brackets denote the averaging over the
+total nuclear spin I with probabilities (S7), and Pm sym-
+bolizes Pm(k) with k = 1, for which the spin squeezing
+reaches its maximum.
+The result of the calculation of the nuclear spin squeez-
+ing degree is shown in Fig. 4(e) by the solid line for the
+same parameters as in panel (d). To find ξs from the ex-
+perimental results, we assume that the nuclear spin dy-
+namics obey the evolution according to Eq. (S2). Then,
+following the same procedure as described above, we find
+the DNSS that corresponds to the experimentally ob-
+served amplitude of the PRC at the given frequency and
+the calculated spin squeezing parameter for this state.
+The results are shown in Fig. 4(e) by the dots.
+For low modulation frequencies we obtain ξs = 0.29,
+which is limited by nuclear spin diffusion due to an in-
+complete hole spin polarization. In the limit of a perfect
+nuclear spin alignment along the z axis, the largest pos-
+sible spin squeezing is ξs = 0.21 because of the quantum
+fluctuations of the transverse components of the total nu-
+clear spin I ∼
+√
+N ∼ 36. This situation can be called
+the Heisenberg limit, and it is only 1.4 times smaller than
+the experimentally reached value.
+The large nuclear spin squeezing suggests deep entan-
+glement between nuclear spins.
+One of the strict evi-
+dences of entanglement is the violation of the generalized
+spin squeezing inequality
+⟨I2
+x⟩ + ⟨I2
+y⟩ + ⟨(Iz − ⟨Iz⟩)2⟩ ≥ N
+2 .
+(S22)
+We note, that this represents a particular case of Eq. (2)
+with M = N, which means that there is at least any
+entanglement in the nuclear spins system.
+The PRC directly measures the transverse nuclear
+spin components ⟨I2
+x⟩ + ⟨I2
+y⟩.
+To check the violation
+of Eq. (S22), we consider the upper limit for the longi-
+tudinal nuclear spin fluctuations ⟨(Iz − ⟨Iz⟩)2⟩ ≤ N/4,
+which corresponds to uncorrelated individual nuclear
+spins. From the definition in Eq. (S20) we obtain that
+states with ξs ≤ 1/
+√
+2 = 0.707 are entangled.
+This
+boundary is shown in Fig. 4(e) by the blue dashed line.
+For the lowest polarization modulation frequency of
+fmod = 100 Hz, the PRC amplitude [L(Bmax) − L(B =
+0)]/L(B = 0) = 0.054 corresponds to I2
+x + I2
+y = 0.054I2,
+which after averaging over I yields ⟨I2
+x + I2
+y⟩ = 0.04N.
+Combined with the upper boundary for the longitudinal
+spin fluctuations we obtain
+⟨I2
+x⟩ + ⟨I2
+y⟩ + ⟨(Iz − ⟨Iz⟩)2⟩ ≤ 0.29N,
+(S23)
+which violates Eq. (S22) almost by a factor of two. In
+particular, using Eq. (2) we find that the achieved DNSS
+is at least equivalent to an M = 0.58N = 750-body en-
+tanglement among the N = 1300 nuclear spins.
+This is a very conservative estimate mainly because of
+the assumption of unsuppressed longitudinal spin fluctu-
+ations. According to our model of DNSS formation, at
+the lowest polarization modulation frequency the DNSS
+steady state is reached, where the depth of entanglement
+is limited by the conservation of I and by nuclear spin
+diffusion. For this state, the calculation of the nuclear
+spin fluctuations yields an M = 113-body entanglement.
+We recall that the smaller M is, the deeper is the entan-
+glement.
+∗ Electronic address: smirnov@mail.ioffe.ru
+[S1] E. Kirstein, D. R. Yakovlev, M. M. Glazov, E. A.
+Zhukov, D. Kudlacik, I. V. Kalitukha, V. F. Sapega,
+G.
+S. Dimitriev,
+M.
+A. Semina,
+M. O. Nestok-
+lon, E. L. Ivchenko, N. E. Kopteva, D. N. Dirin,
+O.
+Nazarenko,
+M.
+V.
+Kovalenko,
+A.
+Baumann,
+J. H¨ocker, V. Dyakonov, and M. Bayer, The Land´e fac-
+tors of electrons and holes in lead halide perovskites:
+universal dependence on the band gap, Nat. Commun.
+13, 3062 (2022)
+[S2] V. V. Belykh, D. R. Yakovlev, M. M. Glazov, P. S.
+Grigoryev, M. Hussain, J. Rautert, D. N. Dirin, M. V.
+Kovalenko, and M. Bayer, Coherent spin dynamics of
+electrons and holes in CsPbBr3 perovskite crystals, Nat.
+Commun. 10 , 673 (2019).
+[S3] E. Kirstein, D. R. Yakovlev, E. A. Zhukov, J. H¨ocker,
+V. Dyakonov, and M. Bayer, Spin Dynamics of Elec-
+trons and Holes Interacting with Nuclei in MAPbI3 Per-
+ovskite Single Crystals, ACS Photon. 9, 1375 (2022).
+[S4] E. Kirstein, D. R. Yakovlev, M. M. Glazov, E. Evers,
+E. A. Zhukov, V. V. Belykh, N. E. Kopteva, D. Kud-
+lacik, O. Nazarenko, D. N. Dirin, M. V. Kovalenko,
+and M. Bayer, Lead-Dominated Hyperfine Interaction
+Impacting the Carrier Spin Dynamics in Halide Per-
+ovskites, Advanced Materials 34, 2105263 (2022).
+[S5] M. M. Glazov, Electron and Nuclear Spin Dynamics
+in Semiconductor Nanostructures (Oxford University
+Press, Oxford, 2018).
+[S6] I. A. Yugova, M. M. Glazov, D. R. Yakovlev, A. A.
+Sokolova, and M. Bayer, Coherent spin dynamics of
+electrons and holes in semiconductor quantum wells and
+quantum dots under periodical optical excitation: Reso-
+nant spin amplification versus spin mode locking, Phys.
+Rev. B 85, 125304 (2012).
+[S7] V. K. Kalevich, Optically induced nuclear magnetic res-
+onance in semiconductors, Sov. Phys. Solid State 28,
+1947 (1986).
+[S8] E. A. Zhukov, A. Greilich, D. R. Yakovlev, K. V. Ka-
+vokin, I. A. Yugova, O. A. Yugov, D. Suter, G. Kar-
+czewski, T. Wojtowicz, J. Kossut, V. V. Petrov, Yu. K.
+Dolgikh, A. Pawlis, and M. Bayer, All-optical NMR in
+semiconductors provided by resonant cooling of nuclear
+spins interacting with electrons in the resonant spin am-
+plification regime, Phys. Rev. B 90, 085311 (2014).
+[S9] R. W. Mocek, V. L. Korenev, M. Bayer, M. Kotur, R. I.
+Dzhioev, D. O. Tolmachev, G. Cascio, K. V. Kavokin,
+and D. Suter, High-efficiency optical pumping of nuclear
+polarization in a GaAs quantum well, Phys. Rev. B 96,
+201303 (2017).
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+S.
+Smirnov,
+E.
+A.
+Zhukov,
+D.
+R.
+Yakovlev,
+E. Kirstein, M. Bayer, and A. Greilich, Spin polarization
+
+S7
+recovery and hanle effect for charge carriers interacting
+with nuclear spins in semiconductors, Phys. Rev. B 102,
+235413 (2020) .
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+relaxation by nuclei in semiconductor quantum dots,
+Phys. Rev. B 65, 205309 (2002).
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+Efros, Pump-probe Faraday rotation and ellipticity in
+an ensemble of singly charged quantum dots, Phys. Rev.
+B 80, 104436 (2009).
+[S13] D. S. Smirnov, E. A. Zhukov, E. Kirstein, D. R.
+Yakovlev,
+D.
+Reuter,
+A.
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+Wieck,
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+Bayer,
+A. Greilich, and M. M. Glazov, Theory of spin iner-
+tia in singly charged quantum dots, Phys. Rev. B 98,
+125306 (2018).
+[S14] J. Ma, X. Wang, C. P. Sun, and F. Nori, Quantum spin
+squeezing, Phys. Rep. 509, 89 (2011).
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+Rev. A 47, 5138 (1993).
+
diff --git a/QNFJT4oBgHgl3EQfJSzf/content/tmp_files/load_file.txt b/QNFJT4oBgHgl3EQfJSzf/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5c0731c2bfab39848aee2b8703fcdaa840e1bb36
--- /dev/null
+++ b/QNFJT4oBgHgl3EQfJSzf/content/tmp_files/load_file.txt
@@ -0,0 +1,1347 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf,len=1346
+page_content='Evidencing the squeezed dark nuclear spin state in lead halide perovskites  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Kirstein,1 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Smirnov,2, ∗ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Zhukov,1, 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Yakovlev,1, 2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Kopteva,1 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Dirin,3 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Hordiichuk,3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Kovalenko,3, 4 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Bayer1, 2  1Experimentelle Physik 2, Technische Universit¨at Dortmund, 44221 Dortmund, Germany  2Ioffe Institute, Russian Academy of Sciences, 194021 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Petersburg, Russia  3Department of Chemistry and Applied Biosciences,  Laboratory of Inorganic Chemistry, ETH Z¨urich, 8093 Z¨urich, Switzerland  4Department of Advanced Materials and Surfaces, Laboratory for Thin Films and Photovoltaics,  Empa - Swiss Federal Laboratories for Materials Science and Technology, 8600 D¨ubendorf, Switzerland  Coherent many-body states are highly promising for robust and scalable quantum information  processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' While far-reaching theoretical predictions have been made for various implementations,  direct experimental evidence of their appealing properties can be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Here, we demon-  strate coherent optical manipulation of the nuclear spin ensemble in the lead halide perovskite  semiconductor FAPbBr3 (FA=formamidinium), targeting a long-postulated collective dark state  that is insensitive to optical pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Via optical orientation of localized hole spins we drive the  nuclear many-body system into an entangled state, requiring a weak magnetic field of only a few  Millitesla strength at cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' During its fast build-up, the nuclear polarization  along the optical axis remains small, while the transverse nuclear spin fluctuations are strongly re-  duced, corresponding to spin squeezing as evidenced by a strong violation of the generalized nuclear  squeezing-inequality with ξs < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The dark state evidenced in this process corresponds to an ap-  proximately 750-body entanglement between the nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Dark nuclear spin states can be exploited  to store quantum information benefiting from their long-lived many-body coherence and to perform  quantum measurements with a precision beyond the standard limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  INTRODUCTION  The range of many-body states in condensed matter is  formidably rich: the fractional quantum Hall [1, 2] and  the Wigner crystal [3–5] states are just two examples [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Quantum coherence is essential for the formation of many  of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The many-body state of localized spins in solid state  is arguably one of the best controlled and potentially  scalable hardware to that end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' For example, an electron  spin can be used to create entanglement between multiple  photons for one-way quantum computing [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' However,  electron and photon spins suffer from fast decoherence  or short lifetime, limiting their applicability for quantum  information storage and many-body unitary operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Nuclear spins, on the other hand, are largely isolated  from their environment resulting in extended coherence  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Thus, they complement electron spins [10–12],  particularly due to possible electron-nuclear spin inter-  facing [13–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Spin-based concepts for solid state quantum comput-  ing and quantum metrology rely on many-body entan-  glement in combination with nuclear spin squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The  latter of which is the ultimate goal of nuclear spin manip-  ulation [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Almost 20 years ago, a coherent many-  body dark nuclear spin state (DNSS) was predicted that  can be formed by orienting optically the spins of resident  charge carriers interacting with the nuclear spin bath in  the host lattice [10, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Mathematically, the DNSS is  the state which vanishes upon action of the rising oper-  ator of the total nuclear spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' That is implemented by  the optical pumping in the experiment: DNSS formation  blocks the optical nuclear spin pumping in analogy with  optically dark states in ensembles of atoms, which are  immune to the light-matter interaction [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The DNSS  belongs to the class of maximally entangled singlet states,  so that it can be exploited not only to suppress the de-  phasing of localized electron and hole spin qubits [22, 23],  but also for storage of quantum information [10, 11] or  for quantum metrology applications [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The DNSS is characterized by destructive interference  of the nuclear spin amplitudes in the collective trans-  verse components, which leads to their strong suppres-  sion without large nuclear spin polarization [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The destructive interference arises from quantum correla-  tions and entanglement between the nuclear spins, some-  what analogous to the spin structure in antiferromagnetic  materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Realization and demonstration of the DNSS  have turned out to be challenging due to threats from  the dipole-dipole and quadrupole nuclear interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' In  prior works on III-V semiconductor quantum dots [25–  27], where the nuclear spin polarization was measured,  the limited polarization value was attributed to DNSS  formation, but without proof of suppression of the trans-  verse spin fluctuations [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Later, detailed investigations  of GaAs quantum dots revealed 80% nuclear polariza-  tion without any signatures of DNSS formation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' the  nuclear spin state in this system could be described by  an effective nuclear spin temperature and the absence of  correlations [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Correlations between nuclear spins was  evidenced for non-thermal nuclear spin states [29], nar-  rowed nuclear spin states [30–32], and nuclear frequency  focusing [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Thus none of these studies evidenced the  DNSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' In a recent study [34], the authors reconstructed  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='11460v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='mtrl-sci]  26 Jan 2023  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Measurement of the nuclear spin polarization in FAPbBr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (a) Formation of the DNSS: after optical orientation  (green light pulse) of a localized hole (magenta arrow), individual nuclear spins Ik (small coral arrows) rotate through the  hyperfine interaction, each by the same angle, so that the total nuclear spin polarization I (large red arrow) is directed along  the optical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  (b) Time-resolved Kerr ellipticity dynamics measured for σ+ and σ− pump pulse polarization measured  at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='191 eV photon energy, in which oscillations associated with electron and hole spin precession about the magnetic field  are superimposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Arrows mark the second oscillation minimum in the hole signal after the pump pulse at time zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (c)  Illustration of nuclear spin polarization in tilted magnetic field for σ+ (left) and σ− (right) pump polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (d) Time-  resolved, measurement of the Overhauser field rise and decay, showing changes on time scales below the 2 ms low pass of the  lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Experimental parameters: T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6 K, P = 2 mW, B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='1 T, θ = 60◦, ∆t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (e) Time-resolved Kerr  ellipticity dynamics measured at different pump powers for fixed pump polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Dashed line indicates the temporal shift  of the hole oscillation for increased pump power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (f,g) Overhauser field for holes and electrons, respectively, extracted from the  Kerr ellipticity dynamics, using the hole and electron g-factors of gh = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4 and ge = +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  the nuclear spin statistics and suggested destructive in-  terference between different nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  However, a  measurement of the transverse spin components, manda-  tory for DNSS demonstration, is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Here we study the intertwined spin dynamics of nu-  clei and charge carriers in a FAPbBr3 lead halide  perovskite crystal, using tailored optical pump-probe  schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' More specifically, we develop a method for as-  sessing the nuclear spin inertia, which allows us to access  the regime of weak magnetic fields in the milliTesla range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  We find that excitation with circularly polarized light  leads to the disappearance of the transverse nuclear spin  fluctuations with simultaneous absence of a significant  longitudinal nuclear spin polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The combination  of these two factors directly evidences the destructive in-  terference of nuclear spins and thus the DNSS formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  This many-body correlated state builds up fast on short  time scales in the ms-range and shows significant spin  squeezing evaluated through the Kitagawa-Ueda param-  eter [35] ξs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='29, corresponding to entanglement of  about 750 nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  RESULTS  The exciton resonance in the hybrid organic-inorganic  lead halide perovskite FAPbBr3 crystals is located at  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='191 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' We have measured the time-resolved Kerr el-  lipticity (TRKE) at this energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' We give basic informa-  tion on the charge carrier spin dynamics and the pho-  toluminescence spectrum in the Supplementary Infor-  mation I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The nuclear spin polarization has been ad-  dressed by the Overhauser field experienced by the res-  ident hole spins, which we observe in TRKE, recorded  at the temperature of T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6 K in tilted magnetic field,  Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The spin dynamics consists of slow and fast  oscillating components related to the hole (h) and elec-  tron (e) spin precession about the total magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The corresponding carrier spin g-factors are gh = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4  and ge = +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4 (Supplementary Information I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  3  −150  −100  −50  0  50  100  150  20  30  40  50  BF (mT)  Kerr Ellipticity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  0  1  2  3  4  5  0  20  40  fmod (kHz)  |KE| (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  0  1  2  3  4  5  0  20  40  fmod (kHz)  HWHM (mT)  −50  0  50  20  30  40  50  BF (mT)  Kerr Ellipticity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  (a)  (b)  (c)  (d)  fmod = 1 kHz  5 kHz  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='1 kHz  2 kHz  1 kHz  fmod =  x10  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Nuclear spin inertia measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (a) Representative PRCs each consisting of a broad (orange filling) and a  narrow (green filling) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (b) PRCs at different polarization modulation frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (c,d) Polarization modulation  frequency dependencies of widths and amplitudes of the broad (red symbols) and narrow PRC components (green symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Lines are guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' T = 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The strong spin-orbit interaction in perovskites pro-  vides a selectivity for light absorption of different circular  polarizations [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' As a result, optical spin orientation  is feasible for resident electrons and holes, localized at  different sites in the sample at low temperatures, which  represents a situation similar to that provided by quan-  tum confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The angular momentum of a σ+ or  σ− photon is transferred to the charge carrier spin po-  larization, and further to the host lattice nuclei through  the hyperfine interaction, which leads to build-up of an  Overhauser field BN parallel or antiparallel to the exter-  nal magnetic field, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' This field can be di-  rectly determined through measurement of the electron  and hole Larmor precession frequencies about the total  magnetic field, to which the external field and the Over-  hauser field contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  With increasing pump power,  the Overhauser field rises [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 1(e)], but saturates at  relatively small values of BN,h = 15 mT for the holes  [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 1(f)] and BN,e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4 mT for the electrons [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 1(g)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Similar to other perovskites [38, 39], the hyperfine inter-  action is much stronger for the holes due to the dominant  s-type orbital contribution to the Bloch wave functions  at the top of the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Thus, we focus on the  holes in our study, which dominate the Kerr ellipticity  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The measured dependence of the Overhauser  field on the direction and strength of the external field is  presented in the Supplementary Information [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The small measured Overhauser field, as compared to  the maximum possible strength of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='3 T evaluated from  the hyperfine coupling constants [40], indicates an un-  usual nuclear spin statistics which can be highlighted fur-  ther by measurement of its rise and decay times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' There-  for, we first close the pump beam to erase the nuclear  spin polarization, and then open the shutter after which  we observe almost immediately a nuclear spin polariza-  tion, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' After closing the shutter again, the  spin polarization decays on the same time scale of about  100 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' This nuclear spin response time is much shorter  than the previously reported longitudinal nuclear spin re-  laxation times of T1,N = 5 s in FA0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='9Cs0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='1PbI2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='8Br0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='2 [40]  and 960 s in MAPbI3 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' In fact, the variation is limited  by the shutter mechanical opening/closing times in ex-  periment, so that the nuclear spin dynamics occur even  faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  To access this surprisingly fast nuclear spin response  time, we develop a technique for measuring the nuclear  spin inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Similarly to the electron spin inertia tech-  nique [42, 43], it monitors the Kerr ellipticity signal  as function of the longitudinal magnetic field (Faraday  geometry) with simultaneous modulation of the pump  helicity at different frequencies fmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  An example for  fmod = 1 kHz is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The application  of a longitudinal magnetic field leads to a spin polariza-  tion recovery due to the suppression of nuclei-induced  spin relaxation of the holes [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Strikingly, the shape  of this polarization recovery curve (PRC) changes with  4  −20  −10  0  10  20  BV (mT)  Kerr Ellipticity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  −40 −20 0  20 40  BV (mT)  Kerr Ellipticity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  25 kHz  50 kHz  75 kHz  100 kHz  0 20 40 60 80 100  0  5  10  15  fmod (kHz)  BNMR (mT)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='56 MHz/T  fmod =  RSA  hole  electron  ZP  NMR  BNMR  (b)  (a)  (c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' ODNMR measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (a) Resonant spin ampli-  fication signals at different modulation frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (b) De-  composition of the RSA signal at fmod = 50 kHz into four  components by fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  From top to bottom:  RSA signal  (blue) together with the sum of the four components (red),  contributions of the hole (green), the electron (dark red), the  zero peak [ZP] (light blue), the nuclear magnetic resonances  [NMR] (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (c) Dependence of the resonance field on the  modulation frequency, in agreement with the 207Pb gyromag-  netic ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' T = 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  the modulation frequency [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 2(b)]: at the high fre-  quency of fmod = 5 kHz, the PRC resembles a wide dip  of Lorentzian shape with the half width at half maximum  (HWHM) of 40 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' With decreasing frequency down to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='1 kHz, this dip gradually disappears and is replaced by  a much narrower dip, having the HWHM of about 1 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Generally, a PRC consists of the broad and the narrow  component, each of which can be phenomenologically de-  scribed by a Lorentzian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Their widths depend weakly on  fmod [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 2(c)], and their amplitudes change in opposite  ways [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 2(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The spin relaxation time of a hole is of the order of  1 µs (as measured using the same spin inertia method  at higher modulation frequencies), so that we attribute  the PRC changes with modulation frequency to fast hole  spin-induced nuclear spin dynamics occurring on time  scales of 1 ms, which is shorter than the typical quantum  decoherence time of nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Further details will be  given in the Theory section, additional pump-power de-  pendencies and measurements for fmod = 0 are presented  in the Supplementary Information [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S3 and S4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Further, we identify the specific nuclear isotope dom-  inating the hyperfine interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' For this, we use the  Voigt geometry with the magnetic field normal to the op-  tical axis, where a nuclear magnetic resonance appears if  the carrier polarization modulation matches the nuclear  Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The carrier polarization modulation  equals the pump modulation fmod [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The Kerr ellip-  ticity signal [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 3(a)] can be separated into an electron-  related component and a hole-related component, both  representing damped oscillations, in addition a nuclei-  related zero field peak and nuclear magnetic resonance  (NMR) appears [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 3(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The latter dominate the sig-  nal, their position is given by BNMR = fmod/γ with  γ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='56 MHz/T [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 3(a,c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' This value closely cor-  responds to the gyromagnetic ratio of 207Pb given by  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='88 MHz [see Supplementary Information, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' As  in previous measurements of optically detected nuclear  magnetic resonance in perovskites, the small chemical  shift may occur due to the presence of a spin-polarized  hole [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  When the light polarization modulation frequency is  high, the nuclear spin distribution remains in equilib-  rium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The polarization recovery effect in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 2(b) at  fmod = 5 kHz then demonstrates nuclei-dominated hole  spin relaxation [44]: In the absence of the external mag-  netic field, the spin precession in the randomly oriented  Overhauser field BN leads to spin dephasing [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4(a)],  while application of a longitudinal field B rotates the  total magnetic field experienced by the holes, Btot =  BN + B, towards the optical axis and suppresses the  spin dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Thus, the HWHM of the PRC in this  case gives the typical fluctuation of the Overhauser field,  ∆B = 40 mT (Supplementary Information II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The PRC directly measures the average angle θ be-  tween the optical z axis and the hole spin precession fre-  quency vector, which includes the Overhauser and exter-  nal field contributions [inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The nuclear  spin inertia measurements reveal the recovery of the hole  spin polarization at magnetic fields above 1 mT, when  decreasing the polarization modulation frequency from  5 kHz down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='1 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' This appears to be a paradox,  since the typical nuclear spin polarization rate is much  smaller than these rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Moreover, the largest nuclear  spin polarization of BN,h = 15 mT shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 1(e)  is considerably smaller than the typical fluctuation of  the Overhauser field ∆B = 40 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' However, the sup-  pression of the transverse Overhauser field fluctuations  in absence of a significant nuclear spin polarization un-  ambiguously evidences the destructive interference be-  tween nuclear spins due to their quantum correlations,  and exactly matches the predictions for the DNSS for-  mation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  THEORY  The established concept of dynamic nuclear spin po-  larization relies on the assumption of an effective nu-  clear spin temperature and the absence of correlations  between the nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' In the DNSS, by contrast, quantum  5  correlations and destructive interference in the collective  transverse nuclear spin components necessitate a purely  quantum mechanical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Thus, to model the co-  herent nuclear spin dynamics, we apply the central spin  box model described by the following Hamiltonian:  H = AIS + ℏΩL,hSz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  (1)  Here I is the total nuclear spin in the hole localization  volume, S is the hole spin, A is the hyperfine coupling  constant of the holes, ΩL is the hole Larmor precession  frequency in the external magnetic field applied along the  optical z axis, and ℏ is the reduced Planck constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The  spin precession about the nuclear field is included in the  renormalization of ΩL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The total nuclear spin is formed by N individual  207Pb 1/2 spins Ik: I = �N  k=1 Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The number of nu-  clear spins can be estimated from the PRC width us-  ing the hyperfine interaction constant A = 33 µeV [40]:  N = [A/(ghµB∆B)]2 = 1300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Using the lattice constant  of a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6 nm, this gives the hole localization length  l = a0(N/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='22)1/3 = 11 nm (with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='22 being the natu-  ral lead spin abundance), which is similar to other per-  ovskites [38, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Since N ≫ 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (1) implies hole spin  precession between two pump pulses with the constant  frequency AI/ℏ + ΩLez (ez is the unit vector along the  z axis), which leads to hole spin dephasing in weak fields  for a randomly oriented total nuclear spin I, close to  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Continuous hole spin pumping and its dephasing lead  to the transfer of angular momentum to the nuclei, as can  be seen from the conservation of the total angular mo-  mentum Iz +Sz by the Hamiltonian (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Simultaneously,  the conservation of the absolute value of the total nuclear  spin I prevents the build-up of a nuclear spin polarization  larger than ∼ 1/  √  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Therefore, the total nuclear spin is  rotated towards the z axis, while its transverse compo-  nents decrease, but its absolute value does not increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The collective nuclear spin dynamics driven by the inter-  action with a single hole spin produces quantum correla-  tions and leads to entanglement between the nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' As a  result, the DNSS is formed [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4(b)], in agreement with  the original theoretical predictions [10, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The  quantum  mechanical  solution  of  the  box  model [46–49] allows us to fit the experimentally mea-  sured amplitude of the broad PRC dip as function of the  polarization modulation frequency, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The  modification of the nuclear spin distribution with de-  creasing modulation frequency is shown in the corre-  sponding insets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The alignment of all nuclear spin fluc-  tuations along the optical axis at low modulation fre-  quencies cancels the hole spin precession [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4(c)] and  restores the hole spin polarization to the same value  as in large magnetic fields, where the hole and nuclear  spins become in effect decoupled (the hole spin dephas-  ing by the nuclei is suppressed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Thereby also the whole  PRC amplitude rises, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' From fit-  ting the nuclear spin inertia we find the DNSS forma-  tion rate (typical frequency of nuclear spin rotation) of  ν0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='9 ms−1 (Supplementary Information II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  ANALYSIS AND DISCUSSION  The ratio of the hole spin polarization at a given mag-  netic field and in saturation (at B ≳ 100 mT) represents  a collective measurement of the total nuclear spin com-  ponents ⟨(I2  x + I2  y)/I2⟩ [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' This allows us to quantify  the suppression of the transverse nuclear spin fluctua-  tions by the Kitagawa and Ueda spin squeezing parame-  ter ξ2  s = 4 ⟨I2  x⟩ /N [19, 35] (⟨I2  x⟩ = ⟨I2  y⟩), which for uncor-  related spins equals to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' This parameter extracted  from the measured PRC amplitude is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4(e)  by the dots, and its simulated frequency dependence with  the same DNSS formation rate is shown by the solid line  (Supplementary Information II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' For almost complete  suppression of the broad PRC dip to values below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='05  at fmod = 100 Hz, the spin squeezing parameter is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='29,  which is limited by nuclear spin diffusion due to incom-  plete hole spin polarization and quantum fluctuations of  the transverse total nuclear spin components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The quantum correlations between the nuclear spins  suggest their entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' This can be shown using the  generalized spin squeezing inequality [19, 50]  ⟨I2  x⟩ + ⟨I2  y⟩ + ⟨(Iz − ⟨Iz⟩)2⟩ ≥ M/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  (2)  Its violation requires M-body entanglement between N  nuclear spins, meaning that there are at least M spins,  which are entangled with the rest of the ensemble [18, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  To show its violation, we use the upper boundary for the  longitudinal nuclear spin fluctuations, ⟨(Iz − ⟨Iz⟩)2⟩ ≤  N/4, which yields the lower limit of ξ(0)  s  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='71 for the  entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' States with ξs < ξ(0)  s  violate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (2)  with M = N and are entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The maximum achieved  DNSS with ξs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='29 is at least M = 750-body entan-  gled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' However, our theoretical simulations of the nuclear  spin fluctuations in the DNSS suggest much deeper en-  tanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  There are good reasons for FAPbBr3 perovskites to be  the material system for experimental observation of the  DNSS: (i) Lead has either nuclear spin 0(206Pb, 208Pb  with abundance of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='9%) or 1/2 (207Pb with abundance  of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='1%), which excludes quadrupole splitting of the  nuclear spin levels due to strain or electric field gradi-  ents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (ii) The abundance of nonzero lead spins is rela-  tively low and the elementary cell is relatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' This  suppresses the nuclear dipole-dipole interactions, which  threaten the DNSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (iii) The magnetic fields that have  to be applied are relatively weak of the order of the fluc-  tuations of the Overhauser field, which accelerates the  DNSS formation ∝ 1/B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  6  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Modeling the nuclear spin inertia for DNSS formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (a) The hole spin precession in the randomly oriented  Overhauser field leads to partial spin dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (b) During the DNSS formation, the Overhauser field rotates towards the  optical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (c) This leads to the suppression of hole spin dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (d) Comparison of the measured (dots) and calculated  (solid line) frequency dependencies of the amplitude of the broad PRC dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The insets show the nuclear spin distribution  functions at the corresponding modulation frequencies normalized to their maxima, evidencing the spin squeezing mostly in  transverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (e) Nuclear spin squeezing parameter ξs calculated from the experimental data (dots) and calculated  numerically (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' States below the blue dashed line are entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (f) Calculated frequency dependence of the PRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Our findings establish lead halide perovskites as a  promising platform for exploration and exploitation of  intertwined hole and nuclear spin dynamics to excite  non-classical collective spin states with quantum corre-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The nuclear spin inertia method is powerful for  clearly demonstrating DNSS formation, especially for ap-  plication to other systems with 1/2 nuclear spins includ-  ing other perovskites or to unstrained quantum dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' In  combination with existing demonstrations of nuclear spin  based quantum registers [17, 52] and collective spin mea-  surements [34, 53], the DNSS may be the most reliable  platform for quantum metrology, quantum information  storage and processing with solid state spins beyond the  standard quantum limit [54], due to the low decoherence  of the DNSS, caused by destructive interference in the  transverse spin fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  ∗ Electronic address: smirnov@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='ioffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content=' de Weerd, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Gomez, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Gregorkiewicz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content=' Matyushkin, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content='  Yassievich, Optical orientation and alignment of excitons  in ensembles of inorganic perovskite nanocrystals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content='  Grigoryev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content=' Rautert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content=' Krieg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content=' Abdelhady, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Maculan, and  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content=' Yakovlev and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Bayer, Coherent Spin Dynamics  of Carriers, in Spin Physics in Semiconductors, Springer  Series in Solid-State Sciences, edited by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Dyakonov  (Springer International Publishing, Cham, 2017) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  155–206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  METHODS  Growth of FAPbBr3 crystals  The FAPbBr3, with FA being formamidinium, per-  ovskite crystal was grown using the inverse temperature  crystallization technique (sample code: OH0071a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  In  essence, the reactant salts (FABr and PbBr2) were dis-  solved in a mixture of DMF:GBL (1:1 v/v), forming the  precursor solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' By rising the temperature of the solu-  tion, the sample crystallized due to retrograde solubility  of the perovskite crystals in the chosen solvent mixture,  see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The studied crystal has a reddish color, see  the inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S1(a), with a size of 5 × 5 × 2 mm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Magneto-optical measurements  The sample was placed in a cryostat at a temperature  variable from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6 K up to 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' For T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6 K the sam-  ple was immersed in superfluid helium, while for temper-  atures in the range from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='2 K to 300 K the sample was  held in cooling helium gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The cryostat is equipped with  a vector magnet composed of three superconducting split  coils orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' This allows us to apply  magnetic fields up to 3 T in any direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' All magnetic  fields, the light vector, and sample surface normal are set  to the horizontal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Note that the 3D vector magnet  allows precise compensation of the residual fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Mag-  netic fields parallel to the light wave vector k are denoted  as BF (Faraday geometry) and magnetic fields perpen-  dicular to k as BV (Voigt geometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The angle θ defines  the tilt of the magnetic field from the Faraday geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Photoluminescence  The  photoluminescence  (PL)  was  excited  by  a  continuous-wave laser with the photon energy of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='06 eV  (405 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The emitted light was coupled into a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='5 m  monochromator equipped with a Peltier cooled charge  coupled device (CCD) via a fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Time-resolved Kerr ellipticity (TRKE)  The coherent spin dynamics of electrons and holes in-  teracting with the nuclear spins were measured by a  degenerate pump-probe setup, where pump and probe  have the same photon energy [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' A titanium-sapphire  (Ti:Sa) laser generates 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='5 ps long pulses with aspectral  width of about 1 nm (about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='5 meV) and pulse repe-  tition rate of 76 MHz (repetition period TR = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='2 ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The Ti:Sa laser beam was fed into an optical paramet-  ric oscillator with an internal frequency doubling and the  output photon energy was adjusted to values around the  exciton resonance to meet the maximum of the Kerr ro-  tation signal at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='191 eV (566 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The laser output was  split into the pump and probe beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The probe pulses  were delayed relative to the pump pulses by a double-  pass mechanical delay line with one meter length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The  pump and probe beams were modulated using a photoe-  lastic modulator (PEM) for the probe and an electroop-  tical modulator (EOM) for the pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The probe beam  was always linearly polarized and amplitude modulated  at a frequency of 84 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The pump beam was either  9  helicity modulated between the σ+/σ− circular polariza-  tions, or amplitude modulated with fixed helicity, either  σ+ or σ−, in the frequency range from 0 to 5 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' In  all cases fmod refers to the helicity modulation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Amplitude modulation can be in effect considered as 0 Hz  helicity modulation, as the signal is independent from the  bare amplitude modulation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' In the experiment  typically 20 Hz to 100 kHz helicity and amplitude mod-  ulation frequencies were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The polarization of the  reflected probe beam was analyzed in respect of the ro-  tation of its elliptical polarization (Kerr ellipticity) with  a balanced photodiode, using a lock-in technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We thankful to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Glazov for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  This work was supported by the Deutsche Forschungsge-  meinschaft through the International Collaborative Re-  search Centre TRR 160 (Project A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The work at ETH  Z¨urich (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=') was financially supported by  the Swiss National Science Foundation (grant agreement  186406, funded in conjunction with SPP2196 through  DFG-SNSF bilateral program) and by the ETH Z¨urich  through the ETH+ Project SynMatLab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' acknowl-  edges the RF President Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  MK-5158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='2,  and the Foundation for the Advancement of Theoreti-  cal Physics and Mathematics “BASIS”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The theoretical  modeling of DNSS formation by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' was supported by  the Russian Science Foundation (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 21-72-10035).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  S1  Supplementary Information  “Evidencing the squeezed dark nuclear spin state in lead halide perovskites”  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' EXPERIMENTAL DETAILS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Photoluminescence and time-resolved Kerr  ellipticity  A photo of the hybrid organic-inorganic FAPbBr3 per-  ovskite crystal under study is shown in the inset of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' It is a crystal with a size of 5 × 5 × 2 mm3  and a glossy semi-transparent reddish color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Its low tem-  perature photoluminescence measured at T = 5 K has  maximum at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='1776 eV, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  In the time-resolved Kerr ellipticity (TRKE) measure-  ments the laser was tuned to the energy of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='191 eV, reso-  nant with the free exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The pump beam was helicity  modulated between σ+/σ− circular polarization at the  frequency of 50 kHz to avoid dynamical nuclear polar-  ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  A typical TRKE dynamics trace measured in  BV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='5 T magnetic field applied in the Voigt geome-  try is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' It is contributed by the co-  herent spin dynamics of charge carriers, which spins are  initially oriented by the circularly polarized pump and  then precess about the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' One can clearly  see two Larmor precession frequencies, corresponding to  electrons (e) and holes (h), which is the typical phe-  nomenology for lead halide perovskite crystals [S1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' We  have shown that the coherent spin dynamics in perovskite  semiconductors are provided by resident electrons and  holes, which are localized in spatially different sites [S2–  S4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The TRKE dynamics were fitted using the function:  TRKE = Ae cos (ΩL,et) exp (−t/T ∗  2,e)  + Ah cos (ΩLt) exp (−t/T ∗  2,h),  (S1)  as shown by the yellow dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Ae(h)  are the Kerr ellipticity amplitudes, reflecting the de-  grees of carrier spin polarization, ΩL = ghµBB/ℏ and  ΩL,e = geµBB/ℏ are the Larmor frequencies with the  carrier g-factors ge(h), and T ∗  2,e(h) are the ensemble spin  dephasing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' There we show the individual compo-  nents of the hole (blue line) and the electron (red line)  spin dynamics obtained form the fit with the parameters  gh = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4 and T ∗  2,h = 780 ps for the holes and ge = +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4  and T ∗  2,e = 1080 ps for the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The assignment of  the components is made based on the universal depen-  dence of their g-factors on the band gap energy that we  have published recently [S1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The time delay is fixed at the negative value of −75 ps,  marked by the arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S1(b), for the measurements  of the resonant spin amplification (RSA) and polarization  recovery curves (PRCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='8  1  Energy (eV)  Photoluminesence (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  FAPbBr3  0  500  1000  1500  0  Time (ps)  Kerr Ellipticity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  gh = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4 T2  = 780 ps  ge = +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4 T2  = 1076 ps  BV = 500 mT, T = 6 K  T = 5 K  (a)  (b)  5 mm  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (a) Photoluminescence spectrum of the FAPbBr3  crystal measured under continuous-wave excitation with the  photon energy of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='06 eV at T = 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Inset shows a photo  of the studied crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (b) Example of the TRKE dynamics  (black line) at the pump-probe energy of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='191 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The pump  beam was helicity modulated between σ+/σ− circular polar-  ization at the frequency of 50 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Its fit with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (S1) is  shown by the yellow dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The electron and hole spin  precession components evaluated from the fit are shown below  by the red and blue lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Nuclear spin polarization in tilted magnetic field  Using constant pump helicity, additional measure-  ments of the nuclear spin polarization were performed  using the same scheme as for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Figures S2(a,b)  show that the Overhauser field for the electrons does not  depend on the strength of the external magnetic field,  as expected for dynamic nuclear spin polarization [S5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The Overhauser field for the holes decreases from 22 mT  at B = 150 mT to 12 mT at B = 250 mT and then  stays about constant at 10 mT up to the magnetic field  of 750 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Nevertheless, it remains small in compari-  son with its typical stochastic fluctuation in equilibrium,  ∆B = 40 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  In the angular dependence of the Overhauser field for  both electron and hole spins, despite a certain scattering  of the achieved field value, a dependence according to a  S2  0  10  20  30  40  50  BN,h (mT)  0  200  400  600  800  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='8  1  B (mT)  BN,e (mT)  θ = 30°  0  5  10  15  20  25  BN,h (mT)  0  30  60  90  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='8  1  Angle θ (°)  BN,e (mT)  B = 125 mT  0  30  60  90  0  5  10  15  20  25  Angle θ (°)  BN,h (mT)  B=  125 mT  250 mT  500 mT  750 mT  (a)  (b)  (c)  (d)  (e)  hole  hole  hole  electron  electron  ΔB  θ = 30°  B = 125 mT  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (a,b) Overhauser field (BN,e(h)) experienced by the hole and electron spins as function of the magnetic field tilted by  θ = 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Lines are guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (c,d) Overhauser fields as function of the tilting angle of the external magnetic field with  strength B = 125 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Lines are fits according to BN ∝ cos (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (e) Same as (c) for different magnetic field strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' For all  panels T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6 K and the pump power is 8 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Isotopes with nonzero nuclear spin in FAPbBr3  with FA = CH2(NH)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Nuclear abundance Spin NMR γN  isotope  [%]  I  [MHz/T]  207Pb  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='10  1/2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='882  79Br  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='69  3/2  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='704  81Br  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='31  3/2  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='538  1H  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='98  1/2  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='577  13C  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='07  1/2  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='708  14N  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='63  1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='078  cosine-function is found, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S2(c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The angular  dependence of the Overhauser field experienced by the  holes for several magnetic fields is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S2(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' PRC at constant helicity pumping  In order to analyze the limit of fmod → 0, we performed  additional PRC measurements for constant pump helic-  ity, σ+ or σ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Figure S3(a) shows the two corresponding  Kerr ellipticity dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' They merge with each other  when reversing amplitude and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  In par-  ticular, when their amplitudes are summed, the narrow  peak around BF = 0 becomes fully compensated and  cannot be seen [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S3(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' We plot also the difference  of the signals, which corresponds to the nuclear spin in-  ertia signal at fmod = 0 and compare it with the case of  fmod = 20 Hz in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Generally, the signals are  similar, however the latter shows two weak maxima at  BF ≈ ±10 mT around the zero-field dip in agreement  with the theoretical modeling [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4(f), green and blue  curves].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' PRC power dependence  Here we present data for the pump power dependence  of the PRC signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' In Figures S4(a,b) we compare the  PRC curves for two different regimes: when the DNSS  is formed (fmod = 20 Hz) and in absence of the DNSS  (fmod = 5 kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  All curves were evaluated by fits with respect to  their amplitude, the ratio of broad dip (Ab) to the  high field (Ahf) amplitude and the dip width, shown in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S4(b,c,d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' To start with 5 kHz mod-  ulation frequency (full symbols), with increasing pump  power the amplitude of the PRC dip and the high field  amplitude rise linearly before showing a saturation trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The narrow dip shows up as a minor peak with magni-  tude and width near the noise level so that is cannot  be evaluated reliably for this frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The ratio of  the PRC dip to the amplitude at high magnetic fields  (≈ 150 mT) [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S4(c)] decreases slightly with increas-  ing pump power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The width stays about constant in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S4(d) for 5 kHz modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  In contrast, for the low modulation frequency of 20 Hz,  the PRC shape changes drastically with varying pump  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Again, the amplitudes at high field (Ahf), of the  broad dip (Ab) and of the narrow dip (An) rise nonlin-  early with a tendency to saturate with increasing pump  S3  σ+  σ-  −100  0  100  −200  0  200  BF (mT)  Kerr Ellipticity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  −100  0  100  −200  0  200  BF (mT)  Kerr Ellipticity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  σ++ σ-  −100  0  100  200  300  400  500  BF (mT)  Kerr Ellipticity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  σ+- σ-  σ+/σ-  fmod=20 Hz  σ+/σ- - alternating  σ+-only  σ--only  (a)  (b)  (c)  (d)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (a) PRC with constant pump helicity σ+ (blue) or σ− (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6 K, pump power 6 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (b) Sum of the two  signals from panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (c) Difference of the two signals from panel (a) (green), corresponding to the nuclear spin inertia signal  at fmod = 0, and nuclear spin inertia signal at fmod = 20 Hz (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (d) Scheme of the applied excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  −100  −50  0  50  100  150  0  100  200  300  BF (mT)  Kerr Ellipticity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='2 mW  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6 mW  1 mW  2 mW  P=6 mW  −100  −50  0  50  100  150  0  100  200  300  BF (mT)  Kerr Ellipticity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='5 mW  1 mW  2 mW  P=6 mW  (a)  (c)  20 Hz  5 kHz  0  1  2  3  4  5  6  7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='5  1  Pump Power (mW)  |Ab/Ahf| (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  0  1  2  3  4  5  6  7  0  20  40  60  80  100  120  Pump Power (mW)  Width (mT)  An  0  1  2  3  4  5  6  7  −200  0  200  400  Pump Power (mW)  Kerr Ellipticity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=')  Ahf  Ab  (e)  (b)  (d)  x10  20 Hz  5 kHz  20 Hz  5 kHz  20 Hz  5 kHz  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' PRC at different pump powers P, for (a) fmod = 20 Hz and (b) fmod = 5 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The curves are not shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (c,d,e)  Parameters evaluated from curves (a,b), full symbols for fmod = 5 kHz and open symbols for fmod = 20 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (c) Pump power  dependence of the amplitudes of the broad Ab and narrow An dips (red & dark red diamonds and green triangles, respectively);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  high field (150 mT) values of the amplitudes Ahf (blue & light blue squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (d) Pump power dependence of the ratio of  amplitude of the broad dip and its high field value (Ab/Ahf) for fmod = 20 Hz (red) and fmod = 5 kHz (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (e) Pump power  dependencies of the HWHM of the broad (open orange diamonds) and narrow (open green triangles) PRC dips at fmod = 20 Hz,  and of the broad dip at fmod = 5 kHz (filled orange diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' For all panels T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' However, by contrast to the previous case, the  ratio Ab/Ahf here drops in a nonlinear fashion much  stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Further, the width of the broad peak broad-  ens significantly with excitation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The width of  the narrow peak remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Note, in both cases  of 5 kHz and 20 Hz, the power dependencies of the spin  signals match each other at the high magnetic field of  BF = ±150 mT, when the hyperfine interaction plays a  minor role, see the blue symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Overall the PRC pump power dependence for the low  time  00  timetime  0S4  modulation frequency [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S4(a)] is similar to the fre-  quency dependence, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' As the DNSS  formation rate is determined by the flux of angular mo-  mentum to the nuclear spin system, this shows that it is  also proportional to the flux of angular momentum from  exciting photons to a hole in agreement with the theo-  retical prediction, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (S19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' DNSS at zero magnetic field  Surprisingly, we do not observe the hole spin polariza-  tion recovery with decreasing modulation frequency at  zero magnetic field, which evidences the fragility of the  DNSS in the field range ≲ 1 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The exact reason for the  DNSS not to form at zero field is unclear so far, but we  note that the ODNMR in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 3 has the same linewidth  of about 1 mT as the narrow PRC dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  This suggests  that the DNSS is destroyed by the non-secular part of  the nuclear dipole-dipole interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Application of the  magnetic field with strengths exceeding that of the local  fields suppresses this interaction and stabilizes the DNSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' SIMULATION OF DNSS FORMATION  We describe the nuclear spin dynamics in the pump-  probe experiments using the central spin box model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The  Hamiltonian of the system reads  H = AIS + ℏΩLSz,  (S2)  where S is the hole spin, A is the joint hyperfine inter-  action constant for all N nuclei in the hole localization  volume, I = �N  n=1 In is the total nuclear spin composed  of the individual spins In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The hyperfine interaction is  assumed here to be isotropic because of the dominant S-  type Bloch wave functions at the top of the valence band  in perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The same description is valid for electrons  in conventional GaAs-like semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' However, in  this work on perovskites the hyperfine interaction of the  holes plays the main role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' PRC without DNSS  In equilibrium or at high enough polarization modu-  lation frequencies, the nuclear spin distribution is Gaus-  sian:  F(BN) = exp(−2B2  N/∆2  B)  (  �  π/2∆B)3  (S3)  with the parameter  ∆B =  √  NA  ghµB  ,  (S4)  determining the dispersion of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' In these  conditions, the PRC is described by [S10]  TRKE = 1 − 2  3  ∆2  B  ∆2  B + B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  (S5)  One can see from this expression that the HWHM of the  PRC is given by ∆B, which in combination with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (S4)  yields  N = (ghµB∆B/A)2,  (S6)  as used in the main text to determine the number of  nuclei N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Nuclear spin dynamics  To describe the nuclear spin dynamics, we note that  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (S2) the total nuclear spin I is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' For N  nuclear spins 1/2, the number of possible realizations of  spin I is CN/2−I  N  − CN/2−I−1  N  , where Ck  n is the binomial  coefficient, and the probability to find the spin I is  PI = (2I + 1)(CN/2−I  N  − CN/2−I−1  N  )  2N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  (S7)  The eigenstates of the system can be labeled by the total  angular momentum Fz = Iz + Sz, and have the form  Ψ±(Fz) = A±(Fz) |Fz + 1/2, ↓⟩ + B±(Fz) |Fz − 1/2, ↑⟩ ,  (S8)  where ↑ (↓) corresponds to a hole with spin Sz = +1/2  (−1/2) and  A+(Fz) = −B−(Fz) =  Ωx  �  2Ω(Ω + Ωz)  ,  (S9a)  A−(Fz) = B+(Fz) =  �  Ω + Ωz  2Ω  ,  (S9b)  with  Ωx = A  ℏ  �  I(I + 1) − F 2z + 1/4,  (S10a)  Ωy = 0,  (S10b)  Ωz = ΩL + A  ℏ Fz  (S10c)  and  Ω = |Ω| = 1  ℏ  �  A2I(I + 1) + ℏ2Ω2  L + 2AFzℏΩL + A2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  (S11)  S5  For a given Iz after initialization of the hole in the  spin-up state, the wave function is |Iz, ↑⟩, which can be  represented as  |Iz, ↑⟩ = αΨ+(Iz + 1/2) + βΨ−(Iz + 1/2),  (S12)  where  α2 =  A2  −(Iz + 1/2)  A2  +(Iz + 1/2) + A2  −(Iz + 1/2),  (S13a)  β2 =  A2  +(Iz + 1/2)  A2  +(Iz + 1/2) + A2  −(Iz + 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  (S13b)  Then the coherence between the states Ψ+(Iz + 1/2)  and Ψ−(Iz + 1/2) is lost [S11] on the timescale T ∗  2,h ∼  ℏ/(AI) ≪ TR, where TR is the repetition period of the  pump pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' As a result, the nuclear spin component Iz  increases by unity with the probability  v(Iz + 1/2) = |αA+(Iz + 1/2)|2 + |βA−(Iz + 1/2)|2  = 2A2  −(Iz + 1/2)A2  +(Iz + 1/2)  A2  −(Iz + 1/2) + A2  +(Iz + 1/2) = 1  2  �Ωx  Ω  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  (S14)  Here the argument Iz + 1/2 is used to denote the total  angular momentum component of the nuclei and the hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  In the limit of classical I, this reduces to v(Iz + 1/2) =  A2(I2  x + I2  y)/(2|AI + ℏΩL|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' For a hole with spin down,  the nuclear spin projection reduces by unity with the  probability v(Iz − 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  In the limit of many nuclear spins, N ≫ 1, it is conve-  nient to introduce the distribution function g(x), where  x = Iz/I, which is normalized such that  1  �  −1  g(x)dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  (S15)  It satisfies the Fokker-Planck equation  I2TR  ∂g  ∂t = Pe(t)I ∂  ∂x (vg) + ∂  ∂x  �v  2  ∂g  ∂x  �  ,  (S16)  where v = (1 − x2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Introducing τ = t/(I2TR) and  Π(t) = Pe(t)I, we obtain  ∂g  ∂τ = Π(t) ∂  ∂x (vg) + ∂  ∂x  �v  2  ∂g  ∂x  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  (S17)  As boundary condition, g(x) should be finite at x = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Nuclear spin inertia  To describe the nuclear spin inertia, we assume that  each pump pulse excites a trion with the probability Γ0  and after its recombination, the hole spin polarization  increases by Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' According to the optical selection rules,  the transverse hole spin components subsequently are  erased [S12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' This allows us to neglect the off-diagonal  components of the hole and nuclei density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' In this  case, the nuclear spin state is described by the probabil-  ities Pm for finding the spin I in the state with Iz = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  We note that after the hole spin decoherence, the hole  spin reduces by a factor of v(Fz), which provides an in-  crease of Iz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Thus, if the polarization modulation period  consists of Nmod pulses, the signal measured by the nu-  clear spin inertia method is given by [S13]  L =  ����Nmod/2  k=1  �I  m=−I Pm(k)v(m + 1/2)  ���  ����Nmod/2  k=1  exp (2πik/Nmod)  ���  ,  (S18)  where Pm(k) describes the distribution of Iz = m after  k pulses, starting from the beginning of the period, and  we assume that it is sufficient to consider only the first  half of the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  This formalism allows us to simulate the evolution of  the nuclear spin distribution function under pulsed exci-  tation with modulation of the light helicity and to cal-  culate the nuclear spin inertia signal as function of the  magnetic field and the modulation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Due to  the numerical difficulties with the boundary conditions  for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (S17), we have used a finite number of nuclei,  namely N = 1300, as estimated in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  To fit the frequency dependence of the PRC amplitude  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4(d), we have used Γ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='03 and Ph = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' The  latter parameter determines the saturation of the PRC  amplitude at low modulation frequencies, and the total  DNSS formation rate  ν0 = Γ0Ph/(TRN) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='9 ms−1  (S19)  determines the characteristic time scale of the nuclear  spin dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Thus, all parameters can be reliably de-  termined from the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Nuclear spin squeezing and entanglement  We characterize the shrinking of the nuclear spin dis-  tribution function along the transverse directions by the  Kitagawa and Ueda spin squeezing parameter ξs [S14,  S15], which is defined by  ξ2  s = 4  N ⟨I2  x⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  (S20)  During the simulation of the nuclear spin inertia, it can  be extracted as  ξ2  s = 4  N  �  I  �  m=−I  Pm  I(I + 1) − m2  2  �  ,  (S21)  S6  where the angular brackets denote the averaging over the  total nuclear spin I with probabilities (S7), and Pm sym-  bolizes Pm(k) with k = 1, for which the spin squeezing  reaches its maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The result of the calculation of the nuclear spin squeez-  ing degree is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4(e) by the solid line for the  same parameters as in panel (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' To find ξs from the ex-  perimental results, we assume that the nuclear spin dy-  namics obey the evolution according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Then,  following the same procedure as described above, we find  the DNSS that corresponds to the experimentally ob-  served amplitude of the PRC at the given frequency and  the calculated spin squeezing parameter for this state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4(e) by the dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  For low modulation frequencies we obtain ξs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='29,  which is limited by nuclear spin diffusion due to an in-  complete hole spin polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' In the limit of a perfect  nuclear spin alignment along the z axis, the largest pos-  sible spin squeezing is ξs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='21 because of the quantum  fluctuations of the transverse components of the total nu-  clear spin I ∼  √  N ∼ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' This situation can be called  the Heisenberg limit, and it is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='4 times smaller than  the experimentally reached value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The large nuclear spin squeezing suggests deep entan-  glement between nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  One of the strict evi-  dences of entanglement is the violation of the generalized  spin squeezing inequality  ⟨I2  x⟩ + ⟨I2  y⟩ + ⟨(Iz − ⟨Iz⟩)2⟩ ≥ N  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  (S22)  We note, that this represents a particular case of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (2)  with M = N, which means that there is at least any  entanglement in the nuclear spins system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  The PRC directly measures the transverse nuclear  spin components ⟨I2  x⟩ + ⟨I2  y⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  To check the violation  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (S22), we consider the upper limit for the longi-  tudinal nuclear spin fluctuations ⟨(Iz − ⟨Iz⟩)2⟩ ≤ N/4,  which corresponds to uncorrelated individual nuclear  spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' From the definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (S20) we obtain that  states with ξs ≤ 1/  √  2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='707 are entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  This  boundary is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 4(e) by the blue dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  For the lowest polarization modulation frequency of  fmod = 100 Hz, the PRC amplitude [L(Bmax) − L(B =  0)]/L(B = 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='054 corresponds to I2  x + I2  y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='054I2,  which after averaging over I yields ⟨I2  x + I2  y⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='04N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Combined with the upper boundary for the longitudinal  spin fluctuations we obtain  ⟨I2  x⟩ + ⟨I2  y⟩ + ⟨(Iz − ⟨Iz⟩)2⟩ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='29N,  (S23)  which violates Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (S22) almost by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' In  particular, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' (2) we find that the achieved DNSS  is at least equivalent to an M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='58N = 750-body en-  tanglement among the N = 1300 nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  This is a very conservative estimate mainly because of  the assumption of unsuppressed longitudinal spin fluctu-  ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' According to our model of DNSS formation, at  the lowest polarization modulation frequency the DNSS  steady state is reached, where the depth of entanglement  is limited by the conservation of I and by nuclear spin  diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' For this state, the calculation of the nuclear  spin fluctuations yields an M = 113-body entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  We recall that the smaller M is, the deeper is the entan-  glement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  ∗ Electronic address: smirnov@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='ioffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='ru  [S1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Kirstein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Yakovlev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Glazov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Zhukov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Kudlacik, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Kalitukha, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Sapega,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Dimitriev,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Semina,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Nestok-  lon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Ivchenko, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Kopteva, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Dirin,  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Nazarenko,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Kovalenko,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Baumann,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' H¨ocker, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Dyakonov, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Bayer, The Land´e fac-  tors of electrons and holes in lead halide perovskites:  universal dependence on the band gap, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Belykh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Glazov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Grigoryev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Hussain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Rautert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Dirin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Kovalenko, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Bayer, Coherent spin dynamics of  electrons and holes in CsPbBr3 perovskite crystals, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 10 , 673 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Zhukov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' H¨ocker,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Dyakonov, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Bayer, Spin Dynamics of Elec-  trons and Holes Interacting with Nuclei in MAPbI3 Per-  ovskite Single Crystals, ACS Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' 9, 1375 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  [S4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Kirstein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Yakovlev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Glazov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Evers,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Zhukov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Belykh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Kopteva, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Kud-  lacik, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Nazarenko, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Dirin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Kovalenko,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Bayer, Lead-Dominated Hyperfine Interaction  Impacting the Carrier Spin Dynamics in Halide Per-  ovskites, Advanced Materials 34, 2105263 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  [S5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Glazov, Electron and Nuclear Spin Dynamics  in Semiconductor Nanostructures (Oxford University  Press, Oxford, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  [S6] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Yugova, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Glazov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' Yakovlev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
+page_content='  Sokolova, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFJT4oBgHgl3EQfJSzf/content/2301.11460v1.pdf'}
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+arXiv:2301.00857v1  [math.FA]  2 Jan 2023
+TOTALLY POSITIVE FUNCTIONS AND GABOR FRAMES
+OVER RATIONAL LATTICES
+KARLHEINZ GR¨OCHENIG
+Abstract. We show that for an arbitrary totally positive function g ∈ L1(R)
+and αβ rational, the Gabor family {e2πiβltg(t − αk) : k, l ∈ Z} is a frame for
+L2(R), if and only if αβ < 1.
+1. Introduction
+Ever since J. von Neumann and D. Gabor claimed that the family of time-
+frequency shifts (phase-space shifts) {e2πilte−π(t−k)2 : k, l ∈ Z} spans L2(R), the
+spanning properties of time-frequency shifts have sparked the curiosity of mathe-
+maticians, physicists, and engineers. Modern versions of this problem replace the
+Gaussian by arbitrary generators and the time-frequency shift over the lattice Z2
+by general lattices or even non-uniform sets.
+To be specific, the time-frequency shift of g ∈ L2(R) by (x, ξ) ∈ R2 is defined as
+(1)
+MξTxf(t) = e2πiξtg(t − x) .
+Given lattice parameters α, β > 0 the set G(g, α, β) = {MβlTαkg : k, l ∈ Z} is called
+Gabor family over the lattice αZ × βZ with window g. Gabor analysis studies the
+question under which conditions on α, β, and g there exist constants A, B > 0,
+such that
+(2)
+A∥f∥2
+2 ≤
+�
+k,l∈Z
+|⟨f, MβlTαkg⟩|2 ≤ B∥f∥2
+2
+∀f ∈ L2(R) .
+If (2) holds, G(g, α, β) is called a Gabor frame. The set
+F(g) = {(α, β) ∈ R2
++ : G(g, α, β) is a frame }
+is called the frame set of g. Knowing F(g) amounts to a complete characterization
+of all (rectangular) lattices αZ × βZ that generate a Gabor frame with a given
+window g.
+Our understanding of the frame set is still very limited. The prototypical result
+is due to Lyubarski [28] and Seip-Wallsten [33,34] and determines the frame set of
+the Gaussian to be F(e−πt2) = {(α, β) ∈ R2
++ : αβ < 1}. This is usually formulated
+by saying that G(e−πt2, α, β) is a frame if and only if αβ < 1. Subsequently, further
+examples of frame sets were found by Janssen and Strohmer [22, 23, 25]. It was
+2010 Mathematics Subject Classification.
+42C15 , 42C40 , 94A20, 42A82.
+Key words and phrases. Totally positive function, Gabor frame, spectral invariance, invertibil-
+ity of totally positive matrices.
+K. G. was supported in part by the project P31887-N32 of the Austrian Science Fund (FWF).
+1
+
+2
+KARLHEINZ GR ¨OCHENIG
+noted in [19] that all examples known until 2010 were based on a totally positive
+window g. This observation led to a more systematic approach to study the frame
+set for totally positive windows.
+To explain the state-of-the-art we recall that a measurable function g on R
+is totally positive, if for every n ∈ N and every two sets of increasing numbers
+x1 < x2 < · · · < xn and y1 < y2 < · · · < yn the matrix
+�
+g(xj − yk)
+�
+j,k=1,...,n has
+non-negative determinant:
+(3)
+det(g(xj − yk)
+�
+j,k=1,...,n ≥ 0 .
+If in addition g is integrable, then g possesses already exponential decay [31,
+Lemma 2]. Often an integrable totally positive function is called a Polya frequency
+function, but we will not make this distinction here.
+By a deep theorem of Schoenberg [31] the Fourier transform of an integrable
+totally positive function g possesses the factorization
+(4)
+ˆg(ξ) = ce−γξ2e2πiνξ
+N
+�
+j=1
+(1 + 2πiνjξ)−1e−2πiνjξ
+with c > 0, ν, νj ∈ R, γ ≥ 0, N ∈ N ∪ {∞} and 0 < γ + �
+j ν2
+j < ∞.
+Among the totally positive functions are the Gaussian e−γt2 for γ > 0 and the
+one-sided exponential functions ηγ(t) = e−γtχR+(γt) with γ ∈ R, γ ̸= 0.
+For the subclass of totally positive functions, for which the factorization (4) is a
+finite product, the results of [18,19] can be summarized as follows:
+Theorem 1.1. Let g ∈ L2(R) be a totally positive function with a finite factoriza-
+tion (4) with N ∈ N other than the one-sided exponentials ηγ, then
+F(g) = {(α, β) ∈ R2
++ : αβ < 1} .
+In other words, the lattice αZ × βZ generates a frame with totally positive
+window g, if and only if αβ < 1.
+Remarkly the proof of Theorem 1.1 required different methods for the cases
+γ = 0 (no Gaussian factor) and γ > 0 (with Gaussian factor). In the former case
+total positivity and the Schoenberg-Whitney conditions were used, in the latter
+case only the factorization was used in conjunction with complex analysis.
+The frame set of the one-sided exponential ηγ is slightly different, since ηγ is not
+smooth enough and the Balian-Low theorem does not apply [20]. In this case the
+frame set is F(ηγ) = {(α, β) ∈ R2
++ : αβ ≤ 1}, as was already known by Janssen [22].
+Theorem 1.1 suggests that the characterization of Gabor frames holds for all
+totally positive generators. Our goal is to prove the following theorem which is a
+step towards the full conjecture formulated in [16].
+Theorem 1.2. Assume that g is totally positive other than a one-sided exponential
+function ηγ. Further assume that αβ is rational. Then G(g, α, β) is a frame for
+L2(R), if and only if αβ < 1.
+
+TOTALLY POSITIVE FUNCTIONS AND GABOR FRAMES OVER RATIONAL LATTICES 3
+Theorem 1.2 implies that the frame set F(g) is an open, dense
+subset of
+{(α, β) ∈ R2
++ : αβ < 1}. This follows from a deformation result of Feichtinger
+and Kaiblinger [11].
+Since every totally positive function is a limit of a totally positive with a finite
+factorization, one would hope that some kind of limiting procedure would work.
+However, so far all our “proofs” contained a gap. Except for the example of the hy-
+perbolic secant (eat +e−at)−1 [25], Theorem 1.2 is the first general statement about
+Gabor frames with a totally positive generator with an infinite factorization (4).
+In this paper we present a new approach that is independent of Schoenberg’s
+factorization (4) and uses only the definition of total positivity. The new proof
+idea comes from a deep theorem of [8] about invertibility of infinite totally positive
+matrices. We also use methods from spectral invariance of matrix algebras and the
+important fact that the Zak transform of a totally positive function has only one
+zero. For rational lattices this method leads also to a new proof of Theorem 1.1.
+To finish the discussion, we mention the superb work [3], which determines the
+frame set of rational functions (Hurwitz functions) g(t) = �N
+j=1
+aj
+t−iwj for wj ∈ C.
+If aj > 0 and wj > 0, then F(g) = {(α, β) ∈ R2
++ : αβ ≤ 1}, for general coefficients
+the frame set contains {(α, β) ∈ R2
++ : αβ ≤ 1} ∩ Qc. Interestingly and in contrast
+to our work, the authors show that all irrational lattices belong to the frame set!
+Other than this the complete frame set has been determined only for the char-
+acteristic function of an interval χ[0,1] [5] and the Haar function [6]. In contrast
+to the results for totally positive windows and rational functions, the frame set is
+extremely complicated and depends on subtle number theoretic properties of the
+product αβ.
+The case of irrational lattices remains open. There is some evidence that gener-
+ically a lattice αZ × βZ with irrational αβ generates a Gabor frame. Important
+results pointing in this direction are contained in [3] and in [6]. In each case an
+underlying dynamical system and some ergodicity are at work. Currently we do
+not know how this works for totally positive windows.
+Of course, much more is known and relevant about Gabor frames. As we are in-
+terested only in the complete characterization of Gabor frames for a given window,
+we refer to the surveys [15,17,20] and monographs [9,14] for the general theory of
+Gabor frames.
+The paper is organized as follows: In Section 2 we review the tools underly-
+ing Gabor analysis. These are Zak transform methods, Banach algebra methods
+concerning spectral invariance, and general characterizations of Gabor frames. All
+these characterizations amount to showing that some (infinite) matrix is invertible.
+In our approach the matrix G under consideration will a sub-matrix of the so-called
+pre-Gramian. This is a technical novelty. In Section 3 we show that this matrix
+is onto ℓ1(Z) by using heavily the properties of spectral invariance and a theorem
+from [8]. In Section 4 we will show that G is also one-to-one. It is only here that
+we will use the rationality of the lattice. Ultimately the proof boils down to the
+fact that a square matrix is invertible, if and only if it is onto.
+
+4
+KARLHEINZ GR ¨OCHENIG
+Acknowledgement. I am deeply indebted to Jose Luis Romero and Joachim
+St¨ockler for comments on earlier versions of this manuscript.
+2. Tools
+2.1. Reduction. Let gβ(x) = β−1/2g(x/β). Then G(g, αZ × βZ) is a frame, if and
+only if G(gβ, αβZ × Z) is a frame. It is immediate from (3) that the set of totally
+positive functions is invariant under dilations. Thus, if g is totally positive, then
+so is gβ. We may therefore always replace g by gβ, α by αβ and β by 1 and thus
+assume without generality that β = 1.
+We note that we may always assume that αβ < 1 and, after the reduction, that
+α < 1, β = 1 because of the density theorem for Gabor frames [20].
+2.2. Spectral invariance. Spectral invariance refers to the phenomenon that an
+invertible, infinite matrix with sufficiently strong off-diagonal decay possesses an
+inverse with the same off-diagonal decay. This topic has a long history with many
+contributions and variations. See [1, 2, 35–38] for a sample of contributions and
+the survey [15].
+The following version can be attributed to Shin-Sun [35] and
+Tessera [38, Thm. 1.8].
+Proposition 2.1 (Stability Theorem). Let G = (Gkl)k,l∈Z be a bi-infinite matrix
+with off-diagonal decay
+|Gkl| ≤ C(1 + |k − l|)−σ
+k, l ∈ Z ,
+for some σ > 1.
+(i) Spectral invariance: If G is invertible on some ℓp0(Z), then G is invertible on
+all ℓp(Z) for 1 ≤ p ≤ ∞.
+(ii) Stability: If G is stable on some ℓp0(Z), i.e., if G satisfies
+∥Gc∥p0 ≥ A∥c∥p0
+for all c ∈ ℓp0(Z),
+then G is stable on all ℓp(Z) for 1 ≤ p ≤ ∞.
+We will need a slight variation of the stability (ii) for surjective maps.
+Proposition 2.2. Let G = (Gkl)k,l∈Z with off-diagonal decay
+|Gkl| ≤ C(1 + |k − l|)−σ
+k, l ∈ Z ,
+for some σ > 1. If G maps ℓ∞(Z) onto ℓ∞(Z), then G also maps ℓ1(Z) onto ℓ1(Z)
+Proof. Note that the off-diagonal decay implies that G defines a bounded operator
+on all ℓp(Z), 1 ≤ p ≤ ∞.
+By the closed range theorem [30, p. 102], the adjoint operator (matrix) on the
+pre-dual G∗ : ℓ1(Z) → ℓ1(Z) is one-to-one with closed range in ℓ1(Z). Consequently,
+G∗ satisfies an inequality of the form
+(5)
+∥G∗c∥1 ≥ A∥c∥1
+for all c ∈ ℓ1(Z)
+with a positive constant A > 0.
+
+TOTALLY POSITIVE FUNCTIONS AND GABOR FRAMES OVER RATIONAL LATTICES 5
+The stability part of Proposition 2.1 then implies that (5) holds for all p-norms,
+in particular, for some A′ > 0 we have
+∥G∗c∥∞ ≥ A′∥c∥∞
+c ∈ ℓ∞(Z) .
+Since G∗ is one-to-one with norm-closed range in ℓ∞(Z), we conclude that G :
+ℓ1(Z) → ℓ1(Z) must be onto ℓ1(Z).
+2.3. Characterizations of Gabor frames. We use the following characterization
+of Gabor frames that goes back to Janssen [21] and the duality theory of Gabor
+frames [29]. This is one of the most successful criteria to verify that a Gabor system
+over a rectangular lattice generates a frame and has been applied many times. We
+refer to [17] for a survey of the known characterizations of Gabor frames and some
+of their uses.
+Theorem 2.3. Assume that g : R → R is continuous and satisfies the condition
+�
+k∈Z supx∈[0,1] |g(x + k)| < ∞. Then the following are equivalent.
+(i) The family G(g, αZ × Z) is a frame for L2(R).
+(ii) For every x ∈ [0, 1), there exist Ax, Bx > 0 such that
+(6)
+Ax∥c∥2
+2 ≤
+�
+j∈Z
+���
+�
+k∈Z
+ckg(x + αj − k)
+���
+2
+≤ Bx∥c∥2
+2
+for all c ∈ ℓ2(Z).
+(iii) There exist A, B > 0 such that, for all x ∈ [0, 1),
+A∥c∥2
+2 ≤
+�
+j∈Z
+���
+�
+k∈Z
+ckg(x + αj − k)
+���
+2
+≤ B∥c∥2
+2
+for all c ∈ ℓ2(Z).
+The matrix P(x) =
+�
+g(x + αj − k)
+�
+j,k∈Z is often called the pre-Gramian of the
+Gabor family.
+Using spectral invariance, we deduce the following sufficient condition for Gabor
+frames.
+Proposition 2.4. Let g be a function on R with polynomial decay |g(t)| ≤ C(1 +
+|t|)−σ for some σ > 1. Assume that for every x ∈ R the set x + αZ contains a
+subset {k + δk : k ∈ Z} with the perturbation δk = δk(x) depending on x, such
+that the matrix G with entries Gkl = g(k + δk − l) is invertible on ℓ∞(Z). Then
+G(g, α, 1) is a Gabor frame.
+Proof. The decay of g implies that G has sufficient off-diagonal decay to apply
+Proposition 2.1(i). Thus spectral invariance asserts that the invertibility of G on
+ℓ∞(Z) implies that G is also invertible on ℓ2(Z). Since x + αZ ⊇ {k + δk : k ∈ Z},
+we have
+�
+j∈Z
+�� �
+l∈Z
+clg(x + αj − l)|2 ≥
+�
+k∈Z
+�� �
+l∈Z
+clg(k + δk − l)|2
+= ∥Gc∥2
+2 ≥ Ax∥c∥2
+2 .
+The upper bound in (6) always holds due to the decay of g. This holds for all x,
+therefore the condition (ii) of Theorem 2.3 implies that G(g, α, 1) is a frame.
+
+6
+KARLHEINZ GR ¨OCHENIG
+Proposition 2.4 says that a suitable submatrix of the pre-Gramian P(x) is in-
+vertible. Proposition 2.4 is related to the characterizations via sampling in shift-
+invariant spaces that figure prominently in [18]. (In an equivalent formulation,
+every set x + αZ contains a perturbation of Z that is interpolating in the shift-
+invariant space associated to g.)
+To prove the main theorem (Theorem 1.2), we will verify the conditions of Propo-
+sition 2.4.
+2.4. The Zak transform. We recall the definition of the Zak transform Zg(x, ξ) =
+�
+k∈Z g(x − k)e2πikξ of a function g ∈ L1(R). More generally, the Zak transform
+with period p > 0 is defined by
+Zpg(x, ξ) =
+�
+k∈Z
+g(x − pk)e2πipkξ .
+The Zak transform is 1
+p-periodic in the second variable and quasi-periodic in the
+first variable, i.e., for all x, ξ ∈ R
+Zpf
+�
+x + pk, ξ + l
+p
+�
+= e2πipkξZpg(x, ξ) .
+Zak transforms of totally positive function possess only few zeros. This important
+property was discovered only recently [26,27,39] and is crucial for our arguments.
+Proposition 2.5. The Zak transform of a totally positive function g possesses a
+unique zero in [0, 1)2 that is assumed at (x0, 1/2) for some x0 ∈ [0, 1).
+3. Surjectivity.
+For the proof of Theorem 1.2 we verify the sufficient condition of Proposition 2.4
+for a totally positive function g. First we prove the surjectivity. Notice that the
+arguments in this section hold for arbitrary values of α ∈ (0, 1).
+Proposition 3.1. Let g be an arbitrary totally positive function in L1(R) and
+assume that its Zak transform has its unique zero at (x0, 1/2). Let ǫ ∈ (0, 1/2),
+M ∈ Z, and (δk)k∈Z ⊆ [x0 + M − 1 + ǫ, x0 + M − ǫ] be a sequence of perturbations.
+Then the matrix G with entries Gkl = g(k + δk − l), k, l ∈ Z, maps ℓ∞(Z) onto
+ℓ∞(Z) and ℓ1(Z) onto ℓ1(Z).
+Proof. For the proof we use the following fundamental result of de Boor, Friedland,
+and Pinkus [8, Cor. 1]. Assume that G is an totally positive matrix that is bounded
+on ℓ∞(Z)1. If the range of G contains a uniformly alternating vector, then G is
+onto ℓ∞(Z).
+This result is the key tool in our proofs. Related results on the
+invertibility of infinite totally positive matrices can be found in [4,7,10].
+Uniformly alternating means that we need to find a sequence c ∈ ℓ∞(Z) such
+that u = Gc satisfies u(k)u(k + 1) < 0 for all k ∈ Z and infk∈Z |u(k)| > 0.
+1The original results are formulated for sign-regular matrices and more general index sets.
+
+TOTALLY POSITIVE FUNCTIONS AND GABOR FRAMES OVER RATIONAL LATTICES 7
+To accomplish this, we take cl = (−1)l and compute (Gc)k. This is
+(Gc)k =
+�
+l∈Z
+g(k + δk − l)(−1)l
+= Zg(k + δk, 1
+2)
+= (−1)kZg(δk, 1
+2) ,
+where the last identity comes from the quasi-periodicity of Zg. Since Zg(x, 1
+2) is
+real-valued and by assumption Zg(x, 1
+2) does not have any zeros on the interval
+[x0 − 1 + ǫ, x0 − ǫ], the numbers Zg(δk, 1
+2) all have the same sign and |Zg(δk, 1
+2)| ≥
+ν > 0 for all k. Consequently u = Gc is uniformly alternating.
+To apply the result of [8], we observe that the matrix G with entries g(k +δk −l)
+is indeed totally positive in the sense that all minors of G of arbitrary dimension
+have a non-negative determinant. But this follows from the definition (3) of total
+positivity.
+Thus by [8, Cor. 1] the matrix G is onto ℓ∞(Z). By Proposition 2.2 G is also
+onto ℓ1(Z).
+To apply the above proposition, we need to select a suitable subset of x + αZ.
+This can always be done with the following lemma.
+Lemma 3.2. Let α < 1, ǫ < (1 − α)/2, M ∈ Z, x0 ∈ [0, 1] fixed, and x ∈ [0, 1]
+arbitrary.
+(i) Then x + αZ contains a subset {k + δk : k ∈ Z} with δk ∈ [x0 + M − 1 +
+ǫ, x0 + M − ǫ] for all k ∈ Z.
+(ii) If α is rational, α = p/q, then the perturbation δk can be chosen to be p-
+periodic, δk+np = δk for all k, n ∈ Z.
+Proof. (i) For arbitrary k ∈ Z the interval [k + x0 − 1 + ǫ, k + x0 − ǫ] has length
+1 − 2ǫ > α and thus contains at least one point of x + αZ, which we can write as
+k + δk.
+(ii) Let α = p/q ∈ Z. For l = 0, . . . , p − 1 choose an integer jl ∈ {0, . . . , q − 1}
+such that
+x + p
+qjl ∈ [l + x0 − 1 + ǫ, l + x0 − ǫ]
+and set
+δl = x + p
+qjl − l .
+Clearly, 0 ≤ j0 < j1 < · · · < jp−1 < q. Now let m = l + np with l ∈ [0, p − 1] ∩ Z
+and n ∈ Z. Then x+ p
+q(jl +nq) = x+ p
+qjl +np ∈ [l+np+x0−1+ǫ, l+np+x0 −ǫ] =
+[m+x0−1+ǫ, m+x0−ǫ], and δm = x0+ p
+q(jl+nq)−m = x0+ p
+q(jl+nq)−(l+np) = δl.
+Thus δk is p-periodic.
+4. Injectivity.
+For the surjectivity of G we have used the fact that g is totally positive, but
+nothing about the parameter α.
+Now we impose in addition that α = p/q is
+
+8
+KARLHEINZ GR ¨OCHENIG
+rational. In this case the action of G is unitarily equivalent to a p × p-matrix-
+valued function. This facilitates the analysis, as for square matrices surjectivity is
+equivalent to injectivity and to invertibility.
+Let again G be the matrix with entries Gkl = g(k + δk − l). Since α = p/q ∈ Q,
+the sequence of perturbations can be chosen to be p-periodic by Lemma 3.2. We
+now use the periodicity of δk to bring in a variant of the Zibulski-Zeevi matrices [41].
+We write k = r +mp and l = s+np with r, s = 0, 1, . . . , p−1, and m, n ∈ Z. Then
+δk = δr and for all m ∈ Z and r = 0, 1, . . . , p − 1 we have
+(Gc)k =
+�
+l∈Z
+clg(k + δk − l)
+=
+p−1
+�
+s=0
+�
+n∈Z
+cs+npg(r + mp + δr − s − np) = dr+mp .
+(7)
+For fixed r, s the sum over n is a convolution, so we may take Fourier series and ob-
+tain a matrix-valued equation of functions. Write xs(ξ) = �
+n∈Z cs+pne2πinpξ for the
+Fourier series of the coefficient in the residue class s and x(ξ) = (x0(ξ), . . . , xp−1(ξ))
+for the associated vector-valued function. Likewise yr(ξ) = �
+m∈Z dr+pme2πimpξ and
+y(ξ) = (y0(ξ), . . . , yp−1(ξ)). The Fourier series of the sequence involving g is pre-
+cisely the Zak transform of g, namely,
+�
+n∈Z
+g(r + δr − s − np)e2πinpξ = Zpg(r + δr − s, ξ) .
+Consequently, after taking Fourier series, (7) can be recast as
+(8)
+p−1
+�
+s=0
+xs(ξ)Zpg(r + δr − s, ξ) = yr(ξ)
+for r = 0, 1, . . . , p − 1 .
+Since c ∈ ℓ1(Z) and G is bounded on ℓ1(Z), (8) holds pointwise for all ξ ∈ [0, 1/p].
+Let A(ξ) be the p × p-matrix-valued function with entries
+Ars(ξ) = Zpg(r + δr − s, ξ)
+r, s = 0, 1, . . . , p − 1 .
+This matrix is related to the Zeevi-Zibulski matrix that occurs in many criteria for
+Gabor frames over rational lattices [17, 40, 41]. The pointwise identity (8) can be
+written as
+(9)
+A(ξ)x(ξ) = y(ξ) .
+We now translate the surjectivity of the infinite matrix G to a property of the
+matrix-valued function det A(ξ).
+Proposition 4.1. Assume that α = p/q is rational, |g(t)| ≤ C(1 + |t|)−σ for some
+C, σ > 1, and that G maps ℓ1(Z) onto ℓ1(Z). Then G is one-to-one on ℓ1(Z) and
+thus invertible on ℓ1(Z).
+Proof. The surjectivity of G on ℓ1(Z) implies in particular that every vector d =
+(d0, d1, . . . , dp−1) ∈ Cp, i.e, d ∈ ℓ1(Z), dk = 0 for k < 0 and k ≥ p is in the
+range of G. In this case yr(ξ) = �
+m∈Z dr+pme2πimpξ = dr and the vector-valued
+
+TOTALLY POSITIVE FUNCTIONS AND GABOR FRAMES OVER RATIONAL LATTICES 9
+function y(ξ) = d is constant. Due to the surjectivity of G there exists a sequence
+c = c(d) ∈ ℓ1(Z) with associated Fourier series x(d)(ξ), such that Gc = d. In view
+of (9)
+A(ξ)x(d)(ξ) = d
+for all ξ ∈ [0, 1/p] .
+Since d ∈ Cp was arbitrary, this means that for fixed ξ the p × p-matrix is onto Cp.
+Consequently, A(ξ) is one-to-one and thus invertible for every ξ ∈ [0, 1/p].
+Now assume that Gc = 0 for some c ∈ ℓ1(Z). Then A(ξ)x(ξ) = 0 for all ξ ∈ [0, 1].
+Since A(ξ) is invertible, this implies that x(ξ) ∈ Cp must be the zero vector for all
+ξ. Consequently, x ≡ 0 and thus c = 0. We have proved that G is one-to-one on
+ℓ1(Z), whence G is invertible on ℓ1(Z).
+We finally summarize the steps that yield the proof of our main theorem.
+Proof of Theorem 1.2. By Lemma 3.2 we can extract a set {k + δk : k ∈ Z} from
+every x + αZ, such that δk avoids the zero x0 of the Zak transform Zg. Then by
+Proposition 3.1 the matrix G =
+�
+g(k + δk − l)
+�
+k,l∈Z is onto ℓ1(Z).
+For rational values of α the matrix G is also one-to-one on ℓ1(Z) by Proposi-
+tion 4.1. Consequently, G is invertible on ℓ1(Z).
+Finally, the spectral invariance guaranteed by Proposition 2.1(i) implies that G
+is invertible on ℓ∞(Z). This is precisely the sufficient condition of Proposition 2.4
+to guarantee that G(g, α, 1) is a frame.
+We are done!
+Alternatively, one could give Proposition 4.1 a more structural touch. We use
+Tp for the translation operator on sequences defined by (Tpc)k = ck−p and observe
+that the underlying matrix G of Proposition 4.1 commutes with Tp.
+Proposition 4.2. Let G = (Gkl)k,l∈Z be a bi-infinite matrix with off-diagonal decay
+|Gkl| ≤ C(1 + |k − l|)−σ
+k, l ∈ Z ,
+for some σ > 1. Assume that G commutes with the translation operator Tp for
+some p ∈ N. If G is onto ℓ1(Z), then G is one-to-one on ℓ1(Z) and thus invertible.
+The proof is similar to the proof of Proposition 4.1.
+References
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+TOTALLY POSITIVE FUNCTIONS AND GABOR FRAMES OVER RATIONAL LATTICES11
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+[40] Y. Y. Zeevi, M. Zibulski, and M. Porat. Multi-window Gabor schemes in signal and im-
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+Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1,
+A-1090 Vienna, Austria
+Email address: karlheinz.groechenig@univie.ac.at
+
diff --git a/T9AyT4oBgHgl3EQf8fpt/content/tmp_files/load_file.txt b/T9AyT4oBgHgl3EQf8fpt/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf,len=672
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='00857v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='FA]  2 Jan 2023  TOTALLY POSITIVE FUNCTIONS AND GABOR FRAMES  OVER RATIONAL LATTICES  KARLHEINZ GR¨OCHENIG  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' We show that for an arbitrary totally positive function g ∈ L1(R)  and αβ rational, the Gabor family {e2πiβltg(t − αk) : k, l ∈ Z} is a frame for  L2(R), if and only if αβ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Introduction  Ever since J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' von Neumann and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Gabor claimed that the family of time-  frequency shifts (phase-space shifts) {e2πilte−π(t−k)2 : k, l ∈ Z} spans L2(R), the  spanning properties of time-frequency shifts have sparked the curiosity of mathe-  maticians, physicists, and engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Modern versions of this problem replace the  Gaussian by arbitrary generators and the time-frequency shift over the lattice Z2  by general lattices or even non-uniform sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  To be specific, the time-frequency shift of g ∈ L2(R) by (x, ξ) ∈ R2 is defined as  (1)  MξTxf(t) = e2πiξtg(t − x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Given lattice parameters α, β > 0 the set G(g, α, β) = {MβlTαkg : k, l ∈ Z} is called  Gabor family over the lattice αZ × βZ with window g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Gabor analysis studies the  question under which conditions on α, β, and g there exist constants A, B > 0,  such that  (2)  A∥f∥2  2 ≤  �  k,l∈Z  |⟨f, MβlTαkg⟩|2 ≤ B∥f∥2  2  ∀f ∈ L2(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  If (2) holds, G(g, α, β) is called a Gabor frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' The set  F(g) = {(α, β) ∈ R2  + : G(g, α, β) is a frame }  is called the frame set of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Knowing F(g) amounts to a complete characterization  of all (rectangular) lattices αZ × βZ that generate a Gabor frame with a given  window g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Our understanding of the frame set is still very limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' The prototypical result  is due to Lyubarski [28] and Seip-Wallsten [33,34] and determines the frame set of  the Gaussian to be F(e−πt2) = {(α, β) ∈ R2  + : αβ < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' This is usually formulated  by saying that G(e−πt2, α, β) is a frame if and only if αβ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Subsequently, further  examples of frame sets were found by Janssen and Strohmer [22, 23, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' It was  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  42C15 , 42C40 , 94A20, 42A82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Totally positive function, Gabor frame, spectral invariance, invertibil-  ity of totally positive matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' was supported in part by the project P31887-N32 of the Austrian Science Fund (FWF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  1  2  KARLHEINZ GR ¨OCHENIG  noted in [19] that all examples known until 2010 were based on a totally positive  window g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' This observation led to a more systematic approach to study the frame  set for totally positive windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  To explain the state-of-the-art we recall that a measurable function g on R  is totally positive, if for every n ∈ N and every two sets of increasing numbers  x1 < x2 < · · · < xn and y1 < y2 < · · · < yn the matrix  �  g(xj − yk)  �  j,k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=',n has  non-negative determinant:  (3)  det(g(xj − yk)  �  j,k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=',n ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  If in addition g is integrable, then g possesses already exponential decay [31,  Lemma 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Often an integrable totally positive function is called a Polya frequency  function, but we will not make this distinction here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  By a deep theorem of Schoenberg [31] the Fourier transform of an integrable  totally positive function g possesses the factorization  (4)  ˆg(ξ) = ce−γξ2e2πiνξ  N  �  j=1  (1 + 2πiνjξ)−1e−2πiνjξ  with c > 0, ν, νj ∈ R, γ ≥ 0, N ∈ N ∪ {∞} and 0 < γ + �  j ν2  j < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Among the totally positive functions are the Gaussian e−γt2 for γ > 0 and the  one-sided exponential functions ηγ(t) = e−γtχR+(γt) with γ ∈ R, γ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  For the subclass of totally positive functions, for which the factorization (4) is a  finite product, the results of [18,19] can be summarized as follows:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Let g ∈ L2(R) be a totally positive function with a finite factoriza-  tion (4) with N ∈ N other than the one-sided exponentials ηγ, then  F(g) = {(α, β) ∈ R2  + : αβ < 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  In other words, the lattice αZ × βZ generates a frame with totally positive  window g, if and only if αβ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Remarkly the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1 required different methods for the cases  γ = 0 (no Gaussian factor) and γ > 0 (with Gaussian factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' In the former case  total positivity and the Schoenberg-Whitney conditions were used, in the latter  case only the factorization was used in conjunction with complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  The frame set of the one-sided exponential ηγ is slightly different, since ηγ is not  smooth enough and the Balian-Low theorem does not apply [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' In this case the  frame set is F(ηγ) = {(α, β) ∈ R2  + : αβ ≤ 1}, as was already known by Janssen [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1 suggests that the characterization of Gabor frames holds for all  totally positive generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Our goal is to prove the following theorem which is a  step towards the full conjecture formulated in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Assume that g is totally positive other than a one-sided exponential  function ηγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Further assume that αβ is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Then G(g, α, β) is a frame for  L2(R), if and only if αβ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  TOTALLY POSITIVE FUNCTIONS AND GABOR FRAMES OVER RATIONAL LATTICES 3  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2 implies that the frame set F(g) is an open, dense  subset of  {(α, β) ∈ R2  + : αβ < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' This follows from a deformation result of Feichtinger  and Kaiblinger [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Since every totally positive function is a limit of a totally positive with a finite  factorization, one would hope that some kind of limiting procedure would work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  However, so far all our “proofs” contained a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Except for the example of the hy-  perbolic secant (eat +e−at)−1 [25], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2 is the first general statement about  Gabor frames with a totally positive generator with an infinite factorization (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  In this paper we present a new approach that is independent of Schoenberg’s  factorization (4) and uses only the definition of total positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' The new proof  idea comes from a deep theorem of [8] about invertibility of infinite totally positive  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' We also use methods from spectral invariance of matrix algebras and the  important fact that the Zak transform of a totally positive function has only one  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' For rational lattices this method leads also to a new proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  To finish the discussion, we mention the superb work [3], which determines the  frame set of rational functions (Hurwitz functions) g(t) = �N  j=1  aj  t−iwj for wj ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  If aj > 0 and wj > 0, then F(g) = {(α, β) ∈ R2  + : αβ ≤ 1}, for general coefficients  the frame set contains {(α, β) ∈ R2  + : αβ ≤ 1} ∩ Qc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Interestingly and in contrast  to our work, the authors show that all irrational lattices belong to the frame set!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Other than this the complete frame set has been determined only for the char-  acteristic function of an interval χ[0,1] [5] and the Haar function [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' In contrast  to the results for totally positive windows and rational functions, the frame set is  extremely complicated and depends on subtle number theoretic properties of the  product αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  The case of irrational lattices remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' There is some evidence that gener-  ically a lattice αZ × βZ with irrational αβ generates a Gabor frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Important  results pointing in this direction are contained in [3] and in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' In each case an  underlying dynamical system and some ergodicity are at work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Currently we do  not know how this works for totally positive windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Of course, much more is known and relevant about Gabor frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' As we are in-  terested only in the complete characterization of Gabor frames for a given window,  we refer to the surveys [15,17,20] and monographs [9,14] for the general theory of  Gabor frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  The paper is organized as follows: In Section 2 we review the tools underly-  ing Gabor analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' These are Zak transform methods, Banach algebra methods  concerning spectral invariance, and general characterizations of Gabor frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' All  these characterizations amount to showing that some (infinite) matrix is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  In our approach the matrix G under consideration will a sub-matrix of the so-called  pre-Gramian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' This is a technical novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' In Section 3 we show that this matrix  is onto ℓ1(Z) by using heavily the properties of spectral invariance and a theorem  from [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' In Section 4 we will show that G is also one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' It is only here that  we will use the rationality of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Ultimately the proof boils down to the  fact that a square matrix is invertible, if and only if it is onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  4  KARLHEINZ GR ¨OCHENIG  Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' I am deeply indebted to Jose Luis Romero and Joachim  St¨ockler for comments on earlier versions of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Tools  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Let gβ(x) = β−1/2g(x/β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Then G(g, αZ × βZ) is a frame, if and  only if G(gβ, αβZ × Z) is a frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' It is immediate from (3) that the set of totally  positive functions is invariant under dilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Thus, if g is totally positive, then  so is gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' We may therefore always replace g by gβ, α by αβ and β by 1 and thus  assume without generality that β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  We note that we may always assume that αβ < 1 and, after the reduction, that  α < 1, β = 1 because of the density theorem for Gabor frames [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Spectral invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Spectral invariance refers to the phenomenon that an  invertible, infinite matrix with sufficiently strong off-diagonal decay possesses an  inverse with the same off-diagonal decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' This topic has a long history with many  contributions and variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' See [1, 2, 35–38] for a sample of contributions and  the survey [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  The following version can be attributed to Shin-Sun [35] and  Tessera [38, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1 (Stability Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Let G = (Gkl)k,l∈Z be a bi-infinite matrix  with off-diagonal decay  |Gkl| ≤ C(1 + |k − l|)−σ  k, l ∈ Z ,  for some σ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  (i) Spectral invariance: If G is invertible on some ℓp0(Z), then G is invertible on  all ℓp(Z) for 1 ≤ p ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  (ii) Stability: If G is stable on some ℓp0(Z), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=', if G satisfies  ∥Gc∥p0 ≥ A∥c∥p0  for all c ∈ ℓp0(Z),  then G is stable on all ℓp(Z) for 1 ≤ p ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  We will need a slight variation of the stability (ii) for surjective maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Let G = (Gkl)k,l∈Z with off-diagonal decay  |Gkl| ≤ C(1 + |k − l|)−σ  k, l ∈ Z ,  for some σ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' If G maps ℓ∞(Z) onto ℓ∞(Z), then G also maps ℓ1(Z) onto ℓ1(Z)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Note that the off-diagonal decay implies that G defines a bounded operator  on all ℓp(Z), 1 ≤ p ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  By the closed range theorem [30, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' 102], the adjoint operator (matrix) on the  pre-dual G∗ : ℓ1(Z) → ℓ1(Z) is one-to-one with closed range in ℓ1(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Consequently,  G∗ satisfies an inequality of the form  (5)  ∥G∗c∥1 ≥ A∥c∥1  for all c ∈ ℓ1(Z)  with a positive constant A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  TOTALLY POSITIVE FUNCTIONS AND GABOR FRAMES OVER RATIONAL LATTICES 5  The stability part of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1 then implies that (5) holds for all p-norms,  in particular, for some A′ > 0 we have  ∥G∗c∥∞ ≥ A′∥c∥∞  c ∈ ℓ∞(Z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Since G∗ is one-to-one with norm-closed range in ℓ∞(Z), we conclude that G :  ℓ1(Z) → ℓ1(Z) must be onto ℓ1(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Characterizations of Gabor frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' We use the following characterization  of Gabor frames that goes back to Janssen [21] and the duality theory of Gabor  frames [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' This is one of the most successful criteria to verify that a Gabor system  over a rectangular lattice generates a frame and has been applied many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' We  refer to [17] for a survey of the known characterizations of Gabor frames and some  of their uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Assume that g : R → R is continuous and satisfies the condition  �  k∈Z supx∈[0,1] |g(x + k)| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  (i) The family G(g, αZ × Z) is a frame for L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  (ii) For every x ∈ [0, 1), there exist Ax, Bx > 0 such that  (6)  Ax∥c∥2  2 ≤  �  j∈Z  ���  �  k∈Z  ckg(x + αj − k)  ���  2  ≤ Bx∥c∥2  2  for all c ∈ ℓ2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  (iii) There exist A, B > 0 such that, for all x ∈ [0, 1),  A∥c∥2  2 ≤  �  j∈Z  ���  �  k∈Z  ckg(x + αj − k)  ���  2  ≤ B∥c∥2  2  for all c ∈ ℓ2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  The matrix P(x) =  �  g(x + αj − k)  �  j,k∈Z is often called the pre-Gramian of the  Gabor family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Using spectral invariance, we deduce the following sufficient condition for Gabor  frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Let g be a function on R with polynomial decay |g(t)| ≤ C(1 +  |t|)−σ for some σ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Assume that for every x ∈ R the set x + αZ contains a  subset {k + δk : k ∈ Z} with the perturbation δk = δk(x) depending on x, such  that the matrix G with entries Gkl = g(k + δk − l) is invertible on ℓ∞(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Then  G(g, α, 1) is a Gabor frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' The decay of g implies that G has sufficient off-diagonal decay to apply  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Thus spectral invariance asserts that the invertibility of G on  ℓ∞(Z) implies that G is also invertible on ℓ2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Since x + αZ ⊇ {k + δk : k ∈ Z},  we have  �  j∈Z  �� �  l∈Z  clg(x + αj − l)|2 ≥  �  k∈Z  �� �  l∈Z  clg(k + δk − l)|2  = ∥Gc∥2  2 ≥ Ax∥c∥2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  The upper bound in (6) always holds due to the decay of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' This holds for all x,  therefore the condition (ii) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='3 implies that G(g, α, 1) is a frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  6  KARLHEINZ GR ¨OCHENIG  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='4 says that a suitable submatrix of the pre-Gramian P(x) is in-  vertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='4 is related to the characterizations via sampling in shift-  invariant spaces that figure prominently in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' (In an equivalent formulation,  every set x + αZ contains a perturbation of Z that is interpolating in the shift-  invariant space associated to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=')  To prove the main theorem (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2), we will verify the conditions of Propo-  sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' The Zak transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' We recall the definition of the Zak transform Zg(x, ξ) =  �  k∈Z g(x − k)e2πikξ of a function g ∈ L1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' More generally, the Zak transform  with period p > 0 is defined by  Zpg(x, ξ) =  �  k∈Z  g(x − pk)e2πipkξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  The Zak transform is 1  p-periodic in the second variable and quasi-periodic in the  first variable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=', for all x, ξ ∈ R  Zpf  �  x + pk, ξ + l  p  �  = e2πipkξZpg(x, ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Zak transforms of totally positive function possess only few zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' This important  property was discovered only recently [26,27,39] and is crucial for our arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' The Zak transform of a totally positive function g possesses a  unique zero in [0, 1)2 that is assumed at (x0, 1/2) for some x0 ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  For the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2 we verify the sufficient condition of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='4  for a totally positive function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' First we prove the surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Notice that the  arguments in this section hold for arbitrary values of α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Let g be an arbitrary totally positive function in L1(R) and  assume that its Zak transform has its unique zero at (x0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Let ǫ ∈ (0, 1/2),  M ∈ Z, and (δk)k∈Z ⊆ [x0 + M − 1 + ǫ, x0 + M − ǫ] be a sequence of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Then the matrix G with entries Gkl = g(k + δk − l), k, l ∈ Z, maps ℓ∞(Z) onto  ℓ∞(Z) and ℓ1(Z) onto ℓ1(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' For the proof we use the following fundamental result of de Boor, Friedland,  and Pinkus [8, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Assume that G is an totally positive matrix that is bounded  on ℓ∞(Z)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' If the range of G contains a uniformly alternating vector, then G is  onto ℓ∞(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  This result is the key tool in our proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Related results on the  invertibility of infinite totally positive matrices can be found in [4,7,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Uniformly alternating means that we need to find a sequence c ∈ ℓ∞(Z) such  that u = Gc satisfies u(k)u(k + 1) < 0 for all k ∈ Z and infk∈Z |u(k)| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  1The original results are formulated for sign-regular matrices and more general index sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  TOTALLY POSITIVE FUNCTIONS AND GABOR FRAMES OVER RATIONAL LATTICES 7  To accomplish this, we take cl = (−1)l and compute (Gc)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' This is  (Gc)k =  �  l∈Z  g(k + δk − l)(−1)l  = Zg(k + δk, 1  2)  = (−1)kZg(δk, 1  2) ,  where the last identity comes from the quasi-periodicity of Zg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Since Zg(x, 1  2) is  real-valued and by assumption Zg(x, 1  2) does not have any zeros on the interval  [x0 − 1 + ǫ, x0 − ǫ], the numbers Zg(δk, 1  2) all have the same sign and |Zg(δk, 1  2)| ≥  ν > 0 for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Consequently u = Gc is uniformly alternating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  To apply the result of [8], we observe that the matrix G with entries g(k +δk −l)  is indeed totally positive in the sense that all minors of G of arbitrary dimension  have a non-negative determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' But this follows from the definition (3) of total  positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Thus by [8, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' 1] the matrix G is onto ℓ∞(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2 G is also  onto ℓ1(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  To apply the above proposition, we need to select a suitable subset of x + αZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  This can always be done with the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Let α < 1, ǫ < (1 − α)/2, M ∈ Z, x0 ∈ [0, 1] fixed, and x ∈ [0, 1]  arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  (i) Then x + αZ contains a subset {k + δk : k ∈ Z} with δk ∈ [x0 + M − 1 +  ǫ, x0 + M − ǫ] for all k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  (ii) If α is rational, α = p/q, then the perturbation δk can be chosen to be p-  periodic, δk+np = δk for all k, n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' (i) For arbitrary k ∈ Z the interval [k + x0 − 1 + ǫ, k + x0 − ǫ] has length  1 − 2ǫ > α and thus contains at least one point of x + αZ, which we can write as  k + δk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  (ii) Let α = p/q ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' For l = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' , p − 1 choose an integer jl ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' , q − 1}  such that  x + p  qjl ∈ [l + x0 − 1 + ǫ, l + x0 − ǫ]  and set  δl = x + p  qjl − l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Clearly, 0 ≤ j0 < j1 < · · · < jp−1 < q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Now let m = l + np with l ∈ [0, p − 1] ∩ Z  and n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Then x+ p  q(jl +nq) = x+ p  qjl +np ∈ [l+np+x0−1+ǫ, l+np+x0 −ǫ] =  [m+x0−1+ǫ, m+x0−ǫ], and δm = x0+ p  q(jl+nq)−m = x0+ p  q(jl+nq)−(l+np) = δl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Thus δk is p-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Injectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  For the surjectivity of G we have used the fact that g is totally positive, but  nothing about the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Now we impose in addition that α = p/q is  8  KARLHEINZ GR ¨OCHENIG  rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' In this case the action of G is unitarily equivalent to a p × p-matrix-  valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' This facilitates the analysis, as for square matrices surjectivity is  equivalent to injectivity and to invertibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Let again G be the matrix with entries Gkl = g(k + δk − l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Since α = p/q ∈ Q,  the sequence of perturbations can be chosen to be p-periodic by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' We  now use the periodicity of δk to bring in a variant of the Zibulski-Zeevi matrices [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  We write k = r +mp and l = s+np with r, s = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' , p−1, and m, n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Then  δk = δr and for all m ∈ Z and r = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' , p − 1 we have  (Gc)k =  �  l∈Z  clg(k + δk − l)  =  p−1  �  s=0  �  n∈Z  cs+npg(r + mp + δr − s − np) = dr+mp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  (7)  For fixed r, s the sum over n is a convolution, so we may take Fourier series and ob-  tain a matrix-valued equation of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Write xs(ξ) = �  n∈Z cs+pne2πinpξ for the  Fourier series of the coefficient in the residue class s and x(ξ) = (x0(ξ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' , xp−1(ξ))  for the associated vector-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Likewise yr(ξ) = �  m∈Z dr+pme2πimpξ and  y(ξ) = (y0(ξ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' , yp−1(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' The Fourier series of the sequence involving g is pre-  cisely the Zak transform of g, namely,  �  n∈Z  g(r + δr − s − np)e2πinpξ = Zpg(r + δr − s, ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Consequently, after taking Fourier series, (7) can be recast as  (8)  p−1  �  s=0  xs(ξ)Zpg(r + δr − s, ξ) = yr(ξ)  for r = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' , p − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Since c ∈ ℓ1(Z) and G is bounded on ℓ1(Z), (8) holds pointwise for all ξ ∈ [0, 1/p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Let A(ξ) be the p × p-matrix-valued function with entries  Ars(ξ) = Zpg(r + δr − s, ξ)  r, s = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' , p − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  This matrix is related to the Zeevi-Zibulski matrix that occurs in many criteria for  Gabor frames over rational lattices [17, 40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' The pointwise identity (8) can be  written as  (9)  A(ξ)x(ξ) = y(ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  We now translate the surjectivity of the infinite matrix G to a property of the  matrix-valued function det A(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Assume that α = p/q is rational, |g(t)| ≤ C(1 + |t|)−σ for some  C, σ > 1, and that G maps ℓ1(Z) onto ℓ1(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Then G is one-to-one on ℓ1(Z) and  thus invertible on ℓ1(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' The surjectivity of G on ℓ1(Z) implies in particular that every vector d =  (d0, d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' , dp−1) ∈ Cp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='e, d ∈ ℓ1(Z), dk = 0 for k < 0 and k ≥ p is in the  range of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' In this case yr(ξ) = �  m∈Z dr+pme2πimpξ = dr and the vector-valued  TOTALLY POSITIVE FUNCTIONS AND GABOR FRAMES OVER RATIONAL LATTICES 9  function y(ξ) = d is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Due to the surjectivity of G there exists a sequence  c = c(d) ∈ ℓ1(Z) with associated Fourier series x(d)(ξ), such that Gc = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' In view  of (9)  A(ξ)x(d)(ξ) = d  for all ξ ∈ [0, 1/p] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Since d ∈ Cp was arbitrary, this means that for fixed ξ the p × p-matrix is onto Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Consequently, A(ξ) is one-to-one and thus invertible for every ξ ∈ [0, 1/p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Now assume that Gc = 0 for some c ∈ ℓ1(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Then A(ξ)x(ξ) = 0 for all ξ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Since A(ξ) is invertible, this implies that x(ξ) ∈ Cp must be the zero vector for all  ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Consequently, x ≡ 0 and thus c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' We have proved that G is one-to-one on  ℓ1(Z), whence G is invertible on ℓ1(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  We finally summarize the steps that yield the proof of our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2 we can extract a set {k + δk : k ∈ Z} from  every x + αZ, such that δk avoids the zero x0 of the Zak transform Zg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Then by  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1 the matrix G =  �  g(k + δk − l)  �  k,l∈Z is onto ℓ1(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  For rational values of α the matrix G is also one-to-one on ℓ1(Z) by Proposi-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Consequently, G is invertible on ℓ1(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Finally, the spectral invariance guaranteed by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1(i) implies that G  is invertible on ℓ∞(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' This is precisely the sufficient condition of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='4  to guarantee that G(g, α, 1) is a frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  We are done!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Alternatively, one could give Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1 a more structural touch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' We use  Tp for the translation operator on sequences defined by (Tpc)k = ck−p and observe  that the underlying matrix G of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1 commutes with Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Let G = (Gkl)k,l∈Z be a bi-infinite matrix with off-diagonal decay  |Gkl| ≤ C(1 + |k − l|)−σ  k, l ∈ Z ,  for some σ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Assume that G commutes with the translation operator Tp for  some p ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' If G is onto ℓ1(Z), then G is one-to-one on ℓ1(Z) and thus invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  The proof is similar to the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Aldroubi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Baskakov, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Krishtal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Slanted matrices, Banach frames, and  sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=', 255:1667–1691, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Baskakov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Wiener’s theorem and asymptotic estimates for elements of inverse matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Funktsional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' i Prilozhen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=', 24(3):64–65, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  [3] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Belov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Kulikov, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Lyubarskii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Gabor frames for rational functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=',  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='1007/s00222-022-01151-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Cavaretta, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
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+page_content=' Dai and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
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+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
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+page_content=' Pinkus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Inverses of infinite sign regular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content='  Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
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+page_content=' Time-frequency analysis of operators, volume 75 of De Gruyter  Studies in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
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+page_content=' The positivity of trans-  lation determinants with an application to the interpolation problem by spline curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AyT4oBgHgl3EQf8fpt/content/2301.00857v1.pdf'}
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+An Improved Method of Estimating the Uncertainty of Air-Shower
+Size at Ultra-High Energies
+A. Colemana,∗, P. Billoirb, O. Delignyc
+aUniversity of Delaware, Department of Physics and Astronomy, Bartol Research Institute, Newark, DE, USA
+bLaboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Sorbonne Université, Université de Paris,
+CNRS-IN2P3, Paris, France
+cLaboratoire de Physique des 2 Infinis Irène Joliot-Curie (IJCLab), CNRS/IN2P3, Université Paris-Saclay, Orsay,
+France
+Abstract
+The collection of a statistically significant number detected of cosmic rays with energy above
+1017 to 1018 eV requires widely-spaced particle detectors at the ground level to detect the ex-
+tensive air showers induced in the atmosphere. The air-shower sizes, proxies of the primary
+energies, are then estimated by fitting the observed signals to a functional form for expecta-
+tions so as to interpolate the signal at a reference distance. The functional form describes
+the rapid falloff of the expected signal with the distance from the shower core, using typi-
+cally two logarithmic slopes to account for the short-range and long-range decreases of signals.
+The uncertainties associated to the air-shower sizes are determined under the assumption of a
+quadratic dependence of the log-likelihood on the fitted parameters around the minimum, so
+that a meaningful variance-covariance matrix is provided. In this paper, we show that for an
+event topology where one signal is much larger than the others, the quadratic dependence of
+the fitted function around the minimum is a poor approximation that leads to an inaccurate
+estimate of the uncertainties. To restore a quadratic shape, we propose to use the polar coordi-
+nates around the detector recording the largest signal, projected onto the plane of the shower
+front, to define the likelihood function in terms of logarithmic polar distances, polar angles and
+logarithmic shower sizes as free parameters. We show that a meaningful variance-covariance
+matrix is then recovered in the new coordinate system, as the dependence of the fitted function
+on the modified parameters is properly approximated by a quadratic function. The use of the
+uncertainties in the new coordinate system for subsequent high-level analyses is illustrated.
+1. Introduction
+Ultra-high energy cosmic rays with energies above 1017 to 1018 eV are observed through the
+extensive air showers they induce in the atmosphere. To collect a statistically significant num-
+ber of events, very-widely-spaced particle detectors at the ground level have been used to max-
+imize the aperture of the observatories. Typical detection methods, such as scintillator panels
+or water-Cherenkov tanks, observe an integrated charge that results from a convolution of the
+detector response with the local particle density, energy distribution, and arrival direction. The
+∗Corresponding author
+Email addresses: alanco@umich.edu (A. Coleman), billoir@lpnhe.in2p3.fr (P. Billoir),
+deligny@ijclab.in2p3.fr (O. Deligny)
+Preprint submitted to Elsevier
+January 5, 2023
+arXiv:2301.01558v1  [hep-ex]  4 Jan 2023
+
+spacing of the particle detectors turns out to be five to ten times larger than the Molière radius,
+which delineates the distance in which more than 90% of the ionizing particles are contained.
+Consequently, it has been an increasingly difficult task to determine the lateral distribution of
+particles on an event-by-event basis, and thus to use the associated total number of particles
+for estimating the energy.
+As an alternative energy estimator, the signal at a distance appropriate to the spacing of the
+shower array is used as a surrogate measurement of the shower size. First proposed by Hillas [1,
+2], the technique, successfully exploited by the Haverah Park Collaboration [3], was reviewed
+in details [4] and adopted by the Pierre Auger Collaboration, the Telescope Array Collaboration,
+and others. The conversion from shower size into primary energy may subsequently include
+several correction factors, calibrations, and/or comparisons with simulations. The key point
+of the technique is to infer the shower size from the amount of signal that is expected to be
+measured by a detector at a reference distance, rref, by fitting a lateral distribution function,
+LDF, to the observed data. At that specific distance, which depends on the topology of the
+surface array, the fluctuations in the age of the showers when reaching the ground, inherited
+from the stochastic variations in the location and character of the leading interactions, reflected
+in the fluctuations of the LDF, are minimised. In this way, the shower size is determined with
+adequate accuracy (< 10%). While a further description of this technique is beyond the scope
+of this paper, it is important to note that if using a shower-size technique, the resolution with
+which the size can be determined ultimately sets the minimum resolution on the energy.
+This estimation is done in practice by fitting the observed signal amplitudes, S(ri), at a dis-
+tance, ri, using an LDF. A common convention is to normalize the LDF using the reference
+distance, rref, such that LDF(rref) ≡ 1 and thus the shower size at the reference distance can be
+directly extracted from S(r) = Sref LDF(r). The fit of such a model to determine the size of the
+air shower involves the determination of at least five parameters, two which describe the orien-
+tation of the shower axis, two which define the intersection of the axis with a plane1, and Sref.
+Additional parameters which describe the exact nature of the exponential decrease in signal
+size with distance from the shower axis may also be fit, but near the triggering threshold, there
+may not be enough degrees-of-freedom and an average LDF is typically used, which is based
+on observed or simulated air showers. The combination of the nearly power-law shape of the
+LDF and the first-order cylindrical symmetry of the signals about the shower axis results in a
+non-trivial phase space for the log-likelihood function (LLH) that can hamper an accurate de-
+termination of the optimal parameters as well as their uncertainties. The aim of this work is to
+explore carefully the LLH phase space for typical event topologies encountered in practice with
+contemporary observatories, i.e. the Pierre Auger Observatory and the Telescope Array.
+We note that while the arrival direction of the air shower is important for e.g., anisotropy
+studies, the two parameters that define the intersection point, the impact point, are generally
+nuisance parameters. This allows for some freedom regarding the choice of parameters to de-
+scribe the location of the impact point. In this work, we show how the estimation of the un-
+certainties, particularly that of the logarithm of Sref, can be determined in a much more stable
+way when employing a “log-polar” coordinate system, rather than a Cartesian one. In section 2,
+we set the stage of the generic framework allowing us to develop an efficient tool to simulate
+and reconstruct events recorded by surface detectors. Using this tool, we are able to extract
+event topologies leading to LLHs that cannot be approximated by quadratic dependencies in
+1Typically this plane is taken to be the ground in the local coordinate system.
+2
+
+the reconstructed parameters. To correct this issue, we propose in section 3 to change the co-
+ordinate system for the fit, switching to a log-polar one, and show that quadratic dependencies
+of the LLHs in the reconstructed parameters more accurately describe the true uncertainties.
+Among a variety of possible high-level analyses, we show in section 4 how the energy calibra-
+tion performed in a manner similar to that used at the Auger Observatory [5, 6] may benefit
+from accurate event-by-event uncertainties. Finally, conclusions are given in section 5.
+2. Simulation and reconstruction of air showers
+We employ a toy model for the simulation of signals in an air-shower array. Rather than
+running e.g., full simulations of the particle cascades, we take the ideal case where the LDF is
+known. Shower-to-shower fluctuations are ignored, as they are nonessential for the purpose
+of this study. In this way, the LDF accurately describes the distribution of signals around the
+shower axis. Showers are randomly sampled on a triangular array with 1.5 km spacing, based
+on the layout of the Pierre Auger Observatory [7], see the left panel of fig. 1. However, we will
+3000
+−
+2000
+−
+1000
+−
+0
+1000 2000 3000
+x / m
+3000
+−
+2000
+−
+1000
+−
+0
+1000
+2000
+3000
+y / m
+2000
+−
+1000
+−
+0
+1000
+2000
+x / m
+2000
+−
+1000
+−
+0
+1000
+2000
+y / m
+Figure 1:
+The layout of the triangular and square arrays (black circles) are shown in the left and right panels,
+respectively. The region over which the impact points are sampled for each layout is outlined in red.
+also show that the effect is the same for a square array with 1200 m spacing, based on the ar-
+rangement of the surface array of Telescope Array [8].
+Without loss of generality, all signals in this work are expressed in Vertical Equivalent Muon
+(VEM) units, regardless of the detector or of the particle species crossing the detector. A VEM
+is defined as the sum of the charge collected for a single muon traversing vertically a detector
+(see e.g. ref. [9]). The value of Sref is sampled randomly from a dN/dSref ∝ S−1
+ref spectrum and
+over a range that is commensurate with energies that correspond to E > 3 EeV for the 1500 m
+surface detector of the Pierre Auger Observatory. In this work, we will focus on air showers with
+a zenith angle of 40◦. A study on the zenith dependence is shown in the appendix and while a
+bias on the shower-size estimation is shown to exist at all angles, there is some dependence on
+the arrival direction in the exact characteristics.
+A signal is assigned to each detector on the array using the LDF,
+S(r) = Sref
+� r
+rref
+�β � r +r0
+rref +r0
+�β+γ
+,
+(1)
+3
+
+with β = −2.5, γ = 0.1, rref = 1000 m, and r0 = 700 m. The falloff of the signal from the shower
+axis is thus described by a logarithmic slope denoted as β, and by a small departure from a
+power-law at large distances governed by γ. The signal assigned to each detector is additionally
+smeared using Gaussian fluctuations of width
+σ(S) = AS
+�
+S.
+(2)
+The scaling in
+�
+S has been shown to describe the Pierre Auger Observatory data [10]. The factor
+AS, chosen to be 1 in this work, is not considering any dependence in zenith angle in this study2.
+Finally, an ideal trigger is implemented which keeps only detectors with a signal above ST = 3.
+While the values and formulas used above are based on those used for water-Cherenkov tanks
+at the Pierre Auger Observatory [11], they are similar in nature to those used by other current
+air-shower experiments [4, 12–15] and the issue being discussed here is ultimately a geometric
+one and is not sensitive to these choices. Also note that in this work, reconstructions will be
+performed with a fixed LDF shape (i.e. β and γ) and thus the choice of rref is arbitrary and has
+no influence on the outcome of this study.
+2.1. Reconstructing the air-shower parameters
+In this study, we mostly study the shower-size estimation. Since air-shower arrays can typi-
+cally be used to reconstruct air showers to an accuracy of better than a few degrees [11, 13–15],
+we will neglect the reconstruction of the shower arrival direction since it is a sub-dominant ef-
+fect when reconstructing the shower size and is determined from arrival times rather than signal
+amplitudes. Thus in this work, MINUIT is given only three parameters, lgSref and two param-
+eters that determine the impact point. In the beginning of this work, the impact point will be
+defined by the (x, y) position in the ground plane, but we will define a better set of coordinates
+in section 3.
+The model of an air shower, with the LDF shape, is applied to the simulated data and the air
+shower parameters are fit by minimizing a negative log-likelihood that describes the probability
+to observe the set of signals, {Si}, in each detector. The likelihood is made of two main terms,
+one that describes the detectors which triggered, and one for those that did not3,
+−2LLH =
+Triggered
+�
+i
+��Si −S(ri)
+σ(S(ri))
+�2
++ln(2πσ2(S(ri)))
+�
+−2
+S j =0
+�
+j
+ln
+�
+P(S(r j))
+�
+.
+(3)
+The second sum describes the probability, P(S), that no signal was observed when a signal,
+S(r j), is expected. For Gaussian fluctuations, this probability is given by,
+P(S) = 1− 1
+2
+�
+1+erf
+� S −ST
+�
+2 σ(S)
+��
+.
+(4)
+2.2. Minimization and error estimation
+The reconstruction of an air shower consists of two tasks, traversing through the LLH-space
+to find the (global, in the ideal case) minimum of eq. (3) and to characterize the curvature near
+2While a given detector may have a zenith-dependent σ(S), here we only study a single zenith angle.
+3Throughout the paper, the notation lg stands for the decimal logarithm, while ln stands for the natural loga-
+rithm.
+4
+
+this minimum to estimate the uncertainty. Ideally, the whole LLH-space could be scanned and
+the global minimum can be directly identified. However, this is typically impractical for a large
+data set and instead most algorithms require a well-estimated starting point and numerically
+calculate and follow the gradient of the LLH-space until a minimum is found. This so-called
+gradient-descent method does not ensure a global minimum is ultimately found and there are
+various other pitfalls, but a discussion of these issues is beyond the scope of this paper. Many
+implementations of this algorithm exist and extensions on this basic concept have been devel-
+oped, but in this work, we focus on MINUIT [16], the standard minimizer that is packaged with
+ROOT [17], which is commonly used in particle and air-shower physics.
+To do the second task, MINUIT provides an estimate of the curvature in the neighborhood
+of the identified minimum in LLH-space. The Hessian matrix of second-derivatives with re-
+spect to all of the free-parameters (impact point, Sref, etc.) is numerically calculated and then
+inverted to produce the covariance matrix. This matrix is an estimate of the local curvature
+assuming that the LLH-space is quadratic near minimum and also provides the correlations
+between parameters. This simplistic definition will be shown to be problematic for the recon-
+struction of a specific class of air showers where the LLH-space does not fulfill the quadratic
+assumption.
+2.3. LLH-space for air-shower events
+Using the LLH defined above, the difficulty for a numerical minimization routine to estimate
+the uncertainties for an air-shower reconstruction can be made evident. Two example events
+of air showers with zenith angles of 40◦ are shown in fig. 2. Each panel shows the results of a
+scan of the potential impact point locations in (x, y) i.e., along the ground. At each location,
+the impact point and arrival direction were held fixed and only the value of lgSref was left free.
+The LLH space is seen to wrap around the detector with the largest signal, indicated by the
+black dot. Exemplified by the 1σ, 2σ, and 3σ contours, the LLH-space is stretched along the red
+lines of constant reconstructed lgSref, which also encircle the detector with the largest signal.
+This curvature is expected, both for the LLH and the lgSref contours since the LDF is dependent
+solely on r.
+For an impact point location sufficiently far from a detector (e.g., top panels of fig. 2), the
+1σ LLH-contour in Cartesian coordinates can be approximately described by an ellipse. In this
+case, the second-derivative matrix calculated by MINUIT is sufficient to estimate the uncer-
+tainty. Both the 1σ LLH-contour and estimated ellipse reside inside the lgSref/lgSbest = {0.8,1.2}
+lines and are in agreement with the estimated uncertainty from MINUIT of σlgSref ≃ 0.16.
+The bottom panels highlight the problem of estimating the uncertainty in lgSref. In this
+case, the impact point is close enough to a detector that the highly-wrapped LLH-space is not
+sufficiently parabolic in Cartesian coordinates for the second-derivative calculation to provide a
+stable estimate of the uncertainty in lgSref. The curvature about the detector is degenerate with
+a gradient of the LLH-space and the relative uncertainty of the shower size is estimated to be
+σlgSref ≃ 0.07 (bottom-right panel) even though the true 1σ contour (bottom-left panel) shows
+that the uncertainty is at least 20%. For an air-shower experiment, this creates a erroneous
+result in the estimated uncertainties on lgSref depending on the impact point.
+To study the phase space of impact points where this effect is most prominent, 50 000 events
+were simulated and reconstructed. Random uniform azimuth angles were chosen and the im-
+pact points were uniformly picked within the boundaries highlighted in fig. 1. The impact point
+(x, y) and lgSref were left as free parameters during the minimization. The errors on lgSref, as
+5
+
+Figure 2: The LLH-space for an event with a zenith angle of 40◦ is shown. The color scale indicates the confidence
+interval of a given impact point being correct. The location with the largest LLH value is indicated with a black
+cross and the location of the detector with the largest signal is indicated by the black circle. The red lines are the
+contours of reconstructed lgSref values with respect to that of the largest LLH value, lgSref/lgSbest = {0.8,1,1.2}. The
+black lines show the 1σ, 2σ, and 3σ contours. In the left panels, these contours represent the true ones while those
+in the right panel correspond to the ellipses given by the covariance matrix from MINUIT. The top and bottom
+panels show events where the shower axis is 227 m and 84 m from the shown detector location, respectively. In the
+panels on the right, the estimated uncertainty of the shower size, σlgSref, is shown in the lower left corner.
+6
+
+300
+4
+3.5
+200
+3
+m
+100
+2.5
+true
+2 6
+0
+1.5
+100
+1
+-200
+0.5
+-30Q
+300
+-200
+-100
+0
+100
+200
+300
+/m
+(x-x
+true300
+4
+3.5
+200
+3
+m
+100
+2.5
+true
+2 6
+1.5
+100
+1
+-200
+0.5
+0.16
+ret
+-300
+300
+-200
+-100
+0
+100
+200
+300
+(x- X
+m
+true300
+4
+3.5
+200
+3
+m
+100
+2.5
+true
+6
+2
+1.5
+7
+-100
+1
+-200
+0.5
+-30Q
+300
+-200
+-100
+0
+100
+200
+300
+(x- x
+/m
+true300
+4
+3.5
+200
+3
+100
+m
+2.5
+true
+2
+6
+0
+1.5
+7
+-100
+1
+-200
+0.5
+007
+ret
+-300
+300
+-200
+-100
+0
+100
+200
+300
+(x-x
+m
+trueFigure 3: Left: Estimated relative uncertainty of the shower-size, as calculated using MINUIT, for 50 000 simulated
+air showers. This distribution is shown as a function of the distance of the detector with the largest signal from the
+shower axis. The values in red are those with Rhot < 200 m. Right: Histogram of the uncertainty for simulations
+with Rhot ≥ 200 m as a function of the true shower size lgSref. The red points are the same events that are shown in
+red in the left panel.
+estimated by the MINUIT algorithm, are shown in fig. 3. As seen in the left panel, the relative
+uncertainty decreases with increasing the distance from the shower axis, Rhot, to the detector
+with the largest signal. However, for small values of Rhot, shown in red, there is an opposite
+trend and the uncertainties are up to an order of magnitude too small. In the right panel, the
+distribution of events is shown versus shower size, again with small values of Rhot shown in red.
+The underestimated shower uncertainties are seen to occur at all shower-size values meaning
+that this effect is independent of the number of triggered detectors.
+3. Fitting in log-polar coordinates
+3.1. Log-polar coordinates
+The issue described above is ultimately related to the non-parabolic nature of the LLH-space
+in terms of the parameters that are given to the minimizer, namely the Cartesian location of the
+impact point. Instead we propose a more natural coordinate system for the task of reconstruct-
+ing the impact point and size. To develop the new parameters, we first define the Cartesian
+ground coordinate system wherein an air shower has zenith and azimuth angles of θ and φ, re-
+spectively, and the impact point is described by the (x, y) intersection of the shower axis with
+the ground. This system is depicted in fig. 4 wherein a sampling area, such as that around the
+black point, is the projection onto the ground, along the Z axis, of a region in the shower plane
+(X ,Y ). The shower frame is thus defined by ns and two perpendicular axes, for example an hor-
+izontal one, nh, and the other one, nv, into the vertical plane containing ns. We consider in the
+following the shower-front coordinates obtained from the ground ones as,
+Xsc = (∆xhot cosφ+∆yhot sinφ)cosθ,
+(5a)
+Ysc = −∆xhot sinφ+∆yhot cosφ.
+(5b)
+7
+
+10
+ref
+S.
+10
+10
+100
+200
+300
+400
+500
+600
+700
+R,180
+160
+140
+120
+100
+10
+80
+60
+10
+40
+20
+0
+10
+2.2 2.4 2.6 2.8
+1.21.41.61.8
+2.
+1g(S
+ref. MY
+Z
+X
+nv
+ns
+nh
+Ψ
+Figure 4: Diagram of the shower frame coordinate system: the Z axis is along the shower axis ns, the X axis is along
+nv in the vertical plane going through the shower axis, and the Y axis is along nh in the horizontal plane. In this
+coordinate system, ψ is used to describe the polar angle about the shower axis, see eq. (6).
+Here, ∆xhot and ∆yhot define a Cartesian location in the ground coordinate system with respect
+to the detector with the largest signal. The values of Xsc and Ysc are the 2D Cartesian locations
+in the shower coordinate system. To reconstruct the air-shower size, we then propose the “log-
+polar” coordinate system defined by the transformation into the shower plane,
+R =
+�
+X 2
+sc +Y 2
+sc,
+(6a)
+Ψ = arctan2(Ysc,Xsc).
+(6b)
+The new variables for reconstructing air showers are given by lgR and Ψ, the log-radius and
+polar angle about a line that is parallel to the shower axis and passes through the detector with
+largest signal. In addition, lgSref, the (decimal) logarithm of Sref, is considered as the third free
+parameter. The choice of anchoring this coordinate system to the detector with the largest
+signal is motivated by the curvature of the LLH-space, as previously seen in fig. 2.
+3.2. Results
+In fig. 5, the same two example events are shown in the log-polar coordinate system defined
+above. For both sample events, several nice properties of the LLH space are seen. Firstly, in
+this space, the lines of constant reconstructed lgSref are almost independent of the Ψ coordun-
+cinate. This is beneficial as this means that the correlation between Ψ and lgSref is small, re-
+gardless of the direction of the shower propagation, which is helpful for attaining a more robust
+estimate of the shower-size uncertainty. Secondly, the 1σ contours are more elliptical mean-
+ing that the assumption when using the Hessian matrix method is more valid and will typically
+return a better error estimate.
+We repeated the study shown in fig. 3 on the same 50 000 simulated events but using the
+log-polar coordinate system described above. The results are shown in fig. 6. In this case, the
+uncertainty estimation is much more well behaved with most of the events with Rhot < 200 m
+appearing now in a narrow band as a function of Sref. This band occurs because shower axes
+that pass very near to a detector are necessarily maximally far from all other detectors. On one
+hand, this will produce less triggered detectors per event. On the other hand, for the detectors
+with signal, the uncertainties, which scale like
+�
+S(r)/S(r) = 1/
+�
+LDF(r) ∝ r −β/2, will be larger
+8
+
+Figure 5:
+The top and bottom panels show the LLH-space for the same two events as those in fig. 2, this time
+calculated in the log-polar coordinate system. The true 1σ, 2σ, and 3σ LLH contours are given in the left panels
+while the right panels are the respective ellipses from the covariance matrix from MINUIT.
+9
+
+4
+2.55H
+3.5
+2.5
+3
+2.45
+m
+2.5
+2.4
+10U
+26
+2.35
+2
+1.5
+2.3
+1
+2.25
+0.5
+2.2
+0
+-3
+-2
+0
+2
+-1
+1
+3
+-W
+true4
+2.55H
+3.5
+2.5
+3
+2.45
+m
+2.5
+2.4
+10U
+26
+2.35
+2
+1.5
+2.3
+1
+2.25
+0.5
+2.2
+0.16
+0
+-3
+-2
+0
+2
+3
+-1
+1
+-W
+true2.1
+3.5
+2.05
+3
+2
+W
+2.5
+1.95
+hot
+2 6
+1.9
+80
+1.5
+1.85
+1
+1.8
+0.5
+1.75
+0
+-3
+-2
+0
+1
+2
+3
+-1
+-
+true4
+2.1
+3.5
+2.05
+3
+2
+m
+2.5
+1.95
+hot
+2 6
+1.9
+ao
+1.5
+1.85
+1
+1.8
+0.5
+O
+1.75
+ref
+0
+-3
+-2
+0
+2
+3
+-1
+1
+-V
+trueFigure 6: The same distributions as shown in fig. 3, using the same 50 000 Monte Carlo trials, but calculating the
+shower-size uncertainty using the log-polar coordinates defined in eq. (6).
+than average for a given shower size. Although the “hottest” detector has the smallest relative
+uncertainty, its weight in the fit is mitigated compared to that of other detectors. This is because
+a small shift δr in the impact point results in a relative change βδr/r for S and is proportional to
+r −β/2−1 in units of σ(S). The fitting procedure can thus change the expected signal of the hottest
+detector at a low cost through a change of the impact point, without affecting the expectation
+for the other detectors. All in all, these effects produce a less constrained LLH space and results
+in a larger uncertainty in the shower size. While there are a few outliers, these make up only
+∼ 0.1% of all events.
+3.3. Dependence on array geometry
+Given that this work is focused on a geometric issue, it is important to understand how the
+layout of the array ultimately impacts the shower-size estimation. We contrast the main results
+shown above after changing to the surface detector layout of Telescope Array, a square grid with
+1200 m spacing between detectors, as shown in the right panel of fig. 1. The results are shown
+in fig. 7.
+For the more dense detector arrangement, more detectors are triggered for a given shower
+size and the precision on lgSref is better. However, it is clear that the same issue is present for this
+array configuration, as seen in the top panels. The radius at which the shower size uncertainty
+undergoes an inflection is even in a similar location given the length scales involved, occurring
+at Rhot/dspacing ≃ 0.125 where dspacing is either 1500 m or 1200 m for the respective array layouts.
+Ultimately, a significant improvement is still observed by using the log-polar coordinates in the
+air shower reconstruction, as seen in the bottom panels.
+3.4. Effects of saturation
+In measured air-shower data, detectors near the shower axis can saturate, providing an un-
+reliable estimate of the signal content at that point. Since it is exactly these types of showers
+that result in non-quadratic LLH contours, it is worthwhile to consider the way that saturated
+stations may impact the results presented above. While the non-linear effects that occur near
+10
+
+10°
+ref
+10-1
+10
+100
+200
+300
+400
+500
+600
+700
+R,180
+160
+10°
+140
+120
+ref
+100
+10
+80
+60
+10
+40
+20
+0
+10
+2.2 2.4 2.6 2.8
+1.21.41.61.8
+2.
+1g(S
+ref. McFigure 7:
+The distributions of the relative estimated shower-size uncertainty are shown for the Telescope Array
+detector layout. In this case, the red points are defined by events with Rhot < 100 m. The top panels correspond to
+reconstructions using Cartesian coordinates (as in fig. 3) while the bottom panels correspond to reconstructions
+using the log-polar coordinates (as in fig. 6).
+11
+
+10
+10
+10-3
+0
+100
+200
+300
+400
+500
+600180
+160
+140
+10
+120
+ret
+100
+10
+80
+60
+10
+40
+20
+0
+10
+2.2 2.4 2.6 2.8
+1.21.41.61.8
+2
+1g(S
+ref. M(10
+ref
+10
+10-
+1(
+0
+100
+200
+300
+400
+500
+600180
+160
+140
+10-
+120
+ref
+100
+b0
+10
+80
+60
+10
+40
+20
+0
+10
+2.2 2.4 2.6 2.8
+1.21.41.61.8
+2
+1g(S
+ref. MFigure 8:
+Left: The uncertainty in the shower size, when using an additional term in the LLH to account for
+saturated stations (eq. (7)) when performing a reconstruction using Cartesian coordinates. Right: The same figure
+when ignoring saturated stations in during the reconstruction of the air-shower size. Both panels in this figure are
+directly comparable to fig. 3.
+the upper end of the dynamic range of a detector will be hardware-dependent, we show a few
+limiting cases for handling saturated detectors during reconstruction.
+In the first case, the underlying signal that would have been measured in the absence of
+saturation is recovered post-hoc. Whether by analytical methods or using machine learning,
+various techniques have been employed to estimate the would-be signal in saturated detectors,
+e.g. [18–20]. In the best case, the signal that would have been measured is precisely estimated
+with the result that effectively nothing has changed with respect to the results shown above. In
+the less ideal case, the recovery introduces an additional uncertainty, i.e. σ(S) −→ σ(S)+∆(S),
+which can simply be taken into account in the reconstruction (eq. (3)). This would widen the
+LLH contours, depending on the relative size of ∆(S), but would not remove the non-quadratic
+behavior close to the saturated detector.
+Alternatively, a corresponding term can be included in the likelihood function to describe
+the probability that a detector observes saturation given an expected signal. In the case of ideal
+saturation, where signals only up to Ssat. can be observed, the corresponding term to include
+in eq. (3) is,
+−2
+saturated
+�
+k
+ln
+�
+1−erf
+� Ssat. −Sk
+�
+2 σ(Sk)
+��
+.
+(7)
+This was tested within the framework of this toy model using Ssat. = 1000 and the results are
+shown in the left panel of fig. 8. In this case, the Cartesian coordinates already present a good
+estimation of the shower size uncertainty. This is a result of the rather non-restrictive nature
+of the additional term in eq. (7) for which expected signals of 1500 and 15 000 (i.e. larger and
+smaller Rhot values) give a probabilities that are indistinguishable from each other within typical
+numerical precision. This term does not have the “repulsive" effect that is observed in fig. 2 and
+thus the LLH-space is already fairly quadratic. However, this comes at the cost of roughly half
+an order of magnitude in statistical precision which can be important for performing a physics
+analysis on the highest-energy cosmic rays.
+In the final case, the saturated detector is completely ignored during the reconstruction of
+12
+
+300
+250
+10
+200
+ref
+10
+150
+100
+10
+50
+0
+10
+1.21.41.61.8
+2.2 2.4 2.6 2.8
+1g(S
+ref. Mc300
+250
+10°
+200
+ref
+10
+150
+100
+10
+50
+10
+0
+1 1.2 1.4 1.6 1.8
+2.2 2.4 2.6 2.8
+1g(S
+ref. Mc E
+lg
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+ S
+ lg
+1
+1.5
+2
+2.5
+a
+-0.75 -0.74 -0.73 -0.72 -0.71 -0.7 -0.69 -0.68 -0.67 -0.66 -0.65
+b 
+0.98
+1
+1.02
+Figure 9:
+Left: Example of correlation between fluorescence-based energy estimate and shower size estimate,
+in terms of (decimal) logarithms. Right: Distribution of reconstructed parameters (blue) that establish the linear
+relationship between lgE and lgSref. The red star indicates the average reconstructed value and the black lines are
+the 1σ and 2σ contours.
+the air-shower size and effectively treated as a hole in the array. This was studied using a satu-
+ration threshold of 1000 and is shown in the right panel of fig. 8. Similar to the previous case,
+the lack of repulsion from the hottest detector does not warp the LLH space in Cartesian coor-
+dinates. The estimations are again robust but at the cost of numerical precision on Sref.
+Note that the exact details of the gap in precision between the events with and without sat-
+uration and where these distributions begin as a function of air-shower size depend on the LDF
+shape, the true air-shower size, and the details of detector system. Shower-to-shower fluctua-
+tions, ignored in this study, should also have impact. However, it can be concluded that in the
+second and third cases, when the saturated detectors do not have any effective influence on the
+preferred core location, the use of Cartesian coordinates does not exhibit the biases in the un-
+certainty of Sref. Finally, note that using the log-polar coordinate system produces equivalent
+uncertainty values for the second and third cases and is thus still a more robust system that can
+account for any of the methods described above.
+4. Application example
+The change of parameters proposed in this study allows for keeping a correct meaning of
+the covariance matrix for any event topology. The use of lgSref instead of Sref may thus be ad-
+vantageous for several high-level analyses requiring an accurate event-by-event uncertainty.
+The price to pay is to redesign the analyses in terms of the variable lgSref. Among other em-
+blematic analyses, we choose to exemplify below such a redesign for the conversion of Sref into
+energy. The general strategy currently employed at both the Pierre Auger Observatory and the
+Telescope Array to carry out this conversion consists in using a set of data with which there
+are reconstructed shower sizes and an energy assignment by another means. For the Pierre
+Auger Observatory, this is done using a special set of air showers that can be reconstructed in-
+dependently by the fluorescence technique and by the surface detector array [6] while for the
+Telescope Array, Monte Carlo simulations are used instead. As an example of how the energy
+conversion can be derived using the estimated uncertainties, we use the former procedure. An
+empirical relationship between the energy measurements from the fluorescence technique and
+shower size measurements then allows for setting a nearly-calorimetric energy scale. To derive
+13
+
+this relationship, we adapt below the likelihood-based strategy from [5, 6] to a setup that, al-
+though simplified, mimics the main features of the Pierre Auger Observatory.
+E and Sref are generally related through a power-low relationship, E = ASB
+ref. The aim of
+the procedure is to infer the a = lg A and b = B parameters governing the linear relationship
+between lgE and lgSref:
+lg(E/eV) = a +b lgSref.
+(8)
+The choices a = −0.7 and b = 1 are fairly representative of those found in the Auger data [6].
+To build the relevant likelihood function, we model the distribution of lg ˆE and lg ˆS values for
+events detected simultaneously by both techniques as
+d2N
+dlg ˆE dlg ˆS
+=
+�
+dlgE dlgS RFD(lg ˆE;E) RSD(lg ˆS;lgS) δ(lgS,lgS(lgE))
+d2N
+dlgE dlgS ,
+(9)
+where quantities with a hat denote estimators and, to render the notations more compact, Sref
+is denoted as S. In this expression, the underlying event rate, d2N/dlgE dlgS, is folded into the
+resolution function of the fluorescence detector, RFD, and that of the surface-array detectors,
+RSD, both expressed in terms of lgE and lgS, respectively. In addition, the Dirac delta function
+guarantees that the underlying relationship given by eq. (8) is satisfied. The energy resolution
+of the fluorescence technique is well described as a Gaussian curve with parameters µE = E
+and σE = 0.08E [6]. To express the function RFD in terms of lg ˆE as the primary variable, we
+thus proceed with the associated Jacobian transformation.4 On the other hand, RSD can be
+considered as Gaussian in terms of lg ˆS (log-normal law in S with parameters µlgS = lgS and
+σlgS extracted from section 3/ fig. 6). Note that to simplify the example application of non-
+essential ingredients for demonstrating the robustness of the method, only the contribution
+to RSD stemming from the detector sampling fluctuations are considered; that is, shower-to-
+shower fluctuations of Sref are here ignored. Carrying out the integration in lgS, the probability
+density function to observe an event with lg ˆE and lg ˆS values is then described as
+p(lg ˆE,lg ˆS) = 1
+N
+�
+dlgE RFD(lg ˆE;E) RSD(lg ˆS;lgS(lgE)) dN
+dlgE ,
+(10)
+where the normalisation factor N is the total number of events detected by the fluorescence
+technique. The likelihood function is finally the product of this expression over the observed
+Nhyb values of lg ˆE and lg ˆS.
+To proceed with the likelihood-based strategy, we make use of the bootstrapping technique
+to substitute the underlying event rate for the observed energies, dN/dlgE → E �
+i δ(E, ˆEi). In
+this way, the final expression of the LLH to maximize reads as
+lnL =
+Nhyb
+�
+k=1
+ln
+�
+��
+1
+2πN
+N
+�
+i=1
+1
+σEi σlgSi
+10lg ˆEk
+10lg ˆSk exp
+�
+��−
+�
+10lg ˆEk −µEi
+�2
+2σ2
+Ei
+�
+��exp
+�
+��−
+�
+10lg ˆSk −µlgSi
+�2
+2σ2
+lgSi
+�
+��
+�
+��, (11)
+using the notation µEi = lg ˆEi, µlgSi = (lg ˆEi − ˆa)/ˆb, and where both uncertainty-related terms
+σEi and σlgSi are estimated by the event-by-event uncertainties.
+4That is, RFD(lg ˆE;E) = 10lg ˆE exp(−(10lg ˆE −µE)2/2σ2
+E)ln10/
+�
+2πσ2
+E.
+14
+
+We illustrate in fig. 9 the statistical performances of the minimisation of −2lnL to recover
+a and b by generating 1,000 mock samples of events above 1 EeV until Nhyb = 3000 events –
+typical of the number of hybrid events recorded at the Pierre Auger Observatory – are drawn
+above 3 EeV. We use the energy spectrum that is reported in ref. [6]. The correlation recovered
+between lgS and lgE is shown as the red line in the left panel for one of the mock samples, while
+the couple of ˆa and ˆb values recovered for the 1,000 realisations are shown as the blue points in
+the right panel. The red star is observed to be consistent with the injected a and b values while
+the expected 68% and 95% elliptic contours are obtained from the average covariance matrix
+returned by MINUIT. The coverage probabilities from the 1,000 samples are 68.5% and 97.0%,
+consistent with the expectations. The uncertainties in 10 ˆa and ˆb are of the same magnitude to
+those in ˆA and ˆB.
+5. Conclusion
+In this paper, we investigated the accuracy of the uncertainty in air-shower size estimation.
+While the uncertainty may be interpreted in the usual way when the errors in the impact point
+are small compared to the distances from the shower axis to the detectors, this is no longer the
+case for event topologies where the shower axis is close to a detector. We showed that in such
+cases, the LLH-space is highly curved in the ground coordinate system. For any algorithm that
+estimates uncertainties using the Hessian matrix formalism i.e., assumes a parabolic shape in
+the LLH-space, this presents a challenge. Using a set of toy Monte Carlo air-shower simulations,
+we showed that this effect is present for both triangular and square array layouts.
+However, this effect can be mitigated by choosing a better set of coordinates to describe the
+shower impact point. We propose the log-polar coordinate system which is centered on the de-
+tector with the largest signal and thus the one around which the LLH-space would be wrapped.
+In this coordinate system, the covariance between the shower size and the polar angle coordi-
+nate, Ψ, is almost decoupled and the LLH-space is better approximated by parabolic curvature
+near the minimum. We note that while this will not result in a systematically better estimation
+of the impact point (nor a worse one), it does produce a better estimation of the shower-size
+uncertainty.
+As this bias is an effect that occurs when the shower axis close to one particular station, we
+also studied the effects of saturation. The Cartesian coordinate system produces acceptable
+estimations of the uncertainty of the air-shower size when the saturated detectors are either
+ignored or simply accounted for in the LLH calculation. This comes at the cost of a degraded
+statistical precision on the air-shower energy and affects the highest-energy events more read-
+ily. When performing a signal recovery on the saturated detectors, the log-polar system is again
+required and the statistical estimation of the air-shower size (as well as the cosmic ray energy) is
+improved. With more modern methods, such as machine learning, or when simply using better
+hardware with a larger dynamic range, saturation effects can be mitigated or removed entirely.
+In any of the cases above, reconstructing using the log-polar coordinate system will produce
+robust results.
+An accurate event-by-event uncertainty of the shower-size estimator may be beneficial for a
+large variety of high-level analyses. As an application example, we have shown for instance that
+the event-by-event uncertainty can be used for the energy calibration of the shower size. While
+this application is emblematic for the use of the shower size, any high-level analysis making use
+of event-by-event estimations of uncertainty will benefit from the technique described here.
+15
+
+Figure A.10:
+The shower-size uncertainty is shown above for four zenith angles. The red entries highlight the
+events with Rhot < 200 m. The plots shown here are equivalent to the 40◦ study described in fig. 3.
+This includes those which make an event selection based on identifying high-quality events via
+their estimated uncertainty. All that is required is to design the analyses in terms of lgS as the
+primary variable.
+Appendix A. Invariance with zenith angle
+The underestimation of the shower-size uncertainty was studied above for a fixed zenith
+angle of 40◦. However, this effect is present for both more- and less-inclined air showers. In
+fig. A.10, we show the bias in the relative shower-size uncertainty versus Rhot for four additional
+zenith angles. The bias for small values of Rhot is obvious for all of these zenith angles. Note
+that the distributions are compressed in the radial direction since, at higher zenith angles, the
+projected spacing between the detectors in the direction of shower propagation shrinks as cosθ.
+Additionally, this projection effect also leads to more detectors close to the shower axis and thus
+having a larger signal with smaller uncertainties. The additional triggered stations that results
+16
+
+θ= 15°
+10°
+ref
+10
+10°
+100
+200
+300
+400
+500
+600
+700
+R,θ= 30°
+10
+ref
+S.
+bs
+10
+10-
+100
+200
+300
+400
+500
+600
+700
+R,0= 45°
+10
+ref
+10
+10-
+100
+200
+300
+400
+500
+600
+700
+R,θ = 60°
+10°
+ref
+S.
+10
+10
+100
+200
+300
+400
+500
+600
+700
+R,from this allows for a more accurate estimation of the shower size which can bee seen by the
+smaller σlgSref at higher zenith angles. While we note that the β and γ parameters of the LDF
+may vary with zenith angle and energy (see e.g. [11]), we neglected this here to highlight only
+the effect of changing the zenith angle.
+References
+[1] A. M. Hillas. In Acta Phys. Acad. Sci. Hung., 29, Suppl. 3, page 355, 1970.
+[2] A. M. Hillas, D. J. Marsden, J. D. Hollows, and H. W. Hunter. Measurement of primary
+energy of air showers in the presence of fluctuations. In 12th International conference on
+cosmic rays, page 1001, 1971.
+[3] M. A. Lawrence, R. J. O. Reid, and A. A. Watson. The Cosmic ray energy spectrum above
+4×1017 eV as measured by the Haverah Park array. J. Phys. G, 17:733–757, 1991.
+[4] David Newton, J. Knapp, and A. A. Watson. The Optimum Distance at which to Determine
+the Size of a Giant Air Shower. Astropart. Phys., 26:414–419, 2007.
+[5] Hans P. Dembinski, Balazs Kégl, Ioana C. Mari¸s, Markus Roth, and Darko Veberiˇc. A likeli-
+hood method to cross-calibrate air-shower detectors. Astropart. Phys., 73:44–51, 2016.
+[6] Alexander Aab et al. Measurement of the cosmic-ray energy spectrum above 2.5×1018 eV
+using the Pierre Auger Observatory. Phys. Rev. D, 102(6):062005, 2020.
+[7] Alexander Aab et al. The Pierre Auger Cosmic Ray Observatory. Nucl. Instrum. Meth. A,
+798:172–213, 2015.
+[8] T. Abu-Zayyad et al. The surface detector array of the Telescope Array experiment. Nucl.
+Instrum. Meth. A, 689:87–97, 2013.
+[9] X. Bertou et al. Calibration of the surface array of the Pierre Auger Observatory. Nucl.
+Instrum. Meth. A, 568:839–846, 2006.
+[10] M. Ave, P. Bauleo, Antonella Castellina, Aaron S. Chou, John L. Harton, R. Knapik, and
+G. Navarra. The accuracy of signal measurement with the water Cherenkov detectors of
+the Pierre Auger Observatory. Nucl. Instrum. Meth. A, 578:180–184, 2007.
+[11] Alexander Aab et al. Reconstruction of events recorded with the surface detector of the
+Pierre Auger Observatory. JINST, 15(10):P10021, 2020.
+[12] S. Yoshida et al. Lateral distribution of charged particles in giant air showers above EeV
+observed by AGASA. J. Phys. G, 20:651–664, 1994.
+[13] T. Abu-Zayyad et al. CORSIKA Simulation of the Telescope Array Surface Detector. 3 2014.
+[14] R. Abbasi et al.
+IceTop: The surface component of IceCube.
+Nucl. Instrum. Meth. A,
+700:188–220, 2013.
+[15] W. D. Apel et al. The KASCADE-Grande experiment. Nucl. Instrum. Meth. A, 620:202–216,
+2010.
+17
+
+[16] F. James and M. Roos. Minuit: A System for Function Minimization and Analysis of the
+Parameter Errors and Correlations. Comput. Phys. Commun., 10:343–367, 1975.
+[17] R. Brun and F. Rademakers. ROOT: An object oriented data analysis framework. Nucl.
+Instrum. Meth. A, 389:81–86, 1997.
+[18] Q. Yue, W. P. Lai, W. C. Chang, H. B. Li, J. Li, S. T. Lin, D. Z. Liu, J. F. Qiu, V. Singh, and H. T.
+Wong. Effective dynamic range in measurements with flash analog to digital convertor.
+Nucl. Instrum. Meth. A, 511:408–416, 2003.
+[19] Darko Veberic. Estimation of the Total Signal in Saturated Stations of Pierre Auger Surface
+Detector. In 33rd International Cosmic Ray Conference, page 0633, 2013.
+[20] Yu Liu, Jing-Jun Zhu, Neil Roberts, Ke-Ming Chen, Yu-Lu Yan, Shuang-Rong Mo, Peng Gu,
+and Hao-Yang Xing. Recovery of saturated signal waveform acquired from high-energy
+particles with artificial neural networks. Nucl. Sci. Tech., 30(10):148, 2019.
+18
+
diff --git a/V9AzT4oBgHgl3EQfmP0G/content/tmp_files/load_file.txt b/V9AzT4oBgHgl3EQfmP0G/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2da65b437711952f198986c6b8ecb64d5e83ae6b
--- /dev/null
+++ b/V9AzT4oBgHgl3EQfmP0G/content/tmp_files/load_file.txt
@@ -0,0 +1,682 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf,len=681
+page_content='An Improved Method of Estimating the Uncertainty of Air-Shower  Size at Ultra-High Energies  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Colemana,∗, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Billoirb, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Delignyc  aUniversity of Delaware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Bartol Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Newark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' DE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' USA  bLaboratoire de Physique Nucléaire et de Hautes Energies (LPNHE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Sorbonne Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Université de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  CNRS-IN2P3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' France  cLaboratoire de Physique des 2 Infinis Irène Joliot-Curie (IJCLab),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' CNRS/IN2P3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Université Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Orsay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  France  Abstract  The collection of a statistically significant number detected of cosmic rays with energy above  1017 to 1018 eV requires widely-spaced particle detectors at the ground level to detect the ex-  tensive air showers induced in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The air-shower sizes, proxies of the primary  energies, are then estimated by fitting the observed signals to a functional form for expecta-  tions so as to interpolate the signal at a reference distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The functional form describes  the rapid falloff of the expected signal with the distance from the shower core, using typi-  cally two logarithmic slopes to account for the short-range and long-range decreases of signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  The uncertainties associated to the air-shower sizes are determined under the assumption of a  quadratic dependence of the log-likelihood on the fitted parameters around the minimum, so  that a meaningful variance-covariance matrix is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In this paper, we show that for an  event topology where one signal is much larger than the others, the quadratic dependence of  the fitted function around the minimum is a poor approximation that leads to an inaccurate  estimate of the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' To restore a quadratic shape, we propose to use the polar coordi-  nates around the detector recording the largest signal, projected onto the plane of the shower  front, to define the likelihood function in terms of logarithmic polar distances, polar angles and  logarithmic shower sizes as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' We show that a meaningful variance-covariance  matrix is then recovered in the new coordinate system, as the dependence of the fitted function  on the modified parameters is properly approximated by a quadratic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The use of the  uncertainties in the new coordinate system for subsequent high-level analyses is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Introduction  Ultra-high energy cosmic rays with energies above 1017 to 1018 eV are observed through the  extensive air showers they induce in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' To collect a statistically significant num-  ber of events, very-widely-spaced particle detectors at the ground level have been used to max-  imize the aperture of the observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Typical detection methods, such as scintillator panels  or water-Cherenkov tanks, observe an integrated charge that results from a convolution of the  detector response with the local particle density, energy distribution, and arrival direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The  ∗Corresponding author  Email addresses: alanco@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='edu (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Coleman), billoir@lpnhe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='fr (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Billoir),  deligny@ijclab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='fr (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Deligny)  Preprint submitted to Elsevier  January 5, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='01558v1  [hep-ex]  4 Jan 2023  spacing of the particle detectors turns out to be five to ten times larger than the Molière radius,  which delineates the distance in which more than 90% of the ionizing particles are contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  Consequently, it has been an increasingly difficult task to determine the lateral distribution of  particles on an event-by-event basis, and thus to use the associated total number of particles  for estimating the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  As an alternative energy estimator, the signal at a distance appropriate to the spacing of the  shower array is used as a surrogate measurement of the shower size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' First proposed by Hillas [1,  2], the technique, successfully exploited by the Haverah Park Collaboration [3], was reviewed  in details [4] and adopted by the Pierre Auger Collaboration, the Telescope Array Collaboration,  and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The conversion from shower size into primary energy may subsequently include  several correction factors, calibrations, and/or comparisons with simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The key point  of the technique is to infer the shower size from the amount of signal that is expected to be  measured by a detector at a reference distance, rref, by fitting a lateral distribution function,  LDF, to the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' At that specific distance, which depends on the topology of the  surface array, the fluctuations in the age of the showers when reaching the ground, inherited  from the stochastic variations in the location and character of the leading interactions, reflected  in the fluctuations of the LDF, are minimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In this way, the shower size is determined with  adequate accuracy (< 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' While a further description of this technique is beyond the scope  of this paper, it is important to note that if using a shower-size technique, the resolution with  which the size can be determined ultimately sets the minimum resolution on the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  This estimation is done in practice by fitting the observed signal amplitudes, S(ri), at a dis-  tance, ri, using an LDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' A common convention is to normalize the LDF using the reference  distance, rref, such that LDF(rref) ≡ 1 and thus the shower size at the reference distance can be  directly extracted from S(r) = Sref LDF(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The fit of such a model to determine the size of the  air shower involves the determination of at least five parameters, two which describe the orien-  tation of the shower axis, two which define the intersection of the axis with a plane1, and Sref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  Additional parameters which describe the exact nature of the exponential decrease in signal  size with distance from the shower axis may also be fit, but near the triggering threshold, there  may not be enough degrees-of-freedom and an average LDF is typically used, which is based  on observed or simulated air showers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The combination of the nearly power-law shape of the  LDF and the first-order cylindrical symmetry of the signals about the shower axis results in a  non-trivial phase space for the log-likelihood function (LLH) that can hamper an accurate de-  termination of the optimal parameters as well as their uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The aim of this work is to  explore carefully the LLH phase space for typical event topologies encountered in practice with  contemporary observatories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' the Pierre Auger Observatory and the Telescope Array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  We note that while the arrival direction of the air shower is important for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=', anisotropy  studies, the two parameters that define the intersection point, the impact point, are generally  nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This allows for some freedom regarding the choice of parameters to de-  scribe the location of the impact point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In this work, we show how the estimation of the un-  certainties, particularly that of the logarithm of Sref, can be determined in a much more stable  way when employing a “log-polar” coordinate system, rather than a Cartesian one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In section 2,  we set the stage of the generic framework allowing us to develop an efficient tool to simulate  and reconstruct events recorded by surface detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Using this tool, we are able to extract  event topologies leading to LLHs that cannot be approximated by quadratic dependencies in  1Typically this plane is taken to be the ground in the local coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  2  the reconstructed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' To correct this issue, we propose in section 3 to change the co-  ordinate system for the fit, switching to a log-polar one, and show that quadratic dependencies  of the LLHs in the reconstructed parameters more accurately describe the true uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  Among a variety of possible high-level analyses, we show in section 4 how the energy calibra-  tion performed in a manner similar to that used at the Auger Observatory [5, 6] may benefit  from accurate event-by-event uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Finally, conclusions are given in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Simulation and reconstruction of air showers  We employ a toy model for the simulation of signals in an air-shower array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Rather than  running e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=', full simulations of the particle cascades, we take the ideal case where the LDF is  known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Shower-to-shower fluctuations are ignored, as they are nonessential for the purpose  of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In this way, the LDF accurately describes the distribution of signals around the  shower axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Showers are randomly sampled on a triangular array with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5 km spacing, based  on the layout of the Pierre Auger Observatory [7], see the left panel of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' However, we will  3000  −  2000  −  1000  −  0  1000 2000 3000  x / m  3000  −  2000  −  1000  −  0  1000  2000  3000  y / m  2000  −  1000  −  0  1000  2000  x / m  2000  −  1000  −  0  1000  2000  y / m  Figure 1:  The layout of the triangular and square arrays (black circles) are shown in the left and right panels,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The region over which the impact points are sampled for each layout is outlined in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  also show that the effect is the same for a square array with 1200 m spacing, based on the ar-  rangement of the surface array of Telescope Array [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  Without loss of generality, all signals in this work are expressed in Vertical Equivalent Muon  (VEM) units, regardless of the detector or of the particle species crossing the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' A VEM  is defined as the sum of the charge collected for a single muon traversing vertically a detector  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The value of Sref is sampled randomly from a dN/dSref ∝ S−1  ref spectrum and  over a range that is commensurate with energies that correspond to E > 3 EeV for the 1500 m  surface detector of the Pierre Auger Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In this work, we will focus on air showers with  a zenith angle of 40◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' A study on the zenith dependence is shown in the appendix and while a  bias on the shower-size estimation is shown to exist at all angles, there is some dependence on  the arrival direction in the exact characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  A signal is assigned to each detector on the array using the LDF,  S(r) = Sref  � r  rref  �β � r +r0  rref +r0  �β+γ  ,  (1)  3  with β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='1, rref = 1000 m, and r0 = 700 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The falloff of the signal from the shower  axis is thus described by a logarithmic slope denoted as β, and by a small departure from a  power-law at large distances governed by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The signal assigned to each detector is additionally  smeared using Gaussian fluctuations of width  σ(S) = AS  �  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  (2)  The scaling in  �  S has been shown to describe the Pierre Auger Observatory data [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The factor  AS, chosen to be 1 in this work, is not considering any dependence in zenith angle in this study2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  Finally, an ideal trigger is implemented which keeps only detectors with a signal above ST = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  While the values and formulas used above are based on those used for water-Cherenkov tanks  at the Pierre Auger Observatory [11], they are similar in nature to those used by other current  air-shower experiments [4, 12–15] and the issue being discussed here is ultimately a geometric  one and is not sensitive to these choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Also note that in this work, reconstructions will be  performed with a fixed LDF shape (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' β and γ) and thus the choice of rref is arbitrary and has  no influence on the outcome of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Reconstructing the air-shower parameters  In this study, we mostly study the shower-size estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Since air-shower arrays can typi-  cally be used to reconstruct air showers to an accuracy of better than a few degrees [11, 13–15],  we will neglect the reconstruction of the shower arrival direction since it is a sub-dominant ef-  fect when reconstructing the shower size and is determined from arrival times rather than signal  amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Thus in this work, MINUIT is given only three parameters, lgSref and two param-  eters that determine the impact point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In the beginning of this work, the impact point will be  defined by the (x, y) position in the ground plane, but we will define a better set of coordinates  in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  The model of an air shower, with the LDF shape, is applied to the simulated data and the air  shower parameters are fit by minimizing a negative log-likelihood that describes the probability  to observe the set of signals, {Si}, in each detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The likelihood is made of two main terms,  one that describes the detectors which triggered, and one for those that did not3,  −2LLH =  Triggered  �  i  ��Si −S(ri)  σ(S(ri))  �2  +ln(2πσ2(S(ri)))  �  −2  S j =0  �  j  ln  �  P(S(r j))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  (3)  The second sum describes the probability, P(S), that no signal was observed when a signal,  S(r j), is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' For Gaussian fluctuations, this probability is given by,  P(S) = 1− 1  2  �  1+erf  � S −ST  �  2 σ(S)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  (4)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Minimization and error estimation  The reconstruction of an air shower consists of two tasks, traversing through the LLH-space  to find the (global, in the ideal case) minimum of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' (3) and to characterize the curvature near  2While a given detector may have a zenith-dependent σ(S), here we only study a single zenith angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  3Throughout the paper, the notation lg stands for the decimal logarithm, while ln stands for the natural loga-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  4  this minimum to estimate the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Ideally, the whole LLH-space could be scanned and  the global minimum can be directly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' However, this is typically impractical for a large  data set and instead most algorithms require a well-estimated starting point and numerically  calculate and follow the gradient of the LLH-space until a minimum is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This so-called  gradient-descent method does not ensure a global minimum is ultimately found and there are  various other pitfalls, but a discussion of these issues is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Many  implementations of this algorithm exist and extensions on this basic concept have been devel-  oped, but in this work, we focus on MINUIT [16], the standard minimizer that is packaged with  ROOT [17], which is commonly used in particle and air-shower physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  To do the second task, MINUIT provides an estimate of the curvature in the neighborhood  of the identified minimum in LLH-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The Hessian matrix of second-derivatives with re-  spect to all of the free-parameters (impact point, Sref, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=') is numerically calculated and then  inverted to produce the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This matrix is an estimate of the local curvature  assuming that the LLH-space is quadratic near minimum and also provides the correlations  between parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This simplistic definition will be shown to be problematic for the recon-  struction of a specific class of air showers where the LLH-space does not fulfill the quadratic  assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' LLH-space for air-shower events  Using the LLH defined above, the difficulty for a numerical minimization routine to estimate  the uncertainties for an air-shower reconstruction can be made evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Two example events  of air showers with zenith angles of 40◦ are shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Each panel shows the results of a  scan of the potential impact point locations in (x, y) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=', along the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' At each location,  the impact point and arrival direction were held fixed and only the value of lgSref was left free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  The LLH space is seen to wrap around the detector with the largest signal, indicated by the  black dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Exemplified by the 1σ, 2σ, and 3σ contours, the LLH-space is stretched along the red  lines of constant reconstructed lgSref, which also encircle the detector with the largest signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  This curvature is expected, both for the LLH and the lgSref contours since the LDF is dependent  solely on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  For an impact point location sufficiently far from a detector (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=', top panels of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 2), the  1σ LLH-contour in Cartesian coordinates can be approximately described by an ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In this  case, the second-derivative matrix calculated by MINUIT is sufficient to estimate the uncer-  tainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Both the 1σ LLH-contour and estimated ellipse reside inside the lgSref/lgSbest = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='8,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='2}  lines and are in agreement with the estimated uncertainty from MINUIT of σlgSref ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  The bottom panels highlight the problem of estimating the uncertainty in lgSref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In this  case, the impact point is close enough to a detector that the highly-wrapped LLH-space is not  sufficiently parabolic in Cartesian coordinates for the second-derivative calculation to provide a  stable estimate of the uncertainty in lgSref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The curvature about the detector is degenerate with  a gradient of the LLH-space and the relative uncertainty of the shower size is estimated to be  σlgSref ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='07 (bottom-right panel) even though the true 1σ contour (bottom-left panel) shows  that the uncertainty is at least 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' For an air-shower experiment, this creates a erroneous  result in the estimated uncertainties on lgSref depending on the impact point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  To study the phase space of impact points where this effect is most prominent, 50 000 events  were simulated and reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Random uniform azimuth angles were chosen and the im-  pact points were uniformly picked within the boundaries highlighted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The impact point  (x, y) and lgSref were left as free parameters during the minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The errors on lgSref, as  5  Figure 2: The LLH-space for an event with a zenith angle of 40◦ is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The color scale indicates the confidence  interval of a given impact point being correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The location with the largest LLH value is indicated with a black  cross and the location of the detector with the largest signal is indicated by the black circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The red lines are the  contours of reconstructed lgSref values with respect to that of the largest LLH value, lgSref/lgSbest = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='8,1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The  black lines show the 1σ, 2σ, and 3σ contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In the left panels, these contours represent the true ones while those  in the right panel correspond to the ellipses given by the covariance matrix from MINUIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The top and bottom  panels show events where the shower axis is 227 m and 84 m from the shown detector location, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In the  panels on the right, the estimated uncertainty of the shower size, σlgSref, is shown in the lower left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  6  300  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  200  3  m  100  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  true  2 6  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  100  1  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  30Q  300  200  100  0  100  200  300  /m  (x-x  true300  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  200  3  m  100  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  true  2 6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  100  1  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='16  ret  300  300  200  100  0  100  200  300  (x- X  m  true300  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  200  3  m  100  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  true  6  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  7  100  1  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  30Q  300  200  100  0  100  200  300  (x- x  /m  true300  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  200  3  100  m  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  true  2  6  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  7  100  1  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  007  ret  300  300  200  100  0  100  200  300  (x-x  m  trueFigure 3: Left: Estimated relative uncertainty of the shower-size, as calculated using MINUIT, for 50 000 simulated  air showers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This distribution is shown as a function of the distance of the detector with the largest signal from the  shower axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The values in red are those with Rhot < 200 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Right: Histogram of the uncertainty for simulations  with Rhot ≥ 200 m as a function of the true shower size lgSref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The red points are the same events that are shown in  red in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  estimated by the MINUIT algorithm, are shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' As seen in the left panel, the relative  uncertainty decreases with increasing the distance from the shower axis, Rhot, to the detector  with the largest signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' However, for small values of Rhot, shown in red, there is an opposite  trend and the uncertainties are up to an order of magnitude too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In the right panel, the  distribution of events is shown versus shower size, again with small values of Rhot shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  The underestimated shower uncertainties are seen to occur at all shower-size values meaning  that this effect is independent of the number of triggered detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Fitting in log-polar coordinates  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Log-polar coordinates  The issue described above is ultimately related to the non-parabolic nature of the LLH-space  in terms of the parameters that are given to the minimizer, namely the Cartesian location of the  impact point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Instead we propose a more natural coordinate system for the task of reconstruct-  ing the impact point and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' To develop the new parameters, we first define the Cartesian  ground coordinate system wherein an air shower has zenith and azimuth angles of θ and φ, re-  spectively, and the impact point is described by the (x, y) intersection of the shower axis with  the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This system is depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 4 wherein a sampling area, such as that around the  black point, is the projection onto the ground, along the Z axis, of a region in the shower plane  (X ,Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The shower frame is thus defined by ns and two perpendicular axes, for example an hor-  izontal one, nh, and the other one, nv, into the vertical plane containing ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' We consider in the  following the shower-front coordinates obtained from the ground ones as,  Xsc = (∆xhot cosφ+∆yhot sinφ)cosθ,  (5a)  Ysc = −∆xhot sinφ+∆yhot cosφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  (5b)  7  10  ref  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  10  10  100  200  300  400  500  600  700  R,180  160  140  120  100  10  80  60  10  40  20  0  10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  1g(S  ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' MY  Z  X  nv  ns  nh  Ψ  Figure 4: Diagram of the shower frame coordinate system: the Z axis is along the shower axis ns, the X axis is along  nv in the vertical plane going through the shower axis, and the Y axis is along nh in the horizontal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In this  coordinate system, ψ is used to describe the polar angle about the shower axis, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  Here, ∆xhot and ∆yhot define a Cartesian location in the ground coordinate system with respect  to the detector with the largest signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The values of Xsc and Ysc are the 2D Cartesian locations  in the shower coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' To reconstruct the air-shower size, we then propose the “log-  polar” coordinate system defined by the transformation into the shower plane,  R =  �  X 2  sc +Y 2  sc,  (6a)  Ψ = arctan2(Ysc,Xsc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  (6b)  The new variables for reconstructing air showers are given by lgR and Ψ, the log-radius and  polar angle about a line that is parallel to the shower axis and passes through the detector with  largest signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In addition, lgSref, the (decimal) logarithm of Sref, is considered as the third free  parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The choice of anchoring this coordinate system to the detector with the largest  signal is motivated by the curvature of the LLH-space, as previously seen in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Results  In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 5, the same two example events are shown in the log-polar coordinate system defined  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' For both sample events, several nice properties of the LLH space are seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Firstly, in  this space, the lines of constant reconstructed lgSref are almost independent of the Ψ coordun-  cinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This is beneficial as this means that the correlation between Ψ and lgSref is small, re-  gardless of the direction of the shower propagation, which is helpful for attaining a more robust  estimate of the shower-size uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Secondly, the 1σ contours are more elliptical mean-  ing that the assumption when using the Hessian matrix method is more valid and will typically  return a better error estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  We repeated the study shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 3 on the same 50 000 simulated events but using the  log-polar coordinate system described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The results are shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In this case, the  uncertainty estimation is much more well behaved with most of the events with Rhot < 200 m  appearing now in a narrow band as a function of Sref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This band occurs because shower axes  that pass very near to a detector are necessarily maximally far from all other detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' On one  hand, this will produce less triggered detectors per event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' On the other hand, for the detectors  with signal, the uncertainties, which scale like  �  S(r)/S(r) = 1/  �  LDF(r) ∝ r −β/2, will be larger  8  Figure 5:  The top and bottom panels show the LLH-space for the same two events as those in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 2, this time  calculated in the log-polar coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The true 1σ, 2σ, and 3σ LLH contours are given in the left panels  while the right panels are the respective ellipses from the covariance matrix from MINUIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  9  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='55H  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='45  m  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='4  10U  26  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
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+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
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+page_content='2  0  3  2  0  2  1  1  3  W  true4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='55H  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
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+page_content='45  m  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='4  10U  26  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='35  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
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+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='16  0  3  2  0  2  3  1  1  W  true2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='05  3  2  W  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='95  hot  2 6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='9  80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='85  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='75  0  3  2  0  1  2  3  1 true4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='05  3  2  m  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='95  hot  2 6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='9  ao  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='85  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  O  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='75  ref  0  3  2  0  2  3  1  1  V  trueFigure 6: The same distributions as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 3, using the same 50 000 Monte Carlo trials, but calculating the  shower-size uncertainty using the log-polar coordinates defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  than average for a given shower size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Although the “hottest” detector has the smallest relative  uncertainty, its weight in the fit is mitigated compared to that of other detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This is because  a small shift δr in the impact point results in a relative change βδr/r for S and is proportional to  r −β/2−1 in units of σ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The fitting procedure can thus change the expected signal of the hottest  detector at a low cost through a change of the impact point, without affecting the expectation  for the other detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' All in all, these effects produce a less constrained LLH space and results  in a larger uncertainty in the shower size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' While there are a few outliers, these make up only  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='1% of all events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Dependence on array geometry  Given that this work is focused on a geometric issue, it is important to understand how the  layout of the array ultimately impacts the shower-size estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' We contrast the main results  shown above after changing to the surface detector layout of Telescope Array, a square grid with  1200 m spacing between detectors, as shown in the right panel of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The results are shown  in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  For the more dense detector arrangement, more detectors are triggered for a given shower  size and the precision on lgSref is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' However, it is clear that the same issue is present for this  array configuration, as seen in the top panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The radius at which the shower size uncertainty  undergoes an inflection is even in a similar location given the length scales involved, occurring  at Rhot/dspacing ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='125 where dspacing is either 1500 m or 1200 m for the respective array layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  Ultimately, a significant improvement is still observed by using the log-polar coordinates in the  air shower reconstruction, as seen in the bottom panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Effects of saturation  In measured air-shower data, detectors near the shower axis can saturate, providing an un-  reliable estimate of the signal content at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Since it is exactly these types of showers  that result in non-quadratic LLH contours, it is worthwhile to consider the way that saturated  stations may impact the results presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' While the non-linear effects that occur near  10  10°  ref  10-1  10  100  200  300  400  500  600  700  R,180  160  10°  140  120  ref  100  10  80  60  10  40  20  0  10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  1g(S  ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' McFigure 7:  The distributions of the relative estimated shower-size uncertainty are shown for the Telescope Array  detector layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In this case, the red points are defined by events with Rhot < 100 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The top panels correspond to  reconstructions using Cartesian coordinates (as in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 3) while the bottom panels correspond to reconstructions  using the log-polar coordinates (as in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  11  10  10  10-3  0  100  200  300  400  500  600180  160  140  10  120  ret  100  10  80  60  10  40  20  0  10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='8  2  1g(S  ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' M(10  ref  10  10-  1(  0  100  200  300  400  500  600180  160  140  10-  120  ref  100  b0  10  80  60  10  40  20  0  10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='8  2  1g(S  ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' MFigure 8:  Left: The uncertainty in the shower size, when using an additional term in the LLH to account for  saturated stations (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' (7)) when performing a reconstruction using Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Right: The same figure  when ignoring saturated stations in during the reconstruction of the air-shower size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Both panels in this figure are  directly comparable to fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  the upper end of the dynamic range of a detector will be hardware-dependent, we show a few  limiting cases for handling saturated detectors during reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  In the first case, the underlying signal that would have been measured in the absence of  saturation is recovered post-hoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Whether by analytical methods or using machine learning,  various techniques have been employed to estimate the would-be signal in saturated detectors,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In the best case, the signal that would have been measured is precisely estimated  with the result that effectively nothing has changed with respect to the results shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In  the less ideal case, the recovery introduces an additional uncertainty, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' σ(S) −→ σ(S)+∆(S),  which can simply be taken into account in the reconstruction (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This would widen the  LLH contours, depending on the relative size of ∆(S), but would not remove the non-quadratic  behavior close to the saturated detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  Alternatively, a corresponding term can be included in the likelihood function to describe  the probability that a detector observes saturation given an expected signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In the case of ideal  saturation, where signals only up to Ssat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' can be observed, the corresponding term to include  in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' (3) is,  −2  saturated  �  k  ln  �  1−erf  � Ssat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' −Sk  �  2 σ(Sk)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  (7)  This was tested within the framework of this toy model using Ssat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' = 1000 and the results are  shown in the left panel of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In this case, the Cartesian coordinates already present a good  estimation of the shower size uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This is a result of the rather non-restrictive nature  of the additional term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' (7) for which expected signals of 1500 and 15 000 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' larger and  smaller Rhot values) give a probabilities that are indistinguishable from each other within typical  numerical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This term does not have the “repulsive" effect that is observed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 2 and  thus the LLH-space is already fairly quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' However, this comes at the cost of roughly half  an order of magnitude in statistical precision which can be important for performing a physics  analysis on the highest-energy cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  In the final case, the saturated detector is completely ignored during the reconstruction of  12  300  250  10  200  ref  10  150  100  10  50  0  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
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+page_content='8  1g(S  ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Mc300  250  10°  200  ref  10  150  100  10  50  10  0  1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
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+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
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+page_content='68 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='67 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='66 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='65  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='98  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='02  Figure 9:  Left: Example of correlation between fluorescence-based energy estimate and shower size estimate,  in terms of (decimal) logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Right: Distribution of reconstructed parameters (blue) that establish the linear  relationship between lgE and lgSref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The red star indicates the average reconstructed value and the black lines are  the 1σ and 2σ contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  the air-shower size and effectively treated as a hole in the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This was studied using a satu-  ration threshold of 1000 and is shown in the right panel of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Similar to the previous case,  the lack of repulsion from the hottest detector does not warp the LLH space in Cartesian coor-  dinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The estimations are again robust but at the cost of numerical precision on Sref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  Note that the exact details of the gap in precision between the events with and without sat-  uration and where these distributions begin as a function of air-shower size depend on the LDF  shape, the true air-shower size, and the details of detector system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Shower-to-shower fluctua-  tions, ignored in this study, should also have impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' However, it can be concluded that in the  second and third cases, when the saturated detectors do not have any effective influence on the  preferred core location, the use of Cartesian coordinates does not exhibit the biases in the un-  certainty of Sref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Finally, note that using the log-polar coordinate system produces equivalent  uncertainty values for the second and third cases and is thus still a more robust system that can  account for any of the methods described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Application example  The change of parameters proposed in this study allows for keeping a correct meaning of  the covariance matrix for any event topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The use of lgSref instead of Sref may thus be ad-  vantageous for several high-level analyses requiring an accurate event-by-event uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  The price to pay is to redesign the analyses in terms of the variable lgSref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Among other em-  blematic analyses, we choose to exemplify below such a redesign for the conversion of Sref into  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The general strategy currently employed at both the Pierre Auger Observatory and the  Telescope Array to carry out this conversion consists in using a set of data with which there  are reconstructed shower sizes and an energy assignment by another means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' For the Pierre  Auger Observatory, this is done using a special set of air showers that can be reconstructed in-  dependently by the fluorescence technique and by the surface detector array [6] while for the  Telescope Array, Monte Carlo simulations are used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' As an example of how the energy  conversion can be derived using the estimated uncertainties, we use the former procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' An  empirical relationship between the energy measurements from the fluorescence technique and  shower size measurements then allows for setting a nearly-calorimetric energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' To derive  13  this relationship, we adapt below the likelihood-based strategy from [5, 6] to a setup that, al-  though simplified, mimics the main features of the Pierre Auger Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  E and Sref are generally related through a power-low relationship, E = ASB  ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The aim of  the procedure is to infer the a = lg A and b = B parameters governing the linear relationship  between lgE and lgSref:  lg(E/eV) = a +b lgSref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  (8)  The choices a = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='7 and b = 1 are fairly representative of those found in the Auger data [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  To build the relevant likelihood function, we model the distribution of lg ˆE and lg ˆS values for  events detected simultaneously by both techniques as  d2N  dlg ˆE dlg ˆS  =  �  dlgE dlgS RFD(lg ˆE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='E) RSD(lg ˆS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='lgS) δ(lgS,lgS(lgE))  d2N  dlgE dlgS ,  (9)  where quantities with a hat denote estimators and, to render the notations more compact, Sref  is denoted as S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In this expression, the underlying event rate, d2N/dlgE dlgS, is folded into the  resolution function of the fluorescence detector, RFD, and that of the surface-array detectors,  RSD, both expressed in terms of lgE and lgS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In addition, the Dirac delta function  guarantees that the underlying relationship given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' (8) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The energy resolution  of the fluorescence technique is well described as a Gaussian curve with parameters µE = E  and σE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='08E [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' To express the function RFD in terms of lg ˆE as the primary variable, we  thus proceed with the associated Jacobian transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='4 On the other hand, RSD can be  considered as Gaussian in terms of lg ˆS (log-normal law in S with parameters µlgS = lgS and  σlgS extracted from section 3/ fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Note that to simplify the example application of non-  essential ingredients for demonstrating the robustness of the method, only the contribution  to RSD stemming from the detector sampling fluctuations are considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' that is, shower-to-  shower fluctuations of Sref are here ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Carrying out the integration in lgS, the probability  density function to observe an event with lg ˆE and lg ˆS values is then described as  p(lg ˆE,lg ˆS) = 1  N  �  dlgE RFD(lg ˆE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='E) RSD(lg ˆS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='lgS(lgE)) dN  dlgE ,  (10)  where the normalisation factor N is the total number of events detected by the fluorescence  technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The likelihood function is finally the product of this expression over the observed  Nhyb values of lg ˆE and lg ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  To proceed with the likelihood-based strategy, we make use of the bootstrapping technique  to substitute the underlying event rate for the observed energies, dN/dlgE → E �  i δ(E, ˆEi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In  this way, the final expression of the LLH to maximize reads as  lnL =  Nhyb  �  k=1  ln  �  ��  1  2πN  N  �  i=1  1  σEi σlgSi  10lg ˆEk  10lg ˆSk exp  �  ��−  �  10lg ˆEk −µEi  �2  2σ2  Ei  �  ��exp  �  ��−  �  10lg ˆSk −µlgSi  �2  2σ2  lgSi  �  ��  �  ��, (11)  using the notation µEi = lg ˆEi, µlgSi = (lg ˆEi − ˆa)/ˆb, and where both uncertainty-related terms  σEi and σlgSi are estimated by the event-by-event uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  4That is, RFD(lg ˆE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='E) = 10lg ˆE exp(−(10lg ˆE −µE)2/2σ2  E)ln10/  �  2πσ2  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  14  We illustrate in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 9 the statistical performances of the minimisation of −2lnL to recover  a and b by generating 1,000 mock samples of events above 1 EeV until Nhyb = 3000 events –  typical of the number of hybrid events recorded at the Pierre Auger Observatory – are drawn  above 3 EeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' We use the energy spectrum that is reported in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The correlation recovered  between lgS and lgE is shown as the red line in the left panel for one of the mock samples, while  the couple of ˆa and ˆb values recovered for the 1,000 realisations are shown as the blue points in  the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The red star is observed to be consistent with the injected a and b values while  the expected 68% and 95% elliptic contours are obtained from the average covariance matrix  returned by MINUIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The coverage probabilities from the 1,000 samples are 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='5% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='0%,  consistent with the expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The uncertainties in 10 ˆa and ˆb are of the same magnitude to  those in ˆA and ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Conclusion  In this paper, we investigated the accuracy of the uncertainty in air-shower size estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  While the uncertainty may be interpreted in the usual way when the errors in the impact point  are small compared to the distances from the shower axis to the detectors, this is no longer the  case for event topologies where the shower axis is close to a detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' We showed that in such  cases, the LLH-space is highly curved in the ground coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' For any algorithm that  estimates uncertainties using the Hessian matrix formalism i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=', assumes a parabolic shape in  the LLH-space, this presents a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Using a set of toy Monte Carlo air-shower simulations,  we showed that this effect is present for both triangular and square array layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  However, this effect can be mitigated by choosing a better set of coordinates to describe the  shower impact point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' We propose the log-polar coordinate system which is centered on the de-  tector with the largest signal and thus the one around which the LLH-space would be wrapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  In this coordinate system, the covariance between the shower size and the polar angle coordi-  nate, Ψ, is almost decoupled and the LLH-space is better approximated by parabolic curvature  near the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' We note that while this will not result in a systematically better estimation  of the impact point (nor a worse one), it does produce a better estimation of the shower-size  uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  As this bias is an effect that occurs when the shower axis close to one particular station, we  also studied the effects of saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The Cartesian coordinate system produces acceptable  estimations of the uncertainty of the air-shower size when the saturated detectors are either  ignored or simply accounted for in the LLH calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' This comes at the cost of a degraded  statistical precision on the air-shower energy and affects the highest-energy events more read-  ily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' When performing a signal recovery on the saturated detectors, the log-polar system is again  required and the statistical estimation of the air-shower size (as well as the cosmic ray energy) is  improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' With more modern methods, such as machine learning, or when simply using better  hardware with a larger dynamic range, saturation effects can be mitigated or removed entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  In any of the cases above, reconstructing using the log-polar coordinate system will produce  robust results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  An accurate event-by-event uncertainty of the shower-size estimator may be beneficial for a  large variety of high-level analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' As an application example, we have shown for instance that  the event-by-event uncertainty can be used for the energy calibration of the shower size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' While  this application is emblematic for the use of the shower size, any high-level analysis making use  of event-by-event estimations of uncertainty will benefit from the technique described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  15  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='10:  The shower-size uncertainty is shown above for four zenith angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The red entries highlight the  events with Rhot < 200 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The plots shown here are equivalent to the 40◦ study described in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  This includes those which make an event selection based on identifying high-quality events via  their estimated uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' All that is required is to design the analyses in terms of lgS as the  primary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Invariance with zenith angle  The underestimation of the shower-size uncertainty was studied above for a fixed zenith  angle of 40◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' However, this effect is present for both more- and less-inclined air showers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' In  fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='10, we show the bias in the relative shower-size uncertainty versus Rhot for four additional  zenith angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The bias for small values of Rhot is obvious for all of these zenith angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' Note  that the distributions are compressed in the radial direction since, at higher zenith angles, the  projected spacing between the detectors in the direction of shower propagation shrinks as cosθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  Additionally, this projection effect also leads to more detectors close to the shower axis and thus  having a larger signal with smaller uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' The additional triggered stations that results  16  θ= 15°  10°  ref  10  10°  100  200  300  400  500  600  700  R,θ= 30°  10  ref  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  bs  10  10-  100  200  300  400  500  600  700  R,0= 45°  10  ref  10  10-  100  200  300  400  500  600  700  R,θ = 60°  10°  ref  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='  10  10  100  200  300  400  500  600  700  R,from this allows for a more accurate estimation of the shower size which can bee seen by the  smaller σlgSref at higher zenith angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content=' While we note that the β and γ parameters of the LDF  may vary with zenith angle and energy (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AzT4oBgHgl3EQfmP0G/content/2301.01558v1.pdf'}
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+1
+Blockchain-aided Secure Semantic Communication
+for AI-Generated Content in Metaverse
+Yijing Lin, Hongyang Du, Dusit Niyato, Fellow, IEEE,
+Jiangtian Nie, Jiayi Zhang, Yanyu Cheng, and Zhaohui Yang
+Abstract—The construction of virtual transportation networks
+requires massive data to be transmitted from edge devices to
+Virtual Service Providers (VSP) to facilitate circulations between
+the physical and virtual domains in Metaverse. Leveraging
+semantic communication for reducing information redundancy,
+VSPs can receive semantic data from edge devices to provide
+varied services through advanced techniques, e.g., AI-Generated
+Content (AIGC), for users to explore digital worlds. But the use of
+semantic communication raises a security issue because attackers
+could send malicious semantic data with similar semantic infor-
+mation but different desired content to break Metaverse services
+and cause wrong output of AIGC. Therefore, in this paper,
+we first propose a blockchain-aided semantic communication
+framework for AIGC services in virtual transportation networks
+to facilitate interactions of the physical and virtual domains
+among VSPs and edge devices. We illustrate a training-based
+targeted semantic attack scheme to generate adversarial semantic
+data by various loss functions. We also design a semantic defense
+scheme that uses the blockchain and zero-knowledge proofs to
+tell the difference between the semantic similarities of adversarial
+and authentic semantic data and to check the authenticity of
+semantic data transformations. Simulation results show that the
+proposed defense method can reduce the semantic similarity of
+the adversarial semantic data and the authentic ones by up to
+30% compared with the attack scheme.
+Index Terms—Metaverse, Blockchain, Semantic Communica-
+tion, Semantic Attacks, Semantic Defenses
+I. INTRODUCTION
+T
+HE word Metaverse was coined in the science-fiction
+novel Snow Crash [1] to describe a virtual reality society
+in which people utilize digital avatars to symbolize themselves
+and experience the world. In recent years, Metaverse has
+received attention from academia and industries as a novel
+Internet application due to the advancement of Augmented
+Reality (AR), Virtual Reality (VR), Artificial Intelligence (AI),
+and Blockchain. Specifically, extending reality (AR/VR) and
+AI technologies can provide users with immersive experiences
+and facilitate continuous data synchronization from the phys-
+ical domain to the virtual domain, which is supported by data
+Corresponding author: Hongyang Du.
+Yijing Lin is with the State Key Laboratory of Networking and Switching
+Technology, Beijing University of Posts and Telecommunications, 100876,
+Beijing, China (e-mail: yjlin@bupt.edu.cn).
+Hongyang Du, Dusit Niyato, Jiangtian Nie, and Yanyu Cheng are
+with
+Nanyang
+Technological
+University,
+639798,
+Singapore
+(e-mail:
+hongyang001@e.ntu.edu.sg;
+dniyato@ntu.edu.sg;
+jnie001@e.ntu.edu.sg;
+yanyu.cheng@ntu.edu.sg).
+Jiayi Zhang is with School of Electronic and Information Engineering, Bei-
+jing Jiaotong University, Beijing 100044, China (jiayizhangg@bjtu.edu.cn).
+Zhaohui Yang is with the College of Information Science and Electronic
+Engineering, Zhejiang University, China (e-mail: yang zhaohui@zju.edu.cn).
+perceived by edge devices and services driven in Metaverse.
+Blockchain can help the physical and virtual domains to share
+information and construct economic systems in a decentralized
+manner. Thus, the Metaverse can be viewed asthe integration
+of multiple technologies supported by massive data interac-
+tions between the physical and virtual domains.
+One of the significant advantages of Metaverse is that people
+can conduct experiments in the virtual world that cannot be
+conducted in the real world. Because the virtual world can
+be created based on real-world data, e.g., images, sensing
+data, and text, the experimental result in Metaverse can be
+used to guide the real world. One example is the virtual
+transportation networks [2], i.e., virtual environments in which
+users can safely train automatic driving algorithms and test
+vehicles. To build such virtual environments, virtual service
+providers (VSPs) can leverage network edge devices to capture
+images from the real world and then render the virtual objects.
+However, this process entails three challenges as follows:
+D1) How to use the data collected from the real world, e.g.,
+images, to achieve fast virtual world building?
+D2) How to improve the efficiency of data transmission to fa-
+cilitate interactions of the physical and virtual domains?
+D3) How to ensure the security of the data received by
+VSP to ensure that the virtual world can be accurately
+synchronized with the real world?
+Since virtual transportation networks require extensive in-
+teractions between the physical domain and Metaverse, edge
+devices can be utilized to capture images of geographical
+landmarks. However, converting these captured images into
+a consistent style for the virtual world is complicated. In
+several virtual service designs, VSPs need to hire digital
+painters to pre-process images [3], which is time-consuming
+and costly. The boom in AI has brought alternative solutions.
+AI-generated content (AIGC) technology [4] allows VSP to
+process quickly images collected from the real world using
+well trained AI models (For D1). However, the collected data
+could still burden the network. For example, the authors in
+[5] state that a pair of sensing devices can generate 3.072
+megabytes of data per second, which challenges conventional
+communication systems. Fortunately, semantic communication
+[6] is introduced to filter out irrelevant information from
+edge devices to reduce information redundancy of VSPs by
+extracting semantic data from raw data and expressing desired
+meanings (For D2). With the help of semantic communica-
+tions, massive amounts of data can be circulated in the virtual
+transportation networks to empower Metaverse services.
+arXiv:2301.11289v1  [cs.CR]  25 Jan 2023
+
+2
+However, the introduction of semantic communication
+brings about higher requirements of the data security. Ef-
+ficient semantic data sharing should be achieved between
+unknown VSPs and edge devices deployed in untrusted en-
+vironments. However, the extracted semantic data can be
+tampered to almost the same descriptors (semantic similarities)
+but different desired meanings. The attacker (edge device)
+can modify the pixels of a sunflower to make it similar to
+the extracted semantic data (snowy mountain) in terms of
+semantic similarities but visually dissimilar [7], which affects
+the output of the AIGC models and is difficult for VSPs to
+detect the difference between the adversarial and authentic
+semantic data. Moreover, it is hard to detect and prevent
+semantic data mutations for virtual transportation networks in
+Metaverse. Since VSPs and edge devices are distributed, it
+is difficult to record, trace, and verify data transformations.
+To solve the aforementioned problems, the blockchain and
+Zero-Knowledge Proof techniques can be used (For D3).
+Thus, we present a blockchain-aided semantic communication
+framework with AIGC services to achieve virtual transporta-
+tion networks in Metaverse. Here, blockchain-aided semantic
+communication can facilitate data circulation and economic
+activities between VSPs and edge devices in a decentralized
+manner [8]. Targeted semantic attacks [7] are utilized to
+generate adversarial semantic data to improve their semantic
+similarities almost up to that of the extracted semantic data by
+training various loss functions. Zero-Knowledge Proof (ZKP)
+[9] is integrated into the proposed framework to process
+semantic data and securely guarantee correct transformations.
+The contributions of this paper are summarized as follows:
+• We propose a blockchain-aided semantic communication
+framework for AIGC in virtual transportation networks
+that ensures the authenticity of semantic data transmitted
+from edge devices to VSPs to facilitate interactions of
+the physical and virtual domains.
+• We illustrate how a training-based targeted semantic at-
+tack scheme generates adversarial semantic data (images)
+without revealing the authentic semantic data. The attack
+semantic data has almost the same semantic similarities
+as the authentic ones but is visually dissimilar to them.
+• We, for the first time, design a blockchain and zero-
+knowledge proof-based semantic defense scheme to as-
+sure the authentication of semantic data. The scheme can
+utilize zero-knowledge proof to record the transforma-
+tions of semantic data, and use blockchain to track and
+verify semantic data mutations.
+The remainder of the paper is described as follows. Section
+II reviews previous works on secure semantic communication.
+Section III demonstrates a blockchain-aided semantic com-
+munication framework for AIGC in Metaverse. Section IV
+illustrates a training-based targeted semantic attack scheme.
+Section V designs a blockchain and zero-knowledge proof-
+based semantic defense scheme. The proposed mechanisms
+are evaluated in Section VI. Section VII concludes the paper
+and elaborates the future work.
+II. RELATED WORK
+In this paper, we consider the integration of blockchain-
+aided semantic communications for Metaverse, which in-
+volves multiple emerging technologies. Therefore, we divide
+the related work into three parts: Blockchain-aided Semantic
+Communications, Semantic Attacks, and Semantic Defenses.
+A. Blockchain-aided Semantic Communications
+Semantic communication [6] can lighten virtual transporta-
+tion network burdens by transmitting relevant semantic data
+to VSPs after processing original data by AI technologies in
+edge devices. Z. Weng et al. [10] utilized deep learning (DL) to
+identify the essential speech information with higher weights
+for semantic communication in dynamic channel conditions.
+To enable edge devices to perform DL-based semantic com-
+munication tasks, H. Xie et al. [11] proposed a lite-distributed
+semantic communication framework for edge devices to trans-
+mit low-complexity texts by optimizing training processes.
+Although semantic communication can help Metaverse re-
+duce information redundancy, it can not handle the challenge
+that how to construct trust among unknown edge devices
+and VSPs to facilitate data sharing and economic activities.
+Blockchain [12] is a peer-to-peer network that can construct
+decentralized ledgers for participants to share data. The inte-
+gration of blockchain and AI-based semantic communication
+can empower Metaverse ecosystems to carry out rich activities
+between the physical and virtual domains [13]. Y. Lin et
+al. [8] proposed a unified blockchain-semantic framework to
+enable Web 3.0 services to implement on-chain and off-chain
+interactions. A proof of semantic mechanism is proposed to
+verify semantic data before adding it to blockchain. However,
+they do not mention the performance indicators of semantic
+data. Y. Lin et al. [14] proposed a blockchain-based seman-
+tic exchange framework that can mint Non-Fungible Tokens
+(NFT) for semantic data, utilize the game theory to facilitate
+exchange, and introduce ZKP to enable privacy-preserving.
+However, they do not consider semantic attacks that may
+reduce exchange efficiency.
+B. Semantic Attacks and Defenses
+The introduction of semantic communication brings about
+security issues for Metaverse. However, current research on
+the security of semantic communication is still in its infancy.
+Q. Hu et al. [15] analyzed semantic noise that causes se-
+mantic data to express misleading meanings. To reduce the
+effects caused by semantic noise, they added weight per-
+turbation to adversarial training processes, suppressed noise-
+related and task-unrelated features, and designed semantic
+similarity-based loss functions to reduce transmitting over-
+heads. X. Luo et al. [16] focused on privacy leakages when
+sharing background knowledge. They introduced symmetric
+encryption to adversarial training processes to encrypt and
+decrypt semantic data to ensure confidentiality. H. Du et al.
+[7] focused on the semantic Internet-of Things (SIoT) and
+proposed new performance indicators for SIoT to quantify
+security issues, including semantic secrecy outage probability
+
+3
+and detection failure probability. They focused on image
+transmission-oriented semantic communication and divided
+semantic attacks into targeted and untargeted semantic attacks.
+The targeted attack can generate adversarial semantic data
+(images) that can be recovered to a given target requested by
+receivers. The adversarial semantic data has almost the same
+descriptors (semantic similarities), but is visually dissimilar
+[17]. The untargeted semantic attack can generate adversarial
+semantic data that minimizes semantic similarities. They do
+not pursue to be recovered to any target by receivers.
+In this paper, we study the targeted semantic attacks and
+introduce ZKP to differ in semantic similarities between
+the adversarial and target images to protect semantic data.
+ZKP has been widely used in blockchain to enable privacy-
+preserving and authenticity. R. Song et al. [9] utilized NFT
+and ZKP to construct a traceable data exchange scheme in
+blockchain, which can protect data privacy and exchange
+fairness. Z. Wang et al. [18] designed a ZKP-based off-chain
+data feed scheme to take in off-chain sensitive data to execute
+the business logic of smart contract-based applications. H.
+Galal et al. [19] leveraged ZKP and smart contracts to hide
+details of NFTs and swap NFTs in a fair manner. Y. Fang et al.
+[20] utilized ZKP to verify the authenticity of model prediction
+processes without leasing private parameters of deep learning
+models. However, the above methods do not consider how
+to use ZKP to prevent targeted semantic attacks to detect
+adversarial semantic data.
+C. Artificial Intelligence Generated Content
+AIGC refers to the use of AI algorithms, natural language
+processing (NLP), and computer version (CV) methods to
+produce a variety of media forms, including text, images,
+and audio. In terms of text content generation, an epoch-
+making AIGC application is the ChatGPT, a conversational
+language model developed by OpenAI [21]. ChatGPT is a
+type of the Generative Pre-trained Transformer (GPT) model
+that is trained on a huge amount of conversational data.
+ChatGPT is able to generate text that sounds as if it were
+written by a human. This makes it helpful for a variety of
+purposes, including chatbots, virtual assistants, and language
+translation. For image generation, diffusion model [22], a new
+class of state-of-the-art generative models, have demonstrated
+exceptional performance in Image Generation tasks and have
+surpassed the performance of generative adversarial networks
+(GANs) [23] in numerous tasks. Stable Diffusion, a AIGC
+model that is released by Stability AI, is an open-source text-
+to-image generator that creates amazing artwork in a matter
+of seconds. It operates with unprecedented speed and quality
+on consumer-grade GPUs. Furthermore, AIGC techniques
+continues to advance in the field of audio generation. The
+authors in [24] propose a deep learning-based method that
+employs contrastive representation learning and clustering to
+automatically derive thematic information from music pieces
+in the training data. The simulation results show that the
+proposed model is capable of generating polyphonic pop piano
+music with repetition and plausible variations.
+One of the primary benefits of AIGC is that it can be
+produced at a number and speed that would be difficult for
+human beings to do alone. It also permits a great degree
+of consistency and precision. Therefore, the development of
+AIGC technologies can provide a strong boost to the evolution
+of the Internet. Notably, despite the potential benefits, there
+are also some worries about the ramifications of AIGC,
+including the possibility for prejudice, the semantic errors
+in the generated content, and the blurring of the boundary
+between human- and machine-generated content. Therefore, it
+is significant to ensure the correctness of the AIGC, avoiding
+attacks from malicious parties that causes resource waste to
+affect the quality of AIGC services.
+III. BLOCKCHAIN-AIDED SEMANTIC COMMUNICATION
+FRAMEWORK FOR AIGC IN METAVERSE
+The implementation of virtual transportation networks re-
+quires the following main steps: 1) Semantic extraction and
+transmission for data interactions between physical and virtual
+domains, 2) Semantic transformation and verification, and 3)
+AIGC for Metaverse, as shown in Fig. 1.
+A. Semantic Extraction and Transmission
+Pedestrians are in danger when auto-driving models in
+vehicles are not trained well enough, which motivates the
+development of virtual transportation networks in the Meta-
+verse. Therefore, it is necessary to digitize physical domains
+using data produced in the real world to simulate environments
+for training vehicles and drivers. VSPs can take pictures
+using edge devices like smartphones, cameras, and sensors
+in physical domains to obtain information about the weather,
+traffic, and geographical landmarks that can be used to train
+and test the detection systems of vehicles in virtual domains.
+Virtual domain output can be fed back into physical domains to
+configure vehicles for better and safer performance. Based on
+the simulated environment, inexperienced drivers and vehicles
+can practice their reactions under unfamiliar weather or traffic
+conditions in Metaverse in a safe way. Therefore, VSPs
+need to frequently interact with edge devices supported by a
+tremendous amount of data to construct virtual transportation
+networks in Metaverse to provide services, which challenges
+the transmission capabilities of edge devices and VSPs.
+Unfortunately, conventional communication systems cannot
+afford such frequent interactions between the physical and vir-
+tual domains, which will cause virtual transportation networks
+in Metaverse to lack enough data to simulate virtual environ-
+ments. Semantic communication, a completely new paradigm
+that extracts semantic meaning from raw data for transmission,
+can be utilized to resolve this challenge. Instead of transmitting
+original data, edge devices can extract the semantic data
+and transmit to VSPs. The VSPs can then use the received
+semantic data to generate simulation environments for drivers
+and autonomous vehicle training on the virtual road. For
+instance, edge devices, e.g., smartphones, positioned at various
+locations along the same road may gather photographs of
+traffic conditions from their perspective. To reduce information
+redundancy, they can utilize semantic segmentation modules
+to crop key components of images as semantic data [2]. Then
+
+4
+Virtual
+Domain
+Physical
+Domain
+Semantic
+Communication
+Blockchain
+Semantic Extraction 
+and Transmission
+Semantic Transformation 
+and Verification 
+AIGC Services
+3D Scenes
+AI Generated Art
+Images from Edge
+devices
+Weather Conditions
+Traffic Conditions
+Landmarks
+Semantic
+Extraction
+Semantic
+Extraction
+Honest Edge
+Devices
+Attacker
+Output
+Input
+Convolutional
+Layers
+Fully Connected
+Layers
+Transformation
+Semantic
+Extraction
+Circuit
+Proof and Transformed
+Semantic Data
+VSPs Verify them on
+Blockchain Smart Contract
+Fig. 1: Blockchain-aided Semantic Communication Framework for AIGC in Metaverse
+edge devices only transmit semantic data to VSPs to report
+traffic conditions on roads.
+However, since VSPs have to collect semantic data from
+multiple edge devices to train virtual transportation networks
+in different situations, malicious edge devices could falsify
+semantic data (images) and corrupt the training process, which
+is dangerous for pedestrians and drivers. In this paper, we
+study the targeted semantic attack [7, 17] in that the adversarial
+and authentic semantic data have almost the same semantic
+similarities but are totally irrelevant. The semantic similarities
+are calculated with the help of high-dimensional descriptors
+extracted by a convolutional neural network (CNN). However,
+malicious edge devices could train a corresponding neural
+network to modify some pixels in attack images to achieve
+similar descriptors as the authentic images to corrupt the
+construction of virtual transportation networks. The training-
+based targeted semantic attack scheme is illustrated in Section
+IV.
+B. AIGC in Metaverse
+After receiving semantic data (images) from edge devices
+deployed in different locations, VSPs can perceive conditions
+or views of landmarks, and render images by AIGC services
+in Metaverse. For example, semantic data of landmarks from
+different perspectives can be utilized by VSPs to render 3D
+scenes to provide users with seamless experiences. VSPs can
+also utilize views of landmarks to generate artworks or avatars
+in Metaverse. Therefore, AIGC services play an important
+role in virtual transportation networks to facilitate the use
+of data resources and the applications of Metaverse. Besides,
+semantic data is important for subsequent AIGC services since
+its quality may affect the content generated by AIGC.
+However, since semantic data circulated in the Metaverse
+may be corrupted by the aforementioned targeted semantic
+attacks produced by malicious edge devices, the AIGC ser-
+vices may do useless work and provide users with hateful
+content. For example, VSPs want to collect images of fa-
+mous landmarks (i.e., Eiffel Tower) while malicious edge
+devices may extract unrelated images of flowers to corrupt
+the subsequent AIGC services that renders 3D scenes with
+
+5
+different perspectives of the landmark. VSPs also want to
+perceive images of landmarks to generate artworks or avatars
+while attackers transmit irrelative semantic data of animals to
+obstacle AIGC services. Since the adversarial semantic data
+(images) modified by attackers have almost the same semantic
+similarities, it is difficult and time-consuming for VSPs to
+verify the authenticity of images. Therefore, it is necessary to
+design a mechanism to ensure the security of data transmission
+to protect the security of AIGC services.
+C. Semantic Transformation and Verification
+Malicious semantic data can affect the security of virtual
+transportation services in Metaverse. Modifying pixels in
+unrelated semantic data increases the semantic similarity score,
+causing the VSPs to use the wrong semantic data as input
+to AIGC and corrupt the virtual environment. Inspired by
+[17], image transformations can be performed to distinguish
+semantic similarities between the adversarial and authentic
+images. However, malicious edge devices can continue ad-
+justing pixels in irrelative images to make them similar to
+transformed semantic data. Moreover, it is impossible for edge
+devices to transform semantic data unlimited times. A possible
+and practical solution is to record and verify transformations
+performed on semantic data.
+Therefore, we propose a blockchain and zero-knowledge
+proof-based semantic defense scheme in Section V. The logic
+of transformations is recorded on the circuit produced by
+the zero-knowledge algorithm. Edge devices utilize extracted
+semantic data as inputs to generate proof of transformations
+and output transformed semantic data. They send the proof and
+semantic data to VSPs for verification. Since VSPs and edge
+devices are distributed in unknown Metaverse environments
+represented by avatars, the blockchain should be used to record
+and verify transformations to prevent data mutations.
+IV. TRAINING-BASED TARGETED SEMANTIC ATTACK
+SCHEME
+Although the blockchain-aided semantic communication
+framework for AIGC in Metaverse can reduce information
+redundancy and establish decentralized trust between un-
+known edge devices and VSPs, malicious edge devices may
+conduct training-based targeted semantic attacks to corrupt
+semantic data (images) circulated in the Metaverse services.
+The targeted semantic attacks refer to transmitting adversarial
+semantic data with almost the same semantic descriptors but
+visually dissimilar to the authentic one [7], which is difficult
+for VSPs to utilize semantic similarities (inner product of
+descriptors among images) evaluation to distinguish them. In
+this section, we illustrate the workflow of the training-based
+targeted semantic attack targeted semantic attacks [7][17].
+Descriptor Extraction. Since extracted semantic data (im-
+ages) is difficult to evaluate, edge devices can utilize a CNN
+to map images to high dimensional descriptors. Then VSPs
+can use the descriptors to distinguish semantic similarities of
+images. The process of descriptor extraction is illustrated as
+follows.
+Semantic Space
+Semantic Feature
+Visual Space
+Visual Feature
+Carrier
+Attack
+Target
+Fig. 2: Relationship among Images
+Let us denote three types of semantic data produced by
+malicious edge devices, including the adversarial semantic
+data xa, the authentic one xt, and the carrier one xc. The
+authentic semantic data xt is extracted from original images by
+semantic extraction in edge devices according to requirements
+and associated with label yt = fc(xt), which has semantic
+similarities with xa. fc(·) is utilized to classify semantic data.
+The carrier semantic data xc is an auxiliary image to help
+malicious edge devices generate xa, which is classified as
+yc = fc(xc) ̸= yt and has visual similarities with xa. The
+adversarial semantic data xa is produced by training networks
+to learn how to modify pixels, which can achieve that xa has
+almost the same descriptors but is visually dissimilar from the
+authentic one xt generated by semantic extraction, as shown
+in Fig. 2. Besides, xa is visually similar to xc but classified
+incorrectly as yt. Therefore, the descriptor extraction process is
+vital for malicious edge devices to produce adversarial images
+xa.
+The input image x should be re-sampled to xs with the same
+dimension s to make xt and xc with the same resolution.
+The image xs is utilized as an input to train a Fully CNN
+given by gxs = g(xs) : RW ×H×3 → Rw×h×d to implement
+feature extraction where W ×H ×3 and w×h×d are weight,
+height, and channel of images. A pooling layer is utilized to
+map the input tensor gxs and the descriptor hxs = h(gxs)
+by the CNN with network parameters θ, which is mapped by
+h : Rw×h×d → Rd. The descriptor is the output of the pooling
+layer, which can be utilized to compare with other descriptors
+to calculate semantic similarities. The l2 normalization is
+utilized to easily compare semantic similarities.
+Loss Function. Considering the goal of malicious edge
+devices is to make xa almost the same as xt in terms of
+semantics but visually similar to xc, the loss function consists
+of the performance loss lts(x, xt) and the distortion loss
+between x and xc, which can be defined as
+Lts(xc, xt; x) = lts(x, xt) + λ∥x − xc∥2,
+(1)
+
+006M3776
+where λ is a hyper-parameter to show the impact of the
+distortion loss.
+Performance Loss. Let us assume that malicious edge
+devices can access to the network structure of the descriptor
+extraction [17] since it is necessary for edge devices to know
+the evaluation standard of VSPs. Considering different scenar-
+ios where targeted semantic attacks generate, referring to [17],
+we introduce the three empirical forms of the performance loss
+as follows.
+Global Descriptor Loss is suitable that malicious edge
+devices even know all parameters of the network of the
+descriptor extraction. Thus, the performance loss function lts
+can be given by calculating the inner product of xa and xt as
+follows:
+lglobal(x, xt) = 1 − h⊤
+x hxt.
+(2)
+Activation Tensor Loss is adapted to the scenario where the
+outputs of the network of the descriptor extraction are the same
+for xa and xt before down-sampling to resolution s, which
+can be denoted by the mean squared difference of gx and gxt
+as follows:
+ltensor(x, xt) = ∥gx − gxt∥2
+w · h · d
+.
+(3)
+Activation Histogram Loss is utilized to preserve first-order
+statistics of activations ∥u(gx, b)i per channel i to achieve
+identical extracted descriptors regardless of spatial information
+in images, which can be denoted as follows:
+lhist(x, xt) = 1
+d
+d
+�
+i=1
+∥u(gx, b)i − u(gxt, b)i∥,
+(4)
+where b is the histogram bin centers.
+Optimization. Our goal is to find the optimal loss function
+that minimizes the semantic similarities of xt and x, which
+equivalently optimizes network parameters θ. We can use the
+Adam algorithm to update θ as
+θt+1 = θt − η
+ρt
+√υt + ϵ,
+(5)
+where η is the learning rate, ρt and υt are the first-order and
+second-order momenta of gradients, and ϵ is utilized to pre-
+vent the √υt + ϵ from being zero. Therefore, the adversarial
+semantic data xa can be expressed by
+xa = arg min
+x
+Lts(xc, xt; x),
+(6)
+where Lts can be replaced by lglobal, ltensor, or lhist accord-
+ing to different scenarios.
+V. BLOCKCHAIN AND ZERO-KNOWLEDGE PROOF-BASED
+SEMANTIC DEFENSE SCHEME
+Since the adversarial semantic data generated by malicious
+edge devices has almost the same descriptors (semantic simi-
+larity) but different desired meanings from the authentic ones
+produced by honest edge devices, inspired by [7][17], we
+utilize blockchain and zero-knowledge proof-based semantic
+defense scheme to help VSPs to identify attack images trans-
+mitted in the Metaverse. Instead of submitting extracted se-
+mantic data directly, edge devices should transform or process
+semantic data by the bilinear interpolation algorithm [25], and
+utilize Zero-Knowledge Proof to record and verify transfor-
+mations. The details of the proposed scheme are elaborated as
+follows, as shown in Fig. 3.
+Transformation. The transformation of semantic data is
+a training-free defense method that uses visual invariance
+to distinguish adversarial and authentic semantic extraction.
+The reason is that attackers adjust some pixels to make
+descriptors of adversarial images similar to the authentic ones
+[7]. As a result, we attempt to increase visual invariance by
+blurring extracted images using the spatial transformation of
+the bilinear interpolation algorithm.
+Let us assume that (x1, y1), (x1, y2), (x2, y1), and (x2, y2)
+are four points in the extracted images. Then the targeted
+points (x, y) can be obtained by the spatial transformation,
+i.e., bilinear interpolation in the x and y directions. The spatial
+transformation can be considered a mapping function f(·, ·).
+Thus, the linear interpolation in the x direction can be derived
+as follows:
+f(x, y1) ≈ x2 − x
+x2 − x1
+f(x1, y1) + x − x1
+x2 − x1
+f(x2, y1)
+(7)
+and
+f(x, y2) ≈ x2 − x
+x2 − x1
+f(x1, y2) + x − x1
+x2 − x1
+f(x2, y2).
+(8)
+The targeted points are transformed by the linear interpolation
+in the y direction as follows:
+f(x, y) ≈ y2 − y
+y2 − y1
+f(x, y1) + y − y1
+y2 − y1
+f(x, y2)
+≈
+(x2 − x)(y2 − y)
+(x2 − x1)(y2 − y1) f(x1, y1) +
+(x − x1)(y2 − y)
+(x2 − x1)(y2 − y1)) f(x2, y1)
++
+(x2 − x)(y − y1)
+(x2 − x1)(y2 − y1) f(x1, y2) +
+(x − x1)(y − y1)
+(x2 − x1)(y2 − y1) f(x2, y2).
+(9)
+Although edge devices can perform transformations to ob-
+tain the blurred semantic data that can distinguish from the
+adversarial one, it is difficult for VSPs to verify whether the
+blurred semantic data is derived from authentic transforma-
+tions. Malicious edge devices may modify some pixels to
+make descriptors of their semantic data close to that of honest
+edge devices after transformations. Therefore, to assure the
+authenticity of the blur transformation for semantic data, we
+utilize ZKP to record transformations and blockchain to verify
+them. The proposed mechanism is a tuple of 3 polynomial-
+time schemes after extracting a circuit mapping the trans-
+formation, including Key Generation, Proof, and Verification,
+which works as follows:
+Extraction. The mechanism initiates the logic in a com-
+putation circuit C using the simple arithmetic expression
+mapping the transformation f to construct the public statement
+s and the private witness w [20]. The circuit implements the
+logic of bilinear interpolation via circom [26], a ZKP circuit
+compiler. The circuit can provide a relation between the inputs
+and outputs semantic data which can be used for verifiable
+
+7
+Blockchain
+Edge Device
+Witness
+Statement
+Transformation
+Bilinear Interpolation
+Extraction
+Circuit
+Proof
+VSP
+Status
+Verification
+Proof
+Verification
+Key
+Evaluation
+Key
+Key
+Generation
+AIGC
+Semantic
+Data
+Proof
+Generation
+Smart Contract
+Render
+Fig. 3: Blockchain and Zero-Knowledge Proof-based Semantic Defense Scheme
+computation on transformations without disclosing the inputs.
+The process of extraction can be illustrated as follows:
+Extract(f) → (s, w).
+(10)
+Key Generation. Edge devices take a security parameter
+1λ and the transformation circuit C as inputs to generate a
+common reference string crs. The common reference string crs
+includes an evaluation key crs.ek and a verification key crs.vk
+for proof and verification. The process of key generation can
+be expressed as follows:
+KeyGen(1λ, C)
+C−→ crs(ek, vk).
+(11)
+Proof. Edge devices take the evaluation key crs.ek, the
+statement s and the witness w related to the transformed
+and original semantic data, which satisfies C(s, w) = 1.
+The statement s and the witness w stand for the public
+and private information corresponding to the transformation
+relation. Thus, a zero-knowledge proof π is generated to reflect
+and verify the relation. The process of proof generation can
+be denoted as follows:
+Prove(crs.ek, s, w)
+C−→ π.
+(12)
+Verification. VSPs utilize smart contracts deployed on
+blockchain to verify the authenticity of transformations and
+implement the business logic of blockchain-aided semantic
+communication for AIGC in Metaverse. Since the verification
+process is implemented in the blockchain, edge devices and
+VSPs can query verification results to construct trust in a
+decentralized manner. The verification key crs.vk, the state-
+ment s, and the proof π are the inputs written into smart
+contracts to determine whether to accept or reject the proof
+according to outputs. When the status of the output is 1,
+the verification succeeds and VSPs accept the semantic data
+provided by edge devices; otherwise, VSPs refuse to accept the
+proof corresponding to semantic data produced by malicious
+edge devices. The process of proof verification can be given
+0
+20
+40
+60
+80
+100
+Iteration
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+ltr(xa, xt)
+Global
+Hist
+Tensor
+Fig. 4: Performance Loss of Attacks
+as follows:
+Verify(crs.vk, s, π) → {0, 1}.
+(13)
+The semantic defense scheme should satisfy the follow-
+ing properties including completeness, soundness, and zero-
+knowledge [9][18][19][20].
+Completeness means that an honest edge device with a
+valid witness w can convince an honest VSP for the au-
+thenticity of transformed semantic data generated from the
+circuit C. If the transformed semantic data extracted by bilinear
+interpolation is correct and edge devices construct the proof
+π correctly through the proof circuit C and the evaluation
+key crs.ek, then the VSPs can use the verification key crs.vk
+generated by the proof circuit C, the proof π, and the public
+information (statement) s to obtain a verification result. For
+each C(s, w) = 1, the proof π generated from an honest
+edge device will be accepted with probability 1, which can
+be denoted as follows:
+Pr
+�
+�
+KeyGen(1λ, C) → crs(ek, vk)
+Prove(crs.ek, s, w) → π
+Verify(crs.vk, s, π) = 1
+�
+� = 1.
+(14)
+
+8
+0
+20
+40
+60
+80
+100
+Iteration
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+h⊤
+xahxt
+Global
+Hist
+Tensor
+(a) Semantic similarity between the
+adversarial image and the authentic
+0
+20
+40
+60
+80
+100
+Iteration
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+h⊤
+xahxc
+Global
+Hist
+Tensor
+(b) Semantic similarity between the
+adversarial image and the carrier
+Fig. 5: Semantic Similarity Performance of Attacks
+Soundness denotes that malicious edge devices cannot
+provide VSPs with authentic transformed semantic data, which
+means that edge devices can provide valid witnesses if they
+can produce valid proofs. If (s, w) is not a valid input to
+the proof circuit C(s, w) = 0, there is a polynomial time
+extractor Ext for a malicious edge device A that makes
+the probabilities of Verify(crs.vk, s, π∗) = 1 are negligible.
+The adversarial advantage of soundness Advsd
+A (λ) can be
+represented as follows:
+Pr
+�
+���
+KeyGen(1λ, C) → crs(ek, vk)
+A(crs.ek, s) → π∗
+Ext(crs.ek, s, π∗) → w
+Verify(crs.vk, s, π∗) = 1
+�
+��� ≤ negl(λ).
+(15)
+Malicious edge devices A can hardly falsify semantic data
+or make honest VSPs V accept invalid proofs about transfor-
+mations f. According to (15), VSPs cannot accept false proof
+π from A except for a negligible probability negl(λ).
+Zero-knowledge represents that VSPs can only verify the
+authenticity of transformed semantic data while they cannot
+obtain the original semantic data unless edge devices send it
+to them. Let us assume that there are probabilistic polynomial
+time simulators S1, S2, and malicious edge devices A1, A2.
+The simulator S1 can produce a common reference string that
+is used by the simulator S2 to generate a simulated proof.
+Then the proposed mechanism has the zero-knowledge prop-
+erty if the adversarial advantage of zero-knowledge Advzk
+A (λ)
+satisfies:
+| Pr(real) − Pr(sim)| ≤ negl(λ),
+(16)
+where Pr(real) and Pr(sim) can be denoted as
+Pr(real) = Pr
+�
+���
+KeyGen(1λ, C) → crs
+A1(f) → (s, w)
+Prove(crs, s, w) → π
+A2(crs, s, w, π) = 1
+�
+���
+(17)
+and
+Pr(sim) = Pr
+�
+���
+S1(1λ, C) → crs
+A1(f) → (s, w)
+S2(crs, s) → π
+A2(crs, s, w, π) = 1
+�
+��� .
+(18)
+VSPs who know public statements mapping with semantic
+data can hardly learn any knowledge about semantic data
+before and after transformation. According to (16), VSPs
+cannot know anything about semantic data other than what
+can be inferred from transformation.
+VI. EXPERIMENTAL EVALUATIONS
+We simulate the proposed mechanism on Ubuntu 20.04LTS,
+Intel Xeon, 8 core, 64G memory, and 25000Mb/s with 2 Tesla
+V100, Go 1.19.4, Node v14.17.0, PyTorch 1.12.1, Torchvi-
+sion 0.13.1, and CUDA 10.2. We perform experiments on a
+standard image benchmark, Revisited Paris [27], to measure
+the attack and defense performance among edge devices and
+VSPs. We exploit pre-trained networks on ImageNet [28]
+and AlexNet [29] to execute the attacks and defenses with
+100 iterations. Unless otherwise mentioned, we use these
+parameters referring to [17]. The learning rate η is set to 0.01.
+The hyper-parameter λ that adjusts the impact of distortion
+loss is 0. The training process performs 100 iterations for our
+experiments.
+A. Performance of Semantic Attack Scheme
+Fig. 4 is the performance loss of the proposed semantic
+attack scheme with different loss functions [Global, Hist, Ten-
+sor] corresponding to the global descriptor, activation tensor,
+and activation histogram as the number of iterations increases.
+Tensor converges much slower than Global and Hist, which
+requires more iterations to reach its plateau.
+Fig. 5 is the semantic similarity attack performance between
+the adversarial semantic data and the authentic or the carrier
+one with three loss functions [Global, Hist, Tensor]. As shown
+in Fig. 5 (a) and Fig. 5 (b), Global and Hist converge around
+the 40th iteration which is much faster than Tensor. The
+semantic similarity performance of attacks is consistent with
+Fig. 4 which shows the same trend in three loss functions.
+B. Performance of Semantic Defense Scheme
+Fig. 6 is the semantic similarity comparison between attack
+and defense schemes by displaying example images. Fig. 7
+is the semantic similarity (descriptors) defense performance
+between the adversarial semantic data and the authentic one or
+the carrier one with three loss functions [Global, Hist, Tensor].
+Compared with Fig. 5, the adversarial image (semantic data)
+can differ in semantic similarities when applying the proposed
+semantic defense scheme. Since the extracted semantic data is
+required to execute the defense scheme and the execution can
+be assured by zero-knowledge proof, the images can differ in
+descriptors. As shown in Fig. 5 (a) and Fig. 7 (a), the semantic
+similarity can reduce by up to 35% between the adversarial and
+the authentic images. Fig. 5 (b) and Fig. 7 (b) show that the
+semantic similarity can decrease 10% between the adversarial
+and the carrier images.
+Fig. 8 (a) is the blockchain consensus overhead for [Attack,
+Authentic, Defense] types of semantic data (image) with [275
+KB,467 KB,10 KB] data sizes when the number of nodes
+is [5,50,5]. As shown in Fig. 8 (a), the proposed scheme can
+
+9
+0
+5
+10
+20
+100
+Number of Iterations
+0.695094
+0.967904
+0.987957
+0.997352
+0.999238
+0.650132
+0.670137
+0.664388
+0.664121
+0.661470
+Target Image
+Attack Image
+Defense Image
+Attack Image
+Transformed
+Carrier Image
+Carrier Image
+0.693920
+0.599826
+0.576982
+0.554904
+0.547588
+1.000000
+0.746711
+0.724178
+0.705464
+0.698031
+Semantic Similarity
+Fig. 6: Semantic Similarity Comparison Between Attack and Defense Schemes
+0
+20
+40
+60
+80
+100
+Iteration
+0.45
+0.50
+0.55
+0.60
+0.65
+h⊤
+xahxt0
+Global
+Hist
+Tensor
+(a) Semantic similarity between the
+adversarial image and the authentic
+0
+20
+40
+60
+80
+100
+Iteration
+0.58
+0.60
+0.62
+0.64
+0.66
+0.68
+0.70
+0.72
+h⊤
+xahxc0
+Global
+Hist
+Tensor
+(b) Semantic similarity between the
+adversarial image and the carrier
+Fig. 7: Semantic Similarity Performance of Defenses
+reduce the time consumed in blockchain to improve consensus
+efficiency due to the attack and defense schemes cropping
+authentic semantic data while differing in semantic similarities
+according to Fig. 7. Fig. 8 (b) is the ZKP computation over-
+head for [GenWitness, GenProof, VerifyProof] operations with
+[10KB,100KB,1MB] data sizes. We can see from Fig. 8 (b)
+that the ZKP computation overhead depends on the GenProof
+operation, while the sizes of semantic data affect little on
+the proposed scheme. Besides, the computation overhead for
+verifying proofs is less than for generating proofs, which can
+be written into smart contracts [9][18][19].
+VII. CONCLUSION AND FUTURE WORK
+In this paper, we first integrated blockchain-aided semantic
+communication into Metaverse to support AIGC services in
+virtual transportation networks. We also presented a training-
+based targeted semantic attack scheme to illustrate potential
+attacks by generating adversarial semantic data with the same
+descriptors but different desired meanings. To prevent the
+10
+20
+30
+40
+50
+Number of Nodes
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Time Consume / s
+Attack
+Authentic
+Defense
+(a) Consensus
+GenWiteness
+GenProof
+VerifyProof
+Type
+0
+20
+40
+60
+80
+100
+120
+Time Consumed / s
+10KB
+100KB
+1MB
+(b) ZKP Computation
+Fig. 8: Consensus and Computation Overheads
+above attack, we designed a blockchain and zero-knowledge
+proof-based semantic defense scheme that records transforma-
+tions of semantic data and verifies mutations in a decentralized
+manner. Simulation results show that the proposed mechanism
+can differ in the descriptors between the adversarial semantic
+data and the authentic one. The defense scheme can identify
+malicious edge devices falsifying semantic data to corrupt
+AIGC services in virtual transportation networks. In future
+research work, we will study how to mitigate malicious
+edge devices and facilitate resource allocation with economic
+incentive mechanisms in the proposed framework.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf,len=716
+page_content='1  Blockchain-aided Secure Semantic Communication  for AI-Generated Content in Metaverse  Yijing Lin, Hongyang Du, Dusit Niyato, Fellow, IEEE,  Jiangtian Nie, Jiayi Zhang, Yanyu Cheng, and Zhaohui Yang  Abstract—The construction of virtual transportation networks  requires massive data to be transmitted from edge devices to  Virtual Service Providers (VSP) to facilitate circulations between  the physical and virtual domains in Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Leveraging  semantic communication for reducing information redundancy,  VSPs can receive semantic data from edge devices to provide  varied services through advanced techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=', AI-Generated  Content (AIGC), for users to explore digital worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' But the use of  semantic communication raises a security issue because attackers  could send malicious semantic data with similar semantic infor-  mation but different desired content to break Metaverse services  and cause wrong output of AIGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Therefore, in this paper,  we first propose a blockchain-aided semantic communication  framework for AIGC services in virtual transportation networks  to facilitate interactions of the physical and virtual domains  among VSPs and edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' We illustrate a training-based  targeted semantic attack scheme to generate adversarial semantic  data by various loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' We also design a semantic defense  scheme that uses the blockchain and zero-knowledge proofs to  tell the difference between the semantic similarities of adversarial  and authentic semantic data and to check the authenticity of  semantic data transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Simulation results show that the  proposed defense method can reduce the semantic similarity of  the adversarial semantic data and the authentic ones by up to  30% compared with the attack scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Index Terms—Metaverse, Blockchain, Semantic Communica-  tion, Semantic Attacks, Semantic Defenses  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' INTRODUCTION  T  HE word Metaverse was coined in the science-fiction  novel Snow Crash [1] to describe a virtual reality society  in which people utilize digital avatars to symbolize themselves  and experience the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' In recent years, Metaverse has  received attention from academia and industries as a novel  Internet application due to the advancement of Augmented  Reality (AR), Virtual Reality (VR), Artificial Intelligence (AI),  and Blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Specifically, extending reality (AR/VR) and  AI technologies can provide users with immersive experiences  and facilitate continuous data synchronization from the phys-  ical domain to the virtual domain, which is supported by data  Corresponding author: Hongyang Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Yijing Lin is with the State Key Laboratory of Networking and Switching  Technology, Beijing University of Posts and Telecommunications, 100876,  Beijing, China (e-mail: yjlin@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Hongyang Du, Dusit Niyato, Jiangtian Nie, and Yanyu Cheng are  with  Nanyang  Technological  University,  639798,  Singapore  (e-mail:  hongyang001@e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  dniyato@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  jnie001@e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  yanyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='cheng@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Jiayi Zhang is with School of Electronic and Information Engineering, Bei-  jing Jiaotong University, Beijing 100044, China (jiayizhangg@bjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Zhaohui Yang is with the College of Information Science and Electronic  Engineering, Zhejiang University, China (e-mail: yang zhaohui@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  perceived by edge devices and services driven in Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Blockchain can help the physical and virtual domains to share  information and construct economic systems in a decentralized  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Thus, the Metaverse can be viewed asthe integration  of multiple technologies supported by massive data interac-  tions between the physical and virtual domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  One of the significant advantages of Metaverse is that people  can conduct experiments in the virtual world that cannot be  conducted in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Because the virtual world can  be created based on real-world data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=', images, sensing  data, and text, the experimental result in Metaverse can be  used to guide the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' One example is the virtual  transportation networks [2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=', virtual environments in which  users can safely train automatic driving algorithms and test  vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' To build such virtual environments, virtual service  providers (VSPs) can leverage network edge devices to capture  images from the real world and then render the virtual objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  However, this process entails three challenges as follows:  D1) How to use the data collected from the real world, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=',  images, to achieve fast virtual world building?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  D2) How to improve the efficiency of data transmission to fa-  cilitate interactions of the physical and virtual domains?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  D3) How to ensure the security of the data received by  VSP to ensure that the virtual world can be accurately  synchronized with the real world?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Since virtual transportation networks require extensive in-  teractions between the physical domain and Metaverse, edge  devices can be utilized to capture images of geographical  landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' However, converting these captured images into  a consistent style for the virtual world is complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' In  several virtual service designs, VSPs need to hire digital  painters to pre-process images [3], which is time-consuming  and costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The boom in AI has brought alternative solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  AI-generated content (AIGC) technology [4] allows VSP to  process quickly images collected from the real world using  well trained AI models (For D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' However, the collected data  could still burden the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' For example, the authors in  [5] state that a pair of sensing devices can generate 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='072  megabytes of data per second, which challenges conventional  communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Fortunately, semantic communication  [6] is introduced to filter out irrelevant information from  edge devices to reduce information redundancy of VSPs by  extracting semantic data from raw data and expressing desired  meanings (For D2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' With the help of semantic communica-  tions, massive amounts of data can be circulated in the virtual  transportation networks to empower Metaverse services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='11289v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='CR]  25 Jan 2023  2  However, the introduction of semantic communication  brings about higher requirements of the data security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Ef-  ficient semantic data sharing should be achieved between  unknown VSPs and edge devices deployed in untrusted en-  vironments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' However, the extracted semantic data can be  tampered to almost the same descriptors (semantic similarities)  but different desired meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The attacker (edge device)  can modify the pixels of a sunflower to make it similar to  the extracted semantic data (snowy mountain) in terms of  semantic similarities but visually dissimilar [7], which affects  the output of the AIGC models and is difficult for VSPs to  detect the difference between the adversarial and authentic  semantic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Moreover, it is hard to detect and prevent  semantic data mutations for virtual transportation networks in  Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Since VSPs and edge devices are distributed, it  is difficult to record, trace, and verify data transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  To solve the aforementioned problems, the blockchain and  Zero-Knowledge Proof techniques can be used (For D3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Thus, we present a blockchain-aided semantic communication  framework with AIGC services to achieve virtual transporta-  tion networks in Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Here, blockchain-aided semantic  communication can facilitate data circulation and economic  activities between VSPs and edge devices in a decentralized  manner [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Targeted semantic attacks [7] are utilized to  generate adversarial semantic data to improve their semantic  similarities almost up to that of the extracted semantic data by  training various loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Zero-Knowledge Proof (ZKP)  [9] is integrated into the proposed framework to process  semantic data and securely guarantee correct transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The contributions of this paper are summarized as follows:  We propose a blockchain-aided semantic communication  framework for AIGC in virtual transportation networks  that ensures the authenticity of semantic data transmitted  from edge devices to VSPs to facilitate interactions of  the physical and virtual domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  We illustrate how a training-based targeted semantic at-  tack scheme generates adversarial semantic data (images)  without revealing the authentic semantic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The attack  semantic data has almost the same semantic similarities  as the authentic ones but is visually dissimilar to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  We, for the first time, design a blockchain and zero-  knowledge proof-based semantic defense scheme to as-  sure the authentication of semantic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The scheme can  utilize zero-knowledge proof to record the transforma-  tions of semantic data, and use blockchain to track and  verify semantic data mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The remainder of the paper is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Section  II reviews previous works on secure semantic communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Section III demonstrates a blockchain-aided semantic com-  munication framework for AIGC in Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Section IV  illustrates a training-based targeted semantic attack scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Section V designs a blockchain and zero-knowledge proof-  based semantic defense scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The proposed mechanisms  are evaluated in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Section VII concludes the paper  and elaborates the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' RELATED WORK  In this paper, we consider the integration of blockchain-  aided semantic communications for Metaverse, which in-  volves multiple emerging technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Therefore, we divide  the related work into three parts: Blockchain-aided Semantic  Communications, Semantic Attacks, and Semantic Defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Blockchain-aided Semantic Communications  Semantic communication [6] can lighten virtual transporta-  tion network burdens by transmitting relevant semantic data  to VSPs after processing original data by AI technologies in  edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Weng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' [10] utilized deep learning (DL) to  identify the essential speech information with higher weights  for semantic communication in dynamic channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  To enable edge devices to perform DL-based semantic com-  munication tasks, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' [11] proposed a lite-distributed  semantic communication framework for edge devices to trans-  mit low-complexity texts by optimizing training processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Although semantic communication can help Metaverse re-  duce information redundancy, it can not handle the challenge  that how to construct trust among unknown edge devices  and VSPs to facilitate data sharing and economic activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Blockchain [12] is a peer-to-peer network that can construct  decentralized ledgers for participants to share data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The inte-  gration of blockchain and AI-based semantic communication  can empower Metaverse ecosystems to carry out rich activities  between the physical and virtual domains [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Lin et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' [8] proposed a unified blockchain-semantic framework to  enable Web 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='0 services to implement on-chain and off-chain  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' A proof of semantic mechanism is proposed to  verify semantic data before adding it to blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' However,  they do not mention the performance indicators of semantic  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' [14] proposed a blockchain-based seman-  tic exchange framework that can mint Non-Fungible Tokens  (NFT) for semantic data, utilize the game theory to facilitate  exchange, and introduce ZKP to enable privacy-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  However, they do not consider semantic attacks that may  reduce exchange efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Semantic Attacks and Defenses  The introduction of semantic communication brings about  security issues for Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' However, current research on  the security of semantic communication is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' [15] analyzed semantic noise that causes se-  mantic data to express misleading meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' To reduce the  effects caused by semantic noise, they added weight per-  turbation to adversarial training processes, suppressed noise-  related and task-unrelated features, and designed semantic  similarity-based loss functions to reduce transmitting over-  heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' [16] focused on privacy leakages when  sharing background knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' They introduced symmetric  encryption to adversarial training processes to encrypt and  decrypt semantic data to ensure confidentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  [7] focused on the semantic Internet-of Things (SIoT) and  proposed new performance indicators for SIoT to quantify  security issues, including semantic secrecy outage probability  3  and detection failure probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' They focused on image  transmission-oriented semantic communication and divided  semantic attacks into targeted and untargeted semantic attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The targeted attack can generate adversarial semantic data  (images) that can be recovered to a given target requested by  receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The adversarial semantic data has almost the same  descriptors (semantic similarities), but is visually dissimilar  [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The untargeted semantic attack can generate adversarial  semantic data that minimizes semantic similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' They do  not pursue to be recovered to any target by receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  In this paper, we study the targeted semantic attacks and  introduce ZKP to differ in semantic similarities between  the adversarial and target images to protect semantic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  ZKP has been widely used in blockchain to enable privacy-  preserving and authenticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' [9] utilized NFT  and ZKP to construct a traceable data exchange scheme in  blockchain, which can protect data privacy and exchange  fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' [18] designed a ZKP-based off-chain  data feed scheme to take in off-chain sensitive data to execute  the business logic of smart contract-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Galal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' [19] leveraged ZKP and smart contracts to hide  details of NFTs and swap NFTs in a fair manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  [20] utilized ZKP to verify the authenticity of model prediction  processes without leasing private parameters of deep learning  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' However, the above methods do not consider how  to use ZKP to prevent targeted semantic attacks to detect  adversarial semantic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Artificial Intelligence Generated Content  AIGC refers to the use of AI algorithms, natural language  processing (NLP), and computer version (CV) methods to  produce a variety of media forms, including text, images,  and audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' In terms of text content generation, an epoch-  making AIGC application is the ChatGPT, a conversational  language model developed by OpenAI [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' ChatGPT is a  type of the Generative Pre-trained Transformer (GPT) model  that is trained on a huge amount of conversational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  ChatGPT is able to generate text that sounds as if it were  written by a human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' This makes it helpful for a variety of  purposes, including chatbots, virtual assistants, and language  translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' For image generation, diffusion model [22], a new  class of state-of-the-art generative models, have demonstrated  exceptional performance in Image Generation tasks and have  surpassed the performance of generative adversarial networks  (GANs) [23] in numerous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Stable Diffusion, a AIGC  model that is released by Stability AI, is an open-source text-  to-image generator that creates amazing artwork in a matter  of seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' It operates with unprecedented speed and quality  on consumer-grade GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Furthermore, AIGC techniques  continues to advance in the field of audio generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The  authors in [24] propose a deep learning-based method that  employs contrastive representation learning and clustering to  automatically derive thematic information from music pieces  in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The simulation results show that the  proposed model is capable of generating polyphonic pop piano  music with repetition and plausible variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  One of the primary benefits of AIGC is that it can be  produced at a number and speed that would be difficult for  human beings to do alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' It also permits a great degree  of consistency and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Therefore, the development of  AIGC technologies can provide a strong boost to the evolution  of the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Notably, despite the potential benefits, there  are also some worries about the ramifications of AIGC,  including the possibility for prejudice, the semantic errors  in the generated content, and the blurring of the boundary  between human- and machine-generated content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Therefore, it  is significant to ensure the correctness of the AIGC, avoiding  attacks from malicious parties that causes resource waste to  affect the quality of AIGC services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' BLOCKCHAIN-AIDED SEMANTIC COMMUNICATION  FRAMEWORK FOR AIGC IN METAVERSE  The implementation of virtual transportation networks re-  quires the following main steps: 1) Semantic extraction and  transmission for data interactions between physical and virtual  domains, 2) Semantic transformation and verification, and 3)  AIGC for Metaverse, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Semantic Extraction and Transmission  Pedestrians are in danger when auto-driving models in  vehicles are not trained well enough, which motivates the  development of virtual transportation networks in the Meta-  verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Therefore, it is necessary to digitize physical domains  using data produced in the real world to simulate environments  for training vehicles and drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' VSPs can take pictures  using edge devices like smartphones, cameras, and sensors  in physical domains to obtain information about the weather,  traffic, and geographical landmarks that can be used to train  and test the detection systems of vehicles in virtual domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Virtual domain output can be fed back into physical domains to  configure vehicles for better and safer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Based on  the simulated environment, inexperienced drivers and vehicles  can practice their reactions under unfamiliar weather or traffic  conditions in Metaverse in a safe way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Therefore, VSPs  need to frequently interact with edge devices supported by a  tremendous amount of data to construct virtual transportation  networks in Metaverse to provide services, which challenges  the transmission capabilities of edge devices and VSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Unfortunately, conventional communication systems cannot  afford such frequent interactions between the physical and vir-  tual domains, which will cause virtual transportation networks  in Metaverse to lack enough data to simulate virtual environ-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Semantic communication, a completely new paradigm  that extracts semantic meaning from raw data for transmission,  can be utilized to resolve this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Instead of transmitting  original data, edge devices can extract the semantic data  and transmit to VSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The VSPs can then use the received  semantic data to generate simulation environments for drivers  and autonomous vehicle training on the virtual road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' For  instance, edge devices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=', smartphones, positioned at various  locations along the same road may gather photographs of  traffic conditions from their perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' To reduce information  redundancy, they can utilize semantic segmentation modules  to crop key components of images as semantic data [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Virtual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Domain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Physical  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Domain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Semantic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Communication  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Semantic Extraction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='and Transmission  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Semantic Transformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='and Verification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='AIGC Services  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='3D Scenes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='AI Generated Art  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Images from Edge  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='devices  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Weather Conditions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Traffic Conditions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Landmarks  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Semantic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Extraction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Semantic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Extraction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Honest Edge  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Devices  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Attacker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Output  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Convolutional  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Layers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Fully Connected  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Layers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Transformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Semantic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Extraction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Circuit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Proof and Transformed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Semantic Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='VSPs Verify them on  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Blockchain Smart Contract  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 1: Blockchain-aided Semantic Communication Framework for AIGC in Metaverse  edge devices only transmit semantic data to VSPs to report  traffic conditions on roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  However, since VSPs have to collect semantic data from  multiple edge devices to train virtual transportation networks  in different situations, malicious edge devices could falsify  semantic data (images) and corrupt the training process, which  is dangerous for pedestrians and drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' In this paper, we  study the targeted semantic attack [7, 17] in that the adversarial  and authentic semantic data have almost the same semantic  similarities but are totally irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The semantic similarities  are calculated with the help of high-dimensional descriptors  extracted by a convolutional neural network (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' However,  malicious edge devices could train a corresponding neural  network to modify some pixels in attack images to achieve  similar descriptors as the authentic images to corrupt the  construction of virtual transportation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The training-  based targeted semantic attack scheme is illustrated in Section  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' AIGC in Metaverse  After receiving semantic data (images) from edge devices  deployed in different locations, VSPs can perceive conditions  or views of landmarks, and render images by AIGC services  in Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' For example, semantic data of landmarks from  different perspectives can be utilized by VSPs to render 3D  scenes to provide users with seamless experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' VSPs can  also utilize views of landmarks to generate artworks or avatars  in Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Therefore, AIGC services play an important  role in virtual transportation networks to facilitate the use  of data resources and the applications of Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Besides,  semantic data is important for subsequent AIGC services since  its quality may affect the content generated by AIGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  However, since semantic data circulated in the Metaverse  may be corrupted by the aforementioned targeted semantic  attacks produced by malicious edge devices, the AIGC ser-  vices may do useless work and provide users with hateful  content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' For example, VSPs want to collect images of fa-  mous landmarks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=', Eiffel Tower) while malicious edge  devices may extract unrelated images of flowers to corrupt  the subsequent AIGC services that renders 3D scenes with  5  different perspectives of the landmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' VSPs also want to  perceive images of landmarks to generate artworks or avatars  while attackers transmit irrelative semantic data of animals to  obstacle AIGC services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Since the adversarial semantic data  (images) modified by attackers have almost the same semantic  similarities, it is difficult and time-consuming for VSPs to  verify the authenticity of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Therefore, it is necessary to  design a mechanism to ensure the security of data transmission  to protect the security of AIGC services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Semantic Transformation and Verification  Malicious semantic data can affect the security of virtual  transportation services in Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Modifying pixels in  unrelated semantic data increases the semantic similarity score,  causing the VSPs to use the wrong semantic data as input  to AIGC and corrupt the virtual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Inspired by  [17], image transformations can be performed to distinguish  semantic similarities between the adversarial and authentic  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' However, malicious edge devices can continue ad-  justing pixels in irrelative images to make them similar to  transformed semantic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Moreover, it is impossible for edge  devices to transform semantic data unlimited times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' A possible  and practical solution is to record and verify transformations  performed on semantic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Therefore, we propose a blockchain and zero-knowledge  proof-based semantic defense scheme in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The logic  of transformations is recorded on the circuit produced by  the zero-knowledge algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Edge devices utilize extracted  semantic data as inputs to generate proof of transformations  and output transformed semantic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' They send the proof and  semantic data to VSPs for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Since VSPs and edge  devices are distributed in unknown Metaverse environments  represented by avatars, the blockchain should be used to record  and verify transformations to prevent data mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' TRAINING-BASED TARGETED SEMANTIC ATTACK  SCHEME  Although the blockchain-aided semantic communication  framework for AIGC in Metaverse can reduce information  redundancy and establish decentralized trust between un-  known edge devices and VSPs, malicious edge devices may  conduct training-based targeted semantic attacks to corrupt  semantic data (images) circulated in the Metaverse services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The targeted semantic attacks refer to transmitting adversarial  semantic data with almost the same semantic descriptors but  visually dissimilar to the authentic one [7], which is difficult  for VSPs to utilize semantic similarities (inner product of  descriptors among images) evaluation to distinguish them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' In  this section, we illustrate the workflow of the training-based  targeted semantic attack targeted semantic attacks [7][17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Descriptor Extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Since extracted semantic data (im-  ages) is difficult to evaluate, edge devices can utilize a CNN  to map images to high dimensional descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Then VSPs  can use the descriptors to distinguish semantic similarities of  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The process of descriptor extraction is illustrated as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Semantic Space  Semantic Feature  Visual Space  Visual Feature  Carrier  Attack  Target  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 2: Relationship among Images  Let us denote three types of semantic data produced by  malicious edge devices, including the adversarial semantic  data xa, the authentic one xt, and the carrier one xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The  authentic semantic data xt is extracted from original images by  semantic extraction in edge devices according to requirements  and associated with label yt = fc(xt), which has semantic  similarities with xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' fc(·) is utilized to classify semantic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The carrier semantic data xc is an auxiliary image to help  malicious edge devices generate xa, which is classified as  yc = fc(xc) ̸= yt and has visual similarities with xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The  adversarial semantic data xa is produced by training networks  to learn how to modify pixels, which can achieve that xa has  almost the same descriptors but is visually dissimilar from the  authentic one xt generated by semantic extraction, as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Besides, xa is visually similar to xc but classified  incorrectly as yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Therefore, the descriptor extraction process is  vital for malicious edge devices to produce adversarial images  xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The input image x should be re-sampled to xs with the same  dimension s to make xt and xc with the same resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The image xs is utilized as an input to train a Fully CNN  given by gxs = g(xs) : RW ×H×3 → Rw×h×d to implement  feature extraction where W ×H ×3 and w×h×d are weight,  height, and channel of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' A pooling layer is utilized to  map the input tensor gxs and the descriptor hxs = h(gxs)  by the CNN with network parameters θ, which is mapped by  h : Rw×h×d → Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The descriptor is the output of the pooling  layer, which can be utilized to compare with other descriptors  to calculate semantic similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The l2 normalization is  utilized to easily compare semantic similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Loss Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Considering the goal of malicious edge  devices is to make xa almost the same as xt in terms of  semantics but visually similar to xc, the loss function consists  of the performance loss lts(x, xt) and the distortion loss  between x and xc, which can be defined as  Lts(xc, xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' x) = lts(x, xt) + λ∥x − xc∥2,  (1)  006M3776  where λ is a hyper-parameter to show the impact of the  distortion loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Performance Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Let us assume that malicious edge  devices can access to the network structure of the descriptor  extraction [17] since it is necessary for edge devices to know  the evaluation standard of VSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Considering different scenar-  ios where targeted semantic attacks generate, referring to [17],  we introduce the three empirical forms of the performance loss  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Global Descriptor Loss is suitable that malicious edge  devices even know all parameters of the network of the  descriptor extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Thus, the performance loss function lts  can be given by calculating the inner product of xa and xt as  follows:  lglobal(x, xt) = 1 − h⊤  x hxt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  (2)  Activation Tensor Loss is adapted to the scenario where the  outputs of the network of the descriptor extraction are the same  for xa and xt before down-sampling to resolution s, which  can be denoted by the mean squared difference of gx and gxt  as follows:  ltensor(x, xt) = ∥gx − gxt∥2  w · h · d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  (3)  Activation Histogram Loss is utilized to preserve first-order  statistics of activations ∥u(gx, b)i per channel i to achieve  identical extracted descriptors regardless of spatial information  in images, which can be denoted as follows:  lhist(x, xt) = 1  d  d  �  i=1  ∥u(gx, b)i − u(gxt, b)i∥,  (4)  where b is the histogram bin centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Our goal is to find the optimal loss function  that minimizes the semantic similarities of xt and x, which  equivalently optimizes network parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' We can use the  Adam algorithm to update θ as  θt+1 = θt − η  ρt  √υt + ϵ,  (5)  where η is the learning rate, ρt and υt are the first-order and  second-order momenta of gradients, and ϵ is utilized to pre-  vent the √υt + ϵ from being zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Therefore, the adversarial  semantic data xa can be expressed by  xa = arg min  x  Lts(xc, xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' x),  (6)  where Lts can be replaced by lglobal, ltensor, or lhist accord-  ing to different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' BLOCKCHAIN AND ZERO-KNOWLEDGE PROOF-BASED  SEMANTIC DEFENSE SCHEME  Since the adversarial semantic data generated by malicious  edge devices has almost the same descriptors (semantic simi-  larity) but different desired meanings from the authentic ones  produced by honest edge devices, inspired by [7][17], we  utilize blockchain and zero-knowledge proof-based semantic  defense scheme to help VSPs to identify attack images trans-  mitted in the Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Instead of submitting extracted se-  mantic data directly, edge devices should transform or process  semantic data by the bilinear interpolation algorithm [25], and  utilize Zero-Knowledge Proof to record and verify transfor-  mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The details of the proposed scheme are elaborated as  follows, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The transformation of semantic data is  a training-free defense method that uses visual invariance  to distinguish adversarial and authentic semantic extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The reason is that attackers adjust some pixels to make  descriptors of adversarial images similar to the authentic ones  [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' As a result, we attempt to increase visual invariance by  blurring extracted images using the spatial transformation of  the bilinear interpolation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Let us assume that (x1, y1), (x1, y2), (x2, y1), and (x2, y2)  are four points in the extracted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Then the targeted  points (x, y) can be obtained by the spatial transformation,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=', bilinear interpolation in the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The spatial  transformation can be considered a mapping function f(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Thus, the linear interpolation in the x direction can be derived  as follows:  f(x, y1) ≈ x2 − x  x2 − x1  f(x1, y1) + x − x1  x2 − x1  f(x2, y1)  (7)  and  f(x, y2) ≈ x2 − x  x2 − x1  f(x1, y2) + x − x1  x2 − x1  f(x2, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  (8)  The targeted points are transformed by the linear interpolation  in the y direction as follows:  f(x, y) ≈ y2 − y  y2 − y1  f(x, y1) + y − y1  y2 − y1  f(x, y2)  ≈  (x2 − x)(y2 − y)  (x2 − x1)(y2 − y1) f(x1, y1) +  (x − x1)(y2 − y)  (x2 − x1)(y2 − y1)) f(x2, y1)  +  (x2 − x)(y − y1)  (x2 − x1)(y2 − y1) f(x1, y2) +  (x − x1)(y − y1)  (x2 − x1)(y2 − y1) f(x2, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  (9)  Although edge devices can perform transformations to ob-  tain the blurred semantic data that can distinguish from the  adversarial one, it is difficult for VSPs to verify whether the  blurred semantic data is derived from authentic transforma-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Malicious edge devices may modify some pixels to  make descriptors of their semantic data close to that of honest  edge devices after transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Therefore, to assure the  authenticity of the blur transformation for semantic data, we  utilize ZKP to record transformations and blockchain to verify  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The proposed mechanism is a tuple of 3 polynomial-  time schemes after extracting a circuit mapping the trans-  formation, including Key Generation, Proof, and Verification,  which works as follows:  Extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The mechanism initiates the logic in a com-  putation circuit C using the simple arithmetic expression  mapping the transformation f to construct the public statement  s and the private witness w [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The circuit implements the  logic of bilinear interpolation via circom [26], a ZKP circuit  compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The circuit can provide a relation between the inputs  and outputs semantic data which can be used for verifiable  7  Blockchain  Edge Device  Witness  Statement  Transformation  Bilinear Interpolation  Extraction  Circuit  Proof  VSP  Status  Verification  Proof  Verification  Key  Evaluation  Key  Key  Generation  AIGC  Semantic  Data  Proof  Generation  Smart Contract  Render  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 3: Blockchain and Zero-Knowledge Proof-based Semantic Defense Scheme  computation on transformations without disclosing the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The process of extraction can be illustrated as follows:  Extract(f) → (s, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  (10)  Key Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Edge devices take a security parameter  1λ and the transformation circuit C as inputs to generate a  common reference string crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The common reference string crs  includes an evaluation key crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='ek and a verification key crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='vk  for proof and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The process of key generation can  be expressed as follows:  KeyGen(1λ, C)  C−→ crs(ek, vk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  (11)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Edge devices take the evaluation key crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='ek, the  statement s and the witness w related to the transformed  and original semantic data, which satisfies C(s, w) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The statement s and the witness w stand for the public  and private information corresponding to the transformation  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Thus, a zero-knowledge proof π is generated to reflect  and verify the relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The process of proof generation can  be denoted as follows:  Prove(crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='ek, s, w)  C−→ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  (12)  Verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' VSPs utilize smart contracts deployed on  blockchain to verify the authenticity of transformations and  implement the business logic of blockchain-aided semantic  communication for AIGC in Metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Since the verification  process is implemented in the blockchain, edge devices and  VSPs can query verification results to construct trust in a  decentralized manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The verification key crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='vk, the state-  ment s, and the proof π are the inputs written into smart  contracts to determine whether to accept or reject the proof  according to outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' When the status of the output is 1,  the verification succeeds and VSPs accept the semantic data  provided by edge devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' otherwise, VSPs refuse to accept the  proof corresponding to semantic data produced by malicious  edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The process of proof verification can be given  0  20  40  60  80  100  Iteration  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='35  ltr(xa, xt)  Global  Hist  Tensor  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 4: Performance Loss of Attacks  as follows:  Verify(crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='vk, s, π) → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  (13)  The semantic defense scheme should satisfy the follow-  ing properties including completeness, soundness, and zero-  knowledge [9][18][19][20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Completeness means that an honest edge device with a  valid witness w can convince an honest VSP for the au-  thenticity of transformed semantic data generated from the  circuit C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' If the transformed semantic data extracted by bilinear  interpolation is correct and edge devices construct the proof  π correctly through the proof circuit C and the evaluation  key crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='ek, then the VSPs can use the verification key crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='vk  generated by the proof circuit C, the proof π, and the public  information (statement) s to obtain a verification result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' For  each C(s, w) = 1, the proof π generated from an honest  edge device will be accepted with probability 1, which can  be denoted as follows:  Pr  �  �  KeyGen(1λ, C) → crs(ek, vk)  Prove(crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='ek, s, w) → π  Verify(crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='vk, s, π) = 1  �  � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  (14)  8  0  20  40  60  80  100  Iteration  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='00  h⊤  xahxt  Global  Hist  Tensor  (a) Semantic similarity between the  adversarial image and the authentic  0  20  40  60  80  100  Iteration  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='00  h⊤  xahxc  Global  Hist  Tensor  (b) Semantic similarity between the  adversarial image and the carrier  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 5: Semantic Similarity Performance of Attacks  Soundness denotes that malicious edge devices cannot  provide VSPs with authentic transformed semantic data, which  means that edge devices can provide valid witnesses if they  can produce valid proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' If (s, w) is not a valid input to  the proof circuit C(s, w) = 0, there is a polynomial time  extractor Ext for a malicious edge device A that makes  the probabilities of Verify(crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='vk, s, π∗) = 1 are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The adversarial advantage of soundness Advsd  A (λ) can be  represented as follows:  Pr  �  ���  KeyGen(1λ, C) → crs(ek, vk)  A(crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='ek, s) → π∗  Ext(crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='ek, s, π∗) → w  Verify(crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='vk, s, π∗) = 1  �  ��� ≤ negl(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  (15)  Malicious edge devices A can hardly falsify semantic data  or make honest VSPs V accept invalid proofs about transfor-  mations f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' According to (15), VSPs cannot accept false proof  π from A except for a negligible probability negl(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Zero-knowledge represents that VSPs can only verify the  authenticity of transformed semantic data while they cannot  obtain the original semantic data unless edge devices send it  to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Let us assume that there are probabilistic polynomial  time simulators S1, S2, and malicious edge devices A1, A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The simulator S1 can produce a common reference string that  is used by the simulator S2 to generate a simulated proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Then the proposed mechanism has the zero-knowledge prop-  erty if the adversarial advantage of zero-knowledge Advzk  A (λ)  satisfies:  | Pr(real) − Pr(sim)| ≤ negl(λ),  (16)  where Pr(real) and Pr(sim) can be denoted as  Pr(real) = Pr  �  ���  KeyGen(1λ, C) → crs  A1(f) → (s, w)  Prove(crs, s, w) → π  A2(crs, s, w, π) = 1  �  ���  (17)  and  Pr(sim) = Pr  �  ���  S1(1λ, C) → crs  A1(f) → (s, w)  S2(crs, s) → π  A2(crs, s, w, π) = 1  �  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  (18)  VSPs who know public statements mapping with semantic  data can hardly learn any knowledge about semantic data  before and after transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' According to (16), VSPs  cannot know anything about semantic data other than what  can be inferred from transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' EXPERIMENTAL EVALUATIONS  We simulate the proposed mechanism on Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='04LTS,  Intel Xeon, 8 core, 64G memory, and 25000Mb/s with 2 Tesla  V100, Go 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='4, Node v14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='0, PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='1, Torchvi-  sion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='1, and CUDA 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' We perform experiments on a  standard image benchmark, Revisited Paris [27], to measure  the attack and defense performance among edge devices and  VSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' We exploit pre-trained networks on ImageNet [28]  and AlexNet [29] to execute the attacks and defenses with  100 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Unless otherwise mentioned, we use these  parameters referring to [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The learning rate η is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  The hyper-parameter λ that adjusts the impact of distortion  loss is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The training process performs 100 iterations for our  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Performance of Semantic Attack Scheme  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 4 is the performance loss of the proposed semantic  attack scheme with different loss functions [Global, Hist, Ten-  sor] corresponding to the global descriptor, activation tensor,  and activation histogram as the number of iterations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Tensor converges much slower than Global and Hist, which  requires more iterations to reach its plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 5 is the semantic similarity attack performance between  the adversarial semantic data and the authentic or the carrier  one with three loss functions [Global, Hist, Tensor].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' As shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 5 (a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 5 (b), Global and Hist converge around  the 40th iteration which is much faster than Tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The  semantic similarity performance of attacks is consistent with  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 4 which shows the same trend in three loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Performance of Semantic Defense Scheme  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 6 is the semantic similarity comparison between attack  and defense schemes by displaying example images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 7  is the semantic similarity (descriptors) defense performance  between the adversarial semantic data and the authentic one or  the carrier one with three loss functions [Global, Hist, Tensor].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Compared with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 5, the adversarial image (semantic data)  can differ in semantic similarities when applying the proposed  semantic defense scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Since the extracted semantic data is  required to execute the defense scheme and the execution can  be assured by zero-knowledge proof, the images can differ in  descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 5 (a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 7 (a), the semantic  similarity can reduce by up to 35% between the adversarial and  the authentic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 5 (b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 7 (b) show that the  semantic similarity can decrease 10% between the adversarial  and the carrier images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 8 (a) is the blockchain consensus overhead for [Attack,  Authentic, Defense] types of semantic data (image) with [275  KB,467 KB,10 KB] data sizes when the number of nodes  is [5,50,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 8 (a), the proposed scheme can  9  0  5  10  20  100  Number of Iterations  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
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+page_content='661470  Target Image  Attack Image  Defense Image  Attack Image  Transformed  Carrier Image  Carrier Image  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
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+page_content=' 6: Semantic Similarity Comparison Between Attack and Defense Schemes  0  20  40  60  80  100  Iteration  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
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+page_content='65  h⊤  xahxt0  Global  Hist  Tensor  (a) Semantic similarity between the  adversarial image and the authentic  0  20  40  60  80  100  Iteration  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
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+page_content='72  h⊤  xahxc0  Global  Hist  Tensor  (b) Semantic similarity between the  adversarial image and the carrier  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 7: Semantic Similarity Performance of Defenses  reduce the time consumed in blockchain to improve consensus  efficiency due to the attack and defense schemes cropping  authentic semantic data while differing in semantic similarities  according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 8 (b) is the ZKP computation over-  head for [GenWitness, GenProof, VerifyProof] operations with  [10KB,100KB,1MB] data sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' We can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 8 (b)  that the ZKP computation overhead depends on the GenProof  operation, while the sizes of semantic data affect little on  the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Besides, the computation overhead for  verifying proofs is less than for generating proofs, which can  be written into smart contracts [9][18][19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' CONCLUSION AND FUTURE WORK  In this paper, we first integrated blockchain-aided semantic  communication into Metaverse to support AIGC services in  virtual transportation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' We also presented a training-  based targeted semantic attack scheme to illustrate potential  attacks by generating adversarial semantic data with the same  descriptors but different desired meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' To prevent the  10  20  30  40  50  Number of Nodes  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content='6  Time Consume / s  Attack  Authentic  Defense  (a) Consensus  GenWiteness  GenProof  VerifyProof  Type  0  20  40  60  80  100  120  Time Consumed / s  10KB  100KB  1MB  (b) ZKP Computation  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' 8: Consensus and Computation Overheads  above attack, we designed a blockchain and zero-knowledge  proof-based semantic defense scheme that records transforma-  tions of semantic data and verifies mutations in a decentralized  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' Simulation results show that the proposed mechanism  can differ in the descriptors between the adversarial semantic  data and the authentic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' The defense scheme can identify  malicious edge devices falsifying semantic data to corrupt  AIGC services in virtual transportation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
+page_content=' In future  research work, we will study how to mitigate malicious  edge devices and facilitate resource allocation with economic  incentive mechanisms in the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFIT4oBgHgl3EQfhytq/content/2301.11289v1.pdf'}
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+Hierarchical Dynamic Masks for Visual
+Explanation of Neural Networks
+Yitao Peng, Longzhen Yang, Yihang Liu, Lianghua He*
+College of Electronic and Information Engineering Tongji University
+4800 Cao’an Highway, Shanghai, China 201804
+{pyt, 2111131, yanglongzhen, helianghua}@tongji.edu.cn
+Abstract
+Saliency methods generating visual explanatory maps representing the
+importance of image pixels for model classification is a popular tech-
+nique for explaining neural network decisions. Hierarchical dynamic masks
+(HDM), a novel explanatory maps generation method, is proposed in this
+paper to enhance the granularity and comprehensiveness of saliency maps.
+First, we suggest the dynamic masks (DM), which enables multiple small-
+sized benchmark mask vectors to roughly learn the critical information in
+the image through an optimization method. Then the benchmark mask
+vectors guide the learning of large-sized auxiliary mask vectors so that
+their superimposed mask can accurately learn fine-grained pixel impor-
+tance information and reduce the sensitivity to adversarial perturbations.
+In addition, we construct the HDM by concatenating DM modules. These
+DM modules are used to find and fuse the regions of interest in the re-
+maining neural network classification decisions in the mask image in a
+learning-based way. Since HDM forces DM to perform importance analy-
+sis in different areas, it makes the fused saliency map more comprehensive.
+The proposed method outperformed previous approaches significantly in
+terms of recognition and localization capabilities when tested on natural
+and medical datasets.
+arXiv:2301.04970v1  [cs.CV]  12 Jan 2023
+
+1
+Introduction
+Neural networks [8, 10, 11] have made remarkable achievements in areas such
+as image recognition [22, 3]. So people began to think about the logic of neural
+network decision-making. Many methods for explaining neural networks have
+been proposed [14]. Among them, the method of generating saliency maps to
+point out the image regions that neural networks pay attention to when making
+decisions has been widely studied.
+The generation of saliency maps [13] is
+divided into three mainstream methods: perturbation-based methods [15, 25],
+gradient-based methods [21], and CAM-based methods [12, 9].
+Gradient-based methods [21, 18] propagate the gradient of the target class
+to the input layer via backpropagation to generate regions that have a large in-
+fluence on the prediction. But the explanations provided by these methods are
+less reliable and sensitive to adversarial perturbations [14]. The method based
+on the input perturbation [4, 15] monitors the change of the prediction proba-
+bility of a specific class while occluding different image regions, and generates
+a saliency map describing the importance of the network prediction probability
+by setting specific perturbation rules and discriminant methods. CAM-based
+method [26] proposes class activation maps (CAM), which provide visual ex-
+planations based on linearly combining activation maps of convolutional neural
+networks. [1] have removed the requirements of the CAM-based methods for
+the network structure and improved the accuracy of positioning.
+In this paper, in order to alleviate the problem that previous saliency map
+methods generate interpretation maps with rough localization and cannot ex-
+press the importance of pixels in a fine-grained manner, we propose DM. It
+trains mask vectors of different sizes through an optimization-based approach.
+A single element of a small-sized mask vector corresponds to a large receptive
+field, and there is less adversarial effect during learning. Large size mask vectors
+have small receptive fields and can extract fine-grained information from images.
+In order to combine the advantages of both, we let the small-size mask vector
+guide the learning of the large-size mask vector, reducing the mask vector’s ad-
+versarial sensitivity while improving its ability to mine fine-grained information.
+This enables the DM’s final fused masks to generate fine-grained explanatory
+map.
+Additionally, we consider how to more completely capture regions of interest
+for neural network classification.
+An image may have multiple regions that
+contribute to the network’s decision, but previous methods often only find one
+main decision region and ignore others. To avoid this problem, we propose HDM
+that exploits DM hierarchically. It masks the salient regions found by the DM
+on the image, re-inputs the masked image into the next DM module to force it
+to find the regions in the image that the network cares about in the remaining
+images, and then fuses the above regions according to their importance through
+a learning-based method. HDM can not only combine the fine search ability of
+DM, but also conduct a comprehensive search for the decision-making area of
+the entire image. This makes the saliency maps it generates not only detailed
+but also comprehensive.
+2
+
+DM
+DM
+Mask   M2
+Input image
+X
+Mask image
+X1
+Mask image
+X2
+Mix mask
+Mh
+Mask   M1
+Network consistency 
+activation
+Timing combined
+Figure 1: Overall architecture of HDM. It is formed by stacking several DM
+blocks. The original image is input into the DM block to generate a mask and
+mask image, and then the mask image is continuously input into the next DM
+block to generate the next mask and mask image. Several masks are obtained
+by repeating the above operations. All the masks are mixed to generate the final
+mixed mask through the learning-based timing combination method, and the
+mixed mask image generated by the dot multiplication of the mixed mask and
+the original image is trained with the original image to calculate the activation
+consistency to form the final mixed mask.
+The key contributions of our work are as follows:
+• We propose a learning-assisted module DM using vectors of various sizes,
+which can analyze the importance of image pixels to model classification
+in detail.
+• We propose a visual interpretation method HDM that leverages DM hier-
+archically to generate fine-grained and comprehensive saliency maps.
+• We conduct multiple experiments on natural and medical datasets and
+verify that our method achieves state-of-the-art in recognition and local-
+ization capabilities.
+2
+Related work
+Existing saliency methods can be divided into three categories, and we briefly
+introduce these methods in this section.
+The first is a perturbation-based method. [4] use the optimization method
+to find the areas in the image that are helpful for the network decision-making,
+and cover the areas in the picture that are not related to the classification. [15]
+detects the importance of various regions in the image in the decision of the
+network by using a randomly masked version of the input image.
+The second is a gradient-based method. Gradient [19] directly computes the
+derivative of a class score with respect to an input image using a first-order
+Taylor expansion. It captures local sensitivity by representing the change at
+3
+
+each input location through gradients. [18] proposed DeepLIFT to decompose
+the neural network’s output prediction. Further, [21] proposed Integrated Gra-
+dients, which uses small changes in the feature space between the interpolated
+image and the input image to measure the correlation between feature changes
+and model predictions.
+The third is a CAM-based approach.Grad-CAM [17] uses the gradient in-
+formation flowing into the last convolutional layer of a convolutional neural
+networks to assign importance values to each neuron. In order to increase the
+positioning accuracy of CAM, [1] proposed Grad-CAM++, which added an ad-
+ditional weight to weigh the elements of the gradient map. FullGrad [20] more
+fully explains model behavior by aggregating full gradient components. [24] pro-
+posed Score-CAM to solve the problem of easily finding false confidence samples.
+Ablation-CAM [16] was present to explore the strength of each factor’s contri-
+bution to overall model. Eigen-CAM [12] makes CAM generation easier and
+more intuitive. XGrad-CAM[6] introduces sensitivity and conservation axioms
+for CAM methods. To generate a more comprehensive saliency map, Layer-
+CAM [9] considers the role of each spatial location in the feature map using
+element-level weights.
+3
+Methodology
+The overall architecture of HDM and generation method are described in Section
+3.1. In addition, Section 3.2 introduces DM to fine study the importance of
+images to network.
+3.1
+Hierarchical Masks Generation
+We propose to use DM hierarchical timing to find the regions of the image that
+neural networks focus on when making decisions. After the current DM finds
+a decision-making area, it masks this area in the image, and then inputs the
+masked image to the next DM module to analyze whether DM can find other
+areas that are helpful for neural network decision-making. Since the decision-
+making area searched by the previous DM module has been shielded, the next
+DM module must search in the unshielded area. Through hierarchical method,
+which forces each DM module to search different regions, it is possible to com-
+prehensively learn the region information in the image that the network pays
+attention to when making decisions.
+Figure 1 is the overall architecture of
+HDM, which consists of many DM modules. DM precisely generates the areas
+of interest to the neural network in the input image. HDM generates compre-
+hensive and refined saliency maps by mixing masks generated by individual DM
+modules through a learning-based timing combination.
+Let the input image be X ∈ RH×W ×C, the DM module is denoted as Q(·).
+After inputting image Xi−1 to DM, DM can calculate the mask Mi (where
+Mi ∈ RH×W ×1) corresponding to the most critical region for decision-making
+in image Xi. We have the following iteration formula, generating S decision
+4
+
+Activation mask
+Original  image 
+Activation image
+Mask vector
+Overlay mask
+Mask map
+Heatmap
+Original  image
+Mask image
+Heatmap image
+Mask1
+Mask2
+MaskD
+Network consistency 
+activation
+Figure 2: The flow of a DM. Above the line, mask vectors of different sizes
+are trained by constraining the original image and activation image to have
+consistent activations on nodes of the neural network. The image below the
+line shows that the upsampled learned mask vectors are stacked to produce an
+overlay image. The mask image is obtained by multiplying the overlay mask
+with the original image.
+The overlay mask uses JET colormap to generate
+heatmap, which is added to the original image to get heatmap image.
+masks, X0 = X, i ∈ {1, 2, ..., S}.
+Mi = Q(Xi−1)
+(1)
+Xi = (1 − N(
+i
+�
+j=1
+Mj))X0
+(2)
+where N(x) =
+x−min(x)
+max(x)−min(x) is a normalization function. The mix mask Mh is
+generated by mixing S masks Mj (j ∈ {1, 2, ..., S}) through the following timing
+combination.
+Mh =
+�S
+j=1 wjMj
+�S
+j=1 wj
+, wj =
+S
+�
+k=j
+v2
+j
+(3)
+5
+
+Note: J(M, X) = ||fp(MX) − fp(X)||2
+2, which represents the consistent
+squared loss generated by MX and X input neural network f at position p.
+In this paper, we let the position p be the node corresponding to the category
+predicted by the neural network. LR(M) = �H
+u=1
+�W
+v=1
+|Mhuv|
+|HW | , which is the
+regularized value corresponding to mask M.
+L(M, X) = J(M, X) + λLR(M)
+(4)
+where λ is a regularization factor. We train the weights of S masks by optimizing
+the following loss in Equation (5) to obtain the mix mask Mh according to
+Equation (3).
+v1, v2, ..., vS ←
+min
+v1,v2,...,vS L(Mh, X0)
+(5)
+Since HDM searches the remaining areas that are conducive to decision-
+making in the current image hierarchically according to time sequence, Mj is
+no less important than Mj+1. Therefor, we constrain wj ≥ wj+1 by Equation
+(3), j ∈ {1, 2, ..., S − 1}.
+3.2
+Dynamic Masks Learning
+DM learns the mask vectors by optimizing the activation consistency between
+the original image and the mask image and the regularization term of the mask
+vectors. The higher the mask value of the region that is more important to
+the classification decision of the neural network, the higher the attention of the
+neural network to this region (refer to supplementary material for explanation).
+We first set a small-sized benchmark mask vector to roughly learn the attention
+of the neural network to each area in the image. The size of the mask vector
+is inversely proportional to its receptive field, and each element of the small-
+sized mask vector corresponds to a large receptive field in the image, so that
+the vector elements can accurately learn the importance of each region in the
+image.
+Benchmark mask vectors {di}D
+i=1, di ∈ Rai×bi×1, di are initialized to τ. For
+any i, j ∈ {1, 2, ..., D}, if i ̸= j then ai ̸= aj or bi ̸= bj. Upsample function
+g(·, h, w) can upsample the original vector to a vector of length h and width w.
+For example, x ∈ Rh1×w1×1, and g(x, h2, w2) ∈ Rh2×w2×1.
+As show in Figure 2, we train {di}D
+i=1 by the activation consistency between
+the mask image and the original image. Mathematically, we optimize the fol-
+lowing loss function.
+L(g(di, H, W), X) = J(g(di, H, W), X)
++ ηiLR(g(di, H, W))
+(6)
+where ηi is a regularization factor. To enable the mask to analyze the importance
+of region of the image at a fine-grained level, we iteratively generate auxiliary
+mask vectors ck
+j (i) on the benchmark mask vectors. ck
+j (i) is defined as follows:
+ck
+j (i) = g(di, tk
+j ai, tk
+j bi)
+(7)
+6
+
+0.2
+0.2
+0.3
+0.3
+0.2
+0.2
+0.3
+0.3
+0.8
+0.8
+0.6
+0.6
+0.8
+0.8
+0.6
+0.6
+0.2
+0.2
+0.3
+0.3
+0.2
+0.2
+0.3
+0.3
+0.8
+0.8
+0.6
+0.6
+0.8
+0.8
+0.6
+0.6
+Network consistency activation
+0.2
+0.3
+0.1
+0.1
+0.1
+0.2
+0.1
+0.4
+0.7
+0.8
+0.2
+0.1
+0.5
+0.1
+0.1
+0.1
+0.2
+0.3
+0.1
+0.1
+0.1
+0.2
+0.1
+0.4
+0.7
+0.8
+0.2
+0.1
+0.5
+0.1
+0.1
+0.1
+0.3
+0.8
+0.6
+0.2
+0.3
+0.8
+0.6
+0.2
+Upsample
+Upsample
+Mask vector Cj
+k(i)
+Mask vector Cj
+k+1(i)
+Upsampling mask vector
+Receptive field
+ in image
+Activation mask
+Activation image
+Original  image
+Mixed mask vector
+0.02 0.04 0.09 0.03
+0.02 0.02 0.06 0.03
+0.32 0.56 0.48 0.12
+0.08 0.40 0.06 0.06
+0.02 0.04 0.09 0.03
+0.02 0.02 0.06 0.03
+0.32 0.56 0.48 0.12
+0.08 0.40 0.06 0.06
+Figure 3: An example of training an auxiliary mask vector. The original image
+size is 224×224. A feature block of the 2×2 and 4×4 mask vector corresponds
+to the 112 × 112 and 56 × 56 receptive field of the original image, respectively.
+The ck+1
+j
+(i) to be trained is multiplied element-wise by the upsampled ck
+j (i) that
+has been trained, and then the multiplication result is upsampled and multiplied
+by the original image to generate the activation image. ck+1
+j
+(i) is trained by
+optimizing the consistency activation of the activation and original image.
+where {tj}T
+j=1 are positive integer hyperparameters, and k ∈ {0, 1, ...., Ki
+j},
+Ki
+j = min{
+�
+lnH−lnai
+lntj
+�
+,
+�
+lnW −lnbi
+lntj
+�
+}, ⌊·⌋ is the floor function.
+As shown in
+Figure 3, the benchmark mask vector is used as the initial guidance vector, and
+the auxiliary mask vector of the upper level guides the auxiliary mask vector of
+the next level to iteratively learn auxiliary vector sets {ck
+j (i)|i ∈ {1, ..., D}, j ∈
+{1, ..., T}, k ∈ {0, ..., Ki
+j}}. The loss function of ck
+j (i) is as the following Equation
+(8).
+L(g(ck
+j (i)g(ck−1
+j
+(i), tk
+j ai, tk
+j bi), H, W), X)
+= J(g(ck
+j (i)g(ck−1
+j
+(i), tk
+j ai, tk
+j bi), H, W), X)
++ η(k,j)
+i
+LR(g(ck
+j (i), H, W))
+(8)
+As shown below the line in Figure 2, Mc is obtained by stacking the auxiliary
+masks, and the redundant information of Mc is removed to obtain the overlay
+mask Mb. Mb is the final output result of DM, which can fine-grainedly describe
+the importance of image regions in neural network decisions.
+Mc =
+D
+�
+i=1
+T
+�
+j=1
+Ki
+j
+�
+k=0
+ck
+j (i)
+(9)
+Mb = N((Mc − γ){Mc ≥ γ})
+(10)
+where γ is the threshold, and {·} represents a truth-valued function, which is 1
+if true; otherwise, it equals 0. N(·) is a normalization function.
+7
+
+Input Image
+HDM(ours)
+Grad-CAM
+Grad-CAM++
+DeepLIFT
+Score-CAM
+XGrad-CAM
+Integrated
+RISE
+FullGrad
+Gradient
+Eigen-CAM
+Layer-CAM
+Ablation-CAM
+DM(ours)
+Mask
+Figure 4: Interpretation results for the ResNet50 network using different tech-
+niques. The saliency maps generated by each method are normalized to the
+range [0,1] and visualized using the JET colormap. The saliency map from blue
+to red indicates a gradual increase in activation probability. Note that the first
+three rows are examples where the prediction is correct, while the last three
+rows are wrong.
+4
+Experiments
+First, the dataset, baseline, and our experimental setup are presented in section
+4.1 and section 4.2 for reproducibility. Second, we present the visual interpreta-
+tion results we generate in section 4.3, qualitatively evaluating the performance
+of our method compared to other previous methods. Furthermore, section 4.4
+visualizes the saliency map generated by the input image in each stage of HDM,
+showing the flow of HDM for inference. In addition, sections 4.5 present the
+results of quantitatively evaluating the fidelity of the interpretation of saliency
+maps. The quality of the saliency maps are measured by localization ability as
+described in section 4.6. Finally, ablation experiments are performed in section
+4.7.
+4.1
+Datasets and Baselines
+Experiments were conducted on two public image recognition datasets, including
+one natural (CUB-200-2011 [23]) and one medical (iChallenge-PM [5]) image
+datasets.
+We compare 13 state-of-the-art saliency map generation methods.
+Gradient-based methods: Gradient [19], DeepLIFT [18], and Integrated [21].
+Perturbation-based methods: RISE [15], and Mask [4]. CAM-based methods:
+Grad-CAM [17], Grad-CAM++ [1], Score-CAM [24], XGrad-CAM [6], FullGrad
+[20], Eigen-CAM [12], Ablation-CAM [16], and Layer-CAM [9]. The saliency
+maps generated by the above-mentioned methods are taken as baselines.
+8
+
+-Input image
+Heatmap image1
+Heatmap image2
+Heatmap image3
+Mixed heatmap image
+Figure 5: The heatmap generated by the whole process of HDM. The first
+column are the input images; the second, third, and fourth column images are
+the heatmap images generated after passing through the DM module for the
+first, second, and third times respectively; the fifth column of images is the
+heatmap image generated by mixing the second, third, and fourth columns of
+heatmaps through a learning-based method, which is also the output of HDM.
+4.2
+Experimental Details
+In this work, ResNet50 [7] was chosen as the neural network for saliency maps
+interpretation. All methods use a ResNet50 model with the same parameters
+and fixed for visual evaluation. The input image is resized to 224×224×3, and
+then normalized using mean vector (0.485, 0.456, 0.406) and variance vector
+(0.229, 0.224, 0.225).
+Our training set contains only images and their class
+labels. ResNet50 is pre-trained on ImageNet [2] before being trained on the
+natural and medical images datasets. Each image is trained for 800 epochs with
+a learning rate set to 1e − 2. We set λ = 1e − 4, ηi = η(k,j)
+i
+= 100. In the CUB-
+200-2011, we set di = i + 5, {di}6
+i=1, {tj}3
+j=1 = {2, 3, 5}; γ is set to the pixel
+value of the top 25% of the pixel value of the salient image; HDM consists of 3
+DM modules. In the iChallenge-PM, we set di = i+5, {di}4
+i=1, {tj}2
+j=1 = {2, 3};
+γ is set to the pixel value of the top 30% of the pixel value of the salient image;
+HDM consists of 1 DM modules. The learning-based mix mask iteration 400
+epochs, and its learning rate is 1e − 1.
+4.3
+Qualitative Evaluations
+Experiments are designed to qualitatively compare the performance of saliency
+maps generated by different methods.
+As shown in Figure 4, saliency maps
+9
+
+point out regions of interest that networks interpret when making decisions,
+and we compare 13 state-of-the-art saliency map generation methods.
+Interpretation results for six different images are reported in Figure 4. In
+each row, the leftmost image is the original image, and the second and third
+rows are the saliency maps generated by our proposed HDM and DM meth-
+ods, respectively. The subsequent columns represent the results obtained using
+different methods. The color of the saliency maps from red to blue represents
+indicates the gradual decrease in the probability of the network’s attention to
+the region.
+As shown in Figure 4, the saliency maps generated by CAM-based methods
+usually only point out the areas of the image that a network pays attention to
+when making decisions. The form of this region is almost always with a center as
+the highest activation, diffuses outward and gradually weakens irregularly, so the
+generated The saliency map has a single structure, lacks integrity, and is not easy
+for humans to understand. In contrast, the saliency map generated by HDM
+can discover multiple decision-making regions and integrate them completely. It
+finely includes the body regions of the bird used for decision making, outlining
+the silhouette of the bird. The HDM method can completely and accurately
+generate the areas of concern when making network decisions, which is more
+comprehensive and easy to understand than the CAM-based method.
+The saliency map generated based on the gradient method can roughly de-
+scribe the contours of the objects concerned by the network decision-making,
+but the pixels of the saliency map are discrete and sharp, and it is difficult for
+people to observe which are the key decision-making areas. By comparison, we
+can see that the saliency map generated by HDM can better describe the outline
+of the object while retaining the smooth change of activation, which makes it
+easier for people to observe the importance of each region.
+The saliency map generated by the perturbation-based method can mark
+multiple regions for decision-making, but such saliency maps cover too large
+areas and have more noise. In contrast, HDM can cover multiple decision regions
+while paying little attention to regions outside the decision region (bird body),
+generating saliency maps with little noise.
+Overall, our method can generate refined and complete saliency maps, which
+are less noisy and easy to understand.
+4.4
+Processes Visualization
+As shown in Figure 5, we show all saliency maps generated by HDM during
+inference on images from the CUB-200-2011 dataset. When researching on the
+CUB-200-2011 dataset, we set HDM to consist of 3 DM modules. The first
+column in Figure 5 is the original input image. The second, third, and fourth
+columns in Figure 5 represent the saliency images of the current most-attended
+regions of the neural network found after inputting the original image or the
+mask image into the DM module in the three stages, respectively. It can be
+seen from Figure 5 that each DM module can accurately and fine-grainedly find
+the remaining regions in the current picture that are helpful for neural network
+10
+
+Evaluation
+Average Drop
+Average Increase
+Percentile
+80%
+70%
+80%
+70%
+Mask
+39.1%
+24.1%
+15.3%
+22.3%
+RISE
+31.2%
+17.4%
+24.2%
+35.4%
+Gradient
+97.8%
+97.4%
+0.4%
+0.8%
+DeepLIFT
+97.7%
+97.5%
+0.3%
+0.6%
+Integrated
+98.4%
+97.8%
+0.3%
+0.5%
+Grad-CAM
+31.5%
+17.0%
+21.3%
+35.7%
+Grad-CAM++
+35.2%
+19.1%
+18.2%
+32.2%
+Score-CAM
+31.2%
+18.2%
+19.7%
+35.3%
+Ablation-CAM
+33.7%
+19.7%
+17.3%
+34.4%
+XGrad-CAM
+33.9%
+18.7%
+18.2%
+36.6%
+Eigen-CAM
+38.5%
+22.9%
+17.1%
+31.8%
+Layer-CAM
+36.6%
+19.3%
+16.3%
+29.6%
+FullGrad
+35.5%
+19.5%
+20.1%
+30.9%
+DM(ours)
+32.6%
+15.7%
+26.4%
+36.9%
+HDM(ours)
+30.1%
+13.3%
+26.7%
+38.8%
+Table 1: Evaluated results on average drop (lower is better) and average increase
+(higher is better).
+decision-making, and the regions of interest found in each stage are almost
+disjoint. This is in line with HDM’s hierarchical search strategy. The mixed
+heatmap images in the last column are the result of mixing the heatmap images
+generated by the first three DM modules via a learning-based approach, which
+makes the final generated saliency maps comprehensive. It can be seen from
+Figure 5 that the final hybrid map generated by HDM is not only comprehensive,
+but also fine-grained.
+4.5
+Faithfulness Evaluation via Image Recognition
+The experiment evaluates the fidelity of the interpretation of the saliency map
+generated by the HDM in the object recognition task, and the average drop
+and average increase [1] are used as evaluation indicators. This experiment sets
+the activation value of a specific percentage of all pixels in the saliency map
+as the threshold value, mutes the pixels below the threshold value, and retains
+the pixel value above the threshold value to generate an interpretation map (in
+the experiment, set 70% and 80% of the pixels are muted). The average drop
+is expressed as �N
+i=1
+max(0,Yi−Oi)
+Yi
+× 100, and the average increase is expressed
+as �N
+i=1
+Sign(Yi<Oi)
+N
+, where Y c
+i
+is the predicted score of image i in category
+c, and Oc
+i is the predicted score of category c with the explanatory map as
+input. Sign(·) is an indicator function that returns 1 if the input is True. In
+the experiment, 10 images are randomly selected for each category on CUB-
+200-2011, and a total of 2000 images are formed for testing. The results are
+reported in Table 1. When the threshold is set to occlude 70% and 80% of
+11
+
+Evaluation
+Deletion Score
+Insertion Score
+Mask
+0.0516
+0.7579
+Grad-CAM
+0.0537
+0.7939
+Grad-CAM++
+0.0578
+0.7937
+Score-CAM
+0.0576
+0.8076
+Ablation-CAM
+0.0643
+0.7724
+XGrad-CAM
+0.0582
+0.7983
+Eigen-CAM
+0.0763
+0.7751
+Layer-CAM
+0.0601
+0.7954
+FullGrad
+0.0537
+0.7958
+DM(ours)
+0.0461
+0.8171
+HDM(ours)
+0.0447
+0.8189
+Table 2: Evaluated results on delecion (lower is better) and insertion (higher is
+better) scores.
+Percentile
+Percentile
+Predict Value
+Predict Value
+Deletion Curve
+Insertion Curve
+Figure 6: Deletion and Insertion curves of the above methods.
+the pixels, HDM achieves 13.3% and 30.1% average drop and 38.8% and 26.7%
+average increase, respectively. HDM outperforms other saliency methods. This
+task shows that HDM can find the most distinguishable regions of the target
+object as much as possible, and can eliminate as much as possible the regions
+irrelevant to the target object distinction.
+To further evaluate the discriminative ability of HDM, we test on the in-
+sertion and deletion metrics proposed in [15]. Deletion and insertion scores are
+the areas under the probability curves described by the predicted probability
+results for deleted or inserted images generated by deleting and inserting pixels
+from the original image in descending order of saliency map activation values,
+12
+
+bGLcGUf!G
+0
+50
+40
+eo
+08
+JOO
+0.0
+5.0
+0'4
+Ea0.0=MAO-noitsldA
+8F20.0=++MA0-b610
+a.0
+820.0=MA0-b610X
+8.0
+Dw(0nL2)=0'04e1
+J'O
+HDW(0L2)=0'044)
+DGJGTIOUCNLAG0
+50
+40
+eo
+08
+JO0
+0.0
+ASr.0=MAO-noitsldA
+aT08.0=MA0-91002
+5.0
+0'4
+IFI8.0=(21Uo)M
+a.0
+HDW(0nL2)=0'8J8a
+8.0
+J'OEvaluation
+Proportion
+Dataset
+CUB-200-2011
+iChallenge-PM
+Mask
+33.31%
+27.31%
+RISE
+29.47%
+14.33%
+Gradient
+31.58%
+13.57%
+DeepLIFT
+29.86%
+13.63%
+Integrated
+31.50%
+13.59%
+Grad-CAM
+50.34%
+20.84%
+Grad-CAM++
+51.37%
+17.36%
+Score-CAM
+54.52%
+24.71%
+Ablation-CAM
+48.33%
+18.14%
+XGrad-CAM
+48.92%
+18.91%
+Eigen-CAM
+57.41%
+21.64%
+Layer-CAM
+56.24%
+20.30%
+FullGrad
+47.61%
+17.94%
+DM(ours)
+53.81%
+32.81%
+HDM(ours)
+57.97%
+32.81%
+Table 3: On CUB-200-2011 and iChallenge-PM datasets, the evaluation results
+of each method in proportion.
+respectively. This experiment adopts the setting in [24], the step size is 1%,
+and the pixel value is set to 0 or 1 to remove or introduce pixels. We compare
+the performance of the methods in Table 2 on the above-mentioned evaluation
+metrics. Here we do not compare other perturbation-based methods, since they
+can produce adversarial effects [15] that interfere with these evaluation metric
+performance. As shown in Figure 6, the deletion and insertion curves produced
+by each model are displayed. The HDM method achieves the largest AUC and
+the steepest curve, which shows that the HDM saliency map can best find the
+area in the current image that has the greatest influence on the network decision.
+Table 2 reports the average results for the above-mentioned 2000 images, and
+our method achieves state-of-the-art on both metrics compared to the respective
+previous methods.
+4.6
+Localization Evaluation
+In this section, the localization ability is employed to measure the quality of
+the generated saliency maps. In the same way as in [24], calculate how much
+energy of the saliency map falls into the segmented foreground region of the
+target object. Binarize the segmented foreground image given in the test set,
+assign the foreground area as 1, and assign the background area as 0, multiply
+the saliency map calculated by each method with the binarized image point
+by point and sum how much energy is obtained in the target foreground. The
+evaluation metric Proportion is expressed as
+� Lc
+(i,j)∈bbox
+� Lc
+(i,j)∈bbox+� Lc
+(i,j)/
+∈bbox . We test
+the localization ability on the CUB-200-2011 and iChallenge-PM datasets, set-
+13
+
+Figure 7: Results of saliency maps generated by HDM on bird and fundus
+images.
+The first and fourth rows are the original images; the second and
+fourth rows are the mixture of the saliency map and the original image; the
+third and sixth rows are the ground-truth.
+ting the foreground region of the image segmentation label as the bbox region.
+In the test set of CUB-200-211, 10 images are randomly selected from each
+category, and a total of 2000 images are formed for experiments; in the test
+set of iChallenge-PM, 200 images are randomly selected as experiments. The
+saliency maps generated by HDM have 57.97% and 32.81% energy falling in the
+ground-truth foreground region of the target in CUB-200-2011 and iChallenge-
+PM, respectively. This verifies that HDM can effectively reduce noisy regions
+in saliency maps that are not relevant to decisions. Our method achieves state-
+of-the-art performance in both natural and medical image localization ability.
+As shown in Figure 7, we visualize the results of the saliency maps generated
+by the HDM on bird and fundus datasets, respectively. In the bird image, the
+decision-making region found in the saliency map of HDM wraps most of the
+14
+
+bird’s body, and carefully evaluates the importance of each position on the body.
+This verifies that HDM is capable of comprehensively locating multiple decision
+regions in natural images and fine-grained division of the importance of each
+region. In the fundus images, HDM can also accurately and completely locate
+pathological areas in the images. The above experiments show that HDM can
+achieve good localization performance in both natural and medical images.
+4.7
+Ablation Study
+In this section, we compare the performance of saliency maps generated using
+the DM module alone and HDM consisting of multiple DM hierarchical timings.
+It can be seen from Table 1 and Table 2 that the saliency map generated by
+DM is superior to all previous methods in the evaluation index of recognition
+ability; Table 3 shows that the localization ability of DM on natural images
+is weaker than some saliency methods, and the localization ability on medical
+images reaches the best. From the above-mentioned tables, it can be seen that
+the hierarchical stacking DM method of HDM can improve the performance of
+the saliency map on the above indicators. Because DM only focuses on finding
+the area that a neural network pays attention to when making decisions, but
+ignores the fact that there may be multiple areas that have an impact on the
+classification of the neural network. HDM makes the saliency map more com-
+prehensive by using DM multiple times to search for decision regions. Therefore,
+both the DM module and the method of hierarchically stacking the DM modules
+are useful and necessary.
+5
+Conclusion
+In this paper, we propose HDM to generate refined and comprehensive visual
+explanatory maps. The DM module is able to generate finely localized saliency
+maps using masks of various sizes that cooperate with each other. HDM an-
+alyzes the masked image hierarchically using the DM module to find areas of
+concern for multiple neural network decision-making classification decisions in
+the image, and combines them well into a comprehensive visual interpretation
+map through a learning-based fusion method. Our method achieves state-of-the-
+art performance in recognition and localization capabilities on natural images
+and medical images, and quantitative evaluation verifies that our method can
+generate more fine-grained and comprehensive saliency maps.
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+
diff --git a/a9E4T4oBgHgl3EQfPAy5/content/tmp_files/load_file.txt b/a9E4T4oBgHgl3EQfPAy5/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf,len=590
+page_content='Hierarchical Dynamic Masks for Visual  Explanation of Neural Networks  Yitao Peng, Longzhen Yang, Yihang Liu, Lianghua He*  College of Electronic and Information Engineering Tongji University  4800 Cao’an Highway, Shanghai, China 201804  {pyt, 2111131, yanglongzhen, helianghua}@tongji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='cn  Abstract  Saliency methods generating visual explanatory maps representing the  importance of image pixels for model classification is a popular tech-  nique for explaining neural network decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Hierarchical dynamic masks  (HDM), a novel explanatory maps generation method, is proposed in this  paper to enhance the granularity and comprehensiveness of saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  First, we suggest the dynamic masks (DM), which enables multiple small-  sized benchmark mask vectors to roughly learn the critical information in  the image through an optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Then the benchmark mask  vectors guide the learning of large-sized auxiliary mask vectors so that  their superimposed mask can accurately learn fine-grained pixel impor-  tance information and reduce the sensitivity to adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  In addition, we construct the HDM by concatenating DM modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' These  DM modules are used to find and fuse the regions of interest in the re-  maining neural network classification decisions in the mask image in a  learning-based way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Since HDM forces DM to perform importance analy-  sis in different areas, it makes the fused saliency map more comprehensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  The proposed method outperformed previous approaches significantly in  terms of recognition and localization capabilities when tested on natural  and medical datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='04970v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='CV]  12 Jan 2023  1  Introduction  Neural networks [8, 10, 11] have made remarkable achievements in areas such  as image recognition [22, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' So people began to think about the logic of neural  network decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Many methods for explaining neural networks have  been proposed [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Among them, the method of generating saliency maps to  point out the image regions that neural networks pay attention to when making  decisions has been widely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  The generation of saliency maps [13] is  divided into three mainstream methods: perturbation-based methods [15, 25],  gradient-based methods [21], and CAM-based methods [12, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Gradient-based methods [21, 18] propagate the gradient of the target class  to the input layer via backpropagation to generate regions that have a large in-  fluence on the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' But the explanations provided by these methods are  less reliable and sensitive to adversarial perturbations [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The method based  on the input perturbation [4, 15] monitors the change of the prediction proba-  bility of a specific class while occluding different image regions, and generates  a saliency map describing the importance of the network prediction probability  by setting specific perturbation rules and discriminant methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' CAM-based  method [26] proposes class activation maps (CAM), which provide visual ex-  planations based on linearly combining activation maps of convolutional neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' [1] have removed the requirements of the CAM-based methods for  the network structure and improved the accuracy of positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  In this paper, in order to alleviate the problem that previous saliency map  methods generate interpretation maps with rough localization and cannot ex-  press the importance of pixels in a fine-grained manner, we propose DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' It  trains mask vectors of different sizes through an optimization-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  A single element of a small-sized mask vector corresponds to a large receptive  field, and there is less adversarial effect during learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Large size mask vectors  have small receptive fields and can extract fine-grained information from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  In order to combine the advantages of both, we let the small-size mask vector  guide the learning of the large-size mask vector, reducing the mask vector’s ad-  versarial sensitivity while improving its ability to mine fine-grained information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  This enables the DM’s final fused masks to generate fine-grained explanatory  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Additionally, we consider how to more completely capture regions of interest  for neural network classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  An image may have multiple regions that  contribute to the network’s decision, but previous methods often only find one  main decision region and ignore others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' To avoid this problem, we propose HDM  that exploits DM hierarchically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' It masks the salient regions found by the DM  on the image, re-inputs the masked image into the next DM module to force it  to find the regions in the image that the network cares about in the remaining  images, and then fuses the above regions according to their importance through  a learning-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' HDM can not only combine the fine search ability of  DM, but also conduct a comprehensive search for the decision-making area of  the entire image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' This makes the saliency maps it generates not only detailed  but also comprehensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  2  DM  DM  Mask   M2  Input image  X  Mask image  X1  Mask image  X2  Mix mask  Mh  Mask   M1  Network consistency  activation  Timing combined  Figure 1: Overall architecture of HDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' It is formed by stacking several DM  blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The original image is input into the DM block to generate a mask and  mask image, and then the mask image is continuously input into the next DM  block to generate the next mask and mask image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Several masks are obtained  by repeating the above operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' All the masks are mixed to generate the final  mixed mask through the learning-based timing combination method, and the  mixed mask image generated by the dot multiplication of the mixed mask and  the original image is trained with the original image to calculate the activation  consistency to form the final mixed mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  The key contributions of our work are as follows:  We propose a learning-assisted module DM using vectors of various sizes,  which can analyze the importance of image pixels to model classification  in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  We propose a visual interpretation method HDM that leverages DM hier-  archically to generate fine-grained and comprehensive saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  We conduct multiple experiments on natural and medical datasets and  verify that our method achieves state-of-the-art in recognition and local-  ization capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  2  Related work  Existing saliency methods can be divided into three categories, and we briefly  introduce these methods in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  The first is a perturbation-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' [4] use the optimization method  to find the areas in the image that are helpful for the network decision-making,  and cover the areas in the picture that are not related to the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' [15]  detects the importance of various regions in the image in the decision of the  network by using a randomly masked version of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  The second is a gradient-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Gradient [19] directly computes the  derivative of a class score with respect to an input image using a first-order  Taylor expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' It captures local sensitivity by representing the change at  3  each input location through gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' [18] proposed DeepLIFT to decompose  the neural network’s output prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Further, [21] proposed Integrated Gra-  dients, which uses small changes in the feature space between the interpolated  image and the input image to measure the correlation between feature changes  and model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  The third is a CAM-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='Grad-CAM [17] uses the gradient in-  formation flowing into the last convolutional layer of a convolutional neural  networks to assign importance values to each neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In order to increase the  positioning accuracy of CAM, [1] proposed Grad-CAM++, which added an ad-  ditional weight to weigh the elements of the gradient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' FullGrad [20] more  fully explains model behavior by aggregating full gradient components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' [24] pro-  posed Score-CAM to solve the problem of easily finding false confidence samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Ablation-CAM [16] was present to explore the strength of each factor’s contri-  bution to overall model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Eigen-CAM [12] makes CAM generation easier and  more intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' XGrad-CAM[6] introduces sensitivity and conservation axioms  for CAM methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' To generate a more comprehensive saliency map, Layer-  CAM [9] considers the role of each spatial location in the feature map using  element-level weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  3  Methodology  The overall architecture of HDM and generation method are described in Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In addition, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='2 introduces DM to fine study the importance of  images to network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='1  Hierarchical Masks Generation  We propose to use DM hierarchical timing to find the regions of the image that  neural networks focus on when making decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' After the current DM finds  a decision-making area, it masks this area in the image, and then inputs the  masked image to the next DM module to analyze whether DM can find other  areas that are helpful for neural network decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Since the decision-  making area searched by the previous DM module has been shielded, the next  DM module must search in the unshielded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Through hierarchical method,  which forces each DM module to search different regions, it is possible to com-  prehensively learn the region information in the image that the network pays  attention to when making decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Figure 1 is the overall architecture of  HDM, which consists of many DM modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' DM precisely generates the areas  of interest to the neural network in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' HDM generates compre-  hensive and refined saliency maps by mixing masks generated by individual DM  modules through a learning-based timing combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Let the input image be X ∈ RH×W ×C, the DM module is denoted as Q(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  After inputting image Xi−1 to DM, DM can calculate the mask Mi (where  Mi ∈ RH×W ×1) corresponding to the most critical region for decision-making  in image Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' We have the following iteration formula, generating S decision  4  Activation mask  Original  image  Activation image  Mask vector  Overlay mask  Mask map  Heatmap  Original  image  Mask image  Heatmap image  Mask1  Mask2  MaskD  Network consistency  activation  Figure 2: The flow of a DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Above the line, mask vectors of different sizes  are trained by constraining the original image and activation image to have  consistent activations on nodes of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The image below the  line shows that the upsampled learned mask vectors are stacked to produce an  overlay image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The mask image is obtained by multiplying the overlay mask  with the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  The overlay mask uses JET colormap to generate  heatmap, which is added to the original image to get heatmap image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  masks, X0 = X, i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=', S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Mi = Q(Xi−1)  (1)  Xi = (1 − N(  i  �  j=1  Mj))X0  (2)  where N(x) =  x−min(x)  max(x)−min(x) is a normalization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The mix mask Mh is  generated by mixing S masks Mj (j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=', S}) through the following timing  combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Mh =  �S  j=1 wjMj  �S  j=1 wj  , wj =  S  �  k=j  v2  j  (3)  5  Note: J(M, X) = ||fp(MX) − fp(X)||2  2, which represents the consistent  squared loss generated by MX and X input neural network f at position p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  In this paper, we let the position p be the node corresponding to the category  predicted by the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' LR(M) = �H  u=1  �W  v=1  |Mhuv|  |HW | , which is the  regularized value corresponding to mask M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  L(M, X) = J(M, X) + λLR(M)  (4)  where λ is a regularization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' We train the weights of S masks by optimizing  the following loss in Equation (5) to obtain the mix mask Mh according to  Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=', vS ←  min  v1,v2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=',vS L(Mh, X0)  (5)  Since HDM searches the remaining areas that are conducive to decision-  making in the current image hierarchically according to time sequence, Mj is  no less important than Mj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Therefor, we constrain wj ≥ wj+1 by Equation  (3), j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=', S − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='2  Dynamic Masks Learning  DM learns the mask vectors by optimizing the activation consistency between  the original image and the mask image and the regularization term of the mask  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The higher the mask value of the region that is more important to  the classification decision of the neural network, the higher the attention of the  neural network to this region (refer to supplementary material for explanation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  We first set a small-sized benchmark mask vector to roughly learn the attention  of the neural network to each area in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The size of the mask vector  is inversely proportional to its receptive field, and each element of the small-  sized mask vector corresponds to a large receptive field in the image, so that  the vector elements can accurately learn the importance of each region in the  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Benchmark mask vectors {di}D  i=1, di ∈ Rai×bi×1, di are initialized to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' For  any i, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=', D}, if i ̸= j then ai ̸= aj or bi ̸= bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Upsample function  g(·, h, w) can upsample the original vector to a vector of length h and width w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  For example, x ∈ Rh1×w1×1, and g(x, h2, w2) ∈ Rh2×w2×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  As show in Figure 2, we train {di}D  i=1 by the activation consistency between  the mask image and the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Mathematically, we optimize the fol-  lowing loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  L(g(di, H, W), X) = J(g(di, H, W), X)  + ηiLR(g(di, H, W))  (6)  where ηi is a regularization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' To enable the mask to analyze the importance  of region of the image at a fine-grained level, we iteratively generate auxiliary  mask vectors ck  j (i) on the benchmark mask vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' ck  j (i) is defined as follows:  ck  j (i) = g(di, tk  j ai, tk  j bi)  (7)  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
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+page_content='6  Network consistency activation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
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+page_content='2  Upsample  Upsample  Mask vector Cj  k(i)  Mask vector Cj  k+1(i)  Upsampling mask vector  Receptive field  in image  Activation mask  Activation image  Original  image  Mixed mask vector  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
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+page_content='06  Figure 3: An example of training an auxiliary mask vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The original image  size is 224×224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' A feature block of the 2×2 and 4×4 mask vector corresponds  to the 112 × 112 and 56 × 56 receptive field of the original image, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  The ck+1  j  (i) to be trained is multiplied element-wise by the upsampled ck  j (i) that  has been trained, and then the multiplication result is upsampled and multiplied  by the original image to generate the activation image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' ck+1  j  (i) is trained by  optimizing the consistency activation of the activation and original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  where {tj}T  j=1 are positive integer hyperparameters, and k ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='., Ki  j},  Ki  j = min{  �  lnH−lnai  lntj  �  ,  �  lnW −lnbi  lntj  �  }, ⌊·⌋ is the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  As shown in  Figure 3, the benchmark mask vector is used as the initial guidance vector, and  the auxiliary mask vector of the upper level guides the auxiliary mask vector of  the next level to iteratively learn auxiliary vector sets {ck  j (i)|i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=', D}, j ∈  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=', T}, k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=', Ki  j}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The loss function of ck  j (i) is as the following Equation  (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  L(g(ck  j (i)g(ck−1  j  (i), tk  j ai, tk  j bi), H, W), X)  = J(g(ck  j (i)g(ck−1  j  (i), tk  j ai, tk  j bi), H, W), X)  + η(k,j)  i  LR(g(ck  j (i), H, W))  (8)  As shown below the line in Figure 2, Mc is obtained by stacking the auxiliary  masks, and the redundant information of Mc is removed to obtain the overlay  mask Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Mb is the final output result of DM, which can fine-grainedly describe  the importance of image regions in neural network decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Mc =  D  �  i=1  T  �  j=1  Ki  j  �  k=0  ck  j (i)  (9)  Mb = N((Mc − γ){Mc ≥ γ})  (10)  where γ is the threshold, and {·} represents a truth-valued function, which is 1  if true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' otherwise, it equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' N(·) is a normalization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  7  Input Image  HDM(ours)  Grad-CAM  Grad-CAM++  DeepLIFT  Score-CAM  XGrad-CAM  Integrated  RISE  FullGrad  Gradient  Eigen-CAM  Layer-CAM  Ablation-CAM  DM(ours)  Mask  Figure 4: Interpretation results for the ResNet50 network using different tech-  niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The saliency maps generated by each method are normalized to the  range [0,1] and visualized using the JET colormap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The saliency map from blue  to red indicates a gradual increase in activation probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Note that the first  three rows are examples where the prediction is correct, while the last three  rows are wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  4  Experiments  First, the dataset, baseline, and our experimental setup are presented in section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='1 and section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='2 for reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Second, we present the visual interpreta-  tion results we generate in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='3, qualitatively evaluating the performance  of our method compared to other previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Furthermore, section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='4  visualizes the saliency map generated by the input image in each stage of HDM,  showing the flow of HDM for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In addition, sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='5 present the  results of quantitatively evaluating the fidelity of the interpretation of saliency  maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The quality of the saliency maps are measured by localization ability as  described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Finally, ablation experiments are performed in section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='1  Datasets and Baselines  Experiments were conducted on two public image recognition datasets, including  one natural (CUB-200-2011 [23]) and one medical (iChallenge-PM [5]) image  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  We compare 13 state-of-the-art saliency map generation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Gradient-based methods: Gradient [19], DeepLIFT [18], and Integrated [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Perturbation-based methods: RISE [15], and Mask [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' CAM-based methods:  Grad-CAM [17], Grad-CAM++ [1], Score-CAM [24], XGrad-CAM [6], FullGrad  [20], Eigen-CAM [12], Ablation-CAM [16], and Layer-CAM [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The saliency  maps generated by the above-mentioned methods are taken as baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  8  Input image  Heatmap image1  Heatmap image2  Heatmap image3  Mixed heatmap image  Figure 5: The heatmap generated by the whole process of HDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The first  column are the input images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' the second, third, and fourth column images are  the heatmap images generated after passing through the DM module for the  first, second, and third times respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' the fifth column of images is the  heatmap image generated by mixing the second, third, and fourth columns of  heatmaps through a learning-based method, which is also the output of HDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='2  Experimental Details  In this work, ResNet50 [7] was chosen as the neural network for saliency maps  interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' All methods use a ResNet50 model with the same parameters  and fixed for visual evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The input image is resized to 224×224×3, and  then normalized using mean vector (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='485, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='456, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='406) and variance vector  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='229, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='224, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='225).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Our training set contains only images and their class  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' ResNet50 is pre-trained on ImageNet [2] before being trained on the  natural and medical images datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Each image is trained for 800 epochs with  a learning rate set to 1e − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' We set λ = 1e − 4, ηi = η(k,j)  i  = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In the CUB-  200-2011, we set di = i + 5, {di}6  i=1, {tj}3  j=1 = {2, 3, 5};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' γ is set to the pixel  value of the top 25% of the pixel value of the salient image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' HDM consists of 3  DM modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In the iChallenge-PM, we set di = i+5, {di}4  i=1, {tj}2  j=1 = {2, 3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  γ is set to the pixel value of the top 30% of the pixel value of the salient image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  HDM consists of 1 DM modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The learning-based mix mask iteration 400  epochs, and its learning rate is 1e − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='3  Qualitative Evaluations  Experiments are designed to qualitatively compare the performance of saliency  maps generated by different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  As shown in Figure 4, saliency maps  9  point out regions of interest that networks interpret when making decisions,  and we compare 13 state-of-the-art saliency map generation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Interpretation results for six different images are reported in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In  each row, the leftmost image is the original image, and the second and third  rows are the saliency maps generated by our proposed HDM and DM meth-  ods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The subsequent columns represent the results obtained using  different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The color of the saliency maps from red to blue represents  indicates the gradual decrease in the probability of the network’s attention to  the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  As shown in Figure 4, the saliency maps generated by CAM-based methods  usually only point out the areas of the image that a network pays attention to  when making decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The form of this region is almost always with a center as  the highest activation, diffuses outward and gradually weakens irregularly, so the  generated The saliency map has a single structure, lacks integrity, and is not easy  for humans to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In contrast, the saliency map generated by HDM  can discover multiple decision-making regions and integrate them completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' It  finely includes the body regions of the bird used for decision making, outlining  the silhouette of the bird.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The HDM method can completely and accurately  generate the areas of concern when making network decisions, which is more  comprehensive and easy to understand than the CAM-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  The saliency map generated based on the gradient method can roughly de-  scribe the contours of the objects concerned by the network decision-making,  but the pixels of the saliency map are discrete and sharp, and it is difficult for  people to observe which are the key decision-making areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' By comparison, we  can see that the saliency map generated by HDM can better describe the outline  of the object while retaining the smooth change of activation, which makes it  easier for people to observe the importance of each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  The saliency map generated by the perturbation-based method can mark  multiple regions for decision-making, but such saliency maps cover too large  areas and have more noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In contrast, HDM can cover multiple decision regions  while paying little attention to regions outside the decision region (bird body),  generating saliency maps with little noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Overall, our method can generate refined and complete saliency maps, which  are less noisy and easy to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='4  Processes Visualization  As shown in Figure 5, we show all saliency maps generated by HDM during  inference on images from the CUB-200-2011 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' When researching on the  CUB-200-2011 dataset, we set HDM to consist of 3 DM modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The first  column in Figure 5 is the original input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The second, third, and fourth  columns in Figure 5 represent the saliency images of the current most-attended  regions of the neural network found after inputting the original image or the  mask image into the DM module in the three stages, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' It can be  seen from Figure 5 that each DM module can accurately and fine-grainedly find  the remaining regions in the current picture that are helpful for neural network  10  Evaluation  Average Drop  Average Increase  Percentile  80%  70%  80%  70%  Mask  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
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+page_content='8%  Table 1: Evaluated results on average drop (lower is better) and average increase  (higher is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  decision-making, and the regions of interest found in each stage are almost  disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' This is in line with HDM’s hierarchical search strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The mixed  heatmap images in the last column are the result of mixing the heatmap images  generated by the first three DM modules via a learning-based approach, which  makes the final generated saliency maps comprehensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' It can be seen from  Figure 5 that the final hybrid map generated by HDM is not only comprehensive,  but also fine-grained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='5  Faithfulness Evaluation via Image Recognition  The experiment evaluates the fidelity of the interpretation of the saliency map  generated by the HDM in the object recognition task, and the average drop  and average increase [1] are used as evaluation indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' This experiment sets  the activation value of a specific percentage of all pixels in the saliency map  as the threshold value, mutes the pixels below the threshold value, and retains  the pixel value above the threshold value to generate an interpretation map (in  the experiment, set 70% and 80% of the pixels are muted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The average drop  is expressed as �N  i=1  max(0,Yi−Oi)  Yi  × 100, and the average increase is expressed  as �N  i=1  Sign(Yi<Oi)  N  , where Y c  i  is the predicted score of image i in category  c, and Oc  i is the predicted score of category c with the explanatory map as  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Sign(·) is an indicator function that returns 1 if the input is True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In  the experiment, 10 images are randomly selected for each category on CUB-  200-2011, and a total of 2000 images are formed for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The results are  reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' When the threshold is set to occlude 70% and 80% of  11  Evaluation  Deletion Score  Insertion Score  Mask  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='0516  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
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+page_content='0537  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
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+page_content='0537  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='7958  DM(ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='0461  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='8171  HDM(ours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='0447  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='8189  Table 2: Evaluated results on delecion (lower is better) and insertion (higher is  better) scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Percentile  Percentile  Predict Value  Predict Value  Deletion Curve  Insertion Curve  Figure 6: Deletion and Insertion curves of the above methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  the pixels, HDM achieves 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='3% and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='1% average drop and 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='8% and 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='7%  average increase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' HDM outperforms other saliency methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' This  task shows that HDM can find the most distinguishable regions of the target  object as much as possible, and can eliminate as much as possible the regions  irrelevant to the target object distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  To further evaluate the discriminative ability of HDM, we test on the in-  sertion and deletion metrics proposed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Deletion and insertion scores are  the areas under the probability curves described by the predicted probability  results for deleted or inserted images generated by deleting and inserting pixels  from the original image in descending order of saliency map activation values,  12  bGLcGUf!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
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+page_content='81%  HDM(ours)  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
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+page_content='81%  Table 3: On CUB-200-2011 and iChallenge-PM datasets, the evaluation results  of each method in proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' This experiment adopts the setting in [24], the step size is 1%,  and the pixel value is set to 0 or 1 to remove or introduce pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' We compare  the performance of the methods in Table 2 on the above-mentioned evaluation  metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Here we do not compare other perturbation-based methods, since they  can produce adversarial effects [15] that interfere with these evaluation metric  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' As shown in Figure 6, the deletion and insertion curves produced  by each model are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The HDM method achieves the largest AUC and  the steepest curve, which shows that the HDM saliency map can best find the  area in the current image that has the greatest influence on the network decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  Table 2 reports the average results for the above-mentioned 2000 images, and  our method achieves state-of-the-art on both metrics compared to the respective  previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='6  Localization Evaluation  In this section, the localization ability is employed to measure the quality of  the generated saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In the same way as in [24], calculate how much  energy of the saliency map falls into the segmented foreground region of the  target object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Binarize the segmented foreground image given in the test set,  assign the foreground area as 1, and assign the background area as 0, multiply  the saliency map calculated by each method with the binarized image point  by point and sum how much energy is obtained in the target foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The  evaluation metric Proportion is expressed as  � Lc  (i,j)∈bbox  � Lc  (i,j)∈bbox+� Lc  (i,j)/  ∈bbox .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' We test  the localization ability on the CUB-200-2011 and iChallenge-PM datasets, set-  13  Figure 7: Results of saliency maps generated by HDM on bird and fundus  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  The first and fourth rows are the original images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' the second and  fourth rows are the mixture of the saliency map and the original image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' the  third and sixth rows are the ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  ting the foreground region of the image segmentation label as the bbox region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  In the test set of CUB-200-211, 10 images are randomly selected from each  category, and a total of 2000 images are formed for experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' in the test  set of iChallenge-PM, 200 images are randomly selected as experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The  saliency maps generated by HDM have 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='97% and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='81% energy falling in the  ground-truth foreground region of the target in CUB-200-2011 and iChallenge-  PM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' This verifies that HDM can effectively reduce noisy regions  in saliency maps that are not relevant to decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Our method achieves state-  of-the-art performance in both natural and medical image localization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  As shown in Figure 7, we visualize the results of the saliency maps generated  by the HDM on bird and fundus datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In the bird image, the  decision-making region found in the saliency map of HDM wraps most of the  14  bird’s body, and carefully evaluates the importance of each position on the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  This verifies that HDM is capable of comprehensively locating multiple decision  regions in natural images and fine-grained division of the importance of each  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In the fundus images, HDM can also accurately and completely locate  pathological areas in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The above experiments show that HDM can  achieve good localization performance in both natural and medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='7  Ablation Study  In this section, we compare the performance of saliency maps generated using  the DM module alone and HDM consisting of multiple DM hierarchical timings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  It can be seen from Table 1 and Table 2 that the saliency map generated by  DM is superior to all previous methods in the evaluation index of recognition  ability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Table 3 shows that the localization ability of DM on natural images  is weaker than some saliency methods, and the localization ability on medical  images reaches the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' From the above-mentioned tables, it can be seen that  the hierarchical stacking DM method of HDM can improve the performance of  the saliency map on the above indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Because DM only focuses on finding  the area that a neural network pays attention to when making decisions, but  ignores the fact that there may be multiple areas that have an impact on the  classification of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' HDM makes the saliency map more com-  prehensive by using DM multiple times to search for decision regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Therefore,  both the DM module and the method of hierarchically stacking the DM modules  are useful and necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  5  Conclusion  In this paper, we propose HDM to generate refined and comprehensive visual  explanatory maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' The DM module is able to generate finely localized saliency  maps using masks of various sizes that cooperate with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' HDM an-  alyzes the masked image hierarchically using the DM module to find areas of  concern for multiple neural network decision-making classification decisions in  the image, and combines them well into a comprehensive visual interpretation  map through a learning-based fusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Our method achieves state-of-the-  art performance in recognition and localization capabilities on natural images  and medical images, and quantitative evaluation verifies that our method can  generate more fine-grained and comprehensive saliency maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
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+page_content=' The caltech-ucsd birds-200-2011 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  [24] Haofan Wang, Zifan Wang, Mengnan Du, Fan Yang, Zijian Zhang, Sirui  Ding, Piotr Mardziel, and Xia Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Score-cam: Score-weighted visual expla-  nations for convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In Proceedings of the IEEE/CVF  conference on computer vision and pattern recognition workshops, pages 24–  25, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  17  [25] Hao Yuan, Lei Cai, Xia Hu, Jie Wang, and Shuiwang Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Interpreting image  classifiers by generating discrete masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  IEEE Transactions on Pattern  Analysis and Machine Intelligence, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  [26] Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Tor-  ralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' Learning deep features for discriminative localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content=' In Proceedings  of the IEEE conference on computer vision and pattern recognition, pages  2921–2929, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
+page_content='  18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfPAy5/content/2301.04970v1.pdf'}
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+Mathematical modelling and numerical simulation of
+reverse-osmosis desalination
+Nicodemo Di Pasquale ∗1, Mayo Akele2, Federico Municchi3, John King2, and Matteo
+Icardi2
+1Department of Chemical Engineering, Brunel University London, Uxbridge, UB8 3PH,
+United Kingdom
+2Department of Mathematics, University of Nottingham, Nottingham, NG7 2RD, United
+Kingdom
+3Department of Mechanical Engineering, Colorado School of Mines, Golden, CO 80401, US
+January 31, 2023
+Special issue on Process Intensification
+Abstract
+The reverse osmosis membrane module is an integral element of a desalination system as it determines the
+overall performance of the desalination plant. The fraction of clean water that can be recovered via this
+process is often limited by salt precipitation which plays a critical role in its sustainability. In this work, we
+present a model to study the complex interplay between flow, transport and precipitation processes in reverse
+osmosis membranes, which together influence recovery and in turn process sustainability. A reactive porous
+interface model describes the membrane with a dynamic evolving porosity and permeability to capture the
+scaling and clogging of the membrane. An open-source finite-volume numerical solver is implemented within
+the OpenFOAM®library and numerical tests are presented here showing the effect of the various parameters
+of the model and the robustness of the model to describe a wide range of operating conditions.
+1
+Introduction
+The demand of freshwater has steadily increased over the last forty decades at a global level, mainly because
+of the increase in population and improving living standards which are leading to an expansion of irrigated
+agriculture and its human consumption. In turn, the increase in consumption is straining the freshwater
+sources in their ability to supply the growing demand of water worldwide with almost two third of the
+total world population experiencing severe water scarcity during at least a part of the year (Mekonnen and
+Hoekstra, 2016; Jones et al., 2019). The current freshwater sources are already overexploited, and even if
+better management is still needed to reduce the misuse of such resources (e.g., wastewater treatments or
+waste reduction) (Najid et al., 2022), these solutions cannot still be enough to meet the future demand of
+freshwater. The ongoing climate change is expected to reduce the availability of freshwater because of the
+receding of glaciers with a subsequent important reduction of the flow in important rivers such as the Mekong
+Yellow, or Gange (Shannon et al., 2008).
+∗Corresponding author: nicodemo.dipasquale@brunel.ac.uk
+1
+arXiv:2301.13160v1  [math.NA]  16 Jan 2023
+
+Almost 98% of the total liquid water on the Earth is not available for the direct use or consumption, as
+it form the total water present in seas and oceans. However, this last fact also means that if the exceedingly
+high saline content in seawater can be reduced or removed, we will have access to the largest source of
+freshwater sources, with which the required amount of freshwater could be delivered without straining the
+natural occurring resources (Jones et al., 2019). Therefore, there is a strong push into developing more effi-
+cient technologies for the desalination of seawater. Among the currently available technologies are thermal
+desalination and membrane processes (Fritzmann et al., 2007; Subramani and Jacangelo, 2015). In thermal
+desalination, seawater is brought to evaporation through multi-effect distillation or multi-stage flash distilla-
+tion (Al-hotmani et al., 2021), and the resulting vapor is subsequently condensed. In membrane technologies,
+a semi-permeable membrane is employed to separate (or filtrate) the solution of salt and water.
+Reverse Osmosis (RO), is a widely employed membrane technology for treating seawater and wastewater
+with salinity up to 70 g/l (Hickenbottom and Cath, 2014) which, due to its relative simplicity and widespread
+diffusion has been among the main topics of research in membrane filtration (Wardeh and Morvan, 2008;
+Luo et al., 2019). One of the main challenges in RO is concentration polarisation (Kim and Hoek, 2005),
+which is the presence, over the membrane, of a solute-rich boundary layer. Concentration polarisation can
+lead to solute precipitation and fouling, significantly reducing the local permeability of the membrane with
+adverse effects on the permeation flux. When the solution contains inorganic salts (such as sodium chloride
+NaCl or calcium sulfate CaSO4 ), the resulting accumulation of crystals on the membrane is also called
+scaling. There is extensive experimental evidence indicating that scaling reduces the membrane performance
+over time (Hu et al., 2014) and therefore, scaling phenomena should play a major role in the mathematical
+modelling of RO systems.
+Computational Fluid Dynamics (CFD) represents a powerful to analyse concentration polarisation and
+scaling, with the earliest attempts dating back to more than two decades ago (Hansen et al., 1998). The
+membrane is usually included as a boundary condition where the flux of the solute is assumed equal to zero
+(complete retention). Previous studies have considered dependence on the solute concentration of properties
+such as the osmotic pressure (Hansen et al., 1998), viscosity, density and diffusion coefficient (Geraldes et al.,
+2001; Wiley and Fletcher, 2002) to include effects of concentration polarisation on the membrane(Wiley and
+Fletcher, 2002; Johnston et al., 2022). In particular, CFD simulations for membranes have been employed to
+analise different geometrical configurations, such as the spacer-filled channels. These include solid elements
+in the feeding flow to increase the shear stress on the surface of the membrane, which in turn increases
+the local mixing and mass transfer across the membrane (Shakaib et al., 2007; Fletcher and Wiley, 2004;
+Fimbres-Weihs and Wiley, 2010; Ghidossi et al., 2006; Lau et al., 2009; Santos et al., 2007; Koutsou et al.,
+2009; Ranade and Kumar, 2006).
+However, the effects of scaling have not been directly included in CFD simulations. In this work, we
+propose a mathematical model and a CFD solver for the analysis of the performances of a RO membrane
+in which we included the possibility for the solute in the feed flow to react (precipitate) on the membrane
+and therefore affecting the membrane permeability. The paper is organised as follows. We describe the
+mathematical framework for the analysis of the membrane, highlighting how the chemical reactions can be
+accounted for in the model. We then discuss the implementation of our model into the widely used open
+source finite volume library OpenFOAM®and we show some results for some typical situations, showing the
+applicability of the whole framework. We then draw some conclusions and we outline the possible extension
+of the model.
+2
+Model
+In this work, we approximate a rectangular 3D membrane module as a 2D channel illustrated in fig. 1. This
+is a common practice employed in other recent CFD studies (Johnston et al., 2022). While the configuration
+we considered can be easily extended to more complex geometries, the focus here is to develop a complete
+mathematical model for the polarization and scaling of the membrane by giving a proof of concept for a
+general-purpose numerical solver. Our main goal is to show the mechanisms governing the evolution of the
+solute at the membrane interface, and a 2D channel geometry allows us to focus on this task.
+2
+
+Figure 1: 2-dimensional domain considered in this work
+Let us therefore assume a rectangular domain Ω ≡ (0, L) × (0, H) with boundary Γ = ∂Ω subdivided in
+three different regions:
+Γm = (0, L) × {0}
+(membrane)
+Γin = {0} × (0, H)
+(inlet)
+Γout = {L} × (0, H)
+(outlet)
+(1)
+where Γm represents the membrane boundary, Γin is the inlet boundary and Γout is the outlet boundary.
+With Γw we represent the remaining part of the boundary constituted by solid boundaries so that Γw =
+Γ\(Γin ∪Γout ∪Γm), as shown in fig. 1. In this study, the permeate flow is not explicitly modelled, therefore
+the membrane represents a boundary condition for the problem.
+The analysis of the filtration process requires the simultaneous solution of the flow field coupled with the
+transport of dissolved chemical species, which can involve one or more chemical reactions. Such chemical
+reactions can lead to solute precipitation, modifying the permeability and porosity of the membrane, thus
+causing a variation in the osmotic pressure and the final yield of the permeate. We summarised the different
+mechanism and their interdependence in fig. 2.
+Our mathematical model is composed of three different and intertwined parts that need to be simulta-
+neously considered to obtain the overall behaviour of the membrane which we are now going to describe in
+detail.
+2.1
+Flow modelling
+The mixture is described as an incompressible Newtonian fluid described by the Navier-Stokes equations:
+∇ · u = 0
+in Ω, t > 0
+(2a)
+ρ
+�∂u
+∂t + (u · ∇u)
+�
+= −∇p + µ∇2u
+in Ω, t > 0
+(2b)
+3
+
+Tw
+in
+ino
+Bulk Feed
+0
+PermeateFigure 2: Sketch of the different mechanisms and models considered in this work.
+where ρ is the density of the fluid u is the velocity vector with components (u, v)T , p is the fluid pressure
+and µ is the dynamic viscosity, ∇ is the gradient operator. We impose the following boundary and initial
+conditions on the system shown in eq. (2) for the velocity and pressure:
+u = 0
+in Ω, t = 0
+(3a)
+u = 0,
+p = Pin
+on Γin
+(3b)
+∇u · n = 0,
+p = Pout
+on Γout
+(3c)
+u = 0
+on Γw
+(3d)
+Finally, the membrane is modelled as a dynamic Dirichlet condition for the velocity v orthogonal to the
+membrane is obtained from the Darcy law as:
+u = 0, v = −k(∆p − ∆π)
+ℓµ
+on Γm
+(4)
+where µ is the fluid viscosity, ℓ is the membrane thickness, k is the membrane permeability, ∆p = pm − pp
+is the pressure difference between the feed side of the membrane, pm, calculated at the membrane boundary
+Γm and the permeate side of the membrane, pp, whereas ∆π is the osmotic pressure gradient. This last
+equation shows that the flux through the membrane is proportional to the difference between the applied
+and osmotic pressure differentials.
+The pressure gradient ∆p = pm − pp is the difference between the pressure on the feed side of the
+membrane pm calculated at the membrane boundary Γm, and the permeate side of the membrane pp. The
+osmotic pressure difference is defined as follows:
+∆π = π − πp = π(φ1, . . . , φN) − π(φp
+1, . . . , φp
+N)
+(5)
+where the feed osmotic pressure π is a function of the N ions concentrations, φi, i = 1, . . . , n, whereas the
+permeate osmotic pressure φp is function of the N ions concentrations in the permeate, φp
+i , i = 1, . . . , n.
+As common in membrane applications (Linares et al., 2014) the osmotic pressure is computed using the
+Van’t Hoff model (Van’t Hoff, 1888):
+π(φ1, . . . , φn) = RTϕ
+N
+�
+i
+φi
+(6)
+where ϕ is the osmotic coefficient, R is the gas constant and T is the temperature we can rewrite eq. (5) as:
+∆π = RTϕ
+N
+�
+i
+(φ − φp
+i ) = RTϕ
+N
+�
+i
+rφ
+i φi
+(7)
+4
+
+Convective and diffusive transport
+Change in osmotic
+Solute buildup
+pressure
+Solute
+Fluid Flow
+Precipitate
+Transport
+Permeability and
+Rate of consumption
+porosity change
+Osmotic Pressurewhere we used the definition of membrane rejection of the species i,
+rφ
+i = 1 − φp
+i
+φi
+, i = 1, . . . , N .
+(8)
+The Van’t Hoff equation assumes a linear relation between concentration and osmotic pressure and is more
+accurate for low concentration of solutes. Different formulations were proposed to take into account the non-
+linear behaviour of solutions, which consider the activity of the solvent (Khraisheh et al., 2019), calculated
+using the Pitzer equation for the electrolyte solutions (Pitzer, 1973). As a proof of concept, we will consider
+here the simplified model since the most complex behaviour can be straightforwardly added to the model
+and the CFD code we will present in the next section. The membrane rejection of the species i expresses
+the amount of solute rejected by the membrane (and therefore not present in the permeate) as a fraction of
+the initial quantity. In this work, we assume rφ
+i = 1 for every ion species, which corresponds to complete
+rejections of the ions at the membrane. In this case, the concentration on the permeate side and the permeate
+osmotic pressure, πp, are both equal to 0 and therefore:
+∆π = RTϕ
+N
+�
+i
+φi .
+(9)
+In the literature, the two most popular models proposed to describe the solute-solvent solute transport
+through the membrane are the solution-diffusion model (Merten, 1963; Lonsdale et al., 1965; Wijmans and
+Baker, 1995) and the Spiegler-Kedem model Spiegler and Kedem. The former expresses the flow through
+the membrane ˙Jv as:
+˙Jv = A(∆p − ∆π)
+(10)
+noting that in our notation v =
+˙Jv
+A(Γm) where Sm is the membrane surface; while for the latter we have
+˙Jv =
+1
+Rmµ(∆p − σ∆π)
+(11)
+with the same identification for ˙Jv (i.e., v =
+˙Jv
+A(Γm)) where Rm is the membrane resistance and σ is the
+reflection coefficients which measure the impermeability of the membrane to the solutes. σ = 1 indicates a
+membrane completely impermeable to solutes and will be the one considered for this work.
+Notice that in eq. (4) we use the Darcy law to rewrite the so-called water permeability of the membrane
+A which appear in the equation in terms of the permeability and thickness of the membrane and the viscosity
+of water as A = Smk
+ℓµ . Using the same reasoning for eq. (11) we obtain that Rm =
+ℓ
+Smk, that is to say, the
+membrane resistance is inversely proportional to the permeability. The identification of the Darcy-related
+terms with the water permeability A or membrane resistance has two main advantages. Firstly, we can
+connect our analysis with the membranes available commercially which are described in terms of water
+permeability or membrane resistance. This last fact allows us to choose the range of the parameters we are
+considering which are appropriate for the description of real membranes. Secondly, using the Darcy derived
+version of the equation for the flow through the membrane allows us to include more detailed mechanisms in
+the model which can take into account more complex phenomena, such as chemical reactions and depositions
+of solids on the membrane as we will show in more details in the next sections. One of the ways considered
+in the literature to include fouling and polarisation of the membrane is to define such contribution as
+additional resistance terms to be added to Rm in eq. (11) (see e.g., (Silva et al., 2011; Lee and Clark, 1998;
+Yeh, 2002)). These additional terms must be derived from experiments or empirical correlations. In contrast,
+our formulation leverages the well-established theory of porous media to include such effects directly in the
+determination of the permeability k.
+The performance of the membrane can be evaluated by using the recovery r, defined as the ratio between
+the permeate flow rate, ˙Qp, and the feed flow rate, ˙Qf:
+r =
+˙Qp
+˙Qf
+=
+vAp
+UinAin
+= vL
+uH
+(12)
+5
+
+where we used the definition of flow rate through a surface, AU/ℓt, to calculate the flux through the in-
+let (subscript in) and the membrane (subscript m) and then we replaced the area in eq. (12) with their
+geometrical expression related to the domain we are considering: Ain = H × Z, Ap = L × Z.
+2.2
+Solute Transport
+In membrane processes, flow and solute transport are tightly coupled through the boundary condition on
+the membrane given by equation eq. (4). The transport equation for the concentration φi of the i−th ion
+species in the bulk liquid is given by:
+∂φi
+∂t + ∇ · (uiφi) = ∇ · (D∇φi) + ξi
+B
+in Ω, t > 0
+(13a)
+φi = 0
+in Ω, t = 0
+(13b)
+φi = φin
+on Γin
+(13c)
+∇φi = 0
+on Γout ∪ Γw
+(13d)
+˙Ji = vφi − Di
+∂φi
+∂y = ξi
+M + ξi
+P
+on Γm
+(13e)
+where ui is the velocity of the i-th chemical species, Di is its diffusion coefficient, ξ is a rate term that
+represents different mechanisms of depletion of the solute, identified by the subscripts B for the chemical
+reaction in the bulk, M, for chemical reaction at the membrane, and P for the fraction of the solute that
+crosses the membrane. The amount of the species i at the membrane, can either precipitate on the membrane
+following a chemical reaction or go through the membrane in the permeate. The sum of these two mechanisms
+must equal the flux of the i-th chemical species at the membrane ˙Ji. We can therefore assume that each of
+these mechanisms corresponds to a fraction of ˙J, and hence that ˙J = κMξM +κP ξP . The coefficients κ have
+the property that:
+κM + κP = 1
+(14)
+If only superficial reaction is present, then κR = 1 and κP = 0, while in the case there is no superficial
+reaction and the solute crosses the membrane then κP = 1 and κR = 0.
+In this work we will make the following assumptions on the behaviour of the system: 1.) The precipitation
+reaction is irreversible and only occurs on the membrane surface, (i.e., ξB = 0) 2.) The transport processes
+are similar for the all the salt ions and the diffusion coefficients are independent of concentration, 3.) The
+salt ions have the same velocity as the fluid (i.e., ui = u for every i), 4.) The porous medium is homogeneous,
+5.) there are only surface reactions at the membrane (i.e., κR = 1).
+2.3
+Chemical kinetics
+Following the principles of mass action, the dynamic behaviour of chemical systems with n components
+involved in m reactions, can be described by a set of first order differential equations with time as the
+independent variable:
+dφ1
+dt = f1(φ1, φ2, . . . , φn, t)
+dφ2
+dt = f2(φ1, φ2, . . . , φn, t)
+...
+dφn
+dt = fn(φ1, φ2, . . . , φn, t)
+(15)
+6
+
+where φi(t), i = 1, . . . , n denotes the volume molar concentration of chemical species Xi at time t. The
+dynamics of the reaction network can be conveniently written in matrix form using the formalism developed
+in (Chellaboina et al., 2009):
+dφi
+dt = ξi
+R = (A − B)T KφA(t), φi(0) = φ0,i, t ≥ 0
+(16)
+where K = diag(K1, . . . , Km) is the diagonal matrix which contains as elements the reaction kinetics Kj,
+j = 1, . . . , m and φ0 is the initial concentration. A and B are the m × n matrices having in each entry the
+stochiometric coefficients of the reactants and products respectively and φA(t) is the matrix obtained by
+replacing each element of A with φalp
+i
+where akj is the element of A in the l-th row and p-th line. The first
+term in the boundary conditions in eq. (13e) takes the form
+ξi
+R =
+�
+(A − B)T KφA(t)
+���
+i ℓ
+(17)
+where the notation [·]|i stands for the i-th component of its (n × 1 vector) argument.
+For a generic first order reaction X1 + X2 → X3 with kinetic constant K, we have therefore
+dφ1
+dt = −Kφ1φ2
+dφ2
+dt = −Kφ1φ2
+dφ3
+dt = Kφ1φ2
+(18)
+and
+ξ1
+R = −Kφ1φ2ℓ
+ξ2
+R = −Kφ1φ2ℓ
+ξ3
+R = Kφ1φ2ℓ
+(19)
+An example of a reaction of this type important in membrane operation is the formation of calcium carbonate,
+through the reaction (Warsinger et al., 2015):
+Ca2
++ + 2 HCO3
+− −−→ CaCO3 + CO2 + H2O
+(20)
+In fact, the scaling caused by the precipitation of calcium carbonate limits the operating condition of desali-
+nation systems for brackish, groundwater, and seawater (Waly et al., 2009).
+One effect to be considered when a chemical reaction is involved is that the reaction between salt ions and
+subsequent precipitation of minerals, often alters the membrane properties, such as porosity and permeability.
+As crystals grow, it is expected that the permeability k, in equation (see eq. (4)) and the porosity, ϵ, will
+decrease, reducing the liquid flow through the membrane (Steefel et al., 2005). To account for the modification
+of the porosity and permeability as a result of mineral precipitation, we employ the Kozeny-Carman model
+(Hommel et al., 2018). This model allows us to quantify the porosity-permeability relations and estimate
+the resulting changes.
+According to the Kozeny-Carman model, the change in permeability is calculated by relating the current
+permeability, k, based on the current porosity ϵ, to the initial permeability k0 corresponding to the initial
+porosity ϵ0 Hommel et al. (2018). These equations consequently follow the form below
+k
+k0
+= f(ϵ)
+f(ϵ0).
+(21)
+7
+
+Thus we can describe the permeability evolution using the following power law (Hommel et al., 2018)
+k
+k0
+= (1 − ϵ0)2
+(1 − ϵ)2
+� ϵ
+ϵ0
+�3
+,
+(22)
+where k is the current permeability, ϵ is the current porosity, k0 is the initial permeability and ϵ0 is the initial
+porosity. The rate at which porosity reduction occurs is given by (Huo et al., 2019; Noiriel et al., 2004):
+ϵ = ϵo − Vs
+ℓ
+t
+�
+t0
+�
+j
+ξj
+Rdt,
+(23)
+where ϵo denotes the initial porosity at t0, Vs is the molar volume of solid precipitate in m3/mol and r(t) is
+the rate of precipitation in mol·m−2 · s−1 and the index j runs over all the possible reactions in the system.
+For the first order reaction we are considering here the eq. (23) simplify as:
+ϵ = ϵo − Vs
+ℓ
+t
+�
+t0
+(Kφ1φ2) dt .
+(24)
+We should again observe the inter-dependencies and feedback mechanisms between fluid flow, transport
+and reaction. Namely, in eq. (24) we see the precipitation reaction leads to a change in porosity which in
+turn affects the permeability in equation eq. (22). The change in permeability impacts the flow via the fluid
+velocity in eq. (4). This, in turn, alters the solute concentration distribution via eq. (13a), which ultimately
+impacts the rate of precipitation again via eq. (13e).
+Moreover, from eq. (24) we can observe that the
+variation of the porosity is proportional to the kinetic reaction constant. This last fact, in turn, simplifies
+the predictions for the clogging of the membrane based on the reactions in the systems. We can expect that
+if there are two chemical reactions in our systems, the first one 10 times slower than the second, then the
+clogging caused by the products of the second reaction will take 10 times the time needed by the products of
+the first reaction to produce the same amount of clogging. One of the strengths of our models is that allow
+these kinds of qualitative analyses even without actually solving the equations.
+3
+Numerical discretisation
+The equations presented in section 2 are solved using the open-source finite volume OpenFOAM 8.0 library
+and the code is available open-source (Icardi, 2022). In order to solve the equation of motion alongside the
+reaction at the membrane we developed a new solver for OpenFOAM called binaryReactionFoam which is
+based on two widely used solvers, pimpleFoam and scalarTransportFoam· The former is a transient solver for
+incompressible flows based on the PIMPLE algorithm while the latter is a concentration transport solver using
+a user-specified velocity field. The solver also includes the possibility of modelling solid precipitation in the
+fluid and a multiphase flow model for the solid particles. The most important element in the computational
+framework, however is represented by the new boundary conditions implemented to model the membrane.
+The membraneVelocity boundary conditions impose the fluid velocity based on the fluid pressure, the permeate
+pressure, and the membrane properties. These are updated in time by linking this boundary condition to
+the one for the scalar concentration, named binaryReaction, which solves for the solid precipitation at the
+boundary and therefore updates the membrane permeability. The equations and boundary conditions are
+coupled iteratively through Picard (fixed point) iteration (through the PIMPLE iterations) until convergence,
+making the whole model fully implicit.
+We simulate a two-dimensional rectangular channel with height h = 0.003 m and length of L = 0.02 m
+discretised on a mesh composed by 600×200 cells. The following discretisation schemes (we direct the reader
+to the OpenFOAM user guide (Ope, 2019) for a detailed description of each scheme) are used to discretise
+the equations:
+8
+
+• advective fluxes (divSchemes Gauss vanLeer) are computed at the faces and the variables interpolated
+with a Total Variation Diminishing scheme;
+• gradient terms (gradSchemes Gauss linear) are approximated with central differencing;
+• surface normal gradients for diffusive fluxes (snGradSchemes orthogonal) are approximated with central
+differencing (the grid is in fact orthogonal and does not need any correction to ensure second order
+accuracy);
+• time derivatives (ddTSchemes backward) are approximated with third order implicit backward Euler
+scheme,
+We specified a fully developed velocity profile at the inlet:
+u(0) = 6uav
+y
+h
+�
+1 − y
+h
+�
+(25)
+where uav is the average velocity along the channel. By specifying the velocity profile at the inlet we need
+only to specify the pressure at the outlet (the value of which is given in table 1). The pressure of the permeate
+through the membrane is assumed constant along the length of the membrane and put equal to zero. For
+longer membranes, this assumption is no longer valid and the permeate flow needs to be modeled explicitly
+(with 1D or 2D models). This will be the subject of future extensions of our framework.
+The initial value of the permeability we consider in the calculations is k = 10−16 m2. Using the expression
+for the water permeability obtained from eq. (4) and eq. (10): A = Smk
+ℓµ , and the viscosity of water at room
+temperature µ = 10−3 Pa s, and the surface of the membrane, Sm = 2·10−6 m2 for the channel configuration
+and ℓ = 10−7m , we obtain a value of A = 2 · 10−12 m (Pa s)−1. This value is in line with the values
+reported for commercial membranes, which are in the range 10−14 to 10−10 (Pa s)−1 (Ruiz-Garc´ıa and de la
+Nuez Pestana, 2019; Lee et al., 1981; Draˇzevi´c et al., 2014).
+Our model is able to include the scaling of the membrane given by the chemical reaction, which can
+modify the membrane permeability through the precipitation of a solid phase obtained as a product of the
+reaction. In this work we considered a range of kinetic reactions, going from very slow to fast reactions,
+i.e. with a value of the kinetic constant spanning four orders of magnitude, from 10−10 to 10−1 m3/mol.
+We considered such a large range of kinetics since our main goal is not to focus on a specific system (and
+reaction) but to give a general description that can be applied to different specific situations.
+4
+Results
+In this work, we employ a fixed flow profile at the inlet, which when considering the properties in table 1,
+gives a Reynolds number equal to 300. Therefore, the system operates in a fully developed laminar flow
+regime. The first property that can be derived from this model is the polarisation of the membrane, which
+represents the accumulation of the solute at the interface of the membrane on the feed side. This is an
+undesired effect since it increases the osmotic pressure reducing the extraction of the permeate per unit of
+energy consumed in the process. We report the variation of the concentration profile in the domain in fig. 3
+for the lowest and highest chemical kinetic rate considered. We can observe in fig. 3 that the concentration
+at the membrane is different from the one in the bulk region in both cases. However, while the case with
+K = 10−15 m3/mol shows a higher concentration with respect to the bulk (see fig. 3a), the case with the
+highest value of the kinetic rate (K = 10−1 m3/mol, see fig. 3b) shows a concentration smaller than the one
+in the bulk.
+The latter observations show that the possible behaviour of the solution near the membrane strongly
+depends on the reaction kinetics. In the first case (the one represented in fig. 3a corresponding to the lowest
+kinetic rate considered K = 10−15 m3/mol), we can observe the “standard” effect of the polarisation of
+the membrane. During the filtration process, there is an accumulation of the solutes molecule on the feed
+side of the membrane, which results in a higher concentration of the solution at the membrane itself. On
+9
+
+Symbol
+definition/value
+units
+k
+10−16
+m2
+ϵ
+0.7
+-
+φa,0
+35
+g/m3
+φb,0
+φa,0
+g/m3
+uin,av
+0.1
+m/s
+D
+0.003
+m
+ℓ
+0.0001
+m
+Vs
+27 · 10−6
+m3/mol
+uav
+0.1
+m/s
+K
+{10−10, 10−5, 10−2, 10−1}
+m3/mol
+ρ
+1000
+kg/m3
+pout
+1000
+kPa
+pperm
+0
+kPa
+µ
+10−3
+Pa s
+Table 1: Summary of the numerical inputs for the physical quantities used in the simulations. Note that
+the units of the kinetic constant K depend on the fact that we considered a binary reaction, whereas for the
+permeability k and porosity ϵ we are considering the initial value (i.e., the value at t = 0 h.
+10
+
+(a) K = 10−10 m3/mol
+(b) K = 10−1 m3/mol
+Figure 3: Contour plot of the concentration within the channel given in units of the initial concentration
+˜φa = φaφin,0. On the left: results reported for the lowest kinetic constant. On the right, results are reported
+for the highest kinetic constant. Note that the starting point of the legend is not zero and is different between
+the pictures to make the results more clear.
+the opposite side, when the reaction rate is almost negligible (as for the case of K = 10−15 m3/mol), the
+solutes are now consumed by the reaction and they accumulate at the membrane interface, leading to the
+concentration profile observed in fig. 3a. In the latter case, the solute is now consumed almost instantly at the
+membrane interface. This result in a transport (convection and diffusion) limited profile of the concentration
+near the interface.
+The two opposite effects just described for the profile of the concentration at the membrane interface give
+an interesting effect on the evolution of the porosity and permeability profiles across the membrane. As the
+reaction proceeds, a new solid phase is formed which precipitates on the membrane modifying its structure
+and therefore its fluid dynamical behaviour. In particular, the solid phase generated during the reaction
+clogs the pores of the membrane, resulting in a variation of the porosity of the membrane with time. On top
+of this, since the concentration along the membrane (in the x-direction) decreases, we can expect a decrease
+in the overall reaction rate (which is proportional to the concentration) and therefore a difference in the
+permeability and porosity over the membrane. When we instead observe polarisation (i.e., in the case of the
+lowest reaction rate) the concentration near the membrane increases with the distance along the membrane
+direction.
+Therefore, we can expect that the reaction velocity (which depend on K and the concentration), at a
+fixed time, will increase along the membrane interface for the case with polarisation (low kinetic reaction)
+and decrease along the membrane for high value of the kinetic reaction rate. We will show quantitatively
+these effects in the next section.
+The description of the properties of the membrane (i.e., porosity, permeability, velocity through the
+membrane) can be given in terms of global quantities, that is to say quantities averaged over all the membrane
+length, which therefore becomes a function of time only. We will start our analysis by giving an account of
+these global properties.
+In fig. 4 we report the variation of the average of the porosity across the membrane as function of time.
+For the lowest kinetic reaction time considered there is no appreciable variation of the porosity after more
+than one day of operations. By increasing the kinetic reaction rate we can start to observe some deviations.
+In particular, for the highest value of the reaction rate the porosity decays to 60% of its original value after
+only one day of operation. This latter kind of results can be useful in determining the operation time that
+we can expect from a membrane given a certain composition of the feed.
+The law of variation of the permeability with the reaction depends on the variation of the porosity, and
+in fact we can expect a similar behaviour. We reported the results for k in fig. 5, where we can see that there
+is an order of magnitude difference between the initial value of the permeability at time t = 0 and after 28
+h of operations.
+The average velocity through the membrane obtained with the conditions specified is 1.8 µs, which
+decreases with the decrease of the permeability of the membrane up to 0.13 µs for the lowest value of k
+and ϵ shown in figs. 4 and 5. In order to maintain the flow across the membrane in the given conditions of
+cross-flow in the channel and for the permeability and porosity given, we have to apply a pressure of approx
+1800 kPa, which is needed to overcome an osmotic pressure of 1000 kPa, which reduces to 978 kPa in the
+11
+
+da
+1.0000
+1.001
+1.002
+1.003
+1.004
+1.0056da
+0.9733
+0.98
+0.985
+0.99
+0.995
+1.0000Figure 4: Plot of the porosity versus time for all the systems considered, K is in mol/m3.
+Figure 5: Plot of the permeability as a function of time for all the systems considered K is in mol/m3.
+12
+
+0.7
+0.7
+3
+0.6
+3
+porosity
+porosity
+-10
+K= 10
+0.6999
+K= 10~5
+0.5
+10
+15
+20
+25
+30
+time (h)
+K= 10
+1
+10
+time (h)(m²)
+10
+K = 10'
+K= 10-5
+K= 10
+K= 10
+0.999
+10
+15
+20
+25
+30
+time (h)
+0.1
+0.1
+1
+10
+100
+time (h)case of the highest reaction rate. The difference in the osmotic pressure for the case K = 10−1 depends on
+the fact that for this case the concentration at the membrane is lower than the bulk (and lower than the
+case with polarisation) because of the very fast reaction (see fig. 3b).
+Despite the fact the osmotic pressure is smaller for the fastest reaction case, this case remains the worse
+in terms of permeate extraction, because the fast scaling of the membrane reduces quickly the porosity until
+the flux stops completely, i.e., we reach a value of v = 0.13 µs when we consider a fast reaction rate, against
+a value one order of magnitude higher for the case of the lowest reaction rate where we do not observe the
+scaling in the simulated time.
+4.1
+Local profiles
+Local profiles at the membrane are analysed for the same range of parameters. Since the smaller value of
+the kinetic constant (K = 10−15, 10−10 m3/mol) gives the same behaviour in the time scale considered,
+we are showing only results for K ≥ 10−5 m3/mol. We summarise our finding in fig. 6 where we reported
+the component v of the velocity of the fluid (i.e., the velocity through the membrane) and the porosity are
+reported.
+According to our analysis in the preceding sections, the flux along the membrane depends on different
+contributions. The first is the frictional pressure drop along the channel, which can cause considerable differ-
+ences in the transmembrane pressure. Secondly, we have the polarisation effects, which increase the osmotic
+pressure (as it is proportional to the concentration difference across the two sides of the membrane), and
+finally, the scaling, which changes the permeability of the membrane itself. While the first contribution can
+be mitigated by a better design of the membrane modules (Krawczyk and J¨onsson, 2014), the contributions
+of the last two effects are difficult to quantify a priori, as it can be seen from fig. 6a. While the frictional
+pressure drop along the channel acts in all the systems in the same way (we are considering the same ge-
+ometry and the same initial conditions for the flow), that is not true for the polarisation and scaling effects.
+In particular, the systems with a lower concentration suffer from polarisation at the membrane, as shown in
+the previous section, which increases the osmotic pressure. The system with higher reaction kinetics instead,
+does not suffer from the polarisation of the membrane (and in fact, the osmotic pressure is lower than the
+case at lower reaction rates, see the previous section). However, the scaling of the membrane, combined
+with the pressure drop in the channel is now dominating, resulting in an overall smaller flux (see blue curves
+against red and black curves in fig. 6a).
+In the variation of the porosity profiles in the membrane, we can observe the qualitative analysis we
+discussed in the previous section. For the larger kinetic rates, the transport-limited boundary layer on the
+membrane is reducing the reaction rate along the membrane. This, in turn, gives a porosity that increases
+along the membrane, with a value of ϵ at the outlet of the membrane, after 28 hours, 3% larger than the
+value at the inlet (see dot-dashed blue curve in fig. 6b). For the lowest reaction rates instead, we obtain
+the opposite behaviour: the polarisation increases the overall velocity along the membrane, which result in
+a reduction of the porosity along the x-direction, even though the low velocity of the reactions results in a
+very small variation (see the red continuous curve in fig. 6b). Near the outlet of the membrane, the porosity
+increases, as a result of the reduction of the polarisation at this point of the domain.
+13
+
+(a) v
+(b) porosity
+Figure 6: Velocity through the membrane and porosity profile along the membrane at different times for
+different value of the kinetic reaction values. The continuous lines are the results at 6 h, the dashed lines at
+12 h and the dot-dashed lines at 28 h. K is in mol/m3.
+5
+Conclusions
+In this work, we presented a comprehensive computational model to describe the solute dynamics ed evolution
+near a membrane for desalination processes. In particular, we included a model to treat the scaling of the
+membrane as solid precipitated following a (general) chemical reaction. We connected the accumulation of
+solids at the membrane with porosity and permeability as described by the Darcy theory of porous media.
+Following this approach, we were able to give a full explicit model to derive the dynamical evolution of the
+filtration process by specifying a few initial parameters (e.g., the property of the solution and the kinetics
+of the reaction).
+The membrane is described as a dynamic boundary condition for the fluid mechanics and solute transport
+equations, which are coupled together through the osmotic pressure term, and therefore the flow through
+the membrane.
+We implemented our model in the widely used software package for CFD calculations
+OpenFOAM®, and performed simulations for a selected range of operating conditions. Results show how
+this model can be used to predict the decay in the flux through the membrane due to the accumulation of
+the precipitated solid originating from the chemical reaction.
+The formulation presented here has two main advantages which make it flexible and powerful in treat-
+ing polarization. First, the proposed formulation can address all the interconnections between the different
+mechanisms (fluid dynamics, solute evolution, chemical reaction, scaling, and fouling) which affect the mem-
+brane performance. The second advantage is that the model can be easily extended to include more complex
+geometries, or models for the osmotic pressure (such as the Pitzer model (Pitzer, 1973; Khraisheh et al.,
+2019)), fluid flow conditions in the system, as well as more complex reactions paths.
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+0
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+X
+K= 102
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+x-direction (mm)1.03
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+1.02
+03/(x)
+3
+0.9999
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+16
+
diff --git a/adFPT4oBgHgl3EQfvjWp/content/tmp_files/load_file.txt b/adFPT4oBgHgl3EQfvjWp/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2873559d8e5062cf50d8792ffc7004e69b56e94c
--- /dev/null
+++ b/adFPT4oBgHgl3EQfvjWp/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf,len=696
+page_content='Mathematical modelling and numerical simulation of  reverse-osmosis desalination  Nicodemo Di Pasquale ∗1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Mayo Akele2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Federico Municchi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' John King2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' and Matteo  Icardi2  1Department of Chemical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Brunel University London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Uxbridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' UB8 3PH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  United Kingdom  2Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' University of Nottingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Nottingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' NG7 2RD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' United  Kingdom  3Department of Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Colorado School of Mines,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Golden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' CO 80401,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' US  January 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 2023  Special issue on Process Intensification  Abstract  The reverse osmosis membrane module is an integral element of a desalination system as it determines the  overall performance of the desalination plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The fraction of clean water that can be recovered via this  process is often limited by salt precipitation which plays a critical role in its sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In this work, we  present a model to study the complex interplay between flow, transport and precipitation processes in reverse  osmosis membranes, which together influence recovery and in turn process sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' A reactive porous  interface model describes the membrane with a dynamic evolving porosity and permeability to capture the  scaling and clogging of the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' An open-source finite-volume numerical solver is implemented within  the OpenFOAM®library and numerical tests are presented here showing the effect of the various parameters  of the model and the robustness of the model to describe a wide range of operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  1  Introduction  The demand of freshwater has steadily increased over the last forty decades at a global level, mainly because  of the increase in population and improving living standards which are leading to an expansion of irrigated  agriculture and its human consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In turn, the increase in consumption is straining the freshwater  sources in their ability to supply the growing demand of water worldwide with almost two third of the  total world population experiencing severe water scarcity during at least a part of the year (Mekonnen and  Hoekstra, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The current freshwater sources are already overexploited, and even if  better management is still needed to reduce the misuse of such resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', wastewater treatments or  waste reduction) (Najid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2022), these solutions cannot still be enough to meet the future demand of  freshwater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The ongoing climate change is expected to reduce the availability of freshwater because of the  receding of glaciers with a subsequent important reduction of the flow in important rivers such as the Mekong  Yellow, or Gange (Shannon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  ∗Corresponding author: nicodemo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='dipasquale@brunel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='uk  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='13160v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='NA]  16 Jan 2023  Almost 98% of the total liquid water on the Earth is not available for the direct use or consumption, as  it form the total water present in seas and oceans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' However, this last fact also means that if the exceedingly  high saline content in seawater can be reduced or removed, we will have access to the largest source of  freshwater sources, with which the required amount of freshwater could be delivered without straining the  natural occurring resources (Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Therefore, there is a strong push into developing more effi-  cient technologies for the desalination of seawater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Among the currently available technologies are thermal  desalination and membrane processes (Fritzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Subramani and Jacangelo, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In thermal  desalination, seawater is brought to evaporation through multi-effect distillation or multi-stage flash distilla-  tion (Al-hotmani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2021), and the resulting vapor is subsequently condensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In membrane technologies,  a semi-permeable membrane is employed to separate (or filtrate) the solution of salt and water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Reverse Osmosis (RO), is a widely employed membrane technology for treating seawater and wastewater  with salinity up to 70 g/l (Hickenbottom and Cath, 2014) which, due to its relative simplicity and widespread  diffusion has been among the main topics of research in membrane filtration (Wardeh and Morvan, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' One of the main challenges in RO is concentration polarisation (Kim and Hoek, 2005),  which is the presence, over the membrane, of a solute-rich boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Concentration polarisation can  lead to solute precipitation and fouling, significantly reducing the local permeability of the membrane with  adverse effects on the permeation flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' When the solution contains inorganic salts (such as sodium chloride  NaCl or calcium sulfate CaSO4 ), the resulting accumulation of crystals on the membrane is also called  scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' There is extensive experimental evidence indicating that scaling reduces the membrane performance  over time (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2014) and therefore, scaling phenomena should play a major role in the mathematical  modelling of RO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Computational Fluid Dynamics (CFD) represents a powerful to analyse concentration polarisation and  scaling, with the earliest attempts dating back to more than two decades ago (Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The  membrane is usually included as a boundary condition where the flux of the solute is assumed equal to zero  (complete retention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Previous studies have considered dependence on the solute concentration of properties  such as the osmotic pressure (Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 1998), viscosity, density and diffusion coefficient (Geraldes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=',  2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Wiley and Fletcher, 2002) to include effects of concentration polarisation on the membrane(Wiley and  Fletcher, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Johnston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In particular, CFD simulations for membranes have been employed to  analise different geometrical configurations, such as the spacer-filled channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' These include solid elements  in the feeding flow to increase the shear stress on the surface of the membrane, which in turn increases  the local mixing and mass transfer across the membrane (Shakaib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Fletcher and Wiley, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Fimbres-Weihs and Wiley, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Ghidossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Lau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Koutsou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=',  2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Ranade and Kumar, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  However, the effects of scaling have not been directly included in CFD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In this work, we  propose a mathematical model and a CFD solver for the analysis of the performances of a RO membrane  in which we included the possibility for the solute in the feed flow to react (precipitate) on the membrane  and therefore affecting the membrane permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We describe the  mathematical framework for the analysis of the membrane, highlighting how the chemical reactions can be  accounted for in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We then discuss the implementation of our model into the widely used open  source finite volume library OpenFOAM®and we show some results for some typical situations, showing the  applicability of the whole framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We then draw some conclusions and we outline the possible extension  of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  2  Model  In this work, we approximate a rectangular 3D membrane module as a 2D channel illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' This  is a common practice employed in other recent CFD studies (Johnston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' While the configuration  we considered can be easily extended to more complex geometries, the focus here is to develop a complete  mathematical model for the polarization and scaling of the membrane by giving a proof of concept for a  general-purpose numerical solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Our main goal is to show the mechanisms governing the evolution of the  solute at the membrane interface, and a 2D channel geometry allows us to focus on this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  2  Figure 1: 2-dimensional domain considered in this work  Let us therefore assume a rectangular domain Ω ≡ (0, L) × (0, H) with boundary Γ = ∂Ω subdivided in  three different regions:  Γm = (0, L) × {0}  (membrane)  Γin = {0} × (0, H)  (inlet)  Γout = {L} × (0, H)  (outlet)  (1)  where Γm represents the membrane boundary, Γin is the inlet boundary and Γout is the outlet boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  With Γw we represent the remaining part of the boundary constituted by solid boundaries so that Γw =  Γ\\(Γin ∪Γout ∪Γm), as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In this study, the permeate flow is not explicitly modelled, therefore  the membrane represents a boundary condition for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  The analysis of the filtration process requires the simultaneous solution of the flow field coupled with the  transport of dissolved chemical species, which can involve one or more chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Such chemical  reactions can lead to solute precipitation, modifying the permeability and porosity of the membrane, thus  causing a variation in the osmotic pressure and the final yield of the permeate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We summarised the different  mechanism and their interdependence in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Our mathematical model is composed of three different and intertwined parts that need to be simulta-  neously considered to obtain the overall behaviour of the membrane which we are now going to describe in  detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='1  Flow modelling  The mixture is described as an incompressible Newtonian fluid described by the Navier-Stokes equations:  ∇ · u = 0  in Ω, t > 0  (2a)  ρ  �∂u  ∂t + (u · ∇u)  �  = −∇p + µ∇2u  in Ω, t > 0  (2b)  3  Tw  in  ino  Bulk Feed  0  PermeateFigure 2: Sketch of the different mechanisms and models considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  where ρ is the density of the fluid u is the velocity vector with components (u, v)T , p is the fluid pressure  and µ is the dynamic viscosity, ∇ is the gradient operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We impose the following boundary and initial  conditions on the system shown in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (2) for the velocity and pressure:  u = 0  in Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' t = 0  (3a)  u = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  p = Pin  on Γin  (3b)  ∇u · n = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  p = Pout  on Γout  (3c)  u = 0  on Γw  (3d)  Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' the membrane is modelled as a dynamic Dirichlet condition for the velocity v orthogonal to the  membrane is obtained from the Darcy law as:  u = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' v = −k(∆p − ∆π)  ℓµ  on Γm  (4)  where µ is the fluid viscosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' ℓ is the membrane thickness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' k is the membrane permeability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' ∆p = pm − pp  is the pressure difference between the feed side of the membrane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' pm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' calculated at the membrane boundary  Γm and the permeate side of the membrane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' pp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' whereas ∆π is the osmotic pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' This last  equation shows that the flux through the membrane is proportional to the difference between the applied  and osmotic pressure differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  The pressure gradient ∆p = pm − pp is the difference between the pressure on the feed side of the  membrane pm calculated at the membrane boundary Γm, and the permeate side of the membrane pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The  osmotic pressure difference is defined as follows:  ∆π = π − πp = π(φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' , φN) − π(φp  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' , φp  N)  (5)  where the feed osmotic pressure π is a function of the N ions concentrations, φi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' , n, whereas the  permeate osmotic pressure φp is function of the N ions concentrations in the permeate, φp  i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  As common in membrane applications (Linares et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2014) the osmotic pressure is computed using the  Van’t Hoff model (Van’t Hoff, 1888):  π(φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' , φn) = RTϕ  N  �  i  φi  (6)  where ϕ is the osmotic coefficient, R is the gas constant and T is the temperature we can rewrite eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (5) as:  ∆π = RTϕ  N  �  i  (φ − φp  i ) = RTϕ  N  �  i  rφ  i φi  (7)  4  Convective and diffusive transport  Change in osmotic  Solute buildup  pressure  Solute  Fluid Flow  Precipitate  Transport  Permeability and  Rate of consumption  porosity change  Osmotic Pressurewhere we used the definition of membrane rejection of the species i,  rφ  i = 1 − φp  i  φi  , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' , N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  (8)  The Van’t Hoff equation assumes a linear relation between concentration and osmotic pressure and is more  accurate for low concentration of solutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Different formulations were proposed to take into account the non-  linear behaviour of solutions, which consider the activity of the solvent (Khraisheh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2019), calculated  using the Pitzer equation for the electrolyte solutions (Pitzer, 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' As a proof of concept, we will consider  here the simplified model since the most complex behaviour can be straightforwardly added to the model  and the CFD code we will present in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The membrane rejection of the species i expresses  the amount of solute rejected by the membrane (and therefore not present in the permeate) as a fraction of  the initial quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In this work, we assume rφ  i = 1 for every ion species, which corresponds to complete  rejections of the ions at the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In this case, the concentration on the permeate side and the permeate  osmotic pressure, πp, are both equal to 0 and therefore:  ∆π = RTϕ  N  �  i  φi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  (9)  In the literature, the two most popular models proposed to describe the solute-solvent solute transport  through the membrane are the solution-diffusion model (Merten, 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Lonsdale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 1965;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Wijmans and  Baker, 1995) and the Spiegler-Kedem model Spiegler and Kedem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The former expresses the flow through  the membrane ˙Jv as:  ˙Jv = A(∆p − ∆π)  (10)  noting that in our notation v =  ˙Jv  A(Γm) where Sm is the membrane surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' while for the latter we have  ˙Jv =  1  Rmµ(∆p − σ∆π)  (11)  with the same identification for ˙Jv (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', v =  ˙Jv  A(Γm)) where Rm is the membrane resistance and σ is the  reflection coefficients which measure the impermeability of the membrane to the solutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' σ = 1 indicates a  membrane completely impermeable to solutes and will be the one considered for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Notice that in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (4) we use the Darcy law to rewrite the so-called water permeability of the membrane  A which appear in the equation in terms of the permeability and thickness of the membrane and the viscosity  of water as A = Smk  ℓµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Using the same reasoning for eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (11) we obtain that Rm =  ℓ  Smk, that is to say, the  membrane resistance is inversely proportional to the permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The identification of the Darcy-related  terms with the water permeability A or membrane resistance has two main advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Firstly, we can  connect our analysis with the membranes available commercially which are described in terms of water  permeability or membrane resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' This last fact allows us to choose the range of the parameters we are  considering which are appropriate for the description of real membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Secondly, using the Darcy derived  version of the equation for the flow through the membrane allows us to include more detailed mechanisms in  the model which can take into account more complex phenomena, such as chemical reactions and depositions  of solids on the membrane as we will show in more details in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' One of the ways considered  in the literature to include fouling and polarisation of the membrane is to define such contribution as  additional resistance terms to be added to Rm in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (11) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', (Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Lee and Clark, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Yeh, 2002)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' These additional terms must be derived from experiments or empirical correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In contrast,  our formulation leverages the well-established theory of porous media to include such effects directly in the  determination of the permeability k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  The performance of the membrane can be evaluated by using the recovery r, defined as the ratio between  the permeate flow rate, ˙Qp, and the feed flow rate, ˙Qf:  r =  ˙Qp  ˙Qf  =  vAp  UinAin  = vL  uH  (12)  5  where we used the definition of flow rate through a surface, AU/ℓt, to calculate the flux through the in-  let (subscript in) and the membrane (subscript m) and then we replaced the area in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (12) with their  geometrical expression related to the domain we are considering: Ain = H × Z, Ap = L × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='2  Solute Transport  In membrane processes, flow and solute transport are tightly coupled through the boundary condition on  the membrane given by equation eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The transport equation for the concentration φi of the i−th ion  species in the bulk liquid is given by:  ∂φi  ∂t + ∇ · (uiφi) = ∇ · (D∇φi) + ξi  B  in Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' t > 0  (13a)  φi = 0  in Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' t = 0  (13b)  φi = φin  on Γin  (13c)  ∇φi = 0  on Γout ∪ Γw  (13d)  ˙Ji = vφi − Di  ∂φi  ∂y = ξi  M + ξi  P  on Γm  (13e)  where ui is the velocity of the i-th chemical species,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Di is its diffusion coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' ξ is a rate term that  represents different mechanisms of depletion of the solute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' identified by the subscripts B for the chemical  reaction in the bulk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' for chemical reaction at the membrane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' and P for the fraction of the solute that  crosses the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The amount of the species i at the membrane, can either precipitate on the membrane  following a chemical reaction or go through the membrane in the permeate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The sum of these two mechanisms  must equal the flux of the i-th chemical species at the membrane ˙Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We can therefore assume that each of  these mechanisms corresponds to a fraction of ˙J, and hence that ˙J = κMξM +κP ξP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The coefficients κ have  the property that:  κM + κP = 1  (14)  If only superficial reaction is present, then κR = 1 and κP = 0, while in the case there is no superficial  reaction and the solute crosses the membrane then κP = 1 and κR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  In this work we will make the following assumptions on the behaviour of the system: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=') The precipitation  reaction is irreversible and only occurs on the membrane surface, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', ξB = 0) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=') The transport processes  are similar for the all the salt ions and the diffusion coefficients are independent of concentration, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=') The  salt ions have the same velocity as the fluid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', ui = u for every i), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=') The porous medium is homogeneous,  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=') there are only surface reactions at the membrane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', κR = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='3  Chemical kinetics  Following the principles of mass action, the dynamic behaviour of chemical systems with n components  involved in m reactions, can be described by a set of first order differential equations with time as the  independent variable:  dφ1  dt = f1(φ1, φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' , φn, t)  dφ2  dt = f2(φ1, φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' , φn, t)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  dφn  dt = fn(φ1, φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' , φn, t)  (15)  6  where φi(t), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' , n denotes the volume molar concentration of chemical species Xi at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The  dynamics of the reaction network can be conveniently written in matrix form using the formalism developed  in (Chellaboina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2009):  dφi  dt = ξi  R = (A − B)T KφA(t), φi(0) = φ0,i, t ≥ 0  (16)  where K = diag(K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' , Km) is the diagonal matrix which contains as elements the reaction kinetics Kj,  j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' , m and φ0 is the initial concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' A and B are the m × n matrices having in each entry the  stochiometric coefficients of the reactants and products respectively and φA(t) is the matrix obtained by  replacing each element of A with φalp  i  where akj is the element of A in the l-th row and p-th line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The first  term in the boundary conditions in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (13e) takes the form  ξi  R =  �  (A − B)T KφA(t)  ���  i ℓ  (17)  where the notation [·]|i stands for the i-th component of its (n × 1 vector) argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  For a generic first order reaction X1 + X2 → X3 with kinetic constant K, we have therefore  dφ1  dt = −Kφ1φ2  dφ2  dt = −Kφ1φ2  dφ3  dt = Kφ1φ2  (18)  and  ξ1  R = −Kφ1φ2ℓ  ξ2  R = −Kφ1φ2ℓ  ξ3  R = Kφ1φ2ℓ  (19)  An example of a reaction of this type important in membrane operation is the formation of calcium carbonate,  through the reaction (Warsinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2015):  Ca2  + + 2 HCO3  − −−→ CaCO3 + CO2 + H2O  (20)  In fact, the scaling caused by the precipitation of calcium carbonate limits the operating condition of desali-  nation systems for brackish, groundwater, and seawater (Waly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  One effect to be considered when a chemical reaction is involved is that the reaction between salt ions and  subsequent precipitation of minerals, often alters the membrane properties, such as porosity and permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  As crystals grow, it is expected that the permeability k, in equation (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (4)) and the porosity, ϵ, will  decrease, reducing the liquid flow through the membrane (Steefel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' To account for the modification  of the porosity and permeability as a result of mineral precipitation, we employ the Kozeny-Carman model  (Hommel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' This model allows us to quantify the porosity-permeability relations and estimate  the resulting changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  According to the Kozeny-Carman model, the change in permeability is calculated by relating the current  permeability, k, based on the current porosity ϵ, to the initial permeability k0 corresponding to the initial  porosity ϵ0 Hommel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' These equations consequently follow the form below  k  k0  = f(ϵ)  f(ϵ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  (21)  7  Thus we can describe the permeability evolution using the following power law (Hommel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2018)  k  k0  = (1 − ϵ0)2  (1 − ϵ)2  � ϵ  ϵ0  �3  ,  (22)  where k is the current permeability, ϵ is the current porosity, k0 is the initial permeability and ϵ0 is the initial  porosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The rate at which porosity reduction occurs is given by (Huo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Noiriel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2004):  ϵ = ϵo − Vs  ℓ  t  �  t0  �  j  ξj  Rdt,  (23)  where ϵo denotes the initial porosity at t0, Vs is the molar volume of solid precipitate in m3/mol and r(t) is  the rate of precipitation in mol·m−2 · s−1 and the index j runs over all the possible reactions in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  For the first order reaction we are considering here the eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (23) simplify as:  ϵ = ϵo − Vs  ℓ  t  �  t0  (Kφ1φ2) dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  (24)  We should again observe the inter-dependencies and feedback mechanisms between fluid flow, transport  and reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Namely, in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (24) we see the precipitation reaction leads to a change in porosity which in  turn affects the permeability in equation eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The change in permeability impacts the flow via the fluid  velocity in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' This, in turn, alters the solute concentration distribution via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (13a), which ultimately  impacts the rate of precipitation again via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (13e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Moreover, from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (24) we can observe that the  variation of the porosity is proportional to the kinetic reaction constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' This last fact, in turn, simplifies  the predictions for the clogging of the membrane based on the reactions in the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We can expect that  if there are two chemical reactions in our systems, the first one 10 times slower than the second, then the  clogging caused by the products of the second reaction will take 10 times the time needed by the products of  the first reaction to produce the same amount of clogging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' One of the strengths of our models is that allow  these kinds of qualitative analyses even without actually solving the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  3  Numerical discretisation  The equations presented in section 2 are solved using the open-source finite volume OpenFOAM 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='0 library  and the code is available open-source (Icardi, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In order to solve the equation of motion alongside the  reaction at the membrane we developed a new solver for OpenFOAM called binaryReactionFoam which is  based on two widely used solvers, pimpleFoam and scalarTransportFoam· The former is a transient solver for  incompressible flows based on the PIMPLE algorithm while the latter is a concentration transport solver using  a user-specified velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The solver also includes the possibility of modelling solid precipitation in the  fluid and a multiphase flow model for the solid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The most important element in the computational  framework, however is represented by the new boundary conditions implemented to model the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  The membraneVelocity boundary conditions impose the fluid velocity based on the fluid pressure, the permeate  pressure, and the membrane properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' These are updated in time by linking this boundary condition to  the one for the scalar concentration, named binaryReaction, which solves for the solid precipitation at the  boundary and therefore updates the membrane permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The equations and boundary conditions are  coupled iteratively through Picard (fixed point) iteration (through the PIMPLE iterations) until convergence,  making the whole model fully implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  We simulate a two-dimensional rectangular channel with height h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='003 m and length of L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='02 m  discretised on a mesh composed by 600×200 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The following discretisation schemes (we direct the reader  to the OpenFOAM user guide (Ope, 2019) for a detailed description of each scheme) are used to discretise  the equations:  8  advective fluxes (divSchemes Gauss vanLeer) are computed at the faces and the variables interpolated  with a Total Variation Diminishing scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  gradient terms (gradSchemes Gauss linear) are approximated with central differencing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  surface normal gradients for diffusive fluxes (snGradSchemes orthogonal) are approximated with central  differencing (the grid is in fact orthogonal and does not need any correction to ensure second order  accuracy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  time derivatives (ddTSchemes backward) are approximated with third order implicit backward Euler  scheme,  We specified a fully developed velocity profile at the inlet:  u(0) = 6uav  y  h  �  1 − y  h  �  (25)  where uav is the average velocity along the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' By specifying the velocity profile at the inlet we need  only to specify the pressure at the outlet (the value of which is given in table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The pressure of the permeate  through the membrane is assumed constant along the length of the membrane and put equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' For  longer membranes, this assumption is no longer valid and the permeate flow needs to be modeled explicitly  (with 1D or 2D models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' This will be the subject of future extensions of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  The initial value of the permeability we consider in the calculations is k = 10−16 m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Using the expression  for the water permeability obtained from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (4) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' (10): A = Smk  ℓµ , and the viscosity of water at room  temperature µ = 10−3 Pa s, and the surface of the membrane, Sm = 2·10−6 m2 for the channel configuration  and ℓ = 10−7m , we obtain a value of A = 2 · 10−12 m (Pa s)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' This value is in line with the values  reported for commercial membranes, which are in the range 10−14 to 10−10 (Pa s)−1 (Ruiz-Garc´ıa and de la  Nuez Pestana, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Draˇzevi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Our model is able to include the scaling of the membrane given by the chemical reaction, which can  modify the membrane permeability through the precipitation of a solid phase obtained as a product of the  reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In this work we considered a range of kinetic reactions, going from very slow to fast reactions,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' with a value of the kinetic constant spanning four orders of magnitude, from 10−10 to 10−1 m3/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  We considered such a large range of kinetics since our main goal is not to focus on a specific system (and  reaction) but to give a general description that can be applied to different specific situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  4  Results  In this work, we employ a fixed flow profile at the inlet, which when considering the properties in table 1,  gives a Reynolds number equal to 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Therefore, the system operates in a fully developed laminar flow  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The first property that can be derived from this model is the polarisation of the membrane, which  represents the accumulation of the solute at the interface of the membrane on the feed side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' This is an  undesired effect since it increases the osmotic pressure reducing the extraction of the permeate per unit of  energy consumed in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We report the variation of the concentration profile in the domain in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 3  for the lowest and highest chemical kinetic rate considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We can observe in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 3 that the concentration  at the membrane is different from the one in the bulk region in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' However, while the case with  K = 10−15 m3/mol shows a higher concentration with respect to the bulk (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 3a), the case with the  highest value of the kinetic rate (K = 10−1 m3/mol, see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 3b) shows a concentration smaller than the one  in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  The latter observations show that the possible behaviour of the solution near the membrane strongly  depends on the reaction kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In the first case (the one represented in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 3a corresponding to the lowest  kinetic rate considered K = 10−15 m3/mol), we can observe the “standard” effect of the polarisation of  the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' During the filtration process, there is an accumulation of the solutes molecule on the feed  side of the membrane, which results in a higher concentration of the solution at the membrane itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' On  9  Symbol  definition/value  units  k  10−16  m2  ϵ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='7 φa,0  35  g/m3  φb,0  φa,0  g/m3  uin,av  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='1  m/s  D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='003  m  ℓ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='0001  m  Vs  27 · 10−6  m3/mol  uav  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='1  m/s  K  {10−10, 10−5, 10−2, 10−1}  m3/mol  ρ  1000  kg/m3  pout  1000  kPa  pperm  0  kPa  µ  10−3  Pa s  Table 1: Summary of the numerical inputs for the physical quantities used in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Note that  the units of the kinetic constant K depend on the fact that we considered a binary reaction, whereas for the  permeability k and porosity ϵ we are considering the initial value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', the value at t = 0 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  10  (a) K = 10−10 m3/mol  (b) K = 10−1 m3/mol  Figure 3: Contour plot of the concentration within the channel given in units of the initial concentration  ˜φa = φaφin,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' On the left: results reported for the lowest kinetic constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' On the right, results are reported  for the highest kinetic constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Note that the starting point of the legend is not zero and is different between  the pictures to make the results more clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  the opposite side, when the reaction rate is almost negligible (as for the case of K = 10−15 m3/mol), the  solutes are now consumed by the reaction and they accumulate at the membrane interface, leading to the  concentration profile observed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In the latter case, the solute is now consumed almost instantly at the  membrane interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' This result in a transport (convection and diffusion) limited profile of the concentration  near the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  The two opposite effects just described for the profile of the concentration at the membrane interface give  an interesting effect on the evolution of the porosity and permeability profiles across the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' As the  reaction proceeds, a new solid phase is formed which precipitates on the membrane modifying its structure  and therefore its fluid dynamical behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In particular, the solid phase generated during the reaction  clogs the pores of the membrane, resulting in a variation of the porosity of the membrane with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' On top  of this, since the concentration along the membrane (in the x-direction) decreases, we can expect a decrease  in the overall reaction rate (which is proportional to the concentration) and therefore a difference in the  permeability and porosity over the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' When we instead observe polarisation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', in the case of the  lowest reaction rate) the concentration near the membrane increases with the distance along the membrane  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Therefore, we can expect that the reaction velocity (which depend on K and the concentration), at a  fixed time, will increase along the membrane interface for the case with polarisation (low kinetic reaction)  and decrease along the membrane for high value of the kinetic reaction rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We will show quantitatively  these effects in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  The description of the properties of the membrane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', porosity, permeability, velocity through the  membrane) can be given in terms of global quantities, that is to say quantities averaged over all the membrane  length, which therefore becomes a function of time only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We will start our analysis by giving an account of  these global properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 4 we report the variation of the average of the porosity across the membrane as function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  For the lowest kinetic reaction time considered there is no appreciable variation of the porosity after more  than one day of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' By increasing the kinetic reaction rate we can start to observe some deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  In particular, for the highest value of the reaction rate the porosity decays to 60% of its original value after  only one day of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' This latter kind of results can be useful in determining the operation time that  we can expect from a membrane given a certain composition of the feed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  The law of variation of the permeability with the reaction depends on the variation of the porosity, and  in fact we can expect a similar behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We reported the results for k in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 5, where we can see that there  is an order of magnitude difference between the initial value of the permeability at time t = 0 and after 28  h of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  The average velocity through the membrane obtained with the conditions specified is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='8 µs, which  decreases with the decrease of the permeability of the membrane up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='13 µs for the lowest value of k  and ϵ shown in figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In order to maintain the flow across the membrane in the given conditions of  cross-flow in the channel and for the permeability and porosity given, we have to apply a pressure of approx  1800 kPa, which is needed to overcome an osmotic pressure of 1000 kPa, which reduces to 978 kPa in the  11  da  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='0000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='001  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='002  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='003  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='004  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='0056da  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='9733  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='985  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='995  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='0000Figure 4: Plot of the porosity versus time for all the systems considered, K is in mol/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Figure 5: Plot of the permeability as a function of time for all the systems considered K is in mol/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='7  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='6  3  porosity  porosity  10  K= 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='6999  K= 10~5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content="5  10  15  20  25  30  time (h)  K= 10  1  10  time (h)(m²)  10  K = 10'  K= 10-5  K= 10  K= 10  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='999  10  15  20  25  30  time (h)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='1  1  10  100  time (h)case of the highest reaction rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The difference in the osmotic pressure for the case K = 10−1 depends on  the fact that for this case the concentration at the membrane is lower than the bulk (and lower than the  case with polarisation) because of the very fast reaction (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Despite the fact the osmotic pressure is smaller for the fastest reaction case, this case remains the worse  in terms of permeate extraction, because the fast scaling of the membrane reduces quickly the porosity until  the flux stops completely, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', we reach a value of v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='13 µs when we consider a fast reaction rate, against  a value one order of magnitude higher for the case of the lowest reaction rate where we do not observe the  scaling in the simulated time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='1  Local profiles  Local profiles at the membrane are analysed for the same range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Since the smaller value of  the kinetic constant (K = 10−15, 10−10 m3/mol) gives the same behaviour in the time scale considered,  we are showing only results for K ≥ 10−5 m3/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We summarise our finding in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 6 where we reported  the component v of the velocity of the fluid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', the velocity through the membrane) and the porosity are  reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  According to our analysis in the preceding sections, the flux along the membrane depends on different  contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The first is the frictional pressure drop along the channel, which can cause considerable differ-  ences in the transmembrane pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Secondly, we have the polarisation effects, which increase the osmotic  pressure (as it is proportional to the concentration difference across the two sides of the membrane), and  finally, the scaling, which changes the permeability of the membrane itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' While the first contribution can  be mitigated by a better design of the membrane modules (Krawczyk and J¨onsson, 2014), the contributions  of the last two effects are difficult to quantify a priori, as it can be seen from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' While the frictional  pressure drop along the channel acts in all the systems in the same way (we are considering the same ge-  ometry and the same initial conditions for the flow), that is not true for the polarisation and scaling effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  In particular, the systems with a lower concentration suffer from polarisation at the membrane, as shown in  the previous section, which increases the osmotic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The system with higher reaction kinetics instead,  does not suffer from the polarisation of the membrane (and in fact, the osmotic pressure is lower than the  case at lower reaction rates, see the previous section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' However, the scaling of the membrane, combined  with the pressure drop in the channel is now dominating, resulting in an overall smaller flux (see blue curves  against red and black curves in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  In the variation of the porosity profiles in the membrane, we can observe the qualitative analysis we  discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' For the larger kinetic rates, the transport-limited boundary layer on the  membrane is reducing the reaction rate along the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' This, in turn, gives a porosity that increases  along the membrane, with a value of ϵ at the outlet of the membrane, after 28 hours, 3% larger than the  value at the inlet (see dot-dashed blue curve in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' For the lowest reaction rates instead, we obtain  the opposite behaviour: the polarisation increases the overall velocity along the membrane, which result in  a reduction of the porosity along the x-direction, even though the low velocity of the reactions results in a  very small variation (see the red continuous curve in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Near the outlet of the membrane, the porosity  increases, as a result of the reduction of the polarisation at this point of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  13  (a) v  (b) porosity  Figure 6: Velocity through the membrane and porosity profile along the membrane at different times for  different value of the kinetic reaction values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The continuous lines are the results at 6 h, the dashed lines at  12 h and the dot-dashed lines at 28 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' K is in mol/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  5  Conclusions  In this work, we presented a comprehensive computational model to describe the solute dynamics ed evolution  near a membrane for desalination processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' In particular, we included a model to treat the scaling of the  membrane as solid precipitated following a (general) chemical reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' We connected the accumulation of  solids at the membrane with porosity and permeability as described by the Darcy theory of porous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  Following this approach, we were able to give a full explicit model to derive the dynamical evolution of the  filtration process by specifying a few initial parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=', the property of the solution and the kinetics  of the reaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  The membrane is described as a dynamic boundary condition for the fluid mechanics and solute transport  equations, which are coupled together through the osmotic pressure term, and therefore the flow through  the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  We implemented our model in the widely used software package for CFD calculations  OpenFOAM®, and performed simulations for a selected range of operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Results show how  this model can be used to predict the decay in the flux through the membrane due to the accumulation of  the precipitated solid originating from the chemical reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  The formulation presented here has two main advantages which make it flexible and powerful in treat-  ing polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' First, the proposed formulation can address all the interconnections between the different  mechanisms (fluid dynamics, solute evolution, chemical reaction, scaling, and fouling) which affect the mem-  brane performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' The second advantage is that the model can be easily extended to include more complex  geometries, or models for the osmotic pressure (such as the Pitzer model (Pitzer, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=' Khraisheh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content=',  2019)), fluid flow conditions in the system, as well as more complex reactions paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='  References  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
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+page_content=' Kouzbour, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
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+page_content='  14  0  (m/s)  2  0  K=10  X  K= 102  V  3  6h  12 h  10  15  20  5  x-direction (mm)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='03  6 h  K= 10  14 h  K=10  28 h  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='0001  (0)3/(x)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='02  03/(x)  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='9999  0  5  10  15  20  3  X-direction (mm)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
+page_content='01  0  5  10  15  20  x-direction (mm)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adFPT4oBgHgl3EQfvjWp/content/2301.13160v1.pdf'}
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+INDIVIDUAL FRAILTY EXCESS HAZARD MODELS IN CANCER
+EPIDEMIOLOGY
+A PREPRINT
+Francisco Javier Rubio
+Department of Statistical Science
+University College London
+London, UK
+f.j.rubio@ucl.ac.uk
+Hein Putter
+Department of Biomedical Data Science
+Leiden University Medical Center
+Leiden, The Netherlands
+h.putter@lumc.nl
+Aurelien Belot
+Inequalities in Cancer Outcomes Network
+Department of Non-Communicable Disease Epidemiology
+London, UK
+aurelien.belot@lshtm.ac.uk
+January 6, 2023
+ABSTRACT
+Unobserved individual heterogeneity is a common challenge in population cancer survival studies.
+This heterogeneity is usually associated with the combination of model misspecification and the failure
+to record truly relevant variables. We investigate the effects of unobserved individual heterogeneity
+in the context of excess hazard models, one of the main tools in cancer epidemiology. We propose an
+individual excess hazard frailty model to account for individual heterogeneity. This represents an
+extension of frailty modelling to the relative survival framework. In order to facilitate the inference
+on the parameters of the proposed model, we select frailty distributions which produce closed-form
+expressions of the marginal hazard and survival functions. The resulting model allows for an intuitive
+interpretation, in which the frailties induce a selection of the healthier individuals among survivors.
+We model the excess hazard using a flexible parametric model with a general hazard structure which
+facilitates the inclusion of time-dependent effects. We illustrate the performance of the proposed
+methodology through a simulation study. We present a real-data example using data from lung cancer
+patients diagnosed in England, and discuss the impact of not accounting for unobserved heterogeneity
+on the estimation of net survival. The methodology is implemented in the R package IFNS.
+Keywords Excess hazard; flexible; frailties; general hazard; net survival.
+1
+Introduction
+Cancer epidemiology has become one of the priorities of many countries and health organisations. The outcomes of
+interest in this area include the time to death since the diagnosis of cancer as well as the survival due to cancer. The
+latter is often interpreted as a proxy of the quality of cancer management. There exist different ways for analysing times
+to event. Briefly, the Overall Survival framework, where the aim is to study the survival function associated with all
+causes of death, and the Competing Risks framework, where the goal is to analyse survival times in the case where the
+death can be attributable to different causes. When the cause of death is known, the Cause-Specific framework allows
+for studying cause-specific quantities, while the Relative Survival framework allows for studying the hazard associated
+to cancer when the cause of death is unavailable or unreliable (e.g. population-based cancer registries data). The main
+idea behind the Relative Survival framework is to separate the hazard associated with cancer from the hazard associated
+with other causes of death, without requiring to know the cause of death but using expected mortality hazard from life
+arXiv:2301.01823v1  [stat.ME]  4 Jan 2023
+
+XXXX
+A PREPRINT
+tables. The overall survival framework is typically avoided in cancer epidemiology as this quantity does not represent
+the survival associated with cancer. Moreover, the survival of patients may be affected if the populations of interest are
+exposed to considerably different hazard mortality rates associated with other causes of death, thus complicating the
+comparison of overall survival functions associated with different populations. Although the cause-specific framework
+is a useful framework for the analysis of cancer survival, the cause of death (e.g. from death certificates) is not available
+or unreliable in most countries, thus limiting its usability in population cancer epidemiology. For this reason, the relative
+survival framework has become the most popular tool for international comparisons of cancer survival [5]. The relative
+survival framework assumes that the individual hazard function can be decomposed as the hazard associated with other
+causes of death plus the “excess” hazard associated with cancer:
+h(t; x) = hP (age + t; year + t, z) + hE(t; x),
+(1)
+where t > 0 is the time-to-event measured since diagnosis (typically in years), hP (age + t; year + t, z) is the expected
+population mortality hazard obtained from a life table based on the characteristics z, “age” represents the age at
+diagnosis of cancer and “year” is the year of diagnosis (so age + t is the age at time t, and year + t is the year at time t).
+The expected population mortality hazard hP (age+t; year+t, z) is considered known, and the excess hazard associated
+with cancer, hE(t; x), is the quantity of interest and is estimated according to the available patient characteristics x.
+Typically, z ⊆ x . A related quantity of interest is the net survival, SN(t; x) = exp{−
+� t
+0 hE(r; x)dr}, which is the
+survival function associated with the excess hazard function hE(t; x) for a patient with characteristics x (see Perme et
+al[28] and Rubio et al[35] for a discussion on these points, and Belot et al[11] for an overview of other measures used
+in cancer survival analysis). Policy-making and international comparisons of cancer management are often based on
+the net survival associated with the entire population or subgroups of interest, which is calculated as the marginal or
+average net survival
+SN(t) =
+�
+SN(t; x)dϕ(x),
+where ϕ is the distribution function of covariates x. In population studies, where the population of interest has covariates
+x1, . . . , xm, this quantity is usually calculated as
+SN(t) =
+m
+�
+i=1
+SN(t; xi).
+Several parametric [13, 16, 17, 25, 29, 34], semi-parametric [36], and nonparametric [28] methods have been proposed
+in the literature for estimating the excess hazard and the net survival.
+One of the challenges in estimating the excess hazard function is that not all of the relevant covariates may be available
+at the population level. Unobserved individual heterogeneity (UIH) refers to the situation where important covariates
+are not recorded, or unavailable in the sample, potentially combined with model misspecification. The effects of not
+accounting for UIH can be substantial. For instance, in the context of overall survival analysis, Aalen[3] and Aalen et
+al[4] show that neglecting UIH induces a bias on the estimation of the parameters, as well as affecting the interpretation
+of some epidemiological measures such as the hazard function and incidence rates. Similar effects are observed by UIH
+produced by model misspecification [21] or missing covariates [24]. In addition, UIH can also affect model selection
+[32] and the estimation of causal effects [37], emphasising the importance of accounting for it.
+Individual frailty models represent a tractable option for accounting for UIH [1, 8, 9, 15, 20, 23, 40, 41]. The idea
+behind frailty models consists of multiplying the individual hazard function by a random correction that follows a
+parametric distribution G (see section 2 for more details on this point). This allows for accounting for non-specific
+departures from the original model (without a frailty term), in the sense that these departures may be a result of model
+misspecification or unavailable relevant covariates. In the relative survival framework, Zahl [42] investigated the use of
+correlated frailty models, which include two random frailties that affect the hazard function associated with other causes
+of death and the excess hazard, separately. However, they encountered a number of inferential issues as his proposal
+added five parameters which cannot be simultaneously estimated, unless an arbitrary restriction of the parameter space
+is introduced. Indeed, jointly modelling the two competing risks (associated to other causes and cancer) is a challenging
+problem that may lead to non-identifiable models,[38] such as that proposed by Zahl [42]. Goeman et al.[19] studied
+the use of frailties in a combination of concurrent and excess hazard models. This is, assuming a further decomposition
+of the excess hazard into a hazard associated with a concurrent disease (assumed to be known) and the hazard associated
+with cancer. They assumed that the frailty term affects both the concurrent disease hazard and the excess hazard equally,
+and focused on modelling the excess hazard using a proportional hazards model with a piecewise baseline hazard and
+without time dependent effects. It is important to notice that assuming a simple model (e.g. without time-dependent
+effects) may induce a variance inflation of the frailty term, as this term captures UIH due to missing covariates or model
+misspecification. Rancoita et al.[30] investigated the use of shared frailty models, which assume that the random frailty
+simultaneously affects the hazard associated with other causes and the excess hazard. Rubio et al.[35] studied the case
+2
+
+XXXX
+A PREPRINT
+where there is a potential mismatch in the estimation of the hazard associated with other causes, as a consequence of
+using an insufficiently stratified life table. They proposed correcting this mismatch using a random frailty affecting only
+the hazard associated with other causes. To our knowledge, the specific case of frailty models that can account for UIH
+on the excess hazard model has not been studied in the literature, neither the consequences of UIH in the estimation of
+net survival, which motivates our study.
+In this paper, we propose a frailty excess hazard model which can account for potential UIH on the excess hazard
+function. This is, we assume that the UIH affects the excess hazard, either as a consequence of missing covariates or
+model misspecification. In order to ameliorate concerns about model misspecification, we model the excess hazard
+using flexible parametric models with a rich hazard structure [34], which can incorporate covariates that affect the
+time scale as well as covariates that act on the hazard scale. Such models include popular hazard structures (such
+as proportional hazards and accelerated failure time) as particular cases. We derive some properties of this model,
+and characterise the marginal hazard and survival functions for several choices of the frailty distribution. We provide
+intuitive interpretations of these functions, and discuss identifiability of the proposed frailty model. The resulting
+models are available in closed form, allowing for a tractable implementation and estimation of the parameters using
+maximum likelihood methods.
+The paper is organised as follows. In section 2, we present the frailty model and derive the marginal hazard and
+survival functions, which are required for writing down the likelihood function, and discuss the interpretation of
+these quantities. We consider several frailty distributions that lead to closed-form expressions of the marginal hazard
+and survival functions. In section 3, we present the flexible parametric models for the excess hazard, and discuss
+interpretation of the parameters. In section 4, we present the expression of the likelihood function and discuss point and
+interval estimation of the parameters. We discuss tools for selecting the models with and without frailties. Section 5
+presents a simulation study that illustrates the performance of the proposed model in scenarios of practical interest,
+and illustrates the ability of the individual frailty model to recover the marginal net survival in the presence of UIH
+(due to the omission of one covariate). Section 6 presents an application using data from patients diagnosed with
+lung cancer in England. Section 7 concludes with a summary of our results and a general discussion about the use of
+frailty models in the relative survival framework. The methodology is implemented in the R package IFNS available at
+https://github.com/FJRubio67/IFNS.
+2
+Excess hazard frailty model
+In this section, we consider a frailty model based on the hazard decomposition (1). This model represents an extension
+of frailty models [9] to the relative survival framework. We calculate the marginal hazard and survival functions, which
+will later be used to construct the likelihood function, and discuss the interpretation of the resulting expressions. We
+also discuss several choices of the frailty distribution that lead to closed-form expressions of the hazard and survival
+functions.
+Consider the following frailty model, where we include a frailty (random effect), λ ∼ G, which multiplies the excess
+hazard function in (1) and is allowed to vary across individuals. More specifically, consider the conditional hazard
+model
+˜h(t | λ; x)
+=
+hP (age + t; year + t, z) + λhE(t; x),
+(2)
+λ
+∼
+G.
+where G is an absolutely continuous cumulative distribution function, with support on R+ and unit mean. In the
+Supplementary material, we present additional details about the identifiability of the proposed model under the hazard
+structure proposed in Section 3, which is guaranteed under the assumption of unit mean of the frailty distribution. The
+conditional survival function is
+˜S(t | λ; x)
+=
+exp{−[HP (age + t; year + t, z) − HP (age; year, z)]} exp [−λHE(t; x)] ,
+(3)
+where HP (·) and HE(·) represent the cumulative hazards associated with hP (·) and hE(·), respectively. Then, after
+integrating out the frailty λ with respect to the distribution G (see Supplementary material), the subgroup-specific
+marginal survival function, for a specific vector of covariates x, can be written as
+˜S(t; x) = exp{−HP (age + t; year + t, z) + HP (age; year, z)}LG{HE(t; x)},
+(4)
+3
+
+XXXX
+A PREPRINT
+where LG{s} =
+� ∞
+0
+e−srdG(r) denotes the Laplace transform of G evaluated at s. Consequently, the marginal hazard
+function is
+˜h(t)
+=
+− d
+dt log ˜S(t; x)
+=
+hP (age + t; year + t, z) − L′
+G{HE(t; x)}
+LG{HE(t; x)}hE(t; x)
+=
+hP (age + t; year + t, z) + E[λ | To ≥ t]hE(t; x),
+(5)
+where To = min {TP , TC} is the observed survival time, TP is the time to death from other causes, and TC is the
+time to death from cancer, and L′
+G{u} =
+∂
+∂z LG{z}
+��
+z=u. A detailed derivation of the last equality is presented
+in the Supplementary material. Equation (5) reveals that the effect of the frailty on the marginal excess hazard is
+time-dependent and also depends on the choice of the frailty distribution G. This implies that the value of the marginal
+excess hazard, for different values of t, may differ for different choices of the frailty distribution (in particular, for larger
+values of the variance of the frailty). This becomes more evident by looking at the resulting expression of the Laplace
+transforms associated with different distributions (e.g. gamma and Inverse Gaussian, shown in the Supplementary
+material). Moreover, this allows us to interpret frailty excess hazard models in a similar way as in Balan et al[9],
+where the frailty is interpreted, at a marginal level, as an element inducing a selection of healthier individuals among
+cancer patients. Moreover, this factor also accounts for unobserved heterogeneity and departures from the fitted model.
+Next, we consider a specific choice for the distribution G: a gamma distribution. This choice allows for obtaining a
+closed-form expression of the marginal survival function, in addition to its appealing flexibility and interpretability
+of parameters. In the Supplementary material, we also present the Laplace transforms associated with the Inverse
+Gaussian and Power Variance Function (PVF) frailties. The PVF is a more general distribution with closed form
+Laplace transform, but with one additional parameter [1]. The gamma distribution is a limit case of the PVF. Other
+three-parameter frailty distributions leading to closed-form Laplace transforms are the compound Poisson distribution
+[2] and the stable distribution [22], which include the gamma, Inverse Gaussian, among other distributions, as particular
+cases. Other frailty distributions, which require numerical integration to calculate the marginal hazard and survival
+functions, are reviewed in Rondeau et al[31].
+Gamma Frailty
+Consider the conditional hazard model (2) and suppose that λ ∼ Ga(µ, b), where Ga(µ, b) denotes a gamma distribution
+with mean parameter µ > 0, scale parameter b > 0, and probability density function g(r; µ, b) =
+r
+µ
+b −1
+Γ
+� µ
+b
+�
+b
+µ
+b exp
+�
+−r
+b
+�
+.
+Under the constraint that the mean parameter µ is set to 1, the scale parameter b of the frailty represents the variance.
+With this assumption (µ = 1) it follows that the subgroup-specific marginal frailty survival function is given by
+˜S(t; x)
+=
+exp {− [HP (age + t; year + t, z) − HP (age; year, z)]}
+{1 + bHE(t; x)}
+1
+b
+.
+(6)
+The subgroup-specific marginal frailty net survival function is
+˜SN(t; x)
+=
+1
+{1 + bHE(t; x)}
+1
+b .
+(7)
+The subgroup-specific marginal frailty net survival (7) is the net survival marginalised with respect to the frailty
+distribution but conditional on the covariates x. This is a corrected version of the classical net survival SN(t; x) =
+exp [−HE(t; x)], and we can see that limb→0 ˜SN(t; x) = SN(t; x).
+The subgroup-specific marginal frailty hazard function is given by
+˜h(t; x) = hP (age + t; year + t, z) +
+hE(t; x)
+1 + bHE(t; x).
+(8)
+Expression (8) provides a nice interpretation of our approach since the observed hazard can be seen as a model with
+a time-dependent weight function ω(t; x, b) =
+1
+1 + bHE(t; x) on the excess hazard, which provides a functional
+form that involves the scale or spread of the correction due to unobserved heterogeneity. Moreover, ω(t; x, b) is a
+decreasing function of HE(t; x) and b. Thus, an increase in these quantities induces a more pronounced frailty effect
+(as 0 < ω(t; x, b) ≤ 1, and ω(t; x, b) = 1 represents no frailty effect). Thus, the correction induced by the weight
+4
+
+XXXX
+A PREPRINT
+function ω(t; x, b) can be interpreted as a reduction in the level of the excess hazard hE(t; x), due to frailties associated
+with unobserved heterogeneity for specific values of the parameters. However, it is important to notice that this does not
+mean that the frailty correction always shrinks the excess hazard hE(·), as the model parameters are estimated jointly
+with the scale parameter b in the weight function. Indeed, the excess hazard in the models with and without frailty are
+not directly comparable. This effect will be illustrated in the simulation study and the real-data application, where we
+observe that the estimates of the parameters of the models with and without frailty do not necessarily coincide when
+there is UIH.
+3
+Flexible parametric models for the excess hazard
+To model the excess hazard, we consider the flexible parametric general hazard (GH) structure [14]:
+hE(t; x, α, β, θ) = h0
+�
+t exp{w⊤α}; θ
+�
+exp
+�
+x⊤β
+�
+.
+(9)
+where x ∈ Rp, w ∈ Rpt, α ∈ Rpt and β ∈ Rp. Typically w ⊆ x. h0(·; θ) represents a parametric baseline hazard
+function with parameters θ ∈ Θ. This is a rich hazard structure which contains, as particular cases, the Proportional
+Hazards (PH) model (α = 0), the Accelerated Hazards (AH) model (β = 0), and the Accelerated Failure Time (AFT)
+model (α = β, w = x). See [34] for an extensive discussion on the GH structure. This structure allows for the
+inclusion of time-scale effects (through w) and effects that act at the hazard level (x). Allowing for a flexible excess
+hazard model helps reducing UIH associated with model misspecification, which in turn is useful to focus our attention
+on capturing the effect of potentially missing covariates. These points have been discussed in the overall survival
+framework in Gasparini et al[18]. The corresponding cumulative hazard function is
+HE(t; x, α, β, θ) = H0
+�
+t exp{w⊤α}; θ
+�
+exp
+�
+x⊤β − w⊤α
+�
+.
+(10)
+The fact that the excess cumulative hazard function can be written in closed form facilitates the implementation of
+the likelihood function (discussed in the next section) as well as simulating survival times from this model. More
+specifically, simulating a random time-to-event from the GH model (9) is simple using the probability integral transform
+(see also Rossell and Rubio32):
+t = F −1
+0
+�
+1 − exp
+�
+log(1 − u) exp(w⊤α − x⊤β)
+�
+; θ
+�
+exp(w⊤α)
+,
+(11)
+where u ∼ (0, 1), and F0 is the cumulative distribution function associated with the baseline hazard h0. Analogously,
+simulating from the frailty model (2) can be done as follows
+t = F −1
+0
+�
+1 − exp
+�
+log(1 − u) exp(w⊤α − x⊤β)/λ
+�
+; θ
+�
+exp(w⊤α)
+,
+where u ∼ (0, 1), and λ ∼ G.
+Regarding the choice of the baseline hazard, Rubio et al[34] presents a discussion on different choices using flexible
+parametric distributions. The desirable properties of this hazard function are: numerical tractability and the ability
+to capture the basic shapes of the hazard (increasing, decreasing, unimodal, bathtub). Rubio et al[34] mention the
+Exponentiated Weibull (EW) and the Generalised gamma (GG), as particular choices for the baseline hazard. Here, we
+consider the use of the Power Generalised Weibull (PGW) distribution [7], but we emphasise that any other flexible
+baseline hazard distribution could be used instead. The PGW distribution is a flexible three-parameter distribution (with
+scale parameter σ > 0 and two shape parameters ν, γ > 0), which can also capture the basic hazard shapes while having
+tractable expressions for the hazard and survival functions [6]. The probability density function, survival function, and
+hazard function of the PGW distribution are presented in the Supplementary material. We could also consider simpler
+(2-parameter) baseline hazards that can capture specific basic shapes such as the Log-Normal distribution (unimodal),
+log-logistic, and the gamma distribution.
+4
+Inference
+Throughout, let (t1, . . . , tn) be the survival times associated with n cancer patients in a population; (δ1, . . . , δn) be the
+corresponding vital status indicators (0-alive, 1-dead); xi ∈ Rp, i =, . . . , n represent the vector of covariates available
+for the ith patient; agei be the age at diagnosis; and yeari be the year of diagnosis. The likelihood function for the
+individual frailty model (4)–(5) is
+˜L(α, β, σ, ν, γ, b)
+∝
+n
+�
+i=1
+�
+hP (agei + ti; yeari + ti, zi) +
+hE(ti; xi, α, β, σ, ν, γ))
+1 + bHE(ti; xi, α, β, σ, ν, γ))
+�δi
+×
+1
+{1 + bHE(ti; xi, α, β, σ, ν, γ)}
+1
+b .
+5
+
+XXXX
+A PREPRINT
+In contrast, the likelihood function associated with the model without frailty (1) is
+L(α, β, σ, ν, γ)
+∝
+n
+�
+i=1
+[hP (agei + ti; yeari + ti, zi) + hE(ti; xi, α, β, σ, ν, γ)]δi × exp {−HE(ti; xi, α, β, σ, ν, γ)}.
+Point estimates of the parameters of these models will be obtained via maximum likelihood estimation. Confidence
+intervals will be calculated using asymptotic normal approximations, which are justified by the typically large samples
+in our applications in cancer epidemiology. Briefly, let ψ be the full vector of parameters for the model of interest, we
+consider the use of confidence intervals of the type �
+ψ ± Z1− τ
+2 diag
+�
+J− 1
+2 ( �
+ψ)
+�
+, where J(ψ) = −
+∂2
+∂ψ∂ψT log L(ψ) is
+the negative of the Hessian matrix of the log-likelihood function under the appropriate parametrisation, and 1−τ ∈ (0, 1)
+is the confidence level.
+We compare these two models using AIC and evaluate their performance using a simulation study. We point out that the
+use of the likelihood ratio test for comparing these two models is more complex as the distribution of the likelihood ratio
+test statistic is not necessarily asymptotically chi-square with one degree of freedom (see, Chapter 8 of 20). Moreover,
+there exist a number of information criteria developed for mixed models that could also be used [26]. In particular, in
+our applications the sample size is large enough that makes the AIC and the marginal AIC [26] very close. A thorough
+exploration of the use of different information criteria is beyond the aims of this paper and we refer the reader to Müller
+et al[26] for a more extensive discussion on this point.
+5
+Simulation Study
+In this section, we present a simulation study where we investigate the performance of the frailty model under several
+scenarios. The proposed scenarios are designed with specific objectives [12]: (Aim 1) to investigate the finite sample
+performance of the proposed approach in different settings, and (Aim 2) to assess the effect of missing covariates (with
+heterogeneous distributions) in subgroups of the population.
+5.1
+Aim 1: finite sample performance
+Data generation and simulation design
+We investigate the impact of (i) different sample sizes and (ii) different values of the regression coefficients on the
+performance of our approach. The different true values of these regression coefficients correspond to 3 different
+scenarios (called Sc1, Sc2 or Sc3 hereafter), and for each scenario, we consider four different sample sizes (n ∈
+{500, 1000, 2000, 5000}). For each scenario, we generate 4 covariates: a continuous covariate (say age) was simulated
+using a mixture of uniform distributions with 0.25 probability for the range (30, 65), 0.35 probability for (65, 75)
+and 0.40 probability for (75, 85) years old; the remaining 3 covariates (say sex, X1 and X2) are binary with equal
+probabilities of being 0 or 1 (i.e. P(sex = 0) = P(sex = 1) = 0.5).
+The simulation of the survival times is based on the additive decomposition of the overall mortality hazard as detailed
+in equation (1). We simulate the “other-causes” time-to event using the UK life tables, and the cancer event time
+(i.e. the time to event from the excess hazard) using the inverse transform method (11), assuming model (9). This is, we
+adopt a GH structure with a PGW baseline excess hazard, a gamma frailty with mean µ∗ = 1 and scale b∗ (its value
+depends on the simulated scenario), and the effects of the covariates age, sex, X1 and X2 (section 3 details the use of
+the inverse transform method with the PGW distribution). We simulate a random drop-out time assuming an exponential
+distribution with a given rate r, the value of this rate being chosen in order to generate around 5% of random drop-out
+in each scenario. Finally, we assume an administrative censoring at 5 years. Both sources of censoring (i.e. random
+drop-out and administrative censoring) induce between 40% and 45% of censoring in total and for all the scenarios. We
+generate and analyse M = 1000 samples in each scenario.
+For Sc1, the true values of the baseline distribution parameters are σ∗ = 0.75, ν∗ = 1.75, γ∗ = 8, the scale of the
+gamma frailty b∗ = 0.5 and the regression coefficients α∗ = β∗ = (1, 1, 1, 1). We emphasise that these are complex
+simulation scenarios where the 4 covariates are included and with both time-dependent and hazard-level effects, in
+combination with a flexible baseline hazard. In practice, one would typically include only some of the covariates
+(e.g. age) as a time-dependent effect, thus reducing the effective number of model parameters. The aim of Sc1 is to
+portray the performance of the proposed frailty model in very challenging scenarios.
+In Sc2 and Sc3, we assumed less complex true models than in Sc1, assuming only 2 covariates with time-dependent
+effects. Moreover, the parameters defining the distribution of the baseline hazard and the variance of the random effect
+6
+
+XXXX
+A PREPRINT
+were also assumed different than in Sc1, in order to cover a suitable range of possible scenarios. The true values used
+for Sc2 and Sc3 can be found in the tables 1 and 2, respectively, in the Supplementary material.
+Analysis of the simulated data
+We analyse the data using model (9) for the excess mortality hazard, assuming the same parametrisation as the one used
+to simulate the data. We report the mean of the estimated regression coefficients, the bias (the difference between the
+mean of the estimated regression coefficients and the true value), the median of the estimated regression coefficients,
+the empirical standard deviation, the mean (estimated) standard error, and the coverage proportions of asymptotic
+confidence intervals.
+Our approach has been fully implemented using R software. The optimisation step was conducted using the R command
+‘nlminb’. The standard errors of regression coefficients were derived from the Hessian matrix obtained with the
+command ‘hessian’ (R package ‘numDeriv’), and the asymptotic 95% confidence intervals were approximated using
+these standard errors. For the initial values for the optimisation step, we used the parameters estimated from a PH model
+with the PGW distribution but without a frailty parameter (so we set to 1 the initial value for the scale of the gamma
+distribution, and for the parameters corresponding to time-dependent effects we used as initial values the ones obtained
+from the PH model). The cases where the optimiser did not converge (as indicated by ‘nlminb’) or when the command
+‘hessian’ produced ‘Inf’ or ‘NaN’ values were excluded: for Sc1, it represents 31, 16, 3, and 0 sets of estimates excluded
+among 1000 estimates with n = 500, 1000, 2000, 5000 observations, respectively. For Sc2, it represents 10, 1, 0, and
+0 sets of estimates excluded among 1000 estimates with n = 500, 1000, 2000, 5000 observations, respectively, while
+none were excluded in Sc3.
+Results
+The results of Sc1 are detailed in Table 1 below, while the results for Sc2 and Sc3 are given in Tables 1 and 2 in the
+Supplementary material, respectively. As expected, the bias decreases and the coverage gets closer to the nominal
+value with increasing sample size. Using different values of regression coefficients and variance of the frailty does
+not alter the performance and the results observed, as shown in Tables 1 and 2 in the Supplementary material. We
+observed a different behaviour in the estimation of the regression coefficients associated with covariates compared to
+those of the baseline hazard parameters and the scale parameter of the frailty. For the regression coefficients associated
+with covariates, the performances are good even with a small sample size of 500 cases. For the baseline distribution
+parameters and the scale of the frailty, the bias is high in situations with sample size of 500 or 1000. When the sample
+size is larger (2000 or 5000), the performances are good, with small bias and coverage near the nominal value.
+For Sc1 with 500 observations (and therefore non negligible bias on the baseline distribution parameters and the scale
+of the frailty), we check the ability of our approach to recover the true subgroup-specific marginal net survival (i.e.
+integrated over the frailty distribution but conditional on covariates, see equation (7). We could see that for the reference
+group (i.e. covariates values set to 0) and despite the bias aforementioned, the true subgroup-specific marginal net
+survival is nicely recovered on average over the set of simulated samples (Figure 1).
+5.2
+Aim 2: impact of missing covariates with heterogeneous distributions
+We now explore two simulation scenarios as follows. The overall aim is to compare the use of the model fitted to the
+entire population for predicting net survival for subgroups of the population, a common strategy in epidemiology, against
+a stratified analysis. In the first scenario, we assume that there exist two subgroups of the population, defined by the
+variable sex, with slightly different PGW baseline hazards (θ1 = (0.5, 1.5, 5) for sex = 1, and θ2 = (0.5, 1.5, 3), for
+sex = 0). The variable sex is binary with probability 0.6 of being 1 and probability 0.4 of being 0. For each subgroup,
+we generate 2 covariates: a continuous covariate (say age) was simulated using a mixture of uniform distributions
+with 0.25 probability for the range (30, 65), 0.35 probability for (65, 75) and 0.40 probability for (75, 85) years old;
+and the variable X1 is also binary, with conditional probability 0.8 of being 1 if sex = 1, an conditional probability
+0.4 of being 1 if sex = 0 (that is, it has a different distribution for each sex). In addition, we assume that the effects
+of the covariates are different for each sex subgroup: for sex = 1, α = (0.7, 0.7, 0.5) and β = (1, 0.5, 1); and for
+sex = 0, α = (0.7, 0.7, 0.25) and β = (0.5, 0.5, 0.25). In the second simulation scenario, we again assume that there
+exist two subgroups of the population, defined by the variable sex, now with markedly different PGW baseline hazards
+(θ1 = (0.5, 1.5, 5) for sex = 1, and θ2 = (0.5, 1.5, 0.75) for sex = 0). The covariates are simulated as in the first
+scenario, and the true values of the regression coefficients α and β are chosen as in the first scenario. For both scenarios,
+we consider sample sizes n = 500, 1000, 5000 and censoring rate of approximately 65%.
+For each scenario, we first fit the classical model without frailty (1) and the frailty model (8) with PGW baseline hazard
+to each simulated data set omitting the variable X1. We calculate the corresponding population net survival curves
+7
+
+XXXX
+A PREPRINT
+Parameter
+True
+MeanMLE
+Bias
+MedianMLE
+Coverage
+Mean StdErr
+EmpSD
+Scenario N=500
+σ
+0.750
+0.974
+0.224
+0.761
+0.817
+0.446
+0.622
+ν
+1.750
+1.983
+0.233
+1.873
+0.932
+0.397
+0.498
+γ
+8.000
+8.690
+0.690
+8.609
+0.787
+4.097
+4.986
+α1
+1.000
+1.097
+0.097
+1.006
+0.928
+0.607
+1.333
+α2
+1.000
+0.984
+-0.016
+0.995
+0.914
+0.273
+0.639
+α3
+1.000
+1.027
+0.027
+0.991
+0.924
+0.565
+1.197
+α4
+1.000
+1.010
+0.010
+0.986
+0.917
+0.608
+1.420
+β1
+1.000
+0.940
+-0.060
+1.016
+0.941
+0.384
+0.846
+β2
+1.000
+1.007
+0.007
+0.996
+0.964
+0.165
+0.369
+β3
+1.000
+0.989
+-0.011
+1.009
+0.957
+0.359
+0.675
+β4
+1.000
+1.003
+0.003
+1.014
+0.946
+0.378
+0.821
+b
+0.500
+1.231
+0.731
+0.549
+0.828
+0.810
+1.675
+Scenario N=1000
+σ
+0.750
+0.870
+0.120
+0.740
+0.885
+0.333
+0.453
+ν
+1.750
+1.847
+0.097
+1.820
+0.937
+0.246
+0.269
+γ
+8.000
+8.364
+0.364
+8.246
+0.869
+3.065
+3.404
+α1
+1.000
+1.029
+0.029
+1.000
+0.957
+0.319
+0.582
+α2
+1.000
+1.023
+0.023
+1.001
+0.955
+0.162
+0.373
+α3
+1.000
+1.004
+0.004
+0.972
+0.953
+0.319
+0.570
+α4
+1.000
+0.969
+-0.031
+0.989
+0.945
+0.326
+0.584
+β1
+1.000
+0.994
+-0.006
+1.008
+0.954
+0.189
+0.263
+β2
+1.000
+0.984
+-0.016
+0.995
+0.948
+0.090
+0.207
+β3
+1.000
+0.997
+-0.003
+1.005
+0.952
+0.191
+0.327
+β4
+1.000
+1.007
+0.007
+0.999
+0.956
+0.194
+0.317
+b
+0.500
+0.836
+0.336
+0.541
+0.923
+0.616
+1.063
+Scenario N=2000
+σ
+0.750
+0.821
+0.071
+0.759
+0.942
+0.227
+0.309
+ν
+1.750
+1.778
+0.028
+1.766
+0.962
+0.165
+0.162
+γ
+8.000
+8.061
+0.061
+8.010
+0.929
+2.170
+2.292
+α1
+1.000
+1.015
+0.015
+0.997
+0.959
+0.192
+0.272
+α2
+1.000
+1.010
+0.010
+1.005
+0.954
+0.093
+0.154
+α3
+1.000
+1.009
+0.009
+0.991
+0.945
+0.197
+0.338
+α4
+1.000
+1.009
+0.009
+0.995
+0.945
+0.193
+0.310
+β1
+1.000
+1.003
+0.003
+1.008
+0.947
+0.113
+0.147
+β2
+1.000
+0.994
+-0.006
+0.998
+0.956
+0.048
+0.087
+β3
+1.000
+0.996
+-0.004
+1.004
+0.946
+0.116
+0.185
+β4
+1.000
+0.997
+-0.003
+1.003
+0.939
+0.114
+0.175
+b
+0.500
+0.660
+0.160
+0.550
+0.964
+0.409
+0.607
+Scenario N=5000
+σ
+0.750
+0.760
+0.010
+0.747
+0.950
+0.123
+0.125
+ν
+1.750
+1.764
+0.014
+1.760
+0.954
+0.102
+0.102
+γ
+8.000
+8.108
+0.108
+8.026
+0.951
+1.352
+1.369
+α1
+1.000
+0.991
+-0.009
+0.987
+0.952
+0.109
+0.110
+α2
+1.000
+1.001
+0.001
+1.002
+0.949
+0.051
+0.052
+α3
+1.000
+0.988
+-0.012
+0.986
+0.948
+0.110
+0.109
+α4
+1.000
+0.987
+-0.013
+0.989
+0.936
+0.110
+0.113
+β1
+1.000
+0.999
+-0.001
+0.997
+0.945
+0.066
+0.066
+β2
+1.000
+1.000
+-0.000
+0.998
+0.949
+0.027
+0.028
+β3
+1.000
+1.001
+0.001
+1.000
+0.943
+0.066
+0.067
+β4
+1.000
+1.002
+0.002
+1.001
+0.938
+0.066
+0.068
+b
+0.500
+0.565
+0.065
+0.543
+0.941
+0.229
+0.233
+Table 1: Simulation results for Aim1, scenario 1. MeanMLE: Mean of the Maximum Likelihood Estimates; MedianMLE:
+median value of the Maximum Likelihood Estimates; Mean StdErr: Mean of the standard errors; EmpSD: Empirical
+standard deviation
+by averaging the individual net survival curves, for both models, associated to each patient. We compare the true
+population net survival curve against the fitted net survival curves obtained with these models. Then, we calculate the
+net survival associated to the subgroups defined by sex = 0, 1 by taking the average of individual net survival curves
+over the corresponding subsets of observations. Our aim here is to compare the net survival curves associated to the
+entire population, and the use of the models fitted to the entire sample for net survival subgroup estimations in the
+presence of UIH.
+In a second stage, we perform a stratified analysis where we fit the classical model without frailty (1) and the frailty
+model (8) to each data subset defined by the variable sex = 0, 1. Then, we calculate the net survival curves associated
+to the subgroups sex = 0, 1 with the average of the individual fitted net survival curves obtained with these models. We
+again compare the corresponding net survival curves against the true net survival for these subgroups.
+8
+
+XXXX
+A PREPRINT
+0
+1
+2
+3
+4
+5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Time
+Net survival
+True individual marginal net survival (reference group)
+Sample−specific individual marginal net survivals
+Mean of the sample−specific individual marginal net survivals
+Figure 1: Scenario Sc1 with 500 observations: subgroup-specific marginal net survival for the reference group
+Results are shown in the Supplementary material (Figures 1–2). In the first scenario and for all sample sizes (Figure 1
+in the Supplementary material), we notice that the average fitted population net survival curves are very close to the true
+population net survival. The subgroup analysis also leads to close net survival curves, with a slightly worst fit from the
+classical model (which remains for all sample sizes). The stratified analysis produces the best results as the average
+fitted net survival curves are virtually the same as the true net survival curves. In the second scenario (Figure 2 in the
+Supplementary material), we notice that the average fitted population net survival curves are close to the true population
+net survival, although a small bias is observed for both models and all sample sizes. The subgroup analysis leads to a
+very large bias for both average curves, and we also notice a discrepancy between the classical and frailty models. This
+bias is largely reduced by using a stratified analysis for all sample sizes.
+6
+Real data application: lung cancer epidemiology
+We analyse a data set obtained from population-based national cancer registry of Non-Small Cell Lung Cancer (NSCLC)
+female patients diagnosed in 2012 in England [10]. We include the following covariates: standardised age at diagnosis
+(agec, continuous); tumour stage at diagnosis (stage, categorical I-IV); the presence of cardiovascular comorbidities
+(CVD, binary); the presence of Chronic Obstructive Pulmonary Disease (COPD, binary); and the standardised Income
+Domain from the 2010 England Indices of Multiple Deprivation (IMD, continuous), defined at the Lower Super Output
+Area level (as a measure of deprivation). Information on stage at diagnosis and comorbidities was obtained from linked
+data (Hospital Episode Statistics -HES- and the Lung Cancer Audit data). The cohort of patients was followed-up until
+the 31st of December 2015, at which time patients alive were right-censored. We restrict the analysis to women with no
+missing covariates. The resulting sample size was n = 14557 patients with complete cases, among which no = 12138
+died before the 31st of December 2015. The 25%, 50% and 75% quantiles of the patients’ age at diagnosis was 64.93,
+72.64, 80.23 while the mean was 72.00. Among these patients, 2434 were diagnosed at Stage I, 1131 at Stage II, 3241
+at Stage III, and 7751 at Stage IV. Finally, 1954 patients were classified with a cardiovascular comorbidity and 3260
+with a chronic obstructive pulmonary disease.
+In order to evaluate the effects of UIH, we compare eight models for this data set. Model C1: (classical) model without
+frailty with time-level effect of agec (wi), and hazard-level (xi) effects of agec, IMD, stage, CVD, and COPD.
+Model F1: frailty model with time-level effect of agec (wi), and hazard-level (xi) effects of agec, IMD, stage,
+CVD, and COPD. Model C2: model without frailty with time-level effect of agec (wi), and hazard-level (xi) effects
+of agec, IMD, CVD, and COPD. Model F2: frailty model with time-level effect of agec (wi), and hazard-level (xi)
+effects of agec, IMD, CVD, and COPD. Model C3: model without frailty with time-level effect of agec (wi), and
+9
+
+XXXX
+A PREPRINT
+hazard-level (xi) effects of agec, IMD. Model F3: frailty model with time-level effect of agec (wi), and hazard-level
+(xi) effects of agec, IMD. Model C4: PGW baseline hazard without covariates. Model F4: frailty model with PGW
+baseline hazard without covariates.
+Figure 2a shows the net survival for the entire cohort obtained with Models C1–C4 and F1–F4. These are obtained
+using the formulas:
+SN(t)
+=
+1
+n
+n
+�
+i=1
+exp{−HE(t; xi, �α, �β, �σ, �ν, �γ)}, Models C1–C4,
+˜SN(t)
+=
+1
+n
+n
+�
+i=1
+1
+�
+1 + �bHE(t; xi, �α, �β, �σ, �ν, �γ)
+� 1
+�b
+, Models F1–F4.
+The AIC values for Models C1–C4 and F1–F4 are shown in Table 2. The overall best model is Model F1, clearly
+favouring a model with all covariates and a frailty. The best model among the classical models (C1–C4) was model C1,
+also favouring the inclusion of all covariates. Some interesting conclusions arise from this model comparison and Figure
+2. Although AIC clearly favours Model F1, Figure 2a shows that the estimated population net survival curves obtained
+for models C1–C4 and F1–F4 are very similar. This indicates that, even though the data favours a frailty model, some
+estimated quantities based on models with and without frailties may be similar. However, this is far from being a general
+conclusion. Figure 2b shows that the net survival stratified by stage for models C1 and F1 exhibit marked differences
+for early tumour stages. These differences are masked in Figure 2a due to the distribution of the tumour stage covariate,
+which mostly contains late-stage patients (see the descriptive analysis). This is, since we are interested in estimating
+quantities based on averages over subgroups of the population, the distribution of the corresponding covariates (patient
+characteristics) plays a key role. Thus, although the estimated models are quite different (see Table 2), the distribution
+of the covariates produces similar marginal net survival curves at the population level. However, this conclusion does
+not apply to the net survival functions stratified by stage. Table 3 presents confidence intervals for the net survival
+at times t = 1, 2, 3, 4 years. These confidence intervals are obtained using a Monte Carlo approximation based on
+the asymptotic normality of the maximum likelihood estimators (see Rubio et al[34] for more details on this). These
+confidence intervals confirm that differences in the estimation of net survival are also present in interval estimates.
+Another point that deserves some attention is the interpretation of parameters. From Table 2, we can see that the
+estimates of the parameters of the baseline hazards markedly differ for models C1 and F1. The reason for this is that
+the estimated baseline hazard associated to model C1 is decreasing, while the estimated baseline hazard associated to
+model F1 is increasing. This explains the differences in the estimates of the parameters, which are simply reflecting
+this contrasting shape. However, we would like to emphasise that the baseline hazards associated to models C1 and F1
+(more generally, models with and without frailty correction) are not directly comparable as (i) the estimates for model
+F1 correspond to a marginal model (marginalised over the frailty term), and (ii) the baseline hazard in the frailty model
+(8) controls both the excess hazard in the numerator and the time-varying weight. Despite this difference in nature,
+comparing them is still useful, as observing differences between the estimates in both models is an indication of UIH as
+such discrepancies only appear when the variance of the frailty term is non-negligible.
+Model
+C1
+F1
+C2
+F2
+C3
+F3
+C4
+F4
+�σ
+0.763
+8.760 (7.072,10.851)
+0.109
+0.769
+0.107
+0.737
+0.099
+0.562
+�ν
+0.987
+1.077 (1.046,1.108)
+1.196
+1.273
+1.199
+1.288
+1.197
+1.407
+�γ
+4.980
+0.813 (0.675,0.980)
+4.185
+0.323
+4.173
+0.314
+4.325
+0.272
+agect
+0.308
+2.633 (1.367,3.900)
+0.271
+0.442
+0.272
+0.436
+–
+–
+agec
+0.328
+0.124 (0.000 0.248)
+0.331
+0.289
+0.340
+0.303
+–
+–
+IMD
+0.530
+0.827 (0.578,1.075)
+0.305
+1.079
+0.319
+1.126
+–
+–
+stage2
+0.706
+0.964 (0.825,1.102)
+–
+–
+–
+–
+–
+–
+stage3
+1.460
+2.085 (1.963,2.207)
+–
+–
+–
+–
+–
+–
+stage4
+2.202
+3.312 (3.169,3.456)
+–
+–
+–
+–
+–
+–
+CVD
+0.214
+0.352 (0.267,0.437)
+0.147
+0.295
+–
+–
+–
+–
+COPD
+0.142
+0.242 (0.172,0.311)
+-0.018
+-0.054
+–
+–
+–
+–
+�b
+–
+0.764 (0.687,0.850)
+–
+4.542
+–
+4.785
+–
+6.836
+AIC
+20548.43
+20140.06
+26328.84
+26311.62
+26306.3
+26302.23
+26864.78
+26808.51
+Table 2: Maximum likelihood estimates and AIC for Models C1–C4 and F1–F4. 95% confidence intervals are also
+presented for the parameters of the selected model, F1.
+10
+
+XXXX
+A PREPRINT
+0
+1
+2
+3
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+time
+Net Survival
+C1
+F1
+C2
+F2
+C3
+F3
+C4
+F4
+0
+1
+2
+3
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+time
+Net Survival
+Stage I
+Stage II
+Stage III
+Stage IV
+(a)
+(b)
+Figure 2: Lung cancer data: (a) net survival curves for the entire population, and (b) stratified net survival curves by
+tumour stage at diagnosis (Model C1 - gray lines, Model F1 - black lines).
+Total population by Stage
+Classical (C1)
+Frailty (F1)
+Stage
+year
+NS
+lower
+upper
+NS
+lower
+upper
+I
+1
+0.814
+0.801
+0.827
+0.859
+0.850
+0.873
+2
+0.722
+0.704
+0.740
+0.737
+0.721
+0.760
+3
+0.661
+0.640
+0.681
+0.639
+0.620
+0.668
+4
+0.615
+0.590
+0.637
+0.560
+0.542
+0.591
+II
+1
+0.664
+0.642
+0.688
+0.706
+0.685
+0.732
+2
+0.524
+0.497
+0.551
+0.524
+0.501
+0.557
+3
+0.440
+0.413
+0.466
+0.409
+0.385
+0.442
+4
+0.382
+0.355
+0.411
+0.331
+0.309
+0.359
+Table 3: Net survival (NS) at t = 1, 2, 3, 4 years and 95% confidence intervals: Classical Model (Model C1) and Frailty
+Model (Model F1).
+Stratified analysis
+The differences in the net survival curves for Stages I–IV in Figure 2 suggest that there is UIH, and that, potentially,
+the unobserved covariates may have a different conditional distribution at each of the strata. To explore this idea, we
+now consider a stratified analysis where we study patients in three different strata: Stage I-II, Stage III, and Stage IV
+tumours. There are clinical and biological reasons for this kind of stratification, as different tumour stages indicate
+the size of the tumour and spread to other organs (metastasis). We compare frailty and non-frailty models for each of
+these groups to evaluate the effects of UIH. For Stages I–II we fit the models: Model C1-I.II: model without frailty
+with time-level effect of agec (wi), and hazard-level (xi) effects of agec, IMD, stage, CVD, and COPD. Model
+F1-I.II: frailty model with time-level effect of agec (wi), and hazard-level (xi) effects of agec, IMD, stage, CVD,
+and COPD. Models C1-III, F1-III, C1-IV, and F1-IV represent the corresponding models for Stage III and Stage IV.
+Results are reported in Table 4. For Stages I-II, the model without frailty is slightly favoured and the estimated frailty
+variance is small. For Stage III, the frailty model is favoured, and we observe differences in the estimates of the model
+parameters. Similarly, for Stage IV, the frailty model is favoured, the estimated frailty variance is large, and there is a
+clear discrepancy in the estimates of the model parameters. Thus, we notice that there are different levels of UIH at
+each of the tumour stage strata. From Figure 3, we can see that the net survival curves for the models with and without
+frailty coincide. This indicates that, after stratifying the data by Stage, UIH has a smaller effect on the estimation of
+net survival, even though there is evidence of UIH for Stages III and IV. It is worth noticing that omitting important
+covariates in the models has an effect on the estimation of the baseline hazard parameters and the regression coefficients
+(see, for instance, the values of the estimates in 2 for the models F1, F2, and F3). This is a well known phenomenon
+11
+
+XXXX
+A PREPRINT
+associated to omitting important covariates in hazard regression models (see Rossell and Rubio[32] for an extensive
+discussion on the effects of omitting covariates in survival regression models).
+C1-I.II
+F1-I.II
+C1-III
+F1-III
+C1-IV
+F1-IV
+�σ
+2.172
+2.839
+0.430
+1.315
+0.090
+0.689
+�ν
+1.099
+1.095
+1.150
+1.074
+1.241
+1.277
+�γ
+3.203
+2.560
+2.286
+0.856
+3.110
+0.235
+agect
+0.399
+0.354
+0.516
+-2.119
+0.287
+0.314
+agec
+0.564
+0.568
+0.363
+0.691
+0.291
+0.277
+IMD
+0.731
+0.758
+0.172
+0.292
+0.631
+1.507
+stage2
+0.770
+0.836
+–
+–
+–
+–
+CVD
+0.446
+0.490
+0.262
+0.359
+0.131
+0.333
+COPD
+0.482
+0.511
+0.092
+0.147
+0.085
+0.260
+�b
+–
+0.211
+–
+0.666
+–
+3.075
+AIC
+7817.53
+7818.49
+6986.52
+6962.63
+5218.65
+5214.64
+Table 4: Maximum likelihood estimates for the models fitted on the stratified data by Stage.
+0
+1
+2
+3
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+time
+Net Survival
+I−12
+II−12
+I−3
+II−3
+I−4
+II−4
+C1−I.II
+F1−I.II
+C1−III
+F1−III
+C1−IV
+F1−IV
+Figure 3: Lung cancer data: net survival curves for the models fitted on the stratified data by Stage. Gray lines represent
+the net survival curves associated to model C1 fitted to the corresponding stratum (Stages I-II, Stage III, or Stage IV).
+Black lines represent the net survival curves associated to model F1 fitted to the corresponding stratum (Stages I-II,
+Stage III, or Stage IV).
+7
+Discussion
+We have developed an extension of frailty models to the relative survival framework, and shown that these models
+induce a selection of the healthier individuals among survivors of cancer. We have shown that the proposed excess
+hazard frailty models are able to capture unobserved individual heterogeneity, a common challenge in population cancer
+epidemiology. We have proposed a general family of parametric excess hazard regression models which guarantee
+identifiability of parameters, and which contains hazard structures of practical interest. The proposed model allows for
+a tractable implementation of the likelihood function as we can include time-level effects while avoiding the need for
+numerical integration. Nonetheless, we point out that other hazard structures could be used as well. Although we have
+focused on the gamma frailty distribution, due to its tractability and interpretability, we have also presented several
+12
+
+XXXX
+A PREPRINT
+alternative choices of the frailty distribution that lead to closed-form expressions (see Supplementary material). A
+thorough comparison of the impact of the choice of the frailty distribution is beyond the aims of this work, but this will
+be considered in future research.
+Our simulations show that the proposed model has good inferential properties. The performance of the proposed
+approach was evaluated in very challenging scenarios, with a small sample size (n = 500), substantial amount of
+censoring (40%), and with a very complicated generating model (the 4 covariates considered had both an effect on
+the time scale and on the hazard level). In real life applications, one would not expect to see all the covariates with
+effects on both the time scale and the hazard level. As shown in Aim 2 of our simulations, unobserved heterogeneity
+may have a non-negligible effect if the model fitted to the entire population is used to obtain net survival curves for
+heterogeneous subgroups. This suggests that discrepancies between net survival curves associated to subgroups of the
+population using the classical and the frailty models may indicate a non-negligible effect of unobserved heterogeneity,
+and that a stratified analysis may help reduce this heterogeneity.
+We have shown that, unsurprisingly, the effects and interpretation of UIH that have been reported in the overall survival
+framework [9] are also present in the relative survival framework. Our work offers an additional perspective, which is
+the exploration of the impact of UIH on the estimation of marginal quantities (such as net survival). We have shown
+that the proposed frailty model is able to detect UIH. However, it is important to keep the practical limitations of doing
+so in mind, as the presence of UIH can also be an indication of model inaccuracy or model misspecification [27]. We
+have proposed a frailty GH model which allows for including time-varying effects. The use of a general and flexible
+model thus helps reduce UIH associated to model misspecification. Detecting UIH beyond model misspecification
+is particularly important when the model fitted to the entire population is also used to estimate marginal quantities
+associated to subgroups of the population [39], as the subgroups of interest may have different generating models, or
+different distributions of the included covariates. Thus, in the presence of UIH, it would be useful to compare the results
+against those obtained with a stratified analysis. This is an interesting finding for analysts interested in estimating
+marginal quantities for a subgroup of patients. Of course, since UIH induces a bias on the estimation of the model
+parameters (for models without individual frailties), the estimation of conditional quantities (such as excess hazards for
+specific covariate values) will also be biased. We have also shown that UIH induces a selection effect over the follow-up
+time. This type of effect has been also discussed by Vaupel and Yashin [39], in the overall survival framework, who
+found that the estimation of conditional (or individual) quantities can be severely affected by UIH in some scenarios.
+We have shown scenarios where a flexible 3-parameter distribution (e.g. PGW) is needed for modelling the shape of
+the baseline hazard. There exist scenarios where a simpler baseline hazard model (e.g. lognormal) or a simpler hazard
+structure (such as the AFT or PH) might be favoured by the data. In such cases, a model selection tool (such as AIC or
+BIC) would help identify the best model for the data (see Rubio et al[34] and Rubio and Drikvandi[33] for a discussion).
+The real data application presents a clear illustration of the fact that discrepancies in the estimation of the net survival
+obtained with models with and without frailty terms depend on the distribution of the patients’ characteristics. This
+is in line with the conclusions obtained in Aim 2 of the simulation study. The net survival is a marginal measure, as
+it is calculated as the average over the observed covariates. Consequently, its estimation naturally depends on the
+distribution of the covariates. In practice, it is important to first compare the different models in order to detect UIH,
+followed by an investigation of the effect of the presence of UIH on the estimation of the quantities of interest (such as
+net survival). Moreover, the real data application indicates that the level of UIH might depend on the observed covariate
+(e.g. tumour stage in our illustration). Therefore, extending our approach with a frailty distribution that depends on the
+observed covariates could be a very interesting research avenue.
+Possible extensions of our work include the use of other parametric baseline hazard distributions, combined with
+alternative hazard structures (e.g. additive hazard structure). In cancer epidemiology, there are other quantities of
+interest which are derived from the estimated excess hazard, such as life years lost, among other quantities [11]. Thus,
+quantifying the impact of UIH on the estimation of these quantities would be of interest. Since our real data application
+involves the use of confidential data, we cannot make it publicly available. However, we have created a repository
+(https://github.com/FJRubio67/IFNS) which contains an application using “The Simulacrum” data set,
+which is a dataset that contains artificial patient-like cancer data. This repository illustrates the proposed methods and
+the software provided.
+To our knowledge, this is the first study where the impact of UIH on estimated quantities is assessed within the relative
+survival framework. Given that the net survival is used for international comparisons and to inform policy-making, a
+careful analysis of factors that may induce bias, such as UIH, on the estimation of this quantity seems appropriate, and
+the methodology proposed in this paper provides a tool for addressing this problem.
+13
+
+XXXX
+A PREPRINT
+Financial disclosure
+AB is funded by a Cancer Research UK programme grant (C7923/A29018).
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+Epidemiology, 28(3):379–386, 2017.
+[38] A. Tsiatis. A nonidentifiability aspect of the problem of competing risks. Proceedings of the National Academy of
+Sciences, 72(1):20–22, 1975.
+[39] J.W. Vaupel and A.I. Yashin. Heterogeneity’s ruses: some surprising effects of selection on population dynamics.
+The American Statistician, 39(3):176–185, 1985.
+[40] J.W. Vaupel, K.G. Manton, and E. Stallard. The impact of heterogeneity in individual frailty on the dynamics of
+mortality. Demography, 16(3):439–454, 1979.
+[41] A. Wienke. Frailty Models in Survival Analysis. Chapman and Hall/CRC, 2010.
+[42] P.H. Zahl. Frailty modelling for the excess hazard. Statistics in Medicine, 16:1573–1585, 1997.
+15
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf,len=1359
+page_content='INDIVIDUAL FRAILTY EXCESS HAZARD MODELS IN CANCER  EPIDEMIOLOGY  A PREPRINT  Francisco Javier Rubio  Department of Statistical Science  University College London  London, UK  f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='rubio@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='uk  Hein Putter  Department of Biomedical Data Science  Leiden University Medical Center  Leiden, The Netherlands  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='putter@lumc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='nl  Aurelien Belot  Inequalities in Cancer Outcomes Network  Department of Non-Communicable Disease Epidemiology  London, UK  aurelien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='belot@lshtm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='uk  January 6, 2023  ABSTRACT  Unobserved individual heterogeneity is a common challenge in population cancer survival studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  This heterogeneity is usually associated with the combination of model misspecification and the failure  to record truly relevant variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We investigate the effects of unobserved individual heterogeneity  in the context of excess hazard models, one of the main tools in cancer epidemiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We propose an  individual excess hazard frailty model to account for individual heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This represents an  extension of frailty modelling to the relative survival framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In order to facilitate the inference  on the parameters of the proposed model, we select frailty distributions which produce closed-form  expressions of the marginal hazard and survival functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The resulting model allows for an intuitive  interpretation, in which the frailties induce a selection of the healthier individuals among survivors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  We model the excess hazard using a flexible parametric model with a general hazard structure which  facilitates the inclusion of time-dependent effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We illustrate the performance of the proposed  methodology through a simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We present a real-data example using data from lung cancer  patients diagnosed in England, and discuss the impact of not accounting for unobserved heterogeneity  on the estimation of net survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The methodology is implemented in the R package IFNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Keywords Excess hazard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' flexible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' frailties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' general hazard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' net survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  1  Introduction  Cancer epidemiology has become one of the priorities of many countries and health organisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The outcomes of  interest in this area include the time to death since the diagnosis of cancer as well as the survival due to cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The  latter is often interpreted as a proxy of the quality of cancer management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' There exist different ways for analysing times  to event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Briefly, the Overall Survival framework, where the aim is to study the survival function associated with all  causes of death, and the Competing Risks framework, where the goal is to analyse survival times in the case where the  death can be attributable to different causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' When the cause of death is known, the Cause-Specific framework allows  for studying cause-specific quantities, while the Relative Survival framework allows for studying the hazard associated  to cancer when the cause of death is unavailable or unreliable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' population-based cancer registries data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The main  idea behind the Relative Survival framework is to separate the hazard associated with cancer from the hazard associated  with other causes of death, without requiring to know the cause of death but using expected mortality hazard from life  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='01823v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='ME]  4 Jan 2023  XXXX  A PREPRINT  tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The overall survival framework is typically avoided in cancer epidemiology as this quantity does not represent  the survival associated with cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Moreover, the survival of patients may be affected if the populations of interest are  exposed to considerably different hazard mortality rates associated with other causes of death, thus complicating the  comparison of overall survival functions associated with different populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Although the cause-specific framework  is a useful framework for the analysis of cancer survival, the cause of death (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' from death certificates) is not available  or unreliable in most countries, thus limiting its usability in population cancer epidemiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' For this reason, the relative  survival framework has become the most popular tool for international comparisons of cancer survival [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The relative  survival framework assumes that the individual hazard function can be decomposed as the hazard associated with other  causes of death plus the “excess” hazard associated with cancer:  h(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x) = hP (age + t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year + t, z) + hE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x),  (1)  where t > 0 is the time-to-event measured since diagnosis (typically in years), hP (age + t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year + t, z) is the expected  population mortality hazard obtained from a life table based on the characteristics z, “age” represents the age at  diagnosis of cancer and “year” is the year of diagnosis (so age + t is the age at time t, and year + t is the year at time t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  The expected population mortality hazard hP (age+t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year+t, z) is considered known, and the excess hazard associated  with cancer, hE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x), is the quantity of interest and is estimated according to the available patient characteristics x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Typically, z ⊆ x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' A related quantity of interest is the net survival, SN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x) = exp{−  � t  0 hE(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)dr}, which is the  survival function associated with the excess hazard function hE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x) for a patient with characteristics x (see Perme et  al[28] and Rubio et al[35] for a discussion on these points, and Belot et al[11] for an overview of other measures used  in cancer survival analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Policy-making and international comparisons of cancer management are often based on  the net survival associated with the entire population or subgroups of interest, which is calculated as the marginal or  average net survival  SN(t) =  �  SN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)dϕ(x),  where ϕ is the distribution function of covariates x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In population studies, where the population of interest has covariates  x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' , xm, this quantity is usually calculated as  SN(t) =  m  �  i=1  SN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Several parametric [13, 16, 17, 25, 29, 34], semi-parametric [36], and nonparametric [28] methods have been proposed  in the literature for estimating the excess hazard and the net survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  One of the challenges in estimating the excess hazard function is that not all of the relevant covariates may be available  at the population level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Unobserved individual heterogeneity (UIH) refers to the situation where important covariates  are not recorded, or unavailable in the sample, potentially combined with model misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The effects of not  accounting for UIH can be substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' For instance, in the context of overall survival analysis, Aalen[3] and Aalen et  al[4] show that neglecting UIH induces a bias on the estimation of the parameters, as well as affecting the interpretation  of some epidemiological measures such as the hazard function and incidence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Similar effects are observed by UIH  produced by model misspecification [21] or missing covariates [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In addition, UIH can also affect model selection  [32] and the estimation of causal effects [37], emphasising the importance of accounting for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Individual frailty models represent a tractable option for accounting for UIH [1, 8, 9, 15, 20, 23, 40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The idea  behind frailty models consists of multiplying the individual hazard function by a random correction that follows a  parametric distribution G (see section 2 for more details on this point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This allows for accounting for non-specific  departures from the original model (without a frailty term), in the sense that these departures may be a result of model  misspecification or unavailable relevant covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In the relative survival framework, Zahl [42] investigated the use of  correlated frailty models, which include two random frailties that affect the hazard function associated with other causes  of death and the excess hazard, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' However, they encountered a number of inferential issues as his proposal  added five parameters which cannot be simultaneously estimated, unless an arbitrary restriction of the parameter space  is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Indeed, jointly modelling the two competing risks (associated to other causes and cancer) is a challenging  problem that may lead to non-identifiable models,[38] such as that proposed by Zahl [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Goeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' [19] studied  the use of frailties in a combination of concurrent and excess hazard models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This is, assuming a further decomposition  of the excess hazard into a hazard associated with a concurrent disease (assumed to be known) and the hazard associated  with cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' They assumed that the frailty term affects both the concurrent disease hazard and the excess hazard equally,  and focused on modelling the excess hazard using a proportional hazards model with a piecewise baseline hazard and  without time dependent effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' It is important to notice that assuming a simple model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' without time-dependent  effects) may induce a variance inflation of the frailty term, as this term captures UIH due to missing covariates or model  misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Rancoita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' [30] investigated the use of shared frailty models, which assume that the random frailty  simultaneously affects the hazard associated with other causes and the excess hazard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Rubio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' [35] studied the case  2  XXXX  A PREPRINT  where there is a potential mismatch in the estimation of the hazard associated with other causes, as a consequence of  using an insufficiently stratified life table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' They proposed correcting this mismatch using a random frailty affecting only  the hazard associated with other causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' To our knowledge, the specific case of frailty models that can account for UIH  on the excess hazard model has not been studied in the literature, neither the consequences of UIH in the estimation of  net survival, which motivates our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  In this paper, we propose a frailty excess hazard model which can account for potential UIH on the excess hazard  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This is, we assume that the UIH affects the excess hazard, either as a consequence of missing covariates or  model misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In order to ameliorate concerns about model misspecification, we model the excess hazard  using flexible parametric models with a rich hazard structure [34], which can incorporate covariates that affect the  time scale as well as covariates that act on the hazard scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Such models include popular hazard structures (such  as proportional hazards and accelerated failure time) as particular cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We derive some properties of this model,  and characterise the marginal hazard and survival functions for several choices of the frailty distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We provide  intuitive interpretations of these functions, and discuss identifiability of the proposed frailty model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The resulting  models are available in closed form, allowing for a tractable implementation and estimation of the parameters using  maximum likelihood methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In section 2, we present the frailty model and derive the marginal hazard and  survival functions, which are required for writing down the likelihood function, and discuss the interpretation of  these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We consider several frailty distributions that lead to closed-form expressions of the marginal hazard  and survival functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In section 3, we present the flexible parametric models for the excess hazard, and discuss  interpretation of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In section 4, we present the expression of the likelihood function and discuss point and  interval estimation of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We discuss tools for selecting the models with and without frailties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Section 5  presents a simulation study that illustrates the performance of the proposed model in scenarios of practical interest,  and illustrates the ability of the individual frailty model to recover the marginal net survival in the presence of UIH  (due to the omission of one covariate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Section 6 presents an application using data from patients diagnosed with  lung cancer in England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Section 7 concludes with a summary of our results and a general discussion about the use of  frailty models in the relative survival framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The methodology is implemented in the R package IFNS available at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='com/FJRubio67/IFNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  2  Excess hazard frailty model  In this section, we consider a frailty model based on the hazard decomposition (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This model represents an extension  of frailty models [9] to the relative survival framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We calculate the marginal hazard and survival functions, which  will later be used to construct the likelihood function, and discuss the interpretation of the resulting expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We  also discuss several choices of the frailty distribution that lead to closed-form expressions of the hazard and survival  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Consider the following frailty model, where we include a frailty (random effect), λ ∼ G, which multiplies the excess  hazard function in (1) and is allowed to vary across individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' More specifically, consider the conditional hazard  model  ˜h(t | λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)  =  hP (age + t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year + t, z) + λhE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x),  (2)  λ  ∼  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  where G is an absolutely continuous cumulative distribution function, with support on R+ and unit mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In the  Supplementary material, we present additional details about the identifiability of the proposed model under the hazard  structure proposed in Section 3, which is guaranteed under the assumption of unit mean of the frailty distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The  conditional survival function is  ˜S(t | λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)  =  exp{−[HP (age + t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year + t, z) − HP (age;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year, z)]} exp [−λHE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)] ,  (3)  where HP (·) and HE(·) represent the cumulative hazards associated with hP (·) and hE(·), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Then, after  integrating out the frailty λ with respect to the distribution G (see Supplementary material), the subgroup-specific  marginal survival function, for a specific vector of covariates x, can be written as  ˜S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x) = exp{−HP (age + t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year + t, z) + HP (age;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year, z)}LG{HE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)},  (4)  3  XXXX  A PREPRINT  where LG{s} =  � ∞  0  e−srdG(r) denotes the Laplace transform of G evaluated at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Consequently, the marginal hazard  function is  ˜h(t)  =  − d  dt log ˜S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)  =  hP (age + t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year + t, z) − L′  G{HE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)}  LG{HE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)}hE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)  =  hP (age + t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year + t, z) + E[λ | To ≥ t]hE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x),  (5)  where To = min {TP , TC} is the observed survival time, TP is the time to death from other causes, and TC is the  time to death from cancer, and L′  G{u} =  ∂  ∂z LG{z}  ��  z=u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' A detailed derivation of the last equality is presented  in the Supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Equation (5) reveals that the effect of the frailty on the marginal excess hazard is  time-dependent and also depends on the choice of the frailty distribution G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This implies that the value of the marginal  excess hazard, for different values of t, may differ for different choices of the frailty distribution (in particular, for larger  values of the variance of the frailty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This becomes more evident by looking at the resulting expression of the Laplace  transforms associated with different distributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' gamma and Inverse Gaussian, shown in the Supplementary  material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Moreover, this allows us to interpret frailty excess hazard models in a similar way as in Balan et al[9],  where the frailty is interpreted, at a marginal level, as an element inducing a selection of healthier individuals among  cancer patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Moreover, this factor also accounts for unobserved heterogeneity and departures from the fitted model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Next, we consider a specific choice for the distribution G: a gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This choice allows for obtaining a  closed-form expression of the marginal survival function, in addition to its appealing flexibility and interpretability  of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In the Supplementary material, we also present the Laplace transforms associated with the Inverse  Gaussian and Power Variance Function (PVF) frailties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The PVF is a more general distribution with closed form  Laplace transform, but with one additional parameter [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The gamma distribution is a limit case of the PVF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Other  three-parameter frailty distributions leading to closed-form Laplace transforms are the compound Poisson distribution  [2] and the stable distribution [22], which include the gamma, Inverse Gaussian, among other distributions, as particular  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Other frailty distributions, which require numerical integration to calculate the marginal hazard and survival  functions, are reviewed in Rondeau et al[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Gamma Frailty  Consider the conditional hazard model (2) and suppose that λ ∼ Ga(µ, b), where Ga(µ, b) denotes a gamma distribution  with mean parameter µ > 0, scale parameter b > 0, and probability density function g(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' µ, b) =  r  µ  b −1  Γ  � µ  b  �  b  µ  b exp  �  −r  b  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Under the constraint that the mean parameter µ is set to 1, the scale parameter b of the frailty represents the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  With this assumption (µ = 1) it follows that the subgroup-specific marginal frailty survival function is given by  ˜S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)  =  exp {− [HP (age + t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year + t, z) − HP (age;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year, z)]}  {1 + bHE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)}  1  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  (6)  The subgroup-specific marginal frailty net survival function is  ˜SN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)  =  1  {1 + bHE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)}  1  b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  (7)  The subgroup-specific marginal frailty net survival (7) is the net survival marginalised with respect to the frailty  distribution but conditional on the covariates x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This is a corrected version of the classical net survival SN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x) =  exp [−HE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)], and we can see that limb→0 ˜SN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x) = SN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  The subgroup-specific marginal frailty hazard function is given by  ˜h(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x) = hP (age + t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' year + t, z) +  hE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x)  1 + bHE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  (8)  Expression (8) provides a nice interpretation of our approach since the observed hazard can be seen as a model with  a time-dependent weight function ω(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x, b) =  1  1 + bHE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x) on the excess hazard, which provides a functional  form that involves the scale or spread of the correction due to unobserved heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Moreover, ω(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x, b) is a  decreasing function of HE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x) and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Thus, an increase in these quantities induces a more pronounced frailty effect  (as 0 < ω(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x, b) ≤ 1, and ω(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x, b) = 1 represents no frailty effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Thus, the correction induced by the weight  4  XXXX  A PREPRINT  function ω(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x, b) can be interpreted as a reduction in the level of the excess hazard hE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x), due to frailties associated  with unobserved heterogeneity for specific values of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' However, it is important to notice that this does not  mean that the frailty correction always shrinks the excess hazard hE(·), as the model parameters are estimated jointly  with the scale parameter b in the weight function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Indeed, the excess hazard in the models with and without frailty are  not directly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This effect will be illustrated in the simulation study and the real-data application, where we  observe that the estimates of the parameters of the models with and without frailty do not necessarily coincide when  there is UIH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  3  Flexible parametric models for the excess hazard  To model the excess hazard, we consider the flexible parametric general hazard (GH) structure [14]:  hE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x, α, β, θ) = h0  �  t exp{w⊤α};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' θ  �  exp  �  x⊤β  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  (9)  where x ∈ Rp, w ∈ Rpt, α ∈ Rpt and β ∈ Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Typically w ⊆ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' h0(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' θ) represents a parametric baseline hazard  function with parameters θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This is a rich hazard structure which contains, as particular cases, the Proportional  Hazards (PH) model (α = 0), the Accelerated Hazards (AH) model (β = 0), and the Accelerated Failure Time (AFT)  model (α = β, w = x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' See [34] for an extensive discussion on the GH structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This structure allows for the  inclusion of time-scale effects (through w) and effects that act at the hazard level (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Allowing for a flexible excess  hazard model helps reducing UIH associated with model misspecification, which in turn is useful to focus our attention  on capturing the effect of potentially missing covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' These points have been discussed in the overall survival  framework in Gasparini et al[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The corresponding cumulative hazard function is  HE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' x, α, β, θ) = H0  �  t exp{w⊤α};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' θ  �  exp  �  x⊤β − w⊤α  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  (10)  The fact that the excess cumulative hazard function can be written in closed form facilitates the implementation of  the likelihood function (discussed in the next section) as well as simulating survival times from this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' More  specifically, simulating a random time-to-event from the GH model (9) is simple using the probability integral transform  (see also Rossell and Rubio32):  t = F −1  0  �  1 − exp  �  log(1 − u) exp(w⊤α − x⊤β)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' θ  �  exp(w⊤α)  ,  (11)  where u ∼ (0, 1), and F0 is the cumulative distribution function associated with the baseline hazard h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Analogously,  simulating from the frailty model (2) can be done as follows  t = F −1  0  �  1 − exp  �  log(1 − u) exp(w⊤α − x⊤β)/λ  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' θ  �  exp(w⊤α)  ,  where u ∼ (0, 1), and λ ∼ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Regarding the choice of the baseline hazard, Rubio et al[34] presents a discussion on different choices using flexible  parametric distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The desirable properties of this hazard function are: numerical tractability and the ability  to capture the basic shapes of the hazard (increasing, decreasing, unimodal, bathtub).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Rubio et al[34] mention the  Exponentiated Weibull (EW) and the Generalised gamma (GG), as particular choices for the baseline hazard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Here, we  consider the use of the Power Generalised Weibull (PGW) distribution [7], but we emphasise that any other flexible  baseline hazard distribution could be used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The PGW distribution is a flexible three-parameter distribution (with  scale parameter σ > 0 and two shape parameters ν, γ > 0), which can also capture the basic hazard shapes while having  tractable expressions for the hazard and survival functions [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The probability density function, survival function, and  hazard function of the PGW distribution are presented in the Supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We could also consider simpler  (2-parameter) baseline hazards that can capture specific basic shapes such as the Log-Normal distribution (unimodal),  log-logistic, and the gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  4  Inference  Throughout, let (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' , tn) be the survival times associated with n cancer patients in a population;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' (δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' , δn) be the  corresponding vital status indicators (0-alive, 1-dead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' xi ∈ Rp, i =, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' , n represent the vector of covariates available  for the ith patient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' agei be the age at diagnosis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' and yeari be the year of diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The likelihood function for the  individual frailty model (4)–(5) is  ˜L(α, β, σ, ν, γ, b)  ∝  n  �  i=1  �  hP (agei + ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' yeari + ti, zi) +  hE(ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' xi, α, β, σ, ν, γ))  1 + bHE(ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' xi, α, β, σ, ν, γ))  �δi  ×  1  {1 + bHE(ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' xi, α, β, σ, ν, γ)}  1  b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  5  XXXX  A PREPRINT  In contrast, the likelihood function associated with the model without frailty (1) is  L(α, β, σ, ν, γ)  ∝  n  �  i=1  [hP (agei + ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' yeari + ti, zi) + hE(ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' xi, α, β, σ, ν, γ)]δi × exp {−HE(ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' xi, α, β, σ, ν, γ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Point estimates of the parameters of these models will be obtained via maximum likelihood estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Confidence  intervals will be calculated using asymptotic normal approximations, which are justified by the typically large samples  in our applications in cancer epidemiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Briefly, let ψ be the full vector of parameters for the model of interest, we  consider the use of confidence intervals of the type �  ψ ± Z1− τ  2 diag  �  J− 1  2 ( �  ψ)  �  , where J(ψ) = −  ∂2  ∂ψ∂ψT log L(ψ) is  the negative of the Hessian matrix of the log-likelihood function under the appropriate parametrisation, and 1−τ ∈ (0, 1)  is the confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  We compare these two models using AIC and evaluate their performance using a simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We point out that the  use of the likelihood ratio test for comparing these two models is more complex as the distribution of the likelihood ratio  test statistic is not necessarily asymptotically chi-square with one degree of freedom (see, Chapter 8 of 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Moreover,  there exist a number of information criteria developed for mixed models that could also be used [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In particular, in  our applications the sample size is large enough that makes the AIC and the marginal AIC [26] very close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' A thorough  exploration of the use of different information criteria is beyond the aims of this paper and we refer the reader to Müller  et al[26] for a more extensive discussion on this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  5  Simulation Study  In this section, we present a simulation study where we investigate the performance of the frailty model under several  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The proposed scenarios are designed with specific objectives [12]: (Aim 1) to investigate the finite sample  performance of the proposed approach in different settings, and (Aim 2) to assess the effect of missing covariates (with  heterogeneous distributions) in subgroups of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='1  Aim 1: finite sample performance  Data generation and simulation design  We investigate the impact of (i) different sample sizes and (ii) different values of the regression coefficients on the  performance of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The different true values of these regression coefficients correspond to 3 different  scenarios (called Sc1, Sc2 or Sc3 hereafter), and for each scenario, we consider four different sample sizes (n ∈  {500, 1000, 2000, 5000}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' For each scenario, we generate 4 covariates: a continuous covariate (say age) was simulated  using a mixture of uniform distributions with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='25 probability for the range (30, 65), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='35 probability for (65, 75)  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='40 probability for (75, 85) years old;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' the remaining 3 covariates (say sex, X1 and X2) are binary with equal  probabilities of being 0 or 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' P(sex = 0) = P(sex = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  The simulation of the survival times is based on the additive decomposition of the overall mortality hazard as detailed  in equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We simulate the “other-causes” time-to event using the UK life tables, and the cancer event time  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' the time to event from the excess hazard) using the inverse transform method (11), assuming model (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This is, we  adopt a GH structure with a PGW baseline excess hazard, a gamma frailty with mean µ∗ = 1 and scale b∗ (its value  depends on the simulated scenario), and the effects of the covariates age, sex, X1 and X2 (section 3 details the use of  the inverse transform method with the PGW distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We simulate a random drop-out time assuming an exponential  distribution with a given rate r, the value of this rate being chosen in order to generate around 5% of random drop-out  in each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Finally, we assume an administrative censoring at 5 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Both sources of censoring (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' random  drop-out and administrative censoring) induce between 40% and 45% of censoring in total and for all the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We  generate and analyse M = 1000 samples in each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  For Sc1, the true values of the baseline distribution parameters are σ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='75, ν∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='75, γ∗ = 8, the scale of the  gamma frailty b∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5 and the regression coefficients α∗ = β∗ = (1, 1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We emphasise that these are complex  simulation scenarios where the 4 covariates are included and with both time-dependent and hazard-level effects, in  combination with a flexible baseline hazard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In practice, one would typically include only some of the covariates  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' age) as a time-dependent effect, thus reducing the effective number of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The aim of Sc1 is to  portray the performance of the proposed frailty model in very challenging scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  In Sc2 and Sc3, we assumed less complex true models than in Sc1, assuming only 2 covariates with time-dependent  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Moreover, the parameters defining the distribution of the baseline hazard and the variance of the random effect  6  XXXX  A PREPRINT  were also assumed different than in Sc1, in order to cover a suitable range of possible scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The true values used  for Sc2 and Sc3 can be found in the tables 1 and 2, respectively, in the Supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Analysis of the simulated data  We analyse the data using model (9) for the excess mortality hazard, assuming the same parametrisation as the one used  to simulate the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We report the mean of the estimated regression coefficients, the bias (the difference between the  mean of the estimated regression coefficients and the true value), the median of the estimated regression coefficients,  the empirical standard deviation, the mean (estimated) standard error, and the coverage proportions of asymptotic  confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Our approach has been fully implemented using R software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The optimisation step was conducted using the R command  ‘nlminb’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The standard errors of regression coefficients were derived from the Hessian matrix obtained with the  command ‘hessian’ (R package ‘numDeriv’), and the asymptotic 95% confidence intervals were approximated using  these standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' For the initial values for the optimisation step, we used the parameters estimated from a PH model  with the PGW distribution but without a frailty parameter (so we set to 1 the initial value for the scale of the gamma  distribution, and for the parameters corresponding to time-dependent effects we used as initial values the ones obtained  from the PH model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The cases where the optimiser did not converge (as indicated by ‘nlminb’) or when the command  ‘hessian’ produced ‘Inf’ or ‘NaN’ values were excluded: for Sc1, it represents 31, 16, 3, and 0 sets of estimates excluded  among 1000 estimates with n = 500, 1000, 2000, 5000 observations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' For Sc2, it represents 10, 1, 0, and  0 sets of estimates excluded among 1000 estimates with n = 500, 1000, 2000, 5000 observations, respectively, while  none were excluded in Sc3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Results  The results of Sc1 are detailed in Table 1 below, while the results for Sc2 and Sc3 are given in Tables 1 and 2 in the  Supplementary material, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' As expected, the bias decreases and the coverage gets closer to the nominal  value with increasing sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Using different values of regression coefficients and variance of the frailty does  not alter the performance and the results observed, as shown in Tables 1 and 2 in the Supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We  observed a different behaviour in the estimation of the regression coefficients associated with covariates compared to  those of the baseline hazard parameters and the scale parameter of the frailty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' For the regression coefficients associated  with covariates, the performances are good even with a small sample size of 500 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' For the baseline distribution  parameters and the scale of the frailty, the bias is high in situations with sample size of 500 or 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' When the sample  size is larger (2000 or 5000), the performances are good, with small bias and coverage near the nominal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  For Sc1 with 500 observations (and therefore non negligible bias on the baseline distribution parameters and the scale  of the frailty), we check the ability of our approach to recover the true subgroup-specific marginal net survival (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  integrated over the frailty distribution but conditional on covariates, see equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We could see that for the reference  group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' covariates values set to 0) and despite the bias aforementioned, the true subgroup-specific marginal net  survival is nicely recovered on average over the set of simulated samples (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='2  Aim 2: impact of missing covariates with heterogeneous distributions  We now explore two simulation scenarios as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The overall aim is to compare the use of the model fitted to the  entire population for predicting net survival for subgroups of the population, a common strategy in epidemiology, against  a stratified analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In the first scenario, we assume that there exist two subgroups of the population, defined by the  variable sex, with slightly different PGW baseline hazards (θ1 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5, 5) for sex = 1, and θ2 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5, 3), for  sex = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The variable sex is binary with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='6 of being 1 and probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='4 of being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' For each subgroup,  we generate 2 covariates: a continuous covariate (say age) was simulated using a mixture of uniform distributions  with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='25 probability for the range (30, 65), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='35 probability for (65, 75) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='40 probability for (75, 85) years old;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  and the variable X1 is also binary, with conditional probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='8 of being 1 if sex = 1, an conditional probability  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='4 of being 1 if sex = 0 (that is, it has a different distribution for each sex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In addition, we assume that the effects  of the covariates are different for each sex subgroup: for sex = 1, α = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5) and β = (1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' and for  sex = 0, α = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='25) and β = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In the second simulation scenario, we again assume that there  exist two subgroups of the population, defined by the variable sex, now with markedly different PGW baseline hazards  (θ1 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5, 5) for sex = 1, and θ2 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='75) for sex = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The covariates are simulated as in the first  scenario, and the true values of the regression coefficients α and β are chosen as in the first scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' For both scenarios,  we consider sample sizes n = 500, 1000, 5000 and censoring rate of approximately 65%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  For each scenario, we first fit the classical model without frailty (1) and the frailty model (8) with PGW baseline hazard  to each simulated data set omitting the variable X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We calculate the corresponding population net survival curves  7  XXXX  A PREPRINT  Parameter  True  MeanMLE  Bias  MedianMLE  Coverage  Mean StdErr  EmpSD  Scenario N=500  σ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='750  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='974  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='622  ν  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='498  γ  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='986  α1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='333  α2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='639  α3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='233  Table 1: Simulation results for Aim1, scenario 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' MeanMLE: Mean of the Maximum Likelihood Estimates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' MedianMLE:  median value of the Maximum Likelihood Estimates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Mean StdErr: Mean of the standard errors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' EmpSD: Empirical  standard deviation  by averaging the individual net survival curves, for both models, associated to each patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We compare the true  population net survival curve against the fitted net survival curves obtained with these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Then, we calculate the  net survival associated to the subgroups defined by sex = 0, 1 by taking the average of individual net survival curves  over the corresponding subsets of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Our aim here is to compare the net survival curves associated to the  entire population, and the use of the models fitted to the entire sample for net survival subgroup estimations in the  presence of UIH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  In a second stage, we perform a stratified analysis where we fit the classical model without frailty (1) and the frailty  model (8) to each data subset defined by the variable sex = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Then, we calculate the net survival curves associated  to the subgroups sex = 0, 1 with the average of the individual fitted net survival curves obtained with these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We  again compare the corresponding net survival curves against the true net survival for these subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  8  XXXX  A PREPRINT  0  1  2  3  4  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='0  Time  Net survival  True individual marginal net survival (reference group)  Sample−specific individual marginal net survivals  Mean of the sample−specific individual marginal net survivals  Figure 1: Scenario Sc1 with 500 observations: subgroup-specific marginal net survival for the reference group  Results are shown in the Supplementary material (Figures 1–2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In the first scenario and for all sample sizes (Figure 1  in the Supplementary material), we notice that the average fitted population net survival curves are very close to the true  population net survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The subgroup analysis also leads to close net survival curves, with a slightly worst fit from the  classical model (which remains for all sample sizes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The stratified analysis produces the best results as the average  fitted net survival curves are virtually the same as the true net survival curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In the second scenario (Figure 2 in the  Supplementary material), we notice that the average fitted population net survival curves are close to the true population  net survival, although a small bias is observed for both models and all sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The subgroup analysis leads to a  very large bias for both average curves, and we also notice a discrepancy between the classical and frailty models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This  bias is largely reduced by using a stratified analysis for all sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  6  Real data application: lung cancer epidemiology  We analyse a data set obtained from population-based national cancer registry of Non-Small Cell Lung Cancer (NSCLC)  female patients diagnosed in 2012 in England [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We include the following covariates: standardised age at diagnosis  (agec, continuous);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' tumour stage at diagnosis (stage, categorical I-IV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' the presence of cardiovascular comorbidities  (CVD, binary);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' the presence of Chronic Obstructive Pulmonary Disease (COPD, binary);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' and the standardised Income  Domain from the 2010 England Indices of Multiple Deprivation (IMD, continuous), defined at the Lower Super Output  Area level (as a measure of deprivation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Information on stage at diagnosis and comorbidities was obtained from linked  data (Hospital Episode Statistics -HES- and the Lung Cancer Audit data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The cohort of patients was followed-up until  the 31st of December 2015, at which time patients alive were right-censored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We restrict the analysis to women with no  missing covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The resulting sample size was n = 14557 patients with complete cases, among which no = 12138  died before the 31st of December 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The 25%, 50% and 75% quantiles of the patients’ age at diagnosis was 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='93,  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='64, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='23 while the mean was 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Among these patients, 2434 were diagnosed at Stage I, 1131 at Stage II, 3241  at Stage III, and 7751 at Stage IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Finally, 1954 patients were classified with a cardiovascular comorbidity and 3260  with a chronic obstructive pulmonary disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  In order to evaluate the effects of UIH, we compare eight models for this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Model C1: (classical) model without  frailty with time-level effect of agec (wi), and hazard-level (xi) effects of agec, IMD, stage, CVD, and COPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Model F1: frailty model with time-level effect of agec (wi), and hazard-level (xi) effects of agec, IMD, stage,  CVD, and COPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Model C2: model without frailty with time-level effect of agec (wi), and hazard-level (xi) effects  of agec, IMD, CVD, and COPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Model F2: frailty model with time-level effect of agec (wi), and hazard-level (xi)  effects of agec, IMD, CVD, and COPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Model C3: model without frailty with time-level effect of agec (wi), and  9  XXXX  A PREPRINT  hazard-level (xi) effects of agec, IMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Model F3: frailty model with time-level effect of agec (wi), and hazard-level  (xi) effects of agec, IMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Model C4: PGW baseline hazard without covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Model F4: frailty model with PGW  baseline hazard without covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Figure 2a shows the net survival for the entire cohort obtained with Models C1–C4 and F1–F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' These are obtained  using the formulas:  SN(t)  =  1  n  n  �  i=1  exp{−HE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' xi, �α, �β, �σ, �ν, �γ)}, Models C1–C4,  ˜SN(t)  =  1  n  n  �  i=1  1  �  1 + �bHE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' xi, �α, �β, �σ, �ν, �γ)  � 1  �b  , Models F1–F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  The AIC values for Models C1–C4 and F1–F4 are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The overall best model is Model F1, clearly  favouring a model with all covariates and a frailty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The best model among the classical models (C1–C4) was model C1,  also favouring the inclusion of all covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Some interesting conclusions arise from this model comparison and Figure  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Although AIC clearly favours Model F1, Figure 2a shows that the estimated population net survival curves obtained  for models C1–C4 and F1–F4 are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This indicates that, even though the data favours a frailty model, some  estimated quantities based on models with and without frailties may be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' However, this is far from being a general  conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Figure 2b shows that the net survival stratified by stage for models C1 and F1 exhibit marked differences  for early tumour stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' These differences are masked in Figure 2a due to the distribution of the tumour stage covariate,  which mostly contains late-stage patients (see the descriptive analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This is, since we are interested in estimating  quantities based on averages over subgroups of the population, the distribution of the corresponding covariates (patient  characteristics) plays a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Thus, although the estimated models are quite different (see Table 2), the distribution  of the covariates produces similar marginal net survival curves at the population level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' However, this conclusion does  not apply to the net survival functions stratified by stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Table 3 presents confidence intervals for the net survival  at times t = 1, 2, 3, 4 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' These confidence intervals are obtained using a Monte Carlo approximation based on  the asymptotic normality of the maximum likelihood estimators (see Rubio et al[34] for more details on this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' These  confidence intervals confirm that differences in the estimation of net survival are also present in interval estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Another point that deserves some attention is the interpretation of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' From Table 2, we can see that the  estimates of the parameters of the baseline hazards markedly differ for models C1 and F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The reason for this is that  the estimated baseline hazard associated to model C1 is decreasing, while the estimated baseline hazard associated to  model F1 is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This explains the differences in the estimates of the parameters, which are simply reflecting  this contrasting shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' However, we would like to emphasise that the baseline hazards associated to models C1 and F1  (more generally, models with and without frailty correction) are not directly comparable as (i) the estimates for model  F1 correspond to a marginal model (marginalised over the frailty term), and (ii) the baseline hazard in the frailty model  (8) controls both the excess hazard in the numerator and the time-varying weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Despite this difference in nature,  comparing them is still useful, as observing differences between the estimates in both models is an indication of UIH as  such discrepancies only appear when the variance of the frailty term is non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='51  Table 2: Maximum likelihood estimates and AIC for Models C1–C4 and F1–F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' 95% confidence intervals are also  presented for the parameters of the selected model, F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='0  time  Net Survival  Stage I  Stage II  Stage III  Stage IV  (a)  (b)  Figure 2: Lung cancer data: (a) net survival curves for the entire population, and (b) stratified net survival curves by  tumour stage at diagnosis (Model C1 - gray lines, Model F1 - black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='359  Table 3: Net survival (NS) at t = 1, 2, 3, 4 years and 95% confidence intervals: Classical Model (Model C1) and Frailty  Model (Model F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Stratified analysis  The differences in the net survival curves for Stages I–IV in Figure 2 suggest that there is UIH, and that, potentially,  the unobserved covariates may have a different conditional distribution at each of the strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' To explore this idea, we  now consider a stratified analysis where we study patients in three different strata: Stage I-II, Stage III, and Stage IV  tumours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' There are clinical and biological reasons for this kind of stratification, as different tumour stages indicate  the size of the tumour and spread to other organs (metastasis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We compare frailty and non-frailty models for each of  these groups to evaluate the effects of UIH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' For Stages I–II we fit the models: Model C1-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='II: model without frailty  with time-level effect of agec (wi), and hazard-level (xi) effects of agec, IMD, stage, CVD, and COPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Model  F1-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='II: frailty model with time-level effect of agec (wi), and hazard-level (xi) effects of agec, IMD, stage, CVD,  and COPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Models C1-III, F1-III, C1-IV, and F1-IV represent the corresponding models for Stage III and Stage IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Results are reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' For Stages I-II, the model without frailty is slightly favoured and the estimated frailty  variance is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' For Stage III, the frailty model is favoured, and we observe differences in the estimates of the model  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Similarly, for Stage IV, the frailty model is favoured, the estimated frailty variance is large, and there is a  clear discrepancy in the estimates of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Thus, we notice that there are different levels of UIH at  each of the tumour stage strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' From Figure 3, we can see that the net survival curves for the models with and without  frailty coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This indicates that, after stratifying the data by Stage, UIH has a smaller effect on the estimation of  net survival, even though there is evidence of UIH for Stages III and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' It is worth noticing that omitting important  covariates in the models has an effect on the estimation of the baseline hazard parameters and the regression coefficients  (see, for instance, the values of the estimates in 2 for the models F1, F2, and F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This is a well known phenomenon  11  XXXX  A PREPRINT  associated to omitting important covariates in hazard regression models (see Rossell and Rubio[32] for an extensive  discussion on the effects of omitting covariates in survival regression models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  C1-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='II  C1-III  F1-III  C1-IV  F1-IV  �σ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='64  Table 4: Maximum likelihood estimates for the models fitted on the stratified data by Stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  0  1  2  3  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='0  time  Net Survival  I−12  II−12  I−3  II−3  I−4  II−4  C1−I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='II  F1−I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='II  C1−III  F1−III  C1−IV  F1−IV  Figure 3: Lung cancer data: net survival curves for the models fitted on the stratified data by Stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Gray lines represent  the net survival curves associated to model C1 fitted to the corresponding stratum (Stages I-II, Stage III, or Stage IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Black lines represent the net survival curves associated to model F1 fitted to the corresponding stratum (Stages I-II,  Stage III, or Stage IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  7  Discussion  We have developed an extension of frailty models to the relative survival framework, and shown that these models  induce a selection of the healthier individuals among survivors of cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We have shown that the proposed excess  hazard frailty models are able to capture unobserved individual heterogeneity, a common challenge in population cancer  epidemiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We have proposed a general family of parametric excess hazard regression models which guarantee  identifiability of parameters, and which contains hazard structures of practical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The proposed model allows for  a tractable implementation of the likelihood function as we can include time-level effects while avoiding the need for  numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Nonetheless, we point out that other hazard structures could be used as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Although we have  focused on the gamma frailty distribution, due to its tractability and interpretability, we have also presented several  12  XXXX  A PREPRINT  alternative choices of the frailty distribution that lead to closed-form expressions (see Supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' A  thorough comparison of the impact of the choice of the frailty distribution is beyond the aims of this work, but this will  be considered in future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Our simulations show that the proposed model has good inferential properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The performance of the proposed  approach was evaluated in very challenging scenarios, with a small sample size (n = 500), substantial amount of  censoring (40%), and with a very complicated generating model (the 4 covariates considered had both an effect on  the time scale and on the hazard level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In real life applications, one would not expect to see all the covariates with  effects on both the time scale and the hazard level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' As shown in Aim 2 of our simulations, unobserved heterogeneity  may have a non-negligible effect if the model fitted to the entire population is used to obtain net survival curves for  heterogeneous subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This suggests that discrepancies between net survival curves associated to subgroups of the  population using the classical and the frailty models may indicate a non-negligible effect of unobserved heterogeneity,  and that a stratified analysis may help reduce this heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  We have shown that, unsurprisingly, the effects and interpretation of UIH that have been reported in the overall survival  framework [9] are also present in the relative survival framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Our work offers an additional perspective, which is  the exploration of the impact of UIH on the estimation of marginal quantities (such as net survival).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We have shown  that the proposed frailty model is able to detect UIH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' However, it is important to keep the practical limitations of doing  so in mind, as the presence of UIH can also be an indication of model inaccuracy or model misspecification [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We  have proposed a frailty GH model which allows for including time-varying effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The use of a general and flexible  model thus helps reduce UIH associated to model misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Detecting UIH beyond model misspecification  is particularly important when the model fitted to the entire population is also used to estimate marginal quantities  associated to subgroups of the population [39], as the subgroups of interest may have different generating models, or  different distributions of the included covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Thus, in the presence of UIH, it would be useful to compare the results  against those obtained with a stratified analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This is an interesting finding for analysts interested in estimating  marginal quantities for a subgroup of patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Of course, since UIH induces a bias on the estimation of the model  parameters (for models without individual frailties), the estimation of conditional quantities (such as excess hazards for  specific covariate values) will also be biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' We have also shown that UIH induces a selection effect over the follow-up  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This type of effect has been also discussed by Vaupel and Yashin [39], in the overall survival framework, who  found that the estimation of conditional (or individual) quantities can be severely affected by UIH in some scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  We have shown scenarios where a flexible 3-parameter distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' PGW) is needed for modelling the shape of  the baseline hazard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' There exist scenarios where a simpler baseline hazard model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' lognormal) or a simpler hazard  structure (such as the AFT or PH) might be favoured by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In such cases, a model selection tool (such as AIC or  BIC) would help identify the best model for the data (see Rubio et al[34] and Rubio and Drikvandi[33] for a discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  The real data application presents a clear illustration of the fact that discrepancies in the estimation of the net survival  obtained with models with and without frailty terms depend on the distribution of the patients’ characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This  is in line with the conclusions obtained in Aim 2 of the simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' The net survival is a marginal measure, as  it is calculated as the average over the observed covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Consequently, its estimation naturally depends on the  distribution of the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In practice, it is important to first compare the different models in order to detect UIH,  followed by an investigation of the effect of the presence of UIH on the estimation of the quantities of interest (such as  net survival).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Moreover, the real data application indicates that the level of UIH might depend on the observed covariate  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' tumour stage in our illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Therefore, extending our approach with a frailty distribution that depends on the  observed covariates could be a very interesting research avenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  Possible extensions of our work include the use of other parametric baseline hazard distributions, combined with  alternative hazard structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' additive hazard structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' In cancer epidemiology, there are other quantities of  interest which are derived from the estimated excess hazard, such as life years lost, among other quantities [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Thus,  quantifying the impact of UIH on the estimation of these quantities would be of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Since our real data application  involves the use of confidential data, we cannot make it publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' However, we have created a repository  (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='com/FJRubio67/IFNS) which contains an application using “The Simulacrum” data set,  which is a dataset that contains artificial patient-like cancer data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' This repository illustrates the proposed methods and  the software provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  To our knowledge, this is the first study where the impact of UIH on estimated quantities is assessed within the relative  survival framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content=' Given that the net survival is used for international comparisons and to inform policy-making, a  careful analysis of factors that may induce bias, such as UIH, on the estimation of this quantity seems appropriate, and  the methodology proposed in this paper provides a tool for addressing this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  13  XXXX  A PREPRINT  Financial disclosure  AB is funded by a Cancer Research UK programme grant (C7923/A29018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
+page_content='  References  [1] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQf2_6M/content/2301.01823v1.pdf'}
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+arXiv:2301.02063v1  [math.AT]  5 Jan 2023
+LINKED SPACES AND EXIT PATHS
+ÖDÜL TETİK
+Abstract. Given a span of ∞-groupoids one of whose legs is a Kan fibration
+and the other a cofibration, we construct an ∞-category, the exit path ∞-
+category of the span. This gives access to stratified geometry by means only
+of unstratified objects.
+Introduction
+This paper proposes a new, simplified, and ‘model-independent’ approach to strati-
+fied geometry. More precisely, we advocate a particular level of resolution at which
+to consider a stratified space that is good enough for many purposes: a linked space
+is a triple M, L, N of spaces together with a fibration π and a cofibration ι as in
+the diagram1
+L
+M
+N
+π
+ι
+To any such span we attach an ∞-category EX. Here, M and N model two strata,
+the former lower than the latter, L is their link, and EX is the exit path ∞-category
+à la Lurie–MacPherson–Treumann. This covers depth 1, and a similar approach to
+higher depth is possible (cf. Remark 1.7), as is a natural definition of maps of linked
+spaces. These will appear elsewhere in a treatment geared towards applications in
+factorisation homology and functorial field theory, but can be readily guessed from
+the depth-1 construction presented here.
+We will mention only a few works from the vast literature related to stratified space
+theory. Some categorical considerations, e.g., in the context of homotopically strati-
+fied spaces à la Quinn [Qui88], and in sister contexts of varying degrees of generality,
+appeared in [Mil09]; [Tre09]; [Woo09], [Lur17, App. A]. The more recent conically-
+smooth variety ([AFT17]), which includes Whitney-stratified spaces ([NV21]) and
+thus in particular algebraic and analytic varieties, has seen categorical treatment in
+a different direction, yet our main construction was originally inspired by a specific
+result in this context: [AFR18a, Lemma 3.3.5], which identifies the space of paths
+(in the exit path ∞-category) that start and end in a given pair of strata as the link
+of that pair. In Quinn’s context, a philosophically similar result is [Mil13, Theorem
+6.3], which characterises equivalences of homotopically stratified spaces by probing
+on strata and (pairwise) homotopy links only. From this point of view, we show
+here that, conversely, the strata and the links are enough to construct the exit path
+∞-category. We should mention that it may be beneficial to explore connections to
+the recent stratified homotopy theory(ies) of [Hai18]; [Nan19]; [Dou21]; [DW21]...
+1The two heads of the arrow π are only suggestive, as it need not surject if, e.g., L = ∅.
+1
+
+2
+ÖDÜL TETİK
+Our approach is quite different methodically and in intent, but completely compat-
+ible in effect. It affords a number of related luxuries, e.g.:
+(1) By asking for strata and links only, we escape the need to introduce strat-
+ified paths and higher paths in order to consider the homotopy theory of
+stratified spaces, as would otherwise be necessary.
+(2) Since we do not need spaces stratified by maps to Alexandrov posets, we
+bypass any and all impositions of regularity or smoothness, besides the
+conditions on the link maps mentioned above.
+(3) Concerning accommodation of examples present in the literature, we re-
+main happily agnostic about the particular definition of stratified space
+that applies.
+We should note with respect to point (1) that stratified paths do appear in our
+construction, but are induced directly by the datum of linked space. That the con-
+ditions imposed on π and ι result directly in the ‘exit path simplicial set’ (Defin-
+ition 2.2), defined for any span of simplicial sets as soon as ι is a cofibration, be-
+coming an ∞-category (Theorem 2.3) explains, in a sense, their incidence in known
+theory.
+Examples are produced whenever spans consisting of a fibration and a cofibration
+are given. We discuss here only one novel infinite-dimensional example concerning
+Grassmannians (Example 2.7).
+Aside from it, we very briefly discuss bordisms
+(Example 2.5) and defects (submanifolds) (Example 2.6).
+Our methods are elementary. We employ only some standard notions concerning
+∞-groupoids and ∞-categories ([Lur23, Part I]). The technical difficulty in the
+adjoining of non-invertible paths is combinatorial, and some auxiliary definitions
+we use to overcome it, although interesting in themselves, do not play a major role
+in practise: one knows the ‘exit index’ of an exit path (Definition 1.6) when one
+sees one. An amusing byproduct, which appears to be new, is an interpretation as
+exit path parametrisations of the Eilenberg–Zilber maps, incarnated here as shuffles
+(Construction 1.5 ff.).
+Acknowledgments. We thank A. S. Cattaneo and K. İ. Berktav for useful conver-
+sations. This research was supported by the NCCR SwissMAP, funded by the Swiss
+National Science Foundation. We acknowledge partial support of SNSF Grant No.
+200020_192080.
+Conventions. The set N of natural numbers includes zero.
+We denote by ∆
+the simplex category, and its objects by [n], n ∈ N.
+Unless stated otherwise,
+∆n = Hom∆(−, [n]) is the standard n-simplex, and we employ the Yoneda Lemma
+without mention. Coface and codegeneracy maps we simply call face and degener-
+acy maps. We say ∞-category to mean a quasicategory, and ∞-groupoid to mean a
+Kan complex. Cartesian products of simplicial sets are defined degreewise. Given
+two simplicial sets C, D, we write CD = Fun(D, C) for the simplicial set whose set
+�
+CD�
+k of k-simplices is the set of maps D×∆k → C of simplicial sets, together with
+the obvious simplicial maps. A cofibration of simplicial sets is a monomorphism.
+
+LINKED SPACES AND EXIT PATHS
+3
+1. Non-invertible paths
+Let M, L and N be ∞-groupoids. We wish to construct an ∞-category that inter-
+prets L as the space of non-invertible paths from M to N, without modifying the
+paths of M and N, and such that vertices remain exactly those of M ∐ N. To this
+end, we first need maps L → M, N, which play the respective roles of source and
+target. For the sake of clarity, we separated the construction into two steps: first,
+in this Section, we discuss of the ‘space’ of non-invertible paths, and then adjoin it
+to M ∐ N in Section 2.
+Definition 1.1. Let ι: L → N be a map of simplicial sets. We call the simplicial
+set P := Pι := L ×N{0} N∆1 the mapping cocylinder of ι.
+Remark. Definition 1.1 is a variation on the under-∞-category construction, and
+reduces to it if L = pt is the constant singleton, in that there is an equivalence
+ι(pt)/N ≃ pt ×N{0} N∆1. Note that otherwise the coslice ι/N does not model a
+space of paths starting in L: its simplices, as simplices of N, are higher-dimensional
+than required to begin with.
+Lemma 1.2. Let P be as in Definition 1.1. If N is an ∞-groupoid, then P ≃ L.
+Proof. We first observe that the source evaluation N∆1 → N{0} is a Kan fibration.
+Now, each fibre N∆1
+p
+≃ p/N, p ∈ N0, is contractible by virtue of being an under-
+∞-groupoid ([Lur23, 018Y]). This verifies condition (4) of [Lur23, 00X2], which
+implies that N∆1 → N{0} is an equivalence, or equivalently (by the same cited
+result), a trivial Kan fibration. As trivial Kan fibrations pull back to trivial Kan
+fibrations, the natural map s: P → L is one such. As it is in particular a Kan
+fibration, the same result implies that s is an equivalence.
+□
+The mapping cocylinder appears in classical topology as follows: in the analogous
+construction with spaces L, N and ι a continuous map, the natural map Pι → N is
+a fibration replacement for ι in view of a homotopy equivalence L ≃ Pι.
+Remark 1.3. There are two induced maps π, τ : P → M, N defined as the composi-
+tions in the diagram
+P
+N∆1
+N{1}
+L
+N{0}
+M
+π
+s
+τ
+⌜
+π
+ι
+where the map N∆1 → N{1} is given by precomposition with ∆{1}×∆k ֒→ ∆1×∆k.
+Ideally, one would adjoin P, using π: L → M, to M∐N as the space of non-invertible
+paths from M to N, by employing π, τ of Remark 1.3 as source and target maps,
+
+4
+ÖDÜL TETİK
+respectively. Unfortunately, for combinatorial reasons, P does not lend itself to this
+directly. Instead, we will extract data out of it that does.
+First, let us delineate the problem in order to motivate the construction to follow.
+Remark 1.4. A vertex of P is a path of N that starts at a point in ι(L). One may
+coherently view this as a path which starts in M, by projecting its source down to
+M via σ0, and which, analogously, ends in N via τ. For higher morphisms, however,
+a direct generalisation requires unnatural choices: for instance, a 1-morphism in P
+may be depicted as
+(1)
+•
+•
+•
+•
+where the bottom edge is in ι(L), and the top edge is in N. (We depict the ∆1-
+coordinate in a k-morphism of N∆1, i.e., in a map ∆1 × ∆k → N of simplicial sets,
+as the upwards vertical coordinate.) Two of the (non-degenerate) 2-simplices of N
+we may extract are
+(2)
+•
+•
+•
+and
+(3)
+•
+•
+•
+corresponding to two (1, 1)-shuffles ∆2 → ∆1×∆1 (à la Eilenberg–Mac Lane–Zilber
+[EZ53]; [EM53]; see also [Lur23, 00RF]).2 If we were to add (1) as a 2-morphism
+to M ∐ N, say with source edge the bottom one, then we would have to choose
+the hypotenuse of the triangle (2) as the target edge, and the vertical edge as the
+intermediate 12-edge. But we may equally well make the analogous choice with
+triangle (3), declaring the left vertical edge the source. The problem is that both
+types of triangles are required for composition: if we wish later to concatenate, say,
+a path in M with a (non-invertible) 1-morphism in P, then we need (assuming there
+is a lift to L) a triangle of the first type. Similarly, if we wish to concatenate a
+non-invertible 1-morphism with a path in N, we need a triangle of the second type.
+Construction 1.5 (exit shuffles). Any pair 1 ≤ j ≤ k of natural numbers determ-
+ines a (1, k − 1)-shuffle Sk
+j = Sj : ∆k → ∆1 × ∆k−1 by setting
+Sj =
+
+
+
+
+
+
+
+
+
+
+
+�
+0
+0
+· · ·
+0
+1j
+1j+1
+1
+· · ·
+1
+0
+1
+· · ·
+j − 1
+j − 1
+j
+j + 1
+· · ·
+k − 1
+�
+,
+j < k
+�
+0
+0
+0
+· · ·
+0
+0
+1
+0
+1
+2
+· · ·
+k − 2
+k − 1
+k − 1
+�
+,
+j = k
+2Triangle (2) is given by the 2-simplex of ∆1 × ∆1 defined by ([2] → [1], [2] → [1]) = ((0, 1 �→
+0; 2 �→ 1), (0 �→ 0; 1, 2 �→ 1)) in ∆. Triangle (3) is given by ((0 �→ 0; 1, 2 �→ 1), (0, 1 �→ 0; 2 �→ 1)).
+The hypotenuse in both triangles is the edge
+�
+[1] id
+−→ [1], [1] id
+−→ [1]
+�
+∈
+�
+∆1 × ∆1�
+1.
+
+LINKED SPACES AND EXIT PATHS
+5
+in path notation, where the subscript j indicates the column number, with column
+count starting at 0. This is the non-degenerate element of
+�
+∆1 × ∆k−1�
+k defined
+by the pair of maps [k] → [1], [k] → [k − 1] given by
+i �→
+�
+(0, i),
+i < j
+(1, i − 1),
+i ≥ j.
+We call Sj an exit shuffle, and j its exit index. It has multiple left inverses, but we
+will use a particular one, Ck
+j = Cj, defined to be postcomposition with the poset
+map [1] × [k − 1] → [k] given by
+(0, i) �→
+�
+i,
+i < j
+j − 1,
+i ≥ j
+(1, i) �→
+�
+j,
+i < j
+i + 1,
+i ≥ j .
+This choice for C is explained by Lemmas 1.10 and 1.11 below.
+Definition 1.6. Let ι: L → N be a map of simplicial sets. For k ≥ 1, we define
+P∆
+k−1 ⊂ Nk × {1, . . ., k}
+to be the subset consisting of pairs (γ, j) such that in the diagram
+∆k
+N
+∆1 × ∆k−1
+Sj
+γ
+Cj
+Γ=γ◦Cj
+the arrow Γ lifts to the mapping cocylinder, i.e., it is in the image of the natural
+map P → N∆1. We call a pair (γ, j) ∈ P∆
+∗ an exit path of index j.
+Remark 1.7 (exit indices at depth 1). In terms of ordinary stratified geometry,
+Construction 1.5 corresponds to the following phenomenon: a stratified k-simplex
+or k-chain ∆k → X of X is a map of stratified spaces, where ∆k = C
+k(pt) is
+the k-fold closed cone on the point.3 The closed cone C(Y ) of a stratified space
+Y → P = PY , where P is the stratifying poset (equipped with the Alexandrov
+topology so that downward-closed subsets are closed) has
+pt
+�
+{0}×Y
+[0, 1] × Y
+as its underlying space, and
+PC(Y ) = P⊳
+Y ,
+i.e., PY with a minimal element adjoined, as its stratifying poset, together with the
+obvious stratification C(Y ) → P⊳
+Y . Now, the stratified map ∆k → X comes with a
+commutative topological square
+∆k
+X
+P∆k
+PX
+f
+sf
+.
+3Within the context of this Remark, ∆k never denotes the standard k-simplex.
+
+6
+ÖDÜL TETİK
+Clearly we have P∆k ≃ [k] as posets. If PX ≃ {a ≺ b}, then the poset map sf
+is determined by a unique minimal ‘exit index’ j ∈ [k]. Namely, let j = 0 if sf
+is constant, or else let j be the smallest number such that sf(j − 1 ≺ j) = a ≺ b
+(referring to sf applied to an arrow). This is well-defined since [k] is connected. As
+we do not refer to stratified paths explicitly, however, the different levels (indices)
+at which a path may exit (from the stratum Xa) give for us different sorts of non-
+invertible paths.
+Note also that we do not consider exit shuffles of index 0, as
+the corresponding k-chains are competely contained within the smooth manifold
+Xb, and similarly we do not consider ‘j = k + 1’, i.e., paths contained within
+Xa. (Besides, these indices do not determine shuffles in the ordinary sense.) The
+generalisation, using multiple exit indices, of the depth-1 Construction 1.5 to higher
+depth is immediate from this analogy, albeit notationally heavy.
+The aim of Definition 1.6 is three-fold.
+• It helps group elements of P∆
+∗ into three classes (Definition 1.8), which will
+play different roles.
+• It ‘fixes orientation’, in the sense that the faces of γ that touch L are
+directed away from L due to the orientation of the accompanying Γ ∈ P∗.
+This precludes ‘paths from N to M’, i.e., enter paths.
+The orientation
+depends on the exit index, so:
+• Unequal pairs (γ, j) ̸= (γ, j′) ∈ P∆
+∗ that share their first coordinate play
+different roles, and this is indispensable, as will become clear.
+Definition 1.8. Let k ≥ 1, (γ, j) ∈ P∆
+k−1, and let di = ∂∗
+i be a face map. Then
+di(γ) is either
+• vertical if it does not factor as follows:
+∆k−1
+∆k
+N
+(∆{0} ∐ ∆{1}) × ∆k−1
+∆1 × ∆k−1
+∄
+∂i
+Sj
+γ
+Γ=γ◦Cj ;
+• or bottom if it factors as follows:
+∆k−1
+∆k
+∆{0} × ∆k−1
+∆1 × ∆k−1
+∃
+∂i
+Sj
+;
+• or top if it factors as follows:
+∆k−1
+∆k
+∆{1} × ∆k−1
+∆1 × ∆k−1
+∃
+∂i
+Sj
+.
+In the exit path ∞-category (Definition 2.2), vertical faces will remain non-invertible,
+bottom faces will become simplices in M, and top faces in N. Writing ‘di(γ) is ver-
+tical’, etc., is slightly abusive, since whether a face is vertical, bottom or top depends
+
+LINKED SPACES AND EXIT PATHS
+7
+stongly on the exit index. This should not cause any confusion because we do not
+use these adjectives in any other context. In [EZ50], ‘low’ and ‘upper’ were used in
+a similar context.
+Definition 1.9. Let k ≥ 1.
+For ∂i and Sj as in Definition 1.8, and for σi a
+degeneracy, we write
+♭k
+j,i = ♭j,i ∈ [k − 1] (resp. ♯k
+j,i = ♯j,i ∈ [k])
+for the smallest number whose image under
+Sj∂i : [k − 1] → [1] × [k − 1] (resp. under Sjσi : [k + 1] → [1] × [k − 1])
+has first coordinate 1. We leave ♭k
+k,k undefined.
+For instance, for k = 5, j = 2, i ≥ 2, we have ♭ = 2, but for i < 2 (with k, j
+unchanged), we have ♭ = 1; in general ♭ ∈ {j, j − 1}, depending on the relative
+positions of i and j in {0, . . . , k}. We will determine ♭ and ♯ explicitly in the proof
+after Definition 2.2: see (9) and (13).
+Lemma 1.10.
+• Let k ≥ 2 and assume (j, i) ̸= (k, k). The composition
+∆1 × ∆k−2
+∆k−1
+∆k
+∆1 × ∆k−1,
+C♭
+∂i
+Sj
+where ♭ = ♭k
+j,i, preserves the first coordinate.
+• The composition
+∆1 × ∆k
+∆k+1
+∆k
+∆1 × ∆k−1,
+C♯
+σi
+Sj
+where ♯ = ♯k
+j,i, preserves the first coordinate.
+Proof. This is a direct check.
+□
+Lemma 1.11. Let k ≥ 2. If (γ, j) ∈ P∆
+k−1 and di(γ) is vertical, then
+di(γ, j) :=
+�
+diγ, ♭k
+j,i
+�
+∈ P∆
+k−2.
+To illustrate, for k = 3,
+(4)
+•
+•
+•
+•
+•
+•
+
+8
+ÖDÜL TETİK
+is a vertical face of exit index ♭ = 2 = j − 1, where (γ, 3) itself, the ‘lower right’
+tetrahedron, is omitted. Similarly,
+(5)
+•
+•
+•
+•
+•
+•
+is a vertical face of index ♭ = 1 = j, where (γ, 1) is the upper left tetrahedron.
+Proof of Lemma 1.11. It suffices to consider the diagram
+(6)
+∆k−1
+∆k
+N
+∆1 × ∆k−1
+∆1 × ∆k−2
+S♭
+∂i
+Sj
+γ
+Cj
+Γ
+C♭
+d′
+Γ′
+,
+which commutes by construction. Lemma 1.10 implies in particular that the re-
+striction of d′ = Si∂iC to ∆{0} × ∆k−2 factors through ∆{0} × ∆k−1, which implies
+that Γ′ = Γd′ lifts to Pk−2, as desired. Note that the case j = i = k is precluded
+by verticality.
+□
+Remark 1.12. Lemma 1.11 does not promote to an if-and-only-if statement. Bottom
+faces also descend to P∆
+k−2, but in a different way. Top faces may or may not. These
+facts will play no role below.
+We close this section by noting the completely analogous fact for degeneracies.
+Lemma 1.13. Let k ≥ 1. If (γ, j) ∈ P∆
+k−1, then si(γ, j) :=
+�
+siγ, ♯k
+j,i
+�
+∈ P∆
+k .
+2. Exit paths
+Remark 2.1. If ι: L ֒→ N is a cofibration, then an exit path (γ, j) ∈ P∆
+k−1 determines
+a canonical (k − 1)-simplex ∆k−1 → L of L, namely (recall Definition 1.6) the
+restriction of Γ = γ ◦ Cj along ∆{0} × ∆k−1 ֒→ ∆1 × ∆k−1 factors then uniquely
+through L.
+We are now ready to give the main construction of this paper.
+Definition 2.2. Let a span
+S =
+�
+M
+π
+←− L
+ι֒−→ N
+�
+of simplicial sets be given, where ι is a cofibration. We define a new simplicial set,
+EX = EX(S), as follows:
+
+LINKED SPACES AND EXIT PATHS
+9
+• EX0 = M0 ∐ N0.
+• EXk = Mk ∐ P∆
+k−1 ∐ Nk for k ≥ 1.
+• Face and degeneracy maps restricted to Mk and Nk are those of M and N.
+• For k = 1 and γ = (γ, 1) ∈ P∆
+0 ⊂ N1, we set4
+d1(γ, 1) = π(d1γ) ∈ M0,
+d0(γ, 1) = τ(d0γ) ∈ N0.
+• For k ≥ 2, (γ, j) ∈ P∆
+k−1, and di a face map:
+– if diγ is vertical,5 then we set di(γ, j) =
+�
+diγ, ♭j,i ∈ P∆
+k−2
+�
+.6
+– if diγ is bottom, then we set di(γ, j) = π(diγ) ∈ Mk−1.
+– if diγ is top, then we set di(γ, j) = τ(diγ) ∈ Nk−1.
+• For k ≥ 1, (γ, j) ∈ P∆
+k−1, and si a degeneracy: si(γ, j) := (siγ, ♯j,i) ∈ P∆
+k .7
+Proof that EX is a simplicial set. We verify the simplicial identities. Below, we as-
+sume k ≥ 2 or k ≥ 3 depending on need, and that (γ, e) ∈ P∆
+k−1.
+didj = dj−1di for i < j: We start by showing that
+(7)
+♭k−1
+♭k
+e,j,i = ♭k−1
+♭k
+e,i,j−1.
+It helps to distinguish the cases
+(8)
+(1) e ≤ i < j, (2) i < e ≤ j, and (3) i < j < e.
+We have (by a direct check)
+(9)
+♭k
+e,j =
+�
+e,
+j ≥ e
+e − 1,
+j < e
+and thus if (1), then L := ♭k−1
+♭k
+e,j,i = ♭k−1
+e,i
+= e and R := ♭k−1
+♭k
+e,i,j−1 = ♭k−1
+e,j−1 = e.
+If (2), then L = ♭k−1
+e,i
+= e − 1 and R = ♭k−1
+e−1,j−1 = e − 1. Finally, if (3), then
+L = ♭k−1
+e−1,i = e − 2 and R = ♭k−1
+e−1,j−1 = e − 2. We should note that in the case (2),
+e is at least 1, and in (3) it is at least 2, so that the expressions make sense. This
+finishes the verification if all involved faces of (γ, e) involved are vertical. Otherwise,
+Lemma 1.10 and Diagram (6) imply the statement; in any of the cases where the
+case excluded in Lemma 1.10 is involved, the face in question is bottom. We will
+give this argument here once and will not repeat it in the verification of the other
+4(noting S = id, C = id if k = 1 (Construction 1.5), and using Remarks 1.3 and 2.1)
+5(Definition 1.8)
+6(Lemma 1.11)
+7(Lemma 1.13)
+
+10
+ÖDÜL TETİK
+simplicial identities below: consider the diagram
+(10)
+∆k−2
+∆k−1
+∆k
+N
+∆1 × ∆k−3
+∆1 × ∆k−2
+∆1 × ∆k−1
+S♭′
+∂i
+S♭
+∂j
+Se
+γ
+C♭′
+C♭
+Ce
+.
+Without loss of generality, say di(dj(γ)) = (∂j∂i)∗γ is bottom, so we need to show
+that so is dj−1di(γ).
+That S♭∂i factors through ∆{0} × ∆k−2 is equivalent to
+Se∂jC♭S♭∂i factoring thusly per Lemma 1.10. Now, Se∂jC♭S♭∂i = Se∂j∂i by the
+construction of C♭, and similarly Se∂j∂i = Se∂j∂iC♭′S♭′. Together with the same
+calculation for ∂i and ∂j replaced respectively by ∂j−1 and ∂i in Diagram (10), we
+see that
+(11)
+Se∂jC♭S♭∂i = Se∂j∂iC♭′S♭′
+and
+Se∂iC♭S♭∂j−1 = Se∂i∂j−1C♭′S♭′.
+The indices ♭(′) in the two equations are a priori not the same (as they are calculated
+for different pairs of indices themselves), but we just showed above in Equation (7)
+that the primed flats on the right hand sides do coincide. Combined with the same
+simplicial identity for N, this means that the right hand sides in (11) agree, which
+implies the statement.
+disj = sj−1di for i < j: Similarly, we first show
+(12)
+L = ♭k−1
+♯k
+e,j,i = ♯k−1
+♭k
+e,i,j−1 = R,
+using the cases (1)–(3) from (8). Note that
+(13)
+♯k
+e,j =
+�
+e,
+j ≥ e
+e + 1,
+j < e
+which, together with (9), implies that if (1), then L = ♭k−1
+e,i
+= e and R = ♯k−1
+e,j−1 = e.
+If (2), then L = ♭k−1
+e,i
+= e − 1 and R = ♯k−1
+e−1,j−1 = e − 1. Finally, if (3), then
+L = ♭k−1
+e+1,i = e and R = ♯k−1
+e−1,j−1 = e. Now, Lemma 1.10 and Diagrams (6) and
+(10) (mutatis mutandis; e.g., using (12) instead of (7) for (11)) again finish the
+verification, analogously to the above. We no longer mention this below.
+disj = id for i = j or i = j + 1: We show
+L = ♭k−1
+♯k
+e,j,i = e.
+If e ≤ j, then L = ♭k−1
+e,i
+= e. If i = j and j < e, then L = ♭k−1
+e+1,i = e. If i = j + 1
+and e ≥ i, then L = ♭k−1
+e+1,i = e. This covers all cases.
+disj = sjdi−1 for i > j + 1: We show
+L = ♭k−1
+♯k
+e,j,i = ♯k−1
+♭k
+e,i−1,j = R.
+If e ≤ j, then L = ♭k−1
+e,i
+= e = ♯k−1
+e,j
+= R. If j + 1 ≤ e < i − 1, then L = ♭k−1
+e+1,i =
+e = ♯k−1
+e−1,j = R. If e = i − 1, then L = ♭k−1
+e+1,i = e + 1 = ♯k−1
+e,j
+= R. If e = i, then
+L = ♭k−1
+e+1,i = e = ♯k−1
+e−1,j = R. Finally, if e > i, both sides are again equal to e.
+
+LINKED SPACES AND EXIT PATHS
+11
+sisj = sj+1si for i ≤ j: Finally, we show
+L = ♯k−1
+♯k
+e,j,i = ♯k−1
+♯k
+e,i,j+1 = R.
+Similarly to the first identity above, it helps to distinguish the cases
+(1) e ≤ i ≤ j, (2) i < e ≤ j, and (3) i ≤ j < e.
+If (1), then L = ♯k−1
+e,i
+= e = ♯k−1
+e,j+1 = R. If (2), then L = ♯k−1
+e,i
+= e+1 = ♯k−1
+e+1,j+1 = R.
+If (3), then L = ♯k−1
+e+1,i = e + 2 = ♯k−1
+e+1,j+1 = R.
+□
+Theorem 2.3. If M, L, N are ∞-groupoids, π: L → M is a Kan fibration, and
+ι: L → N is a cofibration, then EX
+�
+M
+π
+←− L
+ι−→ N
+�
+is an ∞-category.
+Definition 2.4. We call a span M
+π
+←− L
+ι−→ N of ∞-groupoids, with π a Kan
+fibration, and ι a cofibration, a linked ∞-groupoid or linked space, of depth 1. We
+call EX its exit path ∞-category.
+Proof of Theorem 2.3. A direct check of the weak Kan property is possible. We
+first give a verbose proof for inner 2-horns, and then briefly discuss the general
+case, which is analogous.
+Let h: Λ2
+1 → EX be given. There are only two nontrivial cases: (1) where the 01-
+edge is in M1 and the 12-edge in P∆
+0 , (2) where the 01-edge is in P∆
+0 and the 12-edge
+in N1. Note that two edges from P∆
+0 cannot give such an h: by construction, sources
+and targets of 1-simplices from P∆
+0 are all in M0 and N0, respectively. If (1), then
+the 01-edge lifts to L along π by assumption, and via ι embeds into N, so that,
+combining this with the 12-edge, we have a horn like the highlighted 2-simplex in
+(4) (the prism itself being unimportant). The lift in N to a 2-simplex H : ∆2 → N
+induces then by the construction of P∆
+1 an element H ∈ P∆
+1 with exit index 2, i.e.,
+a lift. (Note that since the lift in H of the 01-edge of h is bottom by construction,
+it is hit by d2.) If (2), then, forgetting P∆
+0 → N1, we first pick a lift in N2. The
+02-edge of such a lift has source vertex in ι(L0) by construction and so descends
+to P∆
+0 . The lift itself descends to P∆
+1 with exit index 1 (like the highlighted face
+2-simplex in (4)).
+Let now h: Λn
+i → EX be given, where 0 < i < n. If any faces of h are in Mn−1,
+we lift them along π to Ln−1 and then embed them them into Nn−1 via ι and so
+reduce the problem to the analogue of case (2) above. The lift H : ∆n → N in N
+descends to H ∈ P∆
+n−1, with exit index determined by the number of faces lifted
+from M.
+□
+Note that compatibility with [AFR18a, Lemma 3.3.5] is easily seen in our context
+from the construction of EX and from Lemma 1.2. (The definition of the exit path
+∞-category in [AFR18a] coincides up to equivalence with that in [Lur17, App. A]
+by another result of the former work.)
+We finally discuss a few examples that were motivating for the main construction
+of this paper. We keep the discussion very brief, as they will be subject of future
+work aimed at specific applications.
+
+12
+ÖDÜL TETİK
+Example 2.5 (Bordisms). Since we only explicitly treated depth 1, we restrict
+ourselves to manifolds with boundary, but the higher-depth treatment of corners is
+analogous. The linked space corresponding to a (smooth) manifold with boundary
+(M, ∂M) has lower stratum ∂M, higher stratum M ◦ = M \ ∂M, link L = ∂M,
+π = id∂M, and ι: ∂M ֒→ M ◦ given by the flow along a nowhere-vanishing inward-
+pointing vector field along the boundary for a chosen nonzero time. An equivalent
+way to pick ι is to consider a tubular neighbourhood of the boundary diffeomorphic
+(via such a vector field) to ∂M × [0, 1) ֒→ M, whose restriction ∂M × (0, 1) ֒→ M ◦
+to positive time hits the interior, and take ι to be the restriction to ∂M × { 1
+2}.
+Example 2.6 (Defects). With a smooth submanifold N ⊂ M of positive codimension
+we may associate a linked space with lower stratum N and higher stratum M \ N.
+The link is given by the sphere bundle of the normal bundle of N, with the obvious
+maps π and ι.
+For instance, the link of R ⊂ R3 is an open (infinite) cylinder,
+whereas the link of S1 ⊂ R3 is a closed cylinder, i.e., a torus.
+Example 2.7 (Depth-1 stratified Grassmannians). For n, k ∈ N, consider the span
+BO(n) × BO(k)
+BO(n)
+BO(n + k)
+π
+⊞
+where π is the coordinate projection and ⊞ is induced by direct-summing of vector
+spaces and the choice of a pairing function (bijection) N × N ∼= N. This gives sub-
+∞-categories of the stratified Grassmannian of [AFR18b] as treated in [Tet22].8 A
+higher-depth treatment can reconstruct the full ∞-category, but we leave this for
+future work.
+References
+[AFR18a]
+D. Ayala, J. Francis and N. Rozenblyum. ‘A stratified homotopy hypothesis’. Journal
+of the European Mathematical Society 21.4 (2018), 1071–1178. arXiv: 1502.01713 [math.AT].
+[AFR18b]
+D. Ayala, J. Francis and N. Rozenblyum. ‘Factorization homology I: Higher categor-
+ies’. Advances in Mathematics 333 (2018), 1042–1177. arXiv: 1504.04007 [math.AT].
+[AFT17]
+D. Ayala, J. Francis and H. L. Tanaka. ‘Local structures on stratified spaces’. Advances
+in Mathematics 307 (2017), 903–1028. arXiv: 1409.0501 [math.AT].
+[Dou21]
+S. Douteau. ‘Homotopy theory of stratified spaces’. Algebraic & Geometric Topology
+21.1 (2021), 507–541. arXiv: 1911.04921 [math.AT].
+[DW21]
+S. Douteau and L. Waas. ‘From homotopy links to stratified homotopy theories’
+(2021). arXiv: 2112.02394 [math.AT].
+[EM53]
+S. Eilenberg and S. Mac Lane. ‘On the Groups H(Π, n), I’. Annals of Mathematics
+58.1 (1953), 55–106.
+[EZ50]
+S. Eilenberg and J. A. Zilber. ‘Semi-simplicial complexes and singular homology’.
+Annals of Mathematics 51.3 (1950), 499–513.
+[EZ53]
+S. Eilenberg and J. A. Zilber. ‘On Products of Complexes’. American Journal of
+Mathematics 75.1 (1953), 200–204.
+[Hai18]
+P. J. Haine. ‘On the homotopy theory of stratified spaces’ (2018). arXiv: 1811.01119 [math.AT].
+[Lur17]
+J. Lurie. Higher Algebra. 2017. url: https://www.math.ias.edu/~lurie/papers/HA.pdf.
+[Lur23]
+J. Lurie. Kerodon. 2023. url: https://kerodon.net.
+8We slightly deviated from the map ⊞ used in [Tet22] by using a pairing function, but only up to
+an equivalence induced by it.
+
+REFERENCES
+13
+[Mil09]
+D. A. Miller. ‘Popaths and Holinks’. Journal of Homotopy and Related Structures 4.1
+(2009), 265–273. arXiv: 0909.1201 [math.AT].
+[Mil13]
+D. A. Miller. ‘Strongly stratified homotopy theory’. Transactions of the American
+Mathematical Society 365.9 (2013), 4933–4962.
+[Nan19]
+S. J. Nand-Lal. ‘A simplicial approach to stratified homotopy theory’. PhD thesis.
+The University of Liverpool, 2019.
+[NV21]
+G. Nocera and M. Volpe. ‘Whitney stratifications are conically smooth’ (2021). arXiv:
+2105.09243 [math.DG].
+[Qui88]
+F. Quinn. ‘Homotopically stratified sets’. Journal of the American Mathematical So-
+ciety (1988), 441–499.
+[Tet22]
+Ö. Tetik. ‘The stratified Grassmannian’ (2022). arXiv: 2211.13824 [math.AT].
+[Tre09]
+D. Treumann. ‘Exit paths and constructible stacks’. Compositio Mathematica 145.6
+(2009), 1504–1532. arXiv: 0708.0659 [math.AT].
+[Woo09]
+J. Woolf. ‘The fundamental category of a stratified space’. Journal of Homotopy and
+Related Structures 4.1 (2009), 359–387. arXiv: 0811.2580 [math.AT].
+Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich,
+Switzerland
+Email address: oeduel.tetik@math.uzh.ch
+
diff --git a/fdA0T4oBgHgl3EQfHf_u/content/tmp_files/load_file.txt b/fdA0T4oBgHgl3EQfHf_u/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..82468b61bbfeb25453e29394955e765fdc948053
--- /dev/null
+++ b/fdA0T4oBgHgl3EQfHf_u/content/tmp_files/load_file.txt
@@ -0,0 +1,502 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf,len=501
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='02063v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='AT]  5 Jan 2023  LINKED SPACES AND EXIT PATHS  ÖDÜL TETİK  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Given a span of ∞-groupoids one of whose legs is a Kan fibration  and the other a cofibration, we construct an ∞-category, the exit path ∞-  category of the span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' This gives access to stratified geometry by means only  of unstratified objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Introduction  This paper proposes a new, simplified, and ‘model-independent’ approach to strati-  fied geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' More precisely, we advocate a particular level of resolution at which  to consider a stratified space that is good enough for many purposes: a linked space  is a triple M, L, N of spaces together with a fibration π and a cofibration ι as in  the diagram1  L  M  N  π  ι  To any such span we attach an ∞-category EX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Here, M and N model two strata,  the former lower than the latter, L is their link, and EX is the exit path ∞-category  à la Lurie–MacPherson–Treumann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' This covers depth 1, and a similar approach to  higher depth is possible (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='7), as is a natural definition of maps of linked  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' These will appear elsewhere in a treatment geared towards applications in  factorisation homology and functorial field theory, but can be readily guessed from  the depth-1 construction presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  We will mention only a few works from the vast literature related to stratified space  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Some categorical considerations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=', in the context of homotopically strati-  fied spaces à la Quinn [Qui88], and in sister contexts of varying degrees of generality,  appeared in [Mil09];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' [Tre09];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' [Woo09], [Lur17, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' The more recent conically-  smooth variety ([AFT17]), which includes Whitney-stratified spaces ([NV21]) and  thus in particular algebraic and analytic varieties, has seen categorical treatment in  a different direction, yet our main construction was originally inspired by a specific  result in this context: [AFR18a, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='5], which identifies the space of paths  (in the exit path ∞-category) that start and end in a given pair of strata as the link  of that pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' In Quinn’s context, a philosophically similar result is [Mil13, Theorem  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='3], which characterises equivalences of homotopically stratified spaces by probing  on strata and (pairwise) homotopy links only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' From this point of view, we show  here that, conversely, the strata and the links are enough to construct the exit path  ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We should mention that it may be beneficial to explore connections to  the recent stratified homotopy theory(ies) of [Hai18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' [Nan19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' [Dou21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' [DW21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  1The two heads of the arrow π are only suggestive, as it need not surject if, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=', L = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  1  2  ÖDÜL TETİK  Our approach is quite different methodically and in intent, but completely compat-  ible in effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' It affords a number of related luxuries, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' :  (1) By asking for strata and links only, we escape the need to introduce strat-  ified paths and higher paths in order to consider the homotopy theory of  stratified spaces, as would otherwise be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  (2) Since we do not need spaces stratified by maps to Alexandrov posets, we  bypass any and all impositions of regularity or smoothness, besides the  conditions on the link maps mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  (3) Concerning accommodation of examples present in the literature, we re-  main happily agnostic about the particular definition of stratified space  that applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  We should note with respect to point (1) that stratified paths do appear in our  construction, but are induced directly by the datum of linked space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' That the con-  ditions imposed on π and ι result directly in the ‘exit path simplicial set’ (Defin-  ition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='2), defined for any span of simplicial sets as soon as ι is a cofibration, be-  coming an ∞-category (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='3) explains, in a sense, their incidence in known  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Examples are produced whenever spans consisting of a fibration and a cofibration  are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We discuss here only one novel infinite-dimensional example concerning  Grassmannians (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Aside from it, we very briefly discuss bordisms  (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='5) and defects (submanifolds) (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Our methods are elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We employ only some standard notions concerning  ∞-groupoids and ∞-categories ([Lur23, Part I]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' The technical difficulty in the  adjoining of non-invertible paths is combinatorial, and some auxiliary definitions  we use to overcome it, although interesting in themselves, do not play a major role  in practise: one knows the ‘exit index’ of an exit path (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='6) when one  sees one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' An amusing byproduct, which appears to be new, is an interpretation as  exit path parametrisations of the Eilenberg–Zilber maps, incarnated here as shuffles  (Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='5 ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Cattaneo and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' İ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Berktav for useful conver-  sations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' This research was supported by the NCCR SwissMAP, funded by the Swiss  National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We acknowledge partial support of SNSF Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  200020_192080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' The set N of natural numbers includes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  We denote by ∆  the simplex category, and its objects by [n], n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Unless stated otherwise,  ∆n = Hom∆(−, [n]) is the standard n-simplex, and we employ the Yoneda Lemma  without mention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Coface and codegeneracy maps we simply call face and degener-  acy maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We say ∞-category to mean a quasicategory, and ∞-groupoid to mean a  Kan complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Cartesian products of simplicial sets are defined degreewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Given  two simplicial sets C, D, we write CD = Fun(D, C) for the simplicial set whose set  �  CD�  k of k-simplices is the set of maps D×∆k → C of simplicial sets, together with  the obvious simplicial maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' A cofibration of simplicial sets is a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  LINKED SPACES AND EXIT PATHS  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Non-invertible paths  Let M, L and N be ∞-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We wish to construct an ∞-category that inter-  prets L as the space of non-invertible paths from M to N, without modifying the  paths of M and N, and such that vertices remain exactly those of M ∐ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' To this  end, we first need maps L → M, N, which play the respective roles of source and  target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' For the sake of clarity, we separated the construction into two steps: first,  in this Section, we discuss of the ‘space’ of non-invertible paths, and then adjoin it  to M ∐ N in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Let ι: L → N be a map of simplicial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We call the simplicial  set P := Pι := L ×N{0} N∆1 the mapping cocylinder of ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='1 is a variation on the under-∞-category construction, and  reduces to it if L = pt is the constant singleton, in that there is an equivalence  ι(pt)/N ≃ pt ×N{0} N∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Note that otherwise the coslice ι/N does not model a  space of paths starting in L: its simplices, as simplices of N, are higher-dimensional  than required to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Let P be as in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If N is an ∞-groupoid, then P ≃ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We first observe that the source evaluation N∆1 → N{0} is a Kan fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Now, each fibre N∆1  p  ≃ p/N, p ∈ N0, is contractible by virtue of being an under-  ∞-groupoid ([Lur23, 018Y]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' This verifies condition (4) of [Lur23, 00X2], which  implies that N∆1 → N{0} is an equivalence, or equivalently (by the same cited  result), a trivial Kan fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' As trivial Kan fibrations pull back to trivial Kan  fibrations, the natural map s: P → L is one such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' As it is in particular a Kan  fibration, the same result implies that s is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  □  The mapping cocylinder appears in classical topology as follows: in the analogous  construction with spaces L, N and ι a continuous map, the natural map Pι → N is  a fibration replacement for ι in view of a homotopy equivalence L ≃ Pι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' There are two induced maps π, τ : P → M, N defined as the composi-  tions in the diagram  P  N∆1  N{1}  L  N{0}  M  π  s  τ  ⌜  π  ι  where the map N∆1 → N{1} is given by precomposition with ∆{1}×∆k ֒→ ∆1×∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Ideally, one would adjoin P, using π: L → M, to M∐N as the space of non-invertible  paths from M to N, by employing π, τ of Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='3 as source and target maps,  4  ÖDÜL TETİK  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Unfortunately, for combinatorial reasons, P does not lend itself to this  directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Instead, we will extract data out of it that does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  First, let us delineate the problem in order to motivate the construction to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' A vertex of P is a path of N that starts at a point in ι(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' One may  coherently view this as a path which starts in M, by projecting its source down to  M via σ0, and which, analogously, ends in N via τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' For higher morphisms, however,  a direct generalisation requires unnatural choices: for instance, a 1-morphism in P  may be depicted as  (1) where the bottom edge is in ι(L), and the top edge is in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' (We depict the ∆1-  coordinate in a k-morphism of N∆1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=', in a map ∆1 × ∆k → N of simplicial sets,  as the upwards vertical coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=') Two of the (non-degenerate) 2-simplices of N  we may extract are  (2) and  (3) corresponding to two (1, 1)-shuffles ∆2 → ∆1×∆1 (à la Eilenberg–Mac Lane–Zilber  [EZ53];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' [EM53];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' see also [Lur23, 00RF]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='2 If we were to add (1) as a 2-morphism  to M ∐ N, say with source edge the bottom one, then we would have to choose  the hypotenuse of the triangle (2) as the target edge, and the vertical edge as the  intermediate 12-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' But we may equally well make the analogous choice with  triangle (3), declaring the left vertical edge the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' The problem is that both  types of triangles are required for composition: if we wish later to concatenate, say,  a path in M with a (non-invertible) 1-morphism in P, then we need (assuming there  is a lift to L) a triangle of the first type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Similarly, if we wish to concatenate a  non-invertible 1-morphism with a path in N, we need a triangle of the second type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='5 (exit shuffles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Any pair 1 ≤ j ≤ k of natural numbers determ-  ines a (1, k − 1)-shuffle Sk  j = Sj : ∆k → ∆1 × ∆k−1 by setting  Sj =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  �  0  0  · ·  0  1j  1j+1  1  · ·  1  0  1  · ·  j − 1  j − 1  j  j + 1  · ·  k − 1  �  ,  j < k  �  0  0  0  · ·  0  0  1  0  1  2  · ·  k − 2  k − 1  k − 1  �  ,  j = k  2Triangle (2) is given by the 2-simplex of ∆1 × ∆1 defined by ([2] → [1], [2] → [1]) = ((0, 1 �→  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' 2 �→ 1), (0 �→ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' 1, 2 �→ 1)) in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Triangle (3) is given by ((0 �→ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' 1, 2 �→ 1), (0, 1 �→ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' 2 �→ 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  The hypotenuse in both triangles is the edge  �  [1] id  −→ [1], [1] id  −→ [1]  �  ∈  �  ∆1 × ∆1�  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  LINKED SPACES AND EXIT PATHS  5  in path notation, where the subscript j indicates the column number, with column  count starting at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' This is the non-degenerate element of  �  ∆1 × ∆k−1�  k defined  by the pair of maps [k] → [1], [k] → [k − 1] given by  i �→  �  (0, i),  i < j  (1, i − 1),  i ≥ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  We call Sj an exit shuffle, and j its exit index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' It has multiple left inverses, but we  will use a particular one, Ck  j = Cj, defined to be postcomposition with the poset  map [1] × [k − 1] → [k] given by  (0, i) �→  �  i,  i < j  j − 1,  i ≥ j  (1, i) �→  �  j,  i < j  i + 1,  i ≥ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  This choice for C is explained by Lemmas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='10 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Let ι: L → N be a map of simplicial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' For k ≥ 1, we define  P∆  k−1 ⊂ Nk × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=', k}  to be the subset consisting of pairs (γ, j) such that in the diagram  ∆k  N  ∆1 × ∆k−1  Sj  γ  Cj  Γ=γ◦Cj  the arrow Γ lifts to the mapping cocylinder, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=', it is in the image of the natural  map P → N∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We call a pair (γ, j) ∈ P∆  ∗ an exit path of index j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='7 (exit indices at depth 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' In terms of ordinary stratified geometry,  Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='5 corresponds to the following phenomenon: a stratified k-simplex  or k-chain ∆k → X of X is a map of stratified spaces, where ∆k = C  k(pt) is  the k-fold closed cone on the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='3 The closed cone C(Y ) of a stratified space  Y → P = PY , where P is the stratifying poset (equipped with the Alexandrov  topology so that downward-closed subsets are closed) has  pt  �  {0}×Y  [0, 1] × Y  as its underlying space, and  PC(Y ) = P⊳  Y ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=', PY with a minimal element adjoined, as its stratifying poset, together with the  obvious stratification C(Y ) → P⊳  Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Now, the stratified map ∆k → X comes with a  commutative topological square  ∆k  X  P∆k  PX  f  sf  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  3Within the context of this Remark, ∆k never denotes the standard k-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  6  ÖDÜL TETİK  Clearly we have P∆k ≃ [k] as posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If PX ≃ {a ≺ b}, then the poset map sf  is determined by a unique minimal ‘exit index’ j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Namely, let j = 0 if sf  is constant, or else let j be the smallest number such that sf(j − 1 ≺ j) = a ≺ b  (referring to sf applied to an arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' This is well-defined since [k] is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' As  we do not refer to stratified paths explicitly, however, the different levels (indices)  at which a path may exit (from the stratum Xa) give for us different sorts of non-  invertible paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Note also that we do not consider exit shuffles of index 0, as  the corresponding k-chains are competely contained within the smooth manifold  Xb, and similarly we do not consider ‘j = k + 1’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=', paths contained within  Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' (Besides, these indices do not determine shuffles in the ordinary sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=') The  generalisation, using multiple exit indices, of the depth-1 Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='5 to higher  depth is immediate from this analogy, albeit notationally heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  The aim of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='6 is three-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  It helps group elements of P∆  ∗ into three classes (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='8), which will  play different roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  It ‘fixes orientation’, in the sense that the faces of γ that touch L are  directed away from L due to the orientation of the accompanying Γ ∈ P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  This precludes ‘paths from N to M’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=', enter paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  The orientation  depends on the exit index, so:  Unequal pairs (γ, j) ̸= (γ, j′) ∈ P∆  ∗ that share their first coordinate play  different roles, and this is indispensable, as will become clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Let k ≥ 1, (γ, j) ∈ P∆  k−1, and let di = ∂∗  i be a face map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Then  di(γ) is either  vertical if it does not factor as follows:  ∆k−1  ∆k  N  (∆{0} ∐ ∆{1}) × ∆k−1  ∆1 × ∆k−1  ∄  ∂i  Sj  γ  Γ=γ◦Cj ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  or bottom if it factors as follows:  ∆k−1  ∆k  ∆{0} × ∆k−1  ∆1 × ∆k−1  ∃  ∂i  Sj  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  or top if it factors as follows:  ∆k−1  ∆k  ∆{1} × ∆k−1  ∆1 × ∆k−1  ∃  ∂i  Sj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  In the exit path ∞-category (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='2), vertical faces will remain non-invertible,  bottom faces will become simplices in M, and top faces in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Writing ‘di(γ) is ver-  tical’, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=', is slightly abusive, since whether a face is vertical, bottom or top depends  LINKED SPACES AND EXIT PATHS  7  stongly on the exit index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' This should not cause any confusion because we do not  use these adjectives in any other context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' In [EZ50], ‘low’ and ‘upper’ were used in  a similar context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  For ∂i and Sj as in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='8, and for σi a  degeneracy, we write  ♭k  j,i = ♭j,i ∈ [k − 1] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' ♯k  j,i = ♯j,i ∈ [k])  for the smallest number whose image under  Sj∂i : [k − 1] → [1] × [k − 1] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' under Sjσi : [k + 1] → [1] × [k − 1])  has first coordinate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We leave ♭k  k,k undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  For instance, for k = 5, j = 2, i ≥ 2, we have ♭ = 2, but for i < 2 (with k, j  unchanged), we have ♭ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' in general ♭ ∈ {j, j − 1}, depending on the relative  positions of i and j in {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We will determine ♭ and ♯ explicitly in the proof  after Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='2: see (9) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Let k ≥ 2 and assume (j, i) ̸= (k, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' The composition  ∆1 × ∆k−2  ∆k−1  ∆k  ∆1 × ∆k−1,  C♭  ∂i  Sj  where ♭ = ♭k  j,i, preserves the first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  The composition  ∆1 × ∆k  ∆k+1  ∆k  ∆1 × ∆k−1,  C♯  σi  Sj  where ♯ = ♯k  j,i, preserves the first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' This is a direct check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  □  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Let k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If (γ, j) ∈ P∆  k−1 and di(γ) is vertical, then  di(γ, j) :=  �  diγ, ♭k  j,i  �  ∈ P∆  k−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  To illustrate, for k = 3,  (4) 8  ÖDÜL TETİK  is a vertical face of exit index ♭ = 2 = j − 1, where (γ, 3) itself, the ‘lower right’  tetrahedron, is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Similarly,  (5) is a vertical face of index ♭ = 1 = j, where (γ, 1) is the upper left tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' It suffices to consider the diagram  (6)  ∆k−1  ∆k  N  ∆1 × ∆k−1  ∆1 × ∆k−2  S♭  ∂i  Sj  γ  Cj  Γ  C♭  d′  Γ′  ,  which commutes by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='10 implies in particular that the re-  striction of d′ = Si∂iC to ∆{0} × ∆k−2 factors through ∆{0} × ∆k−1, which implies  that Γ′ = Γd′ lifts to Pk−2, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Note that the case j = i = k is precluded  by verticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  □  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='11 does not promote to an if-and-only-if statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Bottom  faces also descend to P∆  k−2, but in a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Top faces may or may not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' These  facts will play no role below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  We close this section by noting the completely analogous fact for degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If (γ, j) ∈ P∆  k−1, then si(γ, j) :=  �  siγ, ♯k  j,i  �  ∈ P∆  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Exit paths  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If ι: L ֒→ N is a cofibration, then an exit path (γ, j) ∈ P∆  k−1 determines  a canonical (k − 1)-simplex ∆k−1 → L of L, namely (recall Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='6) the  restriction of Γ = γ ◦ Cj along ∆{0} × ∆k−1 ֒→ ∆1 × ∆k−1 factors then uniquely  through L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  We are now ready to give the main construction of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Let a span  S =  �  M  π  ←− L  ι֒−→ N  �  of simplicial sets be given, where ι is a cofibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We define a new simplicial set,  EX = EX(S), as follows:  LINKED SPACES AND EXIT PATHS  9  EX0 = M0 ∐ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  EXk = Mk ∐ P∆  k−1 ∐ Nk for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Face and degeneracy maps restricted to Mk and Nk are those of M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  For k = 1 and γ = (γ, 1) ∈ P∆  0 ⊂ N1, we set4  d1(γ, 1) = π(d1γ) ∈ M0,  d0(γ, 1) = τ(d0γ) ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  For k ≥ 2, (γ, j) ∈ P∆  k−1, and di a face map:  – if diγ is vertical,5 then we set di(γ, j) =  �  diγ, ♭j,i ∈ P∆  k−2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='6  – if diγ is bottom, then we set di(γ, j) = π(diγ) ∈ Mk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  – if diγ is top, then we set di(γ, j) = τ(diγ) ∈ Nk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  For k ≥ 1, (γ, j) ∈ P∆  k−1, and si a degeneracy: si(γ, j) := (siγ, ♯j,i) ∈ P∆  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='7  Proof that EX is a simplicial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We verify the simplicial identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Below, we as-  sume k ≥ 2 or k ≥ 3 depending on need, and that (γ, e) ∈ P∆  k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  didj = dj−1di for i < j: We start by showing that  (7)  ♭k−1  ♭k  e,j,i = ♭k−1  ♭k  e,i,j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  It helps to distinguish the cases  (8)  (1) e ≤ i < j, (2) i < e ≤ j, and (3) i < j < e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  We have (by a direct check)  (9)  ♭k  e,j =  �  e,  j ≥ e  e − 1,  j < e  and thus if (1), then L := ♭k−1  ♭k  e,j,i = ♭k−1  e,i  = e and R := ♭k−1  ♭k  e,i,j−1 = ♭k−1  e,j−1 = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  If (2), then L = ♭k−1  e,i  = e − 1 and R = ♭k−1  e−1,j−1 = e − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Finally, if (3), then  L = ♭k−1  e−1,i = e − 2 and R = ♭k−1  e−1,j−1 = e − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We should note that in the case (2),  e is at least 1, and in (3) it is at least 2, so that the expressions make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' This  finishes the verification if all involved faces of (γ, e) involved are vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Otherwise,  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='10 and Diagram (6) imply the statement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' in any of the cases where the  case excluded in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='10 is involved, the face in question is bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We will  give this argument here once and will not repeat it in the verification of the other  4(noting S = id, C = id if k = 1 (Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='5), and using Remarks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='1)  5(Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='8)  6(Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='11)  7(Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='13)  10  ÖDÜL TETİK  simplicial identities below: consider the diagram  (10)  ∆k−2  ∆k−1  ∆k  N  ∆1 × ∆k−3  ∆1 × ∆k−2  ∆1 × ∆k−1  S♭′  ∂i  S♭  ∂j  Se  γ  C♭′  C♭  Ce  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Without loss of generality, say di(dj(γ)) = (∂j∂i)∗γ is bottom, so we need to show  that so is dj−1di(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  That S♭∂i factors through ∆{0} × ∆k−2 is equivalent to  Se∂jC♭S♭∂i factoring thusly per Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Now, Se∂jC♭S♭∂i = Se∂j∂i by the  construction of C♭, and similarly Se∂j∂i = Se∂j∂iC♭′S♭′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Together with the same  calculation for ∂i and ∂j replaced respectively by ∂j−1 and ∂i in Diagram (10), we  see that  (11)  Se∂jC♭S♭∂i = Se∂j∂iC♭′S♭′  and  Se∂iC♭S♭∂j−1 = Se∂i∂j−1C♭′S♭′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  The indices ♭(′) in the two equations are a priori not the same (as they are calculated  for different pairs of indices themselves), but we just showed above in Equation (7)  that the primed flats on the right hand sides do coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Combined with the same  simplicial identity for N, this means that the right hand sides in (11) agree, which  implies the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  disj = sj−1di for i < j: Similarly, we first show  (12)  L = ♭k−1  ♯k  e,j,i = ♯k−1  ♭k  e,i,j−1 = R,  using the cases (1)–(3) from (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Note that  (13)  ♯k  e,j =  �  e,  j ≥ e  e + 1,  j < e  which, together with (9), implies that if (1), then L = ♭k−1  e,i  = e and R = ♯k−1  e,j−1 = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  If (2), then L = ♭k−1  e,i  = e − 1 and R = ♯k−1  e−1,j−1 = e − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Finally, if (3), then  L = ♭k−1  e+1,i = e and R = ♯k−1  e−1,j−1 = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Now, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='10 and Diagrams (6) and  (10) (mutatis mutandis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=', using (12) instead of (7) for (11)) again finish the  verification, analogously to the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We no longer mention this below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  disj = id for i = j or i = j + 1: We show  L = ♭k−1  ♯k  e,j,i = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  If e ≤ j, then L = ♭k−1  e,i  = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If i = j and j < e, then L = ♭k−1  e+1,i = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If i = j + 1  and e ≥ i, then L = ♭k−1  e+1,i = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' This covers all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  disj = sjdi−1 for i > j + 1: We show  L = ♭k−1  ♯k  e,j,i = ♯k−1  ♭k  e,i−1,j = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  If e ≤ j, then L = ♭k−1  e,i  = e = ♯k−1  e,j  = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If j + 1 ≤ e < i − 1, then L = ♭k−1  e+1,i =  e = ♯k−1  e−1,j = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If e = i − 1, then L = ♭k−1  e+1,i = e + 1 = ♯k−1  e,j  = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If e = i, then  L = ♭k−1  e+1,i = e = ♯k−1  e−1,j = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Finally, if e > i, both sides are again equal to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  LINKED SPACES AND EXIT PATHS  11  sisj = sj+1si for i ≤ j: Finally, we show  L = ♯k−1  ♯k  e,j,i = ♯k−1  ♯k  e,i,j+1 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Similarly to the first identity above, it helps to distinguish the cases  (1) e ≤ i ≤ j, (2) i < e ≤ j, and (3) i ≤ j < e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  If (1), then L = ♯k−1  e,i  = e = ♯k−1  e,j+1 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If (2), then L = ♯k−1  e,i  = e+1 = ♯k−1  e+1,j+1 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  If (3), then L = ♯k−1  e+1,i = e + 2 = ♯k−1  e+1,j+1 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  □  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If M, L, N are ∞-groupoids, π: L → M is a Kan fibration, and  ι: L → N is a cofibration, then EX  �  M  π  ←− L  ι−→ N  �  is an ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We call a span M  π  ←− L  ι−→ N of ∞-groupoids, with π a Kan  fibration, and ι a cofibration, a linked ∞-groupoid or linked space, of depth 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We  call EX its exit path ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' A direct check of the weak Kan property is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We  first give a verbose proof for inner 2-horns, and then briefly discuss the general  case, which is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Let h: Λ2  1 → EX be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' There are only two nontrivial cases: (1) where the 01-  edge is in M1 and the 12-edge in P∆  0 , (2) where the 01-edge is in P∆  0 and the 12-edge  in N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Note that two edges from P∆  0 cannot give such an h: by construction, sources  and targets of 1-simplices from P∆  0 are all in M0 and N0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If (1), then  the 01-edge lifts to L along π by assumption, and via ι embeds into N, so that,  combining this with the 12-edge, we have a horn like the highlighted 2-simplex in  (4) (the prism itself being unimportant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' The lift in N to a 2-simplex H : ∆2 → N  induces then by the construction of P∆  1 an element H ∈ P∆  1 with exit index 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=',  a lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' (Note that since the lift in H of the 01-edge of h is bottom by construction,  it is hit by d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=') If (2), then, forgetting P∆  0 → N1, we first pick a lift in N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' The  02-edge of such a lift has source vertex in ι(L0) by construction and so descends  to P∆  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' The lift itself descends to P∆  1 with exit index 1 (like the highlighted face  2-simplex in (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Let now h: Λn  i → EX be given, where 0 < i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' If any faces of h are in Mn−1,  we lift them along π to Ln−1 and then embed them them into Nn−1 via ι and so  reduce the problem to the analogue of case (2) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' The lift H : ∆n → N in N  descends to H ∈ P∆  n−1, with exit index determined by the number of faces lifted  from M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  □  Note that compatibility with [AFR18a, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='5] is easily seen in our context  from the construction of EX and from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' (The definition of the exit path  ∞-category in [AFR18a] coincides up to equivalence with that in [Lur17, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' A]  by another result of the former work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=')  We finally discuss a few examples that were motivating for the main construction  of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' We keep the discussion very brief, as they will be subject of future  work aimed at specific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  12  ÖDÜL TETİK  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='5 (Bordisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Since we only explicitly treated depth 1, we restrict  ourselves to manifolds with boundary, but the higher-depth treatment of corners is  analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' The linked space corresponding to a (smooth) manifold with boundary  (M, ∂M) has lower stratum ∂M, higher stratum M ◦ = M \\ ∂M, link L = ∂M,  π = id∂M, and ι: ∂M ֒→ M ◦ given by the flow along a nowhere-vanishing inward-  pointing vector field along the boundary for a chosen nonzero time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' An equivalent  way to pick ι is to consider a tubular neighbourhood of the boundary diffeomorphic  (via such a vector field) to ∂M × [0, 1) ֒→ M, whose restriction ∂M × (0, 1) ֒→ M ◦  to positive time hits the interior, and take ι to be the restriction to ∂M × { 1  2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='6 (Defects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' With a smooth submanifold N ⊂ M of positive codimension  we may associate a linked space with lower stratum N and higher stratum M \\ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  The link is given by the sphere bundle of the normal bundle of N, with the obvious  maps π and ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  For instance, the link of R ⊂ R3 is an open (infinite) cylinder,  whereas the link of S1 ⊂ R3 is a closed cylinder, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=', a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='7 (Depth-1 stratified Grassmannians).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' For n, k ∈ N, consider the span  BO(n) × BO(k)  BO(n)  BO(n + k)  π  ⊞  where π is the coordinate projection and ⊞ is induced by direct-summing of vector  spaces and the choice of a pairing function (bijection) N × N ∼= N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' This gives sub-  ∞-categories of the stratified Grassmannian of [AFR18b] as treated in [Tet22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='8 A  higher-depth treatment can reconstruct the full ∞-category, but we leave this for  future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  References  [AFR18a]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Ayala, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Francis and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Rozenblyum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' ‘A stratified homotopy hypothesis’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Journal  of the European Mathematical Society 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='4 (2018), 1071–1178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' arXiv: 1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='01713 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='AT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  [AFR18b]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Ayala, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Francis and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Rozenblyum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' ‘Factorization homology I: Higher categor-  ies’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Advances in Mathematics 333 (2018), 1042–1177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' arXiv: 1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='04007 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='AT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  [AFT17]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Ayala, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Francis and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Tanaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' ‘Local structures on stratified spaces’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Advances  in Mathematics 307 (2017), 903–1028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' arXiv: 1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='0501 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='AT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  [Dou21]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Douteau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' ‘Homotopy theory of stratified spaces’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Algebraic & Geometric Topology  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='1 (2021), 507–541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' arXiv: 1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='04921 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='AT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  [DW21]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Douteau and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Waas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' ‘From homotopy links to stratified homotopy theories’  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' arXiv: 2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='02394 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='AT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  [EM53]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Eilenberg and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Mac Lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' ‘On the Groups H(Π, n), I’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Annals of Mathematics  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
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+page_content=' ‘Semi-simplicial complexes and singular homology’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Annals of Mathematics 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Zilber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' ‘On Products of Complexes’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' American Journal of  Mathematics 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='1 (1953), 200–204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  [Hai18]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Haine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' ‘On the homotopy theory of stratified spaces’ (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' arXiv: 1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='01119 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='AT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  [Lur17]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Lurie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Higher Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='ias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='edu/~lurie/papers/HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  [Lur23]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Lurie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Kerodon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' url: https://kerodon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  8We slightly deviated from the map ⊞ used in [Tet22] by using a pairing function, but only up to  an equivalence induced by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
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+page_content=' Quinn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' ‘Homotopically stratified sets’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
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+page_content='  [Tet22]  Ö.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
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+page_content=' arXiv: 2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
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+page_content=' Compositio Mathematica 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
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+page_content=' arXiv: 0708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
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+page_content='  [Woo09]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
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+page_content=' ‘The fundamental category of a stratified space’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' Journal of Homotopy and  Related Structures 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='1 (2009), 359–387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content=' arXiv: 0811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='2580 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='AT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='  Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich,  Switzerland  Email address: oeduel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='tetik@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
+page_content='ch' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdA0T4oBgHgl3EQfHf_u/content/2301.02063v1.pdf'}
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+arXiv:2301.00428v1  [math.DG]  1 Jan 2023
+Weighted nonlinear flag manifolds as coadjoint orbits
+Stefan Haller1 and Cornelia Vizman2
+Abstract
+A weighted nonlinear flag is a nested set of closed submanifolds, each submanifold
+endowed with a volume density. We study the geometry of Fr´echet manifolds of weighted
+nonlinear flags, in this way generalizing the weighted nonlinear Grassmannians. When the
+ambient manifold is symplectic, we use these nonlinear flags to describe a class of coadjoint
+orbits of the group of Hamiltonian diffeomorphisms, orbits that consist of weighted isotropic
+nonlinear flags.
+1
+Introduction
+In this article we study manifolds of weighted nonlinear flags, motivated by the fact that
+one can use them to describe new coadjoint orbits of the Hamiltonian group. This adds to
+the already known coadjoint orbits described with configuration spaces of points, weighted
+isotropic nonlinear Grassmannians [26, 15, 7], symplectic nonlinear Grassmannians [8], and
+manifolds of symplectic nonlinear flags [9].
+Let M be a smooth manifold, and let S = (S1, . . . , Sr) be a collection of closed manifolds
+of strictly increasing dimensions. We consider the Fr´echet manifold FlagS(M) of nonlinear
+flags of type S, i.e., sequences of nested embedded submanifolds N1 ⊆ · · · ⊆ Nr in M, with
+Ni diffeomorphic to Si. Considering submanifolds equipped with nowhere zero densities, one
+obtains the manifold Flagwt
+S (M) of weighted nonlinear flags. We describe its Fr´echet manifold
+structure in two ways: as a splitting smooth submanifold of the cartesian product of weighted
+nonlinear Grassmannians of type Si in M, and as a locally trivial smooth fiber bundle over
+FlagS(M) associated to the principal bundle of nonlinear frames of type S in M. To each
+weighted nonlinear flag one associates a compactly supported distribution on M with controlled
+singularities, by the Diff(M) equivariant inclusion
+J : Flagwt
+S (M) ֒→ C∞(M)∗,
+�
+J((N1, ν1), . . . , (Nr, νr)), f
+�
+:=
+r
+�
+i=1
+�
+Ni
+fνi.
+The Diff(M) orbits in Flagwt
+S (M) are submanifolds of finite codimension, for which we give an
+explicit homological description, up to connected components, using nonlinear flags decorated
+with cohomology classes (see Theorems 2.11 and 2.15).
+Inspired by the results in [26, 15, 7] on weighted isotropic nonlinear Grassmannians, we
+consider the manifold of weighted isotropic nonlinear flags in a symplectic manifold (M, ω).
+The orbits for the natural action of the Hamiltonian group Hamc(M) are submanifolds of finite
+1Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
+stefan.haller@univie.ac.at
+2Department of Mathematics, West University of Timi¸soara, Bd. V. Parvan 4, 300223 Timi¸soara, Romania.
+cornelia.vizman@e-uvt.ro
+32020 Mathematics Subject Classification. 58D10 (primary); 37K65, 53C30, 53D20, 58D05.
+1
+
+codimension, described as leaves of an isodrastic foliation. Each isodrastic leaf of weighted
+nonlinear flags comes equipped with a canonical weakly non-degenerate symplectic form, and
+the map J restricts to an equivariant moment map for the Hamc(M) action, thus identifying
+the leaf with a coadjoint orbit of the Hamiltonian group (see Theorem 3.14). Moreover, this
+coadjoint orbit is a splitting symplectic submanifold in a product of coadjoint orbits of weighted
+submanifolds of type Si in M.
+The lowest dimensional examples are the coadjoint orbits of Hamc(R2) consisting of pointed
+weighted vortex loops, treated in [3]. We give more examples with nested spheres or tori, and
+provide explicit descriptions of the corresponding coadjoint orbits of the Hamiltonian group.
+Acknowledgments.
+The first author would like to thank the West University of Timi¸soa-
+ra for the warm hospitality. He gratefully acknowledges the support of the Austrian Science
+Fund (FWF) grant P31663. The second author would like to thank the University of Vienna
+for the warm hospitality. She was partially supported by CNCS UEFISCDI, project number
+PN-III-P4-ID-PCE-2020-2888.
+2
+Manifolds of weighted nonlinear flags
+A nonlinear flag is a sequence of nested closed submanifolds N1 ⊆ · · · ⊆ Nr in a smooth
+manifold M. A weighted nonlinear flag is a nonlinear flag together with a volume density νi
+on each submanifold Ni. Integrating against test functions f ∈ C∞(M), a weighted nonlinear
+flag provides a compactly supported distribution on M with mild singularities, �r
+i=1
+�
+Ni fνi.
+We will show that the space of all weighted nonlinear flags in M is a Fr´echet manifold
+in a natural way. In fact, this is the total space of a locally trivial smooth bundle over the
+manifold of nonlinear flags discussed in [9]. The natural Diff(M) action on the base of this
+bundle is locally transitive [9, Proposition 2.9(a)]. The main aim of this section is to describe
+the Diff(M) orbits in the space of weighted nonlinear flags, see Theorem 2.11 below.
+2.1
+Weighted nonlinear Grassmannians
+In this section we recall some basic facts about the manifolds of weighted submanifolds that
+appear in [26, 15, 7]. These weighted nonlinear Grassmannians constitute a special case of the
+weighted nonlinear flags to be introduced in Section 2.2. We present them here in a manner
+that readily generalizes to the setting of nonlinear flags.
+Let S be a closed manifold of dimension k, allowed to be nonconnected and nonorientable.
+For each manifold M, we let GrS(M) denote the nonlinear Grassmannian of type S in M,
+i.e., the space of all smooth submanifolds in M that are diffeomorphic to S. Moreover, we let
+EmbS(M) denote the space of all parametrized submanifolds of type S in M, i.e., the space of
+all smooth embeddings of S into M. Both, EmbS(M) and GrS(M), are Fr´echet manifolds in
+a natural way. Furthermore, the Diff(M) equivariant map
+EmbS(M) → GrS(M),
+ϕ �→ ϕ(S),
+(1)
+is a smooth principal bundle with structure group Diff(S), a Fr´echet Lie group, see [2, 19, 20]
+and [14, Theorem 44.1].
+For each closed k-dimensional manifold S, let
+Den(S) := Γ∞(|Λ|S) = Ωk(S; OS)
+2
+
+denote the space of all smooth densities on S. Here OS denotes the orientation bundle of S
+and |Λ|S = ΛkT ∗S ⊗OS, see e.g. [16]. Densities on S are the geometric quantities which can be
+integrated over S in a coordinate independent way, without specifying an orientation or even
+assuming orientability. We denote by Den×(S) the space of volume densities, i.e., the space of
+nowhere vanishing densities. Clearly, this is an open subset in the Fr´echet space Den(S).
+We define the weighted nonlinear Grassmannian of type S in M by
+Grwt
+S (M) :=
+�
+(N, ν)
+�� N ∈ GrS(M), ν ∈ Den×(N)
+�
+,
+(2)
+that is, the space of all submanifolds of type S in M, decorated with a nowhere zero density.
+We equip this space with the structure of a Fr´echet manifold by declaring the natural bijection
+Grwt
+S (M) = EmbS(M) ×Diff(S) Den×(S),
+�
+ϕ(S), ϕ∗µ
+�
+↔ [ϕ, µ],
+(3)
+to be a diffeomorphism. Here the right hand side denotes the total space of the bundle asso-
+ciated to the nonlinear frame bundle in (1) and the natural Diff(S) action on Den×(S). In
+particular, the canonical forgetful map
+Grwt
+S (M) → GrS(M),
+(N, ν) �→ N,
+(4)
+becomes a locally trivial smooth bundle with typical fiber Den×(S). Indeed, it corresponds to
+the bundle projection of the associated bundle EmbS(M) ×Diff(S) Den×(S) → GrS(M) via the
+identification in (3).
+There is a canonical Diff(M) equivariant map
+J : Grwt
+S (M) → C∞(M)∗,
+⟨J(N, ν), f⟩ :=
+�
+N
+fν.
+This map is injective, and its image consists of compactly supported distributions with mild
+singularities: J(N, ν) is supported on N, and its wave front set coincides with the conormal
+bundle of N.
+Let µ ∈ Den×(S) be a volume density. The space
+Grwt
+S,µ(M) :=
+�
+(N, ν) ∈ Grwt
+S (M)
+�� (S, µ) ∼= (N, ν)
+�
+is called the nonlinear Grassmannian of weighted submanifolds of type (S, µ) in M. It consists
+of all weighted submanifolds (N, ν) in M such that there exists a diffeomorphism S → N
+taking µ to ν. Denoting the Diff(S) orbit of µ by Den(S)µ, the identification in (3) restricts
+to a canonical bijection
+Grwt
+S,µ(M) = EmbS(M) ×Diff(S) Den(S)µ.
+(5)
+It is well known [21] that the Diff(S)0 orbit of µ is a convex subset that consists of all volume
+densities on S that represent the same cohomology class as µ in Hk(S; OS), the de Rham
+cohomology with coefficients in the orientation bundle. Hence, the Diff(S) orbit of µ coincides
+with the set of all volume densities on S that are in the preimage of Hk(S; OS)[µ], the (finite)
+Diff(S) orbit of [µ] in Hk(S; OS), under the Diff(S) equivariant linear map
+hS : Den(S) → Hk(S; OS),
+hS(α) = [α].
+(6)
+More succinctly,
+Den(S)µ = Den×(S) ∩ h−1
+S
+�
+Hk(S; OS)[µ]
+�
+.
+(7)
+3
+
+Hence, Den(S)µ is an open subset in a finite union of parallel closed affine subspaces with finite
+codimension. In particular, Den(S)µ is a splitting smooth submanifold in Den×(S) with finite
+codimension dim Hk(S; OS) and with tangent spaces
+Tα Den(S)µ = ker hS = dΩk−1(S; OS).
+(8)
+Using (5) we conclude that Grwt
+S,µ(M) is a splitting smooth submanifold in Grwt
+S (M) with finite
+codimension dim Hk(S; OS). Moreover, the canonical forgetful map in (4) restricts to a locally
+trivial smooth fiber bundle Grwt
+S,µ(M) → GrS(M) with typical fiber Den(S)µ.
+The space of cohomologically weighted submanifolds of type S in M is defined as
+Grhwt
+S
+(M) :=
+�
+(N, [ν]) : N ∈ GrS(M), [ν] ∈ Hk(N; ON)
+�
+.
+Using the canonical bijection
+Grhwt
+S
+(M) = EmbS(M) ×Diff(S) Hk(S; OS),
+(9)
+we turn Grhwt
+S
+(M) into a smooth vector bundle of finite rank dim Hk(S; OS) over GrS(M).
+The canonical Diff(M) equivariant map
+hGrS(M) : Grwt
+S (M) → Grhwt
+S
+(M),
+(N, ν) �→ (N, [ν]),
+is a smooth bundle map over GrS(M).
+Indeed, via the diffeomorphisms in (5) and (9) it
+corresponds to the map induced by (6).
+The space of cohomologically weighted submanifolds of type (S, [µ]) in M is defined by
+Grhwt
+S,[µ](M) :=
+�
+(N, [ν]) ∈ Grhwt
+S
+(M) : (N, [ν]) ∼= (S, [µ])
+�
+and consists of all cohomologically weighted submanifolds (N, [ν]) such that there exists a
+diffeomorphism S → N taking the cohomology class [µ] to [ν]. As (9) restricts to a bijection
+Grhwt
+S,[µ](M) = EmbS(M) ×Diff(S) Hk(S; OS)[µ],
+we see that Grhwt
+S,[µ](M) is a finite covering of GrS(M). Using (7) we conclude
+Grwt
+S,µ(M) = h−1
+GrS(M)
+�
+Grhwt
+S,[µ](M)
+�
+.
+(10)
+It is well known that the Diffc(M) action on EmbS(M) admits local smooth sections, see
+for instance [9, Lemma 2.1(c)]. Furthermore, the (transitive) Diff(S) action on Den(S)µ also
+admits local smooth sections. The latter can be shown using Moser’s method of proof in [21,
+Section 4], see Lemma 2.12 below.
+Using Lemma A.1 in the Appendix, we conclude that
+the natural Diffc(M) action on Grwt
+S,µ(M) admits local smooth sections. In particular, this
+action is locally transitive.
+Hence, each connected component of Grwt
+S,µ(M) is a Diffc(M)0
+orbit.
+Consequently, each Diffc(M) or Diff(M) orbit in Grwt
+S,µ(M) is a union of connected
+components.
+Remark 2.1. Poincar´e duality provides a canonical Diff(S) equivariant isomorphism
+Hk(S; OS) = H0(S; R).
+Hence, specifying a cohomology class [µ] ∈ Hk(S; OS) amounts to specifying the total volume
+of µ on each connected component of S.
+4
+
+Example 2.2. If S is connected, then Hk(S; OS) = R and the Diff(S) action is trivial on this
+cohomology. Hence, the orbit Hk(S; OS)[µ] is a one-point set, and
+Den(S)µ =
+�
+α ∈ Den×(S) :
+�
+S α =
+�
+S µ
+�
+(11)
+is connected. Correspondingly,
+Grwt
+S,µ(M) =
+�
+(N, ν) ∈ Grwt
+S (M) :
+�
+N ν =
+�
+S µ
+�
+.
+This is the case considered in [26, 15, 7].
+If S is built out of two diffeomorphic connected components, then Hk(S; OS) ∼= R2 and any
+diffeomorphism swapping the two connected components acts nontrivially on this cohomology.
+If µ has equal total volume on the two connected components, then the orbit Hk(S; OS)[µ] is
+a one-point set and Den(S)µ is connected. Otherwise Hk(S; OS)[µ] consists of two points and,
+by (7), Den(S)µ has two connected components.
+Remark 2.3. Suppose µ ∈ Den×(S).
+It is well known that Diff(S, µ), the group of diffeo-
+morphisms preserving µ, is a splitting Lie subgroup in Diff(S), see [11, Theorem III.2.5.3
+on page 203].
+Moreover, the map provided by the action, Diff(S) → Den(S)µ, f �→ f∗µ,
+is a smooth principal bundle with structure group Diff(S, µ). Via (5) this implies that the
+surjective and Diff(M) equivariant map
+EmbS(M) → Grwt
+S,µ(M),
+ϕ �→
+�
+ϕ(S), ϕ∗µ
+�
+,
+is smooth principal bundle with structure group Diff(S, µ).
+Remark 2.4. Suppose (N, ν) ∈ Grwt
+S (M) and let Grwt
+S (M)(N,ν) denote its Diffc(M) orbit.
+Combining the preceeding remark with the fact that the Diffc(M) action on EmbS(M) admits
+local smooth sections [9, Lemma 2.1(c)], we see that the map provided by the action,
+Diffc(M) → Grwt
+S (M)(N,ν),
+f �→
+�
+f(N), f∗ν
+�
+is a smooth principal bundle with structure group Diffc(M, N, ν), the group of diffeomorphisms
+preserving N and ν. The latter is a splitting Lie subgroup in Diffc(M), for it coincides with
+the preimage of Diff(N, ν) under the canonical bundle projection Diffc(M, N) → Diff(N), see
+[9, Lemma 2.1(d)]. Hence, each orbit may be regarded as a homogeneous space,
+Grwt
+S (M)(N,ν) = Diffc(M)/ Diffc(M, N, ν).
+2.2
+Weighted nonlinear flag manifolds
+Fix natural numbers ki such that
+0 ≤ k1 < k2 < · · · < kr
+(12)
+and let S = (S1, . . . , Sr) be a collection of closed smooth manifolds with dim Si = ki.
+For a smooth manifold M we let
+FlagS(M) :=
+�
+�
+N1, . . . , Nr) ∈
+r
+�
+i=1
+GrSi(M)
+�����∀i : Ni ⊆ Ni+1
+�
+denote the space of nonlinear flags of type S in M, and we write
+FrS(M) :=
+�
+(ϕ1, . . . , ϕr) ∈
+r
+�
+i=1
+EmbSi(M)
+�����∀i : ϕi(Si) ⊆ ϕi+1(Si+1)
+�
+5
+
+for the space of the space of nonlinear frames of type S in M. In [9, Proposition 2.3] it has
+been shown that FlagS(M) and FrS(M) are splitting smooth submanifolds of �r
+i=1 GrSi(M)
+and �r
+i=1 EmbSi(M), respectively. Moreover, the canonical Diff(M) equivariant map
+FrS(M) → FlagS(M),
+(ϕ1, . . . , ϕr) �→
+�
+ϕ1(S1), . . . , ϕr(Sr)
+�
+(13)
+is a smooth principal fiber bundle with structure group
+Diff(S) :=
+r
+�
+i=1
+Diff(Si).
+We denote the space of weighted nonlinear flags of type S in M by
+Flagwt
+S (M) :=
+�
+�
+(N1, ν1), . . . , (Nr, νr)
+�
+∈
+r
+�
+i=1
+Grwt
+Si (M)
+�����∀i : Ni ⊆ Ni+1
+�
+.
+(14)
+This is a splitting smooth submanifold in �r
+i=1 Grwt
+Si (M), for it coincides with the preimage
+of the splitting smooth submanifold FlagS(M) under the bundle projection �r
+i=1 Grwt
+Si (M) →
+�r
+i=1 GrSi(M). Moreover, the canonical Diff(M) equivariant forgetful map
+Flagwt
+S (M) → FlagS(M),
+�
+(N1, ν1), . . . , (Nr, νr)
+�
+�→ (N1, . . . , Nr),
+(15)
+is a smooth fiber bundle with typical fiber
+Den×(S) :=
+r
+�
+i=1
+Den×(Si).
+(16)
+The latter is a Diff(S) invariant open subset in the Fr´echet space Den(S) := �r
+i=1 Den(Si).
+Furthermore, the canonical Diff(M) equivariant bijection
+Flagwt
+S (M) = FrS(M) ×Diff(S) Den×(S),
+(17)
+��
+ϕ1(S1), (ϕ1)∗µ1
+�
+, . . . ,
+�
+ϕr(Sr), (ϕr)∗µr
+��
+↔
+�
+(ϕ1, . . . , ϕr), (µ1, . . . , µr)
+�
+,
+is a diffeomorphism between Flagwt
+S (M) and the bundle associated to the nonlinear frame
+bundle in (13) and the canonical Diff(S) action on Den×(S). Indeed, this is just the bundle
+diffeomorphism �r
+i=1 Grwt
+Si (M) = �r
+i=1 EmbSi(M) ×Diff(Si) Den×(Si) obtained by taking the
+product of the diffeomorphisms in (3), restricted over the submanifold FlagS(M) in its base
+�r
+i=1 GrSi(M).
+We have a canonical Diff(M) equivariant map
+J : Flagwt
+S (M) → C∞(M)∗,
+�
+J((N1, ν1), . . . , (Nr, νr)), f
+�
+:=
+r
+�
+i=1
+�
+Ni
+fνi.
+(18)
+The image of J consists of compactly supported distributions on M with mild singularities.
+More precisely, the wave front set of J((N1, ν1), . . . , (Nr, νr)) coincides with the union of the
+conormal bundles of N1, . . . , Nr.
+Lemma 2.5. The map in (18) is injective.
+6
+
+Proof. Suppose J((N1, ν1), . . . , (Nr, νr)) = J((N ′
+1, ν′
+1), . . . , (N ′
+r, ν′
+r)). Proceeding by induction
+on r, it suffices to show Nr = N ′
+r and νr = ν′
+r.
+To show Nr = N ′
+r, we assume by contradiction that there exists x ∈ Nr with x /∈ N ′
+r. Using
+(12), we see that Nr \Nr−1 is dense in Nr. Thus, we may w.l.o.g. assume x /∈ Nr−1. Moreover,
+νr(x) ̸= 0 as νr does not vanish on Nr. Hence, if f is a smooth bump function supported on
+a sufficiently small neighborhood of x, then ⟨J((N1, ν1), . . . , (Nr, νr)), f⟩ =
+�
+Nr fνr ̸= 0 and
+⟨J((N ′
+1, ν′
+1), . . . , (N ′
+r, ν′
+r)), f⟩ = 0. Since this contradicts our assumption, we conclude Nr = N ′
+r.
+To show νr = ν′
+r, we assume by contradiction that there exists x ∈ Nr = N ′
+r with
+νr(x) ̸= ν′
+r(x).
+As before, we may w.l.o.g. assume x /∈ Nr−1 and x /∈ N ′
+r−1.
+Hence, if
+f is a smooth bump function supported in a sufficiently small neighborhood of x, then
+⟨J((N1, ν1), . . . , (Nr, νr)), f⟩ =
+�
+Nr fνr ̸=
+�
+N′r fν′
+r = ⟨J((N ′
+1, ν′
+1), . . . , (N ′
+r, ν′
+r)), f⟩. Since this
+contradicts our assumption, we conclude νr = ν′
+r.
+Remark 2.6. Suppose ω is a symplectic form on M, and let Flagsymp
+S
+(M) denote the manifold
+of symplectic nonlinear flags of type S, cf. [9, Section 4.2]. Recall that this is the open subset
+consisting of all flags (N1, . . . , Nr) ∈ FlagS(M) such that ω restricts to a symplectic form on
+each Ni. Hence, ki must be even and ωki/2 pulls back to a volume form on Ni which in turn
+gives rise to a volume density νi = |ι∗
+Niωki/2| on Ni. Consequently, the symplectic form ω
+provides a Symp(M, ω) equivariant injective smooth map (section)
+Flagsymp
+S
+(M) → Flagwt
+S (M)
+(19)
+which is right inverse to the restriction of the canonical bundle projection in (15). Composing
+the map in (19) with J in (18), we obtain the moment map considered in [9, Eq. (38)].
+Remark 2.7. A Riemannian metric g on M induces a volume density on every submanifold
+of M. Hence, g provides a smooth section FlagS(M) → Flagwt
+S (M) of the canonical bundle
+projection in (15), which is Isom(M, g) equivariant, cf. [26, Section 6].
+2.3
+Reduction of structure group
+It will be convenient to use a reduction of the structure group for the principal frame bundle
+in (13). To this end, we fix embeddings ιi : Si → Si+1 and put ι = (ι1, . . . , ιr−1).
+We begin by recalling some facts from [9, Proposition 2.10]. The space of nonlinear flags
+of type (S, ι) in M,
+FlagS,ι(M) :=
+�
+(N1, . . . , Nr) ∈ FlagS(M)
+����
+�
+S1
+ι1
+−→ S2
+ι2
+−→ · · · → Sr
+�
+∼=
+�
+N1 ⊆ N2 ⊆ · · · ⊆ Nr
+�
+�
+,
+consists of all nonlinear flags (N1, . . . , Nr) in M such that there exist diffeomorphisms Si → Ni,
+1 ≤ i ≤ r intertwining ιi with the canonical inclusion Ni ⊆ Ni+1. This is a Diff(M) invariant
+open and closed subset in FlagS(M). The space of parametrized nonlinear flags (nonlinear
+frames) of type (S, ι) in M,
+FrS,ι(M) := {(ϕ1, . . . , ϕr) ∈ FrS(M)|∀i : ϕi = ϕi+1 ◦ ιi} ,
+is a splitting smooth submanifold of FrS(M).
+Moreover, the map FrS,ι(M) → FlagS,ι(M)
+obtained by restriction of (13) is a smooth principal bundle with structure group
+Diff(S; ι) :=
+�
+(g1, . . . , gr) ∈
+r
+�
+i=1
+Diff(Si)
+�����∀i : gi+1 ◦ ιi = ιi ◦ gi
+�
+.
+(20)
+7
+
+The latter is a splitting Lie subgroup in Diff(S) with Lie algebra
+X(S; ι) =
+�
+(Z1, . . . , Zr) ∈
+r
+�
+i=1
+X(Si)
+�����∀i : Zi+1 ◦ ιi = Tιi ◦ Zi
+�
+.
+(21)
+We obtain a Diff(M) equivariant commutative diagram
+FrS,ι(M)
+Diff(S;ι)
+�
+� �
+� FrS(M)
+Diff(S)
+�
+FlagS,ι(M)� �
+� FlagS(M).
+(22)
+which may be regarded as a reduction of the structure group for (13) along the inclusion
+Diff(S; ι) ⊆ Diff(S) over FlagS,ι(M), see [9, Proposition 2.10] for more details.
+Remark 2.8. The Diff(M) equivariant bijection
+FrS,ι(M) = EmbSr(M),
+(ϕ1, . . . , ϕr) �→ ϕr,
+(23)
+is a diffeomorphism [9, Proposition 2.10(b)]. Correspondingly, we have a group isomorphism
+Diff(S; ι) = Diff(Sr; Σ),
+(g1, . . . , gr) �→ gr,
+(24)
+where Diff(Sr; Σ) denotes the subgroup of all diffeomorphisms of Sr preserving the nonlinear
+flag Σ = (Σ1, . . . , Σr−1) in Sr, where Σi := (ιr−1 ◦ · · · ◦ ιi)(Si). The latter is a splitting Lie
+subgroup of Diff(Sr), see [9, Proposition 2.9(b)], and (24) is a diffeomorphism of Lie groups
+[9, Proposition 2.10(a)]. The Lie algebra of Diff(S; ι), can be identified in a similar way with
+X(Sr; Σ), the Lie algebra of vector fields on Sr that are tangent to Σ1, . . . , Σr−1.
+We are interested in the reduction of structure group (22) because the Diffc(M) action on
+FrS,ι(M) admits local smooth sections. This follows from [9, Lemma 2.1(c)] and the diffeo-
+morphism in (23).
+Let Flagwt
+S,ι(M) denote the preimage of FlagS,ι(M) under the bundle projection in (15).
+Restricting the diffeomorphism in (17) over FlagS,ι(M) and combining this with the diffeomor-
+phism in (23), we obtain a Diff(M) equivariant diffeomorphism of bundles over FlagS,ι(M),
+Flagwt
+S,ι(M) = EmbSr(M) ×Diff(S,ι) Den×(S).
+(25)
+2.4
+The Diff(M) action on the space of weighted nonlinear flags
+In this section we aim at describing the Diff(M) orbits in Flagwt
+S (M), see Theorem 2.11 below.
+Let ι = (ι1, . . . , ιr−1) be a collection of embeddings ιi : Si → Si+1 and suppose µ =
+(µ1, . . . , µr) ∈ Den×(S). We define the space of weighted flags of type (S, ι, µ) in M by
+Flagwt
+S,ι,µ(M) :=
+��
+(N1, ν1), . . . , (Nr, νr)
+�
+∈ Flagwt
+S (M)
+����
+�
+S1
+ι1
+−→ S2
+ι2
+−→ · · · → Sr, µ1, . . . , µr
+�
+∼=
+�
+N1 ⊆ N2 ⊆ · · · ⊆ Nr, ν1, . . . , νr
+�
+�
+,
+that is, the space of all weighted flags
+�
+(N1, ν1), . . . , (Nr, νr)
+�
+in M such that there exist
+diffeomorphisms Si → Ni, 1 ≤ i ≤ r, intertwining ιi with the canonical inclusion Ni ⊆ Ni+1,
+and taking µi to νi.
+8
+
+Denoting the Diff(S, ι) orbit of µ by Den(S)ι,µ, the diffeomorphism in (25) restricts to a
+Diff(M) equivariant bijection
+Flagwt
+S,ι,µ(M) = EmbSr(M) ×Diff(S,ι) Den(S)ι,µ.
+(26)
+Consider the finite dimensional vector space
+H(S, ι) :=
+r
+�
+i=1
+Hki�
+Si, ιi−1(Si−1); OSi
+�
+=
+r
+�
+i=1
+H0
+�
+Si \ ιi−1(Si−1); R
+�
+.
+(27)
+Here the left hand side denotes relative de Rham cohomology with coefficients in the orientation
+bundle, and we are using the convention S0 = ∅. The Diff(S, ι) equivariant identification on the
+right hand side indicates Poincar´e–Lefschetz duality. We have a Diff(S, ι) equivariant linear
+map
+hS,ι : Den(S) → H(S, ι),
+hS,ι(µ1, . . . , µr) :=
+�
+[µ1], . . . , [µr]
+�
+.
+(28)
+Pinning down the class [µ] := hS,ι(µ) thus amounts to specifying the integrals of µi over each
+connected component of Si \ ιi−1(Si−1) for i = 1, . . . , r.
+Remark 2.9 (Large codimensions). If the codimensions dim(Si) − dim(Si−1) are all strictly
+larger than one, then Hki�
+Si, ιi−1(Si−1); OSi
+�
+= Hki�
+Si; OSi
+�
+= H0(Si; R), and
+H(S, ι) =
+r
+�
+i=1
+Hki�
+Si; OSi
+�
+=
+r
+�
+i=1
+H0
+�
+Si; R
+�
+.
+Hence, in this case, the cohomology space H(S, ι) does not depend on the embeddings ι.
+Proposition 2.10. In this situation the following hold true:
+(a) The Diff(S, ι)0 orbit of µ coincides with the convex set Den×(S) ∩ h−1
+S,ι([µ]). In particular,
+this orbit is a splitting smooth submanifold in Den×(S) with finite codimension dim H(S, ι).
+(b) The Diff(S, ι)0 action on Den×(S) ∩ h−1
+S,ι([µ]) admits local smooth sections.
+(c) Denoting the (finite) Diff(S; ι) orbit of [µ] by H(S, ι)[µ], the Diff(S; ι) orbit of µ is
+Den(S)ι,µ = Den×(S) ∩ h−1
+S,ι
+�
+H(S, ι)[µ]
+�
+.
+(29)
+In particular, Den(S)ι,µ is a splitting smooth submanifold in Den×(S) with finite codimen-
+sion dim H(S, ι) and with tangent spaces
+Tα Den(S)ι,µ = ker hS,ι =
+�
+(dγ1, . . . , dγr) : γi ∈ Ωki−1(Si; OSi), ι∗
+i−1γi = 0
+�
+.
+(30)
+Moreover, the Diff(S, ι) action on Den(S)ι,µ admits local smooth sections.
+(d) The canonical inclusion Den(S)ι,µ ⊆ �r
+i=1 Den(Si)µi is a splitting smooth submanifold of
+finite codimension.
+We postpone the proof of this proposition and proceed with the main result in this section:
+Theorem 2.11. In this situation the following hold true:
+(a) The space Flagwt
+S,ι,µ(M) is a splitting smooth submanifold in Flagwt
+S,ι(M) with finite codi-
+mension dim H(S, ι).
+9
+
+(b) The canonical Diff(M) equivariant forgetful map Flagwt
+S,ι,µ(M) → FlagS,ι(M) is a locally
+trivial smooth fiber bundle with typical fiber Den(S)ι,µ.
+(c) The canonical inclusion Flagwt
+S,ι,µ(M) ⊆ �r
+i=1 Grwt
+Si,µi(M) is a splitting smooth submanifold.
+(d) The Diffc(M) action on Flagwt
+S,ι,µ(M) admits local smooth sections. In particular, each
+connected component of Flagwt
+S,ι,µ(M) is a Diffc(M)0 orbit. Furthermore, every Diff(M)
+or Diffc(M) orbit in Flagwt
+S,ι,µ(M) is a union of connected components.
+Proof. Parts (a) and (b) follow by combining (25) and (26) with Proposition 2.10(c).
+Part (c) follows from Proposition 2.10(d) and the reduction of structure groups in (22) via
+the diffeomorphisms in (5) and (26), see also (23).
+Let us finally turn to part (d). By Proposition 2.10(c), the (transitive) Diff(S; ι) action
+on Den(S)ι,µ admits local smooth sections. The Diffc(M) action on EmbSr(M) admits local
+smooth sections too, cf. [9, Lemma 2.1(c)]. Using Lemma A.1, we conclude that the Diffc(M)
+action on Flagwt
+S,ι,µ(M) admits local smooth sections, cf. (25).
+We will prove Proposition 2.10 by induction on the depth of the flags, using the following
+crucial lemma whose proof we postpone.
+Lemma 2.12. Let S be a closed submanifold of N such that dim(S) < dim(N) = n, and
+consider the Diff(N, S) equivariant linear map
+h : Den(N) → Hn(N, S; ON),
+h(µ) = [µ].
+Then, for each κ ∈ Hn(N, S; ON), the natural Diff(N, S)0 action on
+Diff(S) ×
+�
+Den×(N) ∩ h−1(κ)
+�
+(31)
+admits local smooth sections.
+Proof of Proposition 2.10. We proceed by induction on r using Lemma 2.12. Let us denote
+the truncated sequences by S′ := (S1, . . . , Sr−1), µ′ := (µ1, . . . , µr−1), and ι′ := (ι1, . . . , ιr−2).
+By induction, the Diff(S′, ι′)0 action on Den×(S′) ∩ h−1
+S′,ι′([µ′]) admits local smooth sections.
+Hence, there exist an open neighborhood U ′ of µ′ in Den×(S′) ∩ h−1
+S′,ι′([µ′]) and a smooth map
+Den×(S′) ∩ h−1
+S′,ι′([µ′]) ⊇ U ′ σ′
+−→ Diff(S′; ι′)0,
+such that for all ˜µ′ ∈ U ′ we have
+�
+σ′(˜µ′)
+�
+∗µ′ = ˜µ′
+and
+σ′(µ′) = id .
+(32)
+Recall that Diff(S′, ι′) is a splitting Lie subgroup in Diff(Sr−1), cf. [9, Propositions 2.9(b)
+and 2.10(a)]. Using [9, Lemma 2.1(d)] this implies that Diff(S, ι) is a splitting Lie subgroup in
+Diff(Sr, ιr−1(Sr−1)). Hence, restricting a local smooth section as in Lemma 2.12, we see that
+the Diff(S, ι)0 action on Diff(S′, ι′) ×
+�
+Den×(Sr) ∩ h−1([µr])
+�
+admits local smooth sections.
+In other words, there exist an open neighborhood V of the identity in Diff(S′; ι′), an open
+neighborhood U ′′ of µr in Den×(Sr) ∩ h−1([µr]), and a smooth map
+Diff(S′; ι′) ×
+�
+Den×(Sr) ∩ h−1([µr])
+�
+⊇ V × U ′′ σ′′
+−→ Diff(S; ι)0,
+10
+
+such that for all g ∈ V and ˜µr ∈ U ′′ we have
+σ′′(g, ˜µr) · (id, µr) = (g, ˜µr)
+and
+σ′′(id, µr) = id .
+(33)
+Hence, U := (σ′)−1(V ) × U ′′ is an open neighborhood of µ in Den×(S) ∩ h−1
+S,ι([µ]), and
+Den×(S) ∩ h−1
+S,ι([µ]) ⊇ U
+σ−→ Diff(S; ι)0,
+σ(˜µ1, . . . , ˜µr) := σ′′(σ′(˜µ1, . . . , ˜µr−1), ˜µr)
+is a local smooth section for the Diff(S; ι)0 action on Den×(S) ∩ h−1
+S,ι([µ]), i.e.,
+σ(µ) = id
+and
+σ(˜µ)∗µ = ˜µ,
+for all ˜µ ∈ U. By convexity, Den×(S) ∩ h−1
+S,ι([µ]) is connected. Therefore, the Diff(S; ι)0 action
+is transitive on Den×(S) ∩ h−1
+S,ι([µ]). This shows (a) and (b). Part (c) follows immediately.
+To see (d), let A denote the preimage of �r
+i=1 Hki(Si; OSi)[µi] under the canonical linear
+surjection H(S, ι) → �r
+i=1 Hki(Si; OSi).
+Hence, A is a finite union of affine subspaces in
+H(S, ι). In view of (7) we have �r
+i=1 Den(Si)µi = Den×(S) ∩ h−1
+S,ι(A). Combining this with
+(29), we conclude that Den(S)ι,µ is a splitting smooth submanifold in �r
+i=1 Den(Si)µi with
+finite codimension dim A.
+Let us next establish the following infinitesimal version of Lemma 2.12:
+Lemma 2.13. Let S be a closed submanifold of N such that dim(S) < dim(N) = n. Suppose
+µ ∈ Den×(N), γ ∈ Ωn−1(N, S; ON) := {α ∈ Ω(N; ON) : ι∗
+Sα = 0}, and Z ∈ X(S). Then there
+exists a vector field X ∈ X(N) such that LXµ = dγ and X|S = Z.
+Proof. Let ˜Z ∈ X(N) be any extension of Z, i.e. ˜Z|S = Z. Note that i ˜Zµ ∈ Ωn−1(N; ON)
+vanishes when pulled back to S, hence the same holds for β := γ − i ˜Zµ ∈ Ωn−1(N; ON).
+Let us first construct ˜γ ∈ Ωn−1(N; ON) such that d˜γ = dβ and ˜γ|S = 0. To this end, we fix
+a smooth homotopy h: N × [0, 1] → N such that h1 = idN, ht|S = idS, and such that h0 maps
+a neighborhood of S into S. Consider the corresponding chain homotopy φ: Ω∗(N; ON) →
+Ω∗−1(N; ON) defined by φ(α) :=
+� 1
+0 ι∗
+t i∂th∗α dt, where ιt : N → N ×[0, 1] denotes the inclusion
+at t, that is, ιt(x) := (x, t). Then φ(α)|S = 0 and d(φ(α)) + φ(dα) = h∗
+1α − h∗
+0α, for all forms
+α ∈ Ω∗(N; ON). In particular, dα = d
+�
+h∗
+0α + φ(dα)
+�
+. Defining ˜γ := h∗
+0β + φ(dβ), we obtain
+d˜γ = dβ. Moreover, h∗
+0β|S = 0 because β vanishes when pulled back to S, hence ˜γ|S = 0 as
+desired.
+Defining a vector field Y ∈ X(N) by iY µ := ˜γ, we obtain Y |S = 0 and LY µ = diY µ = d˜γ =
+d(γ − i ˜Zµ) = dγ − L ˜Zµ. Hence, the vector field X := ˜Z + Y has the desired properties.
+Proof of Lemma 2.12. Recall that the infinitesimal Diff(N, S) action on Diff(S) × Den(N) is
+ζX(f, µ) =
+�
+RX|S(f), −LXµ
+�
+,
+where X ∈ X(N, S), f ∈ Diff(S), and µ ∈ Den(S). Here, for Z ∈ Tid Diff(S) = X(S) we let
+RZ(f) denote the right invariant vector field on Diff(S) such that RZ(id) = Z.
+Note that the vector field ˜Z in the proof of Lemma 2.13 can be chosen to depend smoothly
+(and linearly) on Z. Hence, the proof of said lemma actually provides a smooth map
+˜σ : Tid Diff(S) × T
+�
+Den×(N) ∩ h−1(κ)
+�
+→ X(N, S)
+11
+
+such that ζ˜σ(Z,dγ)(id, µ) = (Z, dγ) for all Z ∈ X(S) = Tid Diff(S), µ ∈ Den×(N) ∩ h−1(κ), and
+dγ ∈ dΩn−1(N, S; ON) = Tµ
+�
+Den×(N) ∩ h−1(κ)
+�
+. Combining this with the right trivialization
+of T Diff(S), we obtain a smooth map
+σ : T
+�
+Diff(S) ×
+�
+Den×(N) ∩ h−1(κ)
+��
+→ X(N, S)
+such that ζσ(Z,ξ)(f, µ) = (Z, ξ) for all f ∈ Diff(S), Z ∈ Tf Diff(S), µ ∈ Den×(N) ∩ h−1(κ), and
+ξ ∈ Tµ
+�
+Den×(N) ∩ h−1(κ)
+�
+. As Diff(N, S) is a regular Lie group, we may apply Lemma A.2
+to conclude that the Diff(N, S)0 action on (31) admits local smooth sections.
+This completes the proof of Theorem 2.11.
+Remark 2.14. In view of Remark 2.3, we expect that the isotropy subgroup
+Diff(S; ι, µ) :=
+�
+(g1, . . . , gr) ∈ Diff(S, ι)
+�� ∀i : g∗
+i µi = µi
+�
+(34)
+is a splitting Lie subgroup of Diff(S; ι) with Lie algebra
+X(S, ι, µ) =
+�
+(Z1, . . . , Zr) ∈ X(S, ι)
+�� ∀i : LZiµi = 0
+�
+,
+(35)
+and the surjective map provided by the action, Diff(S, ι) → Den(S)ι,µ, is a locally trivial
+smooth principal fiber bundle with structure group Diff(S; ι, µ). This would follow in a rather
+straigthforward manner, via induction on the depth of the flags, if one could show that the
+isotropy group {f ∈ Diff(N, S) : f|S = id, f ∗µ = µ} is a splitting Lie subgroup in Diff(N, S),
+whenever S is a closed submanifold of N and µ is a volume density on N. The proof in [11,
+Theorem III.2.5.3 on page 203] covers the case S = ∅. However, the adaptation of said proof
+to nontrivial S is not entirely straigthforward, and we will not attempt to prove this here.
+Note that via the diffeomorphism in (24) the group Diff(S; ι, µ) corresponds to the subgroup
+of Diff(Sr; Σ) consisting of all diffeomorphisms that preserve µr and whose restriction to Σi
+preserves (ιr−1 ◦ · · · ◦ ιi)∗µi, for 1 ≤ i ≤ r − 1. Similarly, the Lie algebra X(S; ι, µ) can be
+identified to the corresponding subalgebra of X(Sr; Σ).
+If the expectation formulated in the preceding paragraph were indeed true, then the sur-
+jective and Diff(M) equivariant map
+EmbSr(M) = FrS,ι(M) → Flagwt
+S,ι,µ(M),
+(36)
+(ϕ1, . . . , ϕr) �→
+��
+ϕ1(S1), (ϕ1)∗µ1
+�
+, . . . ,
+�
+ϕr(Sr), (ϕr)∗µr
+��
+,
+(37)
+would be a locally trivial smooth principal fiber bundle with structure group Diff(S; ι, µ),
+see (26). Moreover, generalizing Remark 2.4, the isotropy group of a weighted flag (N, ν),
+Diffc(M; N, ν) :=
+�
+g ∈ Diffc(M)
+�� ∀i : g(Ni) = Ni, g|∗
+Niνi = νi
+�
+,
+would be a splitting Lie subgroup of Diffc(M), for it coincides with the preimage of Diff(S; ι, µ)
+under the bundle projection Diffc(M; N) → Diff(N, ιN ), cf. [9, Lemma 2.1(d), Proposi-
+tion 2.9(b), and Proposition 2.10(a)]. Furthermore, the orbit map Diffc(M) → Flagwt
+S (M)(N ,ν)
+provided by the action would be a locally trivial smooth principal bundle with structure group
+Diffc(M; N, ν). Hence, the Diffc(M) orbit of (N, ν) could be regarded as a homogeneous space
+Flagwt
+S (M)(N ,ν) = Diffc(M)/ Diffc(M; N, ν).
+12
+
+2.5
+A homological description
+In this section we give a more explicit description of the manifold Flagwt
+S,ι,µ(M).
+If N = (N1, . . . , Nr) is a nonlinear flag of type S in M, we put
+H(N) :=
+r
+�
+i=1
+Hki�
+Ni, Ni−1; ONi
+�
+=
+r
+�
+i=1
+H0
+�
+Ni \ Ni−1; R
+�
+,
+with the convention that N0 = ∅.
+We define the space of homologically weighted flags of type S in M by
+Flaghwt
+S
+(M) :=
+�
+(N, [ν]) : N ∈ Flagwt
+S (M), [ν] ∈ H(N)
+�
+Note that we have a Diff(M) equivariant forgetful map
+Flaghwt
+S
+(M) → FlagS(M),
+(38)
+as well as a Diff(M) equivariant map
+hFlagS(M) : Flagwt
+S (M) → Flaghwt
+S
+(M),
+(N, ν) �→ (N, [ν]).
+(39)
+Let Flaghwt
+S,ι (M) denote the preimage of the open subset FlagS,ι(M) under the projection
+in (38). Using the canonical Diff(M) equivariant identifications
+Flaghwt
+S,ι (M) = EmbSr(M) ×Diff(S,ι) H(S, ι),
+(40)
+we equip Flaghwt
+S
+(M) with the structure of a smooth vector bundle of finite (possibly noncon-
+stant) rank over FlagS(M) and with projection (38). The map in (39) is a smooth bundle map
+over FlagS(M). Indeed, via the identifications in (25) and (40), the map hS,ι in (28) induces
+a bundle map Flagwt
+S,ι(M) → Flaghwt
+S,ι (M) which coincides with the restriction of (39).
+We define the space of homologically weighted flags of type (S, ι, [µ]) in M by
+Flaghwt
+S,ι,[µ](M) :=
+�
+(N, [ν]) ∈ Flaghwt
+S
+(M)
+����
+�
+S1
+ι1
+−→ S2
+ι2
+−→ · · · → Sr, [µ]
+�
+∼=
+�
+N1 ⊆ N2 ⊆ · · · ⊆ Nr, [ν]
+�
+�
+,
+that is, the space of all homologically weighted flags (N, [ν]) in M such that there exist dif-
+feomorphisms Si → Ni, 1 ≤ i ≤ r, intertwining ιi with the canonical inclusion Ni ⊆ Ni+1 and
+taking [µ] ∈ H(S, ι) to [ν] ∈ H(N). The diffeomorphism in (40) restricts to a bijection
+Flaghwt
+S,ι,[µ](M) = EmbSr(M) ×Diff(S;ι) H(S, ι)[µ].
+(41)
+Hence, since the orbit H(S, ι)[µ] is finite, Flaghwt
+S,ι,[µ](M) is a Diff(M) invariant closed subman-
+ifold in Flaghwt
+S,ι (M) of codimension dim H(S, ι) by (40). Moreover, the forgetful map
+Flaghwt
+S,ι,[µ](M) → FlagS,ι(M)
+(42)
+is a Diff(M) equivariant covering map with finite fibers.
+We complement Theorem 2.11 with the following:
+Theorem 2.15. In this situation we have
+Flagwt
+S,ι,µ(M) = h−1
+FlagS(M)
+�
+Flaghwt
+S,ι,[µ](M)
+�
+.
+13
+
+Proof. This follows from the identity (29) in Proposition 2.10(c), using (26) and (41).
+Remark 2.16. If the action of Diff(S, ι) on H(S, ι) is trivial, then the Diff(S, ι)0 and Diff(S, ι)
+orbits in Den×(S) coincide in view of Proposition 2.10, and the forgetful (covering) map in
+(42) is a diffeomorphism, Flaghwt
+S,ι,[µ](M) = FlagS,ι(M). This happens in particular when all
+Si \ι(Si−1) are connected, as in Examples 2.17 and 2.18 below. Under the latter connectedness
+assumption, H(S, ι) = Rr via the isomorphism [µ] �→
+��
+S1 µ1, . . . ,
+�
+Sr µr
+�
+and
+Den(S)ι,µ =
+�
+α ∈ Den×(S) :
+�
+Si αi =
+�
+Si µi
+�
+.
+Moreover, Theorem 2.15 ensures that
+Flagwt
+S,ι,µ(M) =
+�
+(N, ν) ∈ Flagwt
+S,ι(M) :
+�
+Ni νi =
+�
+Si µi
+�
+.
+(43)
+This also applies in the situation of Remark 2.9, provided each model manifold Si is connected.
+A description for nested spheres in the same vein can be found in Example 2.19.
+Example 2.17 (Nested tori). If (S, ι) denotes the standard (meridional) embeddings between
+tori, T0 ⊆ T1 ⊆ · · · ⊆ Tr, then H(S, ι) = Rr+1 via the isomorphism [µ] �→ (
+�
+Ti µi).
+Example 2.18 (Nested projective spaces). If (S, ι) denotes the standard embeddings between
+projective spaces, P0 ⊆ P1 ⊆ · · · ⊆ Pr, then H(S, ι) = Rr+1 via the isomorphism [µ] �→ (
+�
+Pi µi).
+Example 2.19 (Nested spheres [13]). If (S, ι) denotes the standard equatorial embeddings be-
+tween spheres, S0 ⊆ S1 ⊆ · · · ⊆ Sr, then H(S, ι) = R2(r+1). The 2(r + 1) numbers assigned to
+[µ] ∈ H(S, ι) by this isomorphism are
+(a+
+0 , a−
+0 , a+
+1 , a−
+1 , . . . , a+
+r , a−
+r ) =
+��
+S0
++ µ0,
+�
+S0
+− µ0,
+�
+S1
++ µ1,
+�
+S1
+− µ1, . . . ,
+�
+Sr
++ µr,
+�
+Sr
+− µr
+�
+,
+where Si
++ and Si
+− denote the northern and southern hemispheres of Si, respectively. Consid-
+ering reflections on hyperplanes, we see that for each 0 ≤ i ≤ r there exists a diffeomorphism
+in Diff(S, ι) swapping Si
++ with Si
+−, but leaving all other hemispheres Sk
+± invariant. Such a
+diffeomorphism interchanges a+
+i with a−
+i , but leaves all other numbers a±
+k unchanged. Hence,
+the Diff(S, ι) orbit H(S, ι)[µ] has 2s elements, where s is the number of 0 ≤ i ≤ r with a+
+i ̸= a−
+i .
+Actually the Diff(S, ι) action on H(S, ι) ∼= R2(r+1) factorizes through an (Z2)2(r+1) action by
+switching or not the numbers a+
+i and a−
+i . We obtain a description similar to the one in (43):
+Flagwt
+S,ι,µ(M) =
+�
+(N, ν) ∈ Flagwt
+S,ι(M)
+�����
+{
+�
+N+
+i νi,
+�
+N−
+i νi} = {a+
+i , a−
+i }, where N +
+i
+and N −
+i
+denote the connected components of Ni \ Ni−1
+�
+.
+(44)
+3
+Coadjoint orbits of the Hamiltonian group
+Throughout this section (M, ω) denotes a symplectic manifold. The nonlinear Grassmannian
+of all isotropic submanifolds of type Sr in M, denoted by Griso
+Sr (M), is a splitting smooth
+submanifold in GrSr(M) which is invariant under the Hamiltonian group. In fact the Ham(M)
+orbits provide a smooth foliation of finite codimension in Griso
+Sr (M) which is called the isodrastic
+foliation [26, 15].
+Suppose L is an isodrastic leaf in Griso
+Sr (M) and let Flagwt iso
+S,ι,µ (M)|L denote the preimage
+of L under the canonical bundle projection Flagwt
+S,ι,µ(M) → GrSr(M). We will show that the
+14
+
+natural Hamc(M) action on Flagwt iso
+S,ι,µ (M)|L admits local smooth sections. In particular, each
+connected component of the latter space is an orbit of Hamc(M).
+The space Flagwt iso
+S,ι,µ (M)|L comes equipped with a canonical weakly non-degenerate sym-
+plectic form, and the restriction of (18) provides an Ham(M) equivariant injective moment
+map for the Hamc(M) action,
+J : Flagwt iso
+S,ι,µ (M)|L ֒→ hamc(M)∗,
+⟨J(N, ν), Xf⟩ =
+r
+�
+i=1
+�
+Ni
+fνi.
+This moment map J maps each connected component of Flagwt iso
+S,ι,µ (M)|L one-to-one onto the
+corresponding coadjoint orbit, see Theorem 3.14. Thereby, we identify coadjoint orbits of the
+Hamiltonian group Hamc(M) that can be modeled on weighted nonlinear flags.
+The material in this section is inspired by the results in [26, 15, 7] on weighted isotropic
+nonlinear Grassmannians.
+3.1
+Isodrasts as Ham(M) orbits
+In view of the tubular neighborhood theorem for isotropic embeddings [23, 24], the space
+Griso
+S (M) of all isotropic submanifolds of type S in M is a splitting smooth submanifold of
+GrS(M), see for instance [15, Section 8]. The tangent space at an isotropic submanifold N is
+TN Griso
+S (M) = {uN ∈ Γ(TN ⊥)|ι∗
+NiuN ω ∈ Ω1(N) closed},
+where TN ⊥ := TM|N/TN denotes the normal bundle to N.
+Weinstein’s [26] isodrastic distribution D on Griso
+S (M) is given by
+DN := {uN ∈ Γ(TN ⊥)|ι∗
+NiuN ω ∈ dC∞(N)}
+(45)
+and has finite codimension dim H1(S; R). This is an integrable distribution [26, 15] whose
+leaves are orbits of Ham(M). It gives rise to a smooth foliation of Griso
+S (M) called the isodrastic
+foliation. In particular, the Ham(M) orbits in Griso
+S (M) are splitting smooth submanifolds.
+Restricting the fundamental frame bundle in (1), we obtain a principal Diff(S) bundle
+Embiso
+S (M) → Griso
+S (M) with total space Embiso
+S (M), the splitting smooth submanifold of
+EmbS(M) consisting of all isotropic embeddings. On Embiso
+S (M) we consider the pullback of
+the isodrastic distribution D:
+Dϕ :=
+�
+uϕ ∈ Γ(ϕ∗TM) : ϕ∗iuϕω ∈ dC∞(S)
+�
+.
+(46)
+This is an integrable distribution with the same codimension, dim H1(S; R), and the leaves
+of D are connected components in the preimage of a leaf of D. According to [26, 15], the
+group Ham(M) acts transitively on the leaves of D. We need the subsequent slightly stronger
+statement in Proposition 3.1.
+The Lie algebra of compactly supported Hamiltonian vector fields will be denoted by
+hamc(M) = {Xf : f ∈ C∞
+c (M)}.
+We let Hamc(M) denote the group of diffeomorphisms
+obtained by integrating time dependent vector fields in hamc(M). For our purpose it will not
+be necessary to consider Hamc(M) as an infinite dimensional Lie group, cf. [14, Section 43.13].
+By a smooth map (section) into Hamc(M) we will simply mean a smooth map into Diffc(M)
+that takes values in Hamc(M).
+Proposition 3.1. The Hamc(M) action on each leaf E ⊆ Embiso
+S (M) of the isodrastic foliation
+D admits local smooth sections.
+15
+
+Proof. To show infinitesimal transitivity, suppose ϕ ∈ Embiso
+S (M) and uϕ ∈ Dϕ, cf. (46).
+Hence, there exists ¯f ∈ C∞(S) such that d ¯f = ϕ∗iuϕω. We extend ¯f to a function f1 ∈ C∞
+c (M)
+such that ¯f = f1 ◦ ϕ. The 1-form β on M along S defined by
+β = df1 ◦ ϕ − iuϕω ∈ Γ(ϕ∗T ∗M)
+(47)
+vanishes on vectors tangent to ϕ(S) ⊆ M. Hence, β can be seen as a fiberwise linear function
+on the normal bundle Tϕ(S)⊥ whose differential along the zero section ϕ(S) coincides with
+β itself. Thus, with the help of a tubular neighborhood of ϕ(S) in M and a suitable bump
+function, we get f2 ∈ C∞
+c (M) such that β = df2 ◦ ϕ. It follows from (47) that df ◦ ϕ = iuϕω
+for f = f1 − f2. We conclude that uϕ = Xf ◦ ϕ is the infinitesimal generator at ϕ for the
+Hamiltonian vector field Xf ∈ hamc(M).
+Using tubular neighborhoods constructed with the help of a Riemannian metric, say, we see
+that the function f may be chosen to depend smoothly on ϕ und uϕ, for ϕ in a sufficiently small
+open neighborhood of a fixed isotropic embedding ϕ0 ∈ E. Hence, we may apply Lemma A.2
+and conclude that the Hamc(M) action admits local smooth sections.
+Corollary 3.2. The Hamc(M) action on each leaf L ⊆ Griso
+S (M) of the isodrastic foliation D
+admits local smooth sections.
+Example 3.3. Every embedded closed curve in the plane is a Lagrangian submanifold of (R2, ω),
+where ω is the canonical area form, thus an element of Griso
+S1(R2). The isodrastic distribution
+D has codimension one. The enclosed area a singles out one isodrast La ⊆ Griso
+S1(R2), i.e. one
+orbit of Hamc(R2).
+A similar phenomena happens for Lagrangian k-tori in R2k, i.e. elements of Griso
+Tk(R2k),
+where Tk := (S1)k. To any ϕ ∈ Embiso
+Tk(R2k) we assign the symplectic area ai of the surface
+in R2k enclosed by the ith meridian ϕi(θ) = ϕ(1, . . . , θ, . . . , 1) of the embedded k-torus. These
+numbers are independent of the choice of the meridian in its homotopy class and of the surface
+having the meridian as boundary. The k-tuple (a1, . . . , ak) is an invariant under isodrastic
+deformations. Actually ai is the action integral of the ith meridian, as defined in [26].
+Let Ea1,...,ak ⊆ Embiso
+Tk(R2k) be the space of all isotropic embeddings having symplectic areas
+a1, . . . , ak. It is a union of isodrastic leaves of Lagrangian embeddings, but it is not necessarily
+Diff(Tk) saturated.
+The Diff(Tk) action on the k-tuples (a1, . . . , ak) factorizes through an
+GL(k, Z) action.
+Let [a1, . . . , ak] denote the orbit of (a1, . . . , ak).
+We define L[a1,...,ak] ⊆
+Griso
+Tk(R2k) to be the image of Ea1,...,ak under the principal Diff(Tk) bundle projection (1).
+Thus L[a1,...,ak] is a union of isodrastic leaves of Lagrangian k-tori in R2k.
+There is also a direct description of L[a1,...,ak]. Given a Lagrangian torus N ∈ GrS
+Tk(R2k),
+we choose a base {[γ1], . . . , [γk]} of H1(N, Z), where γi are loops in N with action integrals ai.
+We observe that the GL(k, Z) orbit [a1, . . . , ak] is independent of the choices and L[a1,...,ak] is
+the space of all Lagrangian tori in GrS
+Tk(R2k) such that the orbit of these action integrals is
+[a1, . . . , ak].
+We will also need the following observation.
+Lemma 3.4. If L is an isotropic submanifold in M, then the canonical inclusion GrS(L) ⊆
+Griso
+S (M) is a splitting smooth submanifold. Moreover, each connected component of GrS(L)
+is a splitting smooth submanifold in an isodrastic leaf in Griso
+S (M).
+Proof. Suppose N ∈ GrS(L), i.e., N ∼= S is a closed submanifold in L.
+By the tubular
+neighborhood theorem, we may w.l.o.g. assume that L is the total space of a vector bundle
+p: L → N, the normal bundle of N in L, and identify N with the zero section in L. We
+16
+
+have a canonical short exact sequence 0 → p∗L → TL → p∗TN → 0 of vector bundles
+over L. Choosing a linear connection on L, we obtain a splitting of this sequence and thus an
+isomorphism TL ∼= p∗TN ⊕p∗L of vector bundles over L. Dualizing, we obtain an isomorphism
+T ∗L ∼= p∗T ∗N ⊕ p∗L∗ of vector bundles over L. We regard this as a diffeomorphism
+T ∗L ∼= T ∗N ⊕ L ⊕ L∗
+that maps the zero section L ⊆ T ∗L identically onto the summand L on the right hand side.
+Via this isomorphism we have
+θL = π∗
+1θN + π∗
+2κ,
+where π1 and π2 denote the projections from T ∗N ⊕L⊕L∗ onto T ∗N and L⊕L∗, respectively,
+θL ∈ Ω1(T ∗L) and θN ∈ Ω1(T ∗N) denote the tautological 1-forms, and κ ∈ Ω1(L ⊕ L∗).
+Indeed, a straightforward computation yields κ(ξ) = ℓ′(C(Tq · ξ)) where x ∈ N, ℓ ∈ Lx,
+ℓ′ ∈ L∗
+x, ξ ∈ T(ℓ,ℓ′)(L ⊕ L∗), q: L ⊕ L∗ → L denotes the projection and C denotes the linear
+connection on L, viewed as a fiberwise linear map C : TL → p∗L over L.
+By the tubular neighborhood theorem for isotropic embeddings [23, 24], we may assume
+M = T ∗L ⊕ p∗E and ω = ˜π∗
+1dθL + ˜π∗
+2ρ, where E is a vector bundle over N, ˜π1 and ˜π2 denote
+the projections from T ∗L ⊕ p∗E onto T ∗L and p∗E, respectively, and ρ is a closed 2-form on
+the total space of p∗E. Combining this with the diffeomorphism in the previous paragraph,
+we obtain a diffeomorphism
+M = T ∗N ⊕ L ⊕ L∗ ⊕ E,
+(48)
+mapping L ⊆ M identically onto the summand L on the right hand side, and such that
+ω = π∗
+1dθN + π∗
+2σ,
+where π1 and π2 denote the projections from T ∗N ⊕ L ⊕ L∗ ⊕ E onto T ∗N and L ⊕ L∗ ⊕ E,
+respectively, and σ is a closed 2-form on the total space of L ⊕ L∗ ⊕ E.
+The diffeomorphism in (48) provides a (standard) chart for the smooth structure on GrS(M)
+centered at N,
+Γ(T ∗N ⊕ L ⊕ L∗ ⊕ E) → GrS(M),
+φ �→ φ(N).
+In this chart, the inclusions GrS(L) ⊆ Griso
+S (M) ⊆ GrS(M) become
+Γ(L) ⊆ {(α, ξ) ∈ Ω1(N) × Γ(L ⊕ L∗ ⊕ E) : dα + ξ∗σ = 0} ⊆ Ω1(N) × Γ(L ⊕ L∗ ⊕ E).
+(49)
+As L is isotropic, σ vanishes when pulled back to L. Hence, by the Poincar´e lemma, σ = dβ for
+a 1-form β on the total space of L ⊕ L∗ ⊕ E which vanishes along L. Via the diffeomorphism
+of Ω1(N) × Γ(L ⊕ L∗ ⊕ E) given by (α, ξ) �→ (α − ξ∗β, ξ), the inclusions in (49) become linear
+inclusions,
+Γ(L) ⊆ Z1(N) × Γ(L ⊕ L∗ ⊕ E) ⊆ Ω1(N) × Γ(L ⊕ L∗ ⊕ E),
+where Z1(N) denotes the space of closed 1-forms on N. Clearly, both inclusions admit com-
+plementary subspaces. In particular, GrS(L) is a splitting smooth submanifold in Griso
+S (M).
+The second assertion follows from the fact that the isodrastic leaf through N corresponds to
+the subspace B1(N) × Γ(L ⊕ L∗ ⊕ E), see [15, Section 8].
+17
+
+3.2
+Weighted isotropic nonlinear Grassmannians as coadjoint orbits
+In this section we recall the results in [26, 15, 7] about coadjoint orbits of the Hamiltonian
+group Hamc(M) modeled on weighted isotropic nonlinear Grassmannians of type S in M, and
+extend them to a possibly nonconnected model manifold S. Here we present them in a manner
+that readily generalizes to manifolds of weighted nonlinear flags.
+Let S be a closed k-dimensional manifold. The preimage of Griso
+S (M) under the canonical
+bundle projection Grwt
+S (M) → GrS(M) is a splitting smooth submanifold in Grwt
+S (M) which
+will be denoted by Grwt iso
+S
+(M). The diffeomorphism in (3) restricts to a diffeomorphism of
+bundles over Griso
+S (M),
+Grwt iso
+S
+(M) = Embiso
+S (M) ×Diff(S) Den×(S).
+The preimage of Griso
+S (M) under the canonical bundle projection Grwt
+S,µ(M) → GrS(M) is
+a splitting smooth submanifold in Grwt
+S,µ(M) which will be denoted by Grwt iso
+S,µ (M) and will be
+referred to as the nonlinear Grassmannian of weighted isotropic submanifolds of type (S, µ) in
+M. The diffeomorphism in (5) restricts to a diffeomorphism of bundles over Griso
+S (M),
+Grwt iso
+S,µ (M) = Embiso
+S (M) ×Diff(S) Den(S)µ.
+We recall the Diff(S) equivariant linear map hS : Den(S) → Hk(S, OS), hS(α) = [α]
+in (6), with kernel dΩk−1(S, OS), which we restrict to Den×(S). The product of integrable
+distributions D × ker ThS on Embiso
+S (M) × Den×(S) descends to an integrable distribution
+¯D := D ×Diff(S) ker ThS
+on Grwt iso
+S
+(M), of codimension dim H1(S; R) + dim Hk(S; OS). The image of ¯D under the
+forgetful map Grwt iso
+S
+(M) → Griso
+S (M) is the isodrastic distribution D in (45). In view of (7),
+each leaf G of ¯D is a connected component in the preimage of an isodrast L in Griso
+S (M) under
+the bundle projection Grwt iso
+S,µ (M) → Griso
+S (M), for some volume density µ on S. Hence, each
+leaf G of ¯D is a splitting smooth submanifold of codimension dim H1(S; R) in Grwt iso
+S,µ (M).
+According to [26, 15], the group Ham(M) acts transitively on the leaves of ¯D. We need the
+following slightly stronger statement.
+Proposition 3.5. The Hamc(M) action on each leaf G ⊆ Grwt iso
+S
+(M) of the isodrastic foliation
+¯D admits local smooth sections.
+Proof. Suppose L is an isodrastic leaf in Griso
+S (M) and let Grwt iso
+S,µ (M)|L denote its preimage
+under the bundle projection Grwt iso
+S,µ (M) → Griso
+S (M). It suffices to show that the Hamc(M)
+action on Grwt iso
+S,µ (M)|L admits local smooth sections. The diffeomorphism in (5) restricts to
+a diffeomorphism
+Grwt iso
+S,µ (M)|L = Embiso
+S (M)|L ×Diff(S) Den(S)µ.
+(50)
+By Proposition 3.1, the Hamc(M) action on Embiso
+S (M)|L admits local smooth sections. Ac-
+cording to Proposition 2.10(b), the Diff(S) action on Den(S)µ admits local smooth sections too.
+Using Lemma A.1 in the Appendix, we conclude that the Hamc(M) action on the associated
+bundle in (50) admits local smooth sections.
+Lemma 3.6 ([15]). The leafwise differential 2-form on (Embiso
+S (M) × Den×(S), D × ker ThS),
+given by
+Ω(ϕ,α)((uϕ, dλ), (vϕ, dγ)) :=
+�
+S
+(ω(uϕ, vϕ)α − ϕ∗iuϕω ∧ γ + ϕ∗ivϕω ∧ λ),
+(51)
+18
+
+is closed and Diff(S) invariant. Moreover, its kernel is spanned by the infinitesimal generators
+of the Diff(S) action.
+By Lemma 3.6, the leafwise differential 2-form Ω in (51) descends to a leafwise symplectic
+form ¯Ω on (Grwt iso
+S
+(M), ¯D). Thus every leaf G of the isodrastic distribution ¯D in Grwt iso
+S
+(M)
+is endowed with a symplectic form, the restriction of ¯Ω, which we denote by the same letter.
+By Proposition 3.5, G is an orbit for the Hamc(M) action on the weighted isotropic nonlinear
+Grassmannian Grwt iso
+S
+(M). This action is Hamiltonian with injective and Ham(M) equivariant
+moment map [15]
+J : G ⊆ Grwt iso
+S
+(M) → hamc(M)∗,
+⟨J(N, ν), Xf ⟩ =
+�
+N
+fν.
+(52)
+Here we use the Symp(M) equivariant isomorphism hamc(M) = C∞
+0 (M) where the latter
+denotes the Lie algebra of all compactly supported functions on M for which the integral with
+respect to the Liouville form vanishes on all closed connected components of M.
+For connected S, the subsequent theorem is due to Weinstein [26] in the Lagrangian case
+and due to Lee [15] for isotropic submanifolds.
+Theorem 3.7 ([26, 15]). The moment map J : (G, ¯Ω) → hamc(M)∗ in (52) is one-to-one onto
+a coadjoint orbit of the Hamiltonian group Hamc(M). The Kostant-Kirillov-Souriau symplectic
+form ωKKS on the coadjoint orbit satisfies J∗ωKKS = ¯Ω.
+In this generality the theorem follows from the discussion above and the following folklore
+result (see for instance the Appendix in [9]):
+Proposition 3.8. Suppose the action of G on (M, Ω) is transitive with injective equivariant
+moment map J : M → g∗. Then J is one-to-one onto a coadjoint orbit of G. Moreover, it
+pulls back the Kostant–Kirillov–Souriau symplectic form ωKKS on the coadjoint orbit to the
+symplectic form Ω.
+The same coadjoint orbits of the Hamiltonian group, under the additional restriction
+H1(S; R) = 0 on the closed connected manifold S, can be obtained via symplectic reduction
+in the Marsden-Weinstein ideal fluid dual pair, as shown in [7].
+3.3
+Manifolds of isotropic nonlinear flags
+In this section we extend the constructions of the previous section to nonlinear flags.
+As in Section 2.3, we let ι = (ι1, . . . , ιr−1) denote a collection of fixed embeddings ιi : Si →
+Si+1. We let Flagiso
+S,ι(M) denote the preimage of Griso
+Sr (M) under the canonical bundle pro-
+jection FlagS,ι(M) → GrSr(M), cf. [9, Remark 2.11]. This is a splitting smooth submanifold
+in FlagS,ι(M) called the manifold of isotropic nonlinear flags of type (S, ι) in M.
+Using
+Lemma 3.4, and proceeding by induction on the depth of the flag, one readily shows that the
+canonical inclusion
+Flagiso
+S,ι(M) ⊆
+r
+�
+i=1
+Griso
+Si (M)
+(53)
+is a splitting smooth submanifold.
+Let Flagwt iso
+S,ι
+(M) denote the preimage of Flagiso
+S,ι(M) under the canonical bundle projection
+Flagwt
+S,ι(M) → FlagS,ι(M). This is a splitting smooth submanifold in Flagwt
+S,ι(M) called the
+19
+
+manifold of weighted isotropic nonlinear flags of type (S, ι) in M. The diffeomorphism in (25)
+restricts to a diffeomorphism of bundles over Flagiso
+S,ι(M),
+Flagwt iso
+S,ι
+(M) = Embiso
+Sr (M) ×Diff(S;ι) Den×(S).
+The canonical inclusion Flagwt iso
+S,ι
+(M) ⊆ �r
+i=1 Grwt iso
+Si
+(M) is a splitting smooth submanifold in
+view of (53).
+As in Section 2.4, we fix µ = (µ1, . . . , µr) ∈ Den×(S), and let Flagwt iso
+S,ι,µ (M) denote the
+preimage of Flagiso
+S,ι(M) under the canonical bundle projection Flagwt
+S,ι,µ(M) → FlagS,ι(M).
+This is a splitting smoth submanifold in Flagwt
+S,ι,µ(M) called the manifold of weighted isotropic
+nonlinear flags of type (S, ι, µ) in M. The diffeomorphism in (26) restricts to a diffeomorphism
+Flagwt iso
+S,ι,µ (M) = Embiso
+Sr (M) ×Diff(S;ι) Den(S)ι,µ.
+(54)
+Using (53) and proceeding as in the proof of Theorem 2.11(c), one readily shows that the
+canonical inclusion Flagwt iso
+S,ι,µ (M) ⊆ �r
+i=1 Grwt iso
+Si,µi (M) is a splitting smooth submanifold.
+Recall the Diff(S; ι) equivariant linear map hS,ι : Den(S) → H(S, ι) in (28) with kernel
+ker ThS,ι = {(dγ1, . . . , dγr) : γi ∈ Ωki−1(Si; OSi), ι∗
+i−1γi = 0},
+whose restriction to Den×(S) we also denote by hS,ι. The product with the isodrastic dis-
+tribution gives the integrable distribution D × ker ThS,ι on Embiso
+Sr (M) × Den×(S), of finite
+codimension dim H1(Sr; R)+dim H(S, ι). This Diff(S; ι) invariant distribution descends to an
+integrable distribution
+¯D = D ×Diff(S;ι) ker ThS,ι
+of the same codimension on Flagwt iso
+S,ι
+(M). The image of ¯D under the forgetful map is an
+integrable distribution on Flagiso
+S,ι(M) of codimension dim H1(Sr; R), which coincides with the
+distribution that descends from the Diff(S; ι) invariant isodrastic distribution D on Embiso
+Sr (M)
+by the principal bundle projection. Using Proposition 2.10(c) and (54), we see that each leaf F
+of ¯D is a connected component in the preimage of an isodrast L in Griso
+Sr (M) under the bundle
+projection Flagwt iso
+S,ι,µ (M) → Griso
+Sr (M), for some volume density µ on S. Hence, each leaf F of
+¯D is a splitting smooth submanifold of codimension dim H1(Sr; R) in Flagwt iso
+S,ι,µ (M).
+Remark 3.9. If H1(Sr; R) = 0, then the leaves of the isodrastic foliation ¯D are the connected
+components of Flagwt iso
+S,ι,µ (M).
+Proposition 3.10. The Hamc(M) action on each leaf F ⊆ Flagwt iso
+S,ι
+(M) of the isodrastic
+foliation ¯D admits local smooth sections.
+Proof. Suppose L is an isodrastic leaf in Griso
+Sr (M) and let Flagwt iso
+S,ι,µ (M)|L denote its preimage
+under the canonical projection Flagwt
+S,ι,µ(M) → GrSr(M). It suffices to show that the Hamc(M)
+action on Flagwt iso
+S,ι,µ (M)|L admits local smooth sections. The diffeomorphism in (54) restricts
+to a diffeomorphism
+Flagwt iso
+S,ι,µ (M)|L = Embiso
+Sr (M)|L ×Diff(S;ι) Den(S)ι,µ.
+(55)
+By Proposition 3.1, the Hamc(M) action on Embiso
+Sr (M)|L admits local smooth sections. Ac-
+cording to Proposition 2.10(c), the Diff(S; ι) action on Den(S)ι,µ admits local smooth sections
+too. With the help of Lemma A.1, we conclude that the Hamc(M) action on the associated
+bundle (55) admits local smooth sections.
+Corollary 3.11. The Hamc(M) action on each leaf of the isodrastic foliation on Flagiso
+S,ι(M)
+admits local smooth sections.
+20
+
+3.4
+Weighted isotropic nonlinear flag manifolds as coadjoint orbits
+In this section we describe coadjoint orbits of the Hamiltonian group consisting of weighted
+isotropic nonlinear flags.
+We aim at defining a leafwise symplectic form ¯Ω on (Flagwt iso
+S,ι
+(M), ¯D).
+We start with
+leafwise differential 2-forms Ωi on (Embiso
+Si (M) × Den×(Si), Di × ker ThSi), for i = 1, . . . , r,
+defined as in Lemma 3.6. Let ji := ιr−1 ◦ · · · ◦ ιi ∈ Emb(Si, Sr). We consider
+qi := j∗
+i × pri : Embiso
+Sr (M) × Den×(S) → Embiso
+Si (M) × Den×(Si),
+which maps the distribution D × ker ThS,ι to Di × ker ThSi. Then
+Ω :=
+r
+�
+i=1
+q∗
+i Ωi
+(56)
+is a closed leafwise differential 2-form on (Embiso
+Sr (M) × Den×(S), D × ker ThS,ι).
+Remark 3.12. The image under Tqi of the distribution D × ker ThS,ι is, in general, strictly
+included in the distribution Di × ker ThSi. The reason is that the projection on the ith factor
+pri : ker ThS,ι → ker ThSi is not surjective in general. This happens for instance in the setting
+of Example 3.18, where ι : {1, . . . , k} → S1 with k ≥ 2: the projection on the second factor is
+d{γ ∈ C∞(S1) : γ ◦ ι = 0}, strictly included in ker ThS1 = dC∞(S1).
+The Diff(S; ι) action on Embiso
+Sr (M) × Den×(S),
+g · (ϕ, (αi)) = (ϕ ◦ gr, (g∗
+i αi)), for g = (g1, . . . , gr) ∈ Diff(S; ι),
+(57)
+has infinitesimal generators of the form
+ζZ(ϕ, (αi)) = (Tϕ ◦ Zr, (diZiαi)), for Z = (Z1, . . . , Zr) ∈ X(S; ι).
+(58)
+They belong to the integrable distribution D×ker ThS,ι. Notice that qi = j∗
+i ×pri is equivariant
+over the homomorphism g ∈ Diff(S; ι) �→ gi ∈ Diff(Si), because ji ◦ gi = gr ◦ ji for all i. Now,
+from the Diff(Si) invariance of Ωi and (56), we deduce the Diff(S; ι) invariance of Ω.
+The next lemma is a generalization of the Lemma 3.6:
+Lemma 3.13. The kernel of the leafwise differential form Ω is spanned by the infinitesimal
+generators of the Diff(S; ι) action (57) on Embiso
+Sr (M) × Den×(S).
+Proof. The formula for the leafwise 2-form in Lemma 3.6 provides a similar formula for Ω:
+Ω(ϕ,(αi))((uϕ, (dλi)), (vϕ, (dγi))) =
+r
+�
+i=1
+�
+Si
+j∗
+i ω(uϕ, vϕ)αi
+−
+r
+�
+i=1
+�
+Si
+j∗
+i (ϕ∗iuϕω) ∧ γi +
+r
+�
+i=1
+�
+Si
+j∗
+i (ϕ∗ivϕω) ∧ λi.
+(59)
+The contraction of Ω with an infinitesimal generator ζZ, given in (58), vanishes for all Z ∈
+X(S; ι). This follows from the analogous statement for Ωi and the infinitesimal generators ζZi
+on Embiso
+Si (M) × Den×(Si), together with the fact that the infinitesimal generators ζZ and ζZi
+are qi related.
+For the converse statement, we start with an arbitrary tangent vector
+(uϕ, (dλi)) ∈ ker Ω(ϕ,(αi)) ⊆ Dϕ × ker hS,ι.
+21
+
+We need to find Z = (Zi) ∈ X(S; ι) such that uϕ = Tϕ ◦ Zr and dλi = diZiαi. In particular
+the rth component Zr has to be tangent to the nonlinear flag Σ = (Σ1, . . . , Σr−1) in Sr, where
+Σi := ji(Si).
+The condition that (uϕ, (dλi)) lies in the kernel of Ω is equivalent to
+r
+�
+i=1
+�
+Si
+j∗
+i (ϕ∗iuϕω) ∧ γi = 0, for all γi ∈ Ωki−1(Si; OSi), ι∗
+i−1γi = 0,
+(60)
+and
+r
+�
+i=1
+�
+Si
+j∗
+i ω(uϕ, vϕ)αi +
+r
+�
+i=1
+�
+Si
+j∗
+i (ϕ∗ivϕω) ∧ λi = 0, for all vϕ ∈ Dϕ.
+(61)
+Choosing differential forms γi = 0 for i = 1, . . . , r − 1 in (60), we get that
+�
+Sr ϕ∗iuϕω ∧ γr = 0,
+for all γr ∈ Ωkr−1(Sr; OSr) with ι∗
+r−1γr = 0. This ensures that ϕ∗iuϕω = 0, meaning that uϕ
+takes values in the symplectic orthogonal to ϕ(Sr) in M.
+We assume by contradiction that uϕ ∈ Dϕ is not everywhere tangent to the isotropic
+submanifold ϕ(Sr) ⊆ M. The subset Sr \ Σr−1 is dense in Sr (because dim Sr−1 < dim Sr), so
+there exists x ∈ Sr \Σr−1 such that uϕ(x) is not tangent to ϕ(Sr). We choose another tangent
+vector vϕ ∈ Dϕ taking values in the symplectic orthogonal to ϕ(Sr) in M, i.e. ϕ∗ivϕω = 0,
+supported in an open neighborhood of x in Sr \ Σr−1.
+The isotropic embedding theorem
+of Weinstein [24] allows to choose vϕ such that ω(uϕ, vϕ) ∈ C∞(Sr) is a nonzero nowhere
+negative function. Plugging vϕ in (61), all terms are zero except
+�
+Sr ω(uϕ, vϕ)αr > 0, leading
+to a contradiction. Thus there exists Zr ∈ X(Sr) with uϕ = Tϕ ◦ Zr.
+The identity (61), rewritten with our uϕ = Tϕ ◦ Zr and arbitrary vϕ ∈ Dϕ, so ϕ∗ivϕω = dh
+with arbitrary h ∈ C∞(S), becomes:
+r
+�
+i=1
+�
+Si
+j∗
+i (iZrdh)αi =
+r
+�
+i=1
+�
+Si
+j∗
+i (dh) ∧ λi, for all h ∈ C∞(S).
+(62)
+We need to show that Zr ∈ X(Sr) is tangent to all the submanifolds of the nonlinear flag
+Σ1 ⊆ · · · ⊆ Σr−1 ⊆ Sr. We do it successively, starting with Σr−1 and ending with Σ1.
+We assume by contradiction that Zr is not tangent to Σr−1 ⊆ Sr.
+Since dim Σr−2 <
+dim Σr−1, we can find a point x ∈ Σr−1 \ Σr−2 such that Zr(x) is not tangent to Σr−1. We
+choose h ∈ C∞(S) supported in a tubular neighorhod of Σr−1 with the following properties:
+h vanishes on Σr−1 (so that j∗
+i dh = 0 for i = 1, . . . , r − 1) and iZrdh restricted to Σr−1 is a
+positive bump function at x that vanishes on Σr−2 (so that j∗
+i (iZrdh) = 0 for i = 1, . . . , r − 2).
+Inserting it in (62) only three terms survive, giving
+�
+Sr−1
+j∗
+r−1(iZrdh)αr−1 =
+�
+Sr
+hd(iZrαr − λr).
+The integral over Sr−1 on the left doesn’t change when cutting h with a function supported
+in a tubular ε-neighborhood of Σr−1 which is constant in an ε/2-neighborhood of Σr−1, while
+the absolute value of the integral over Sr on the right can be made as small as we need by
+making ε small enough. This leads to a contradiction. Thus Zr must be tangent to Σr−1, so
+there exists Zr−1 ∈ X(Sr−1) which is jr−1 related to Zr.
+In a similar way, step by step, we address all the remaining submanifolds Σr−2, . . . , Σ1 of
+the nonlinear flag Σ, finding Zi ∈ X(Si) that are ji related to Zr. Knowing this, the identity
+(62) becomes
+r
+�
+i=1
+�
+Si
+j∗
+i dh ∧ (λi − iZiαi) = 0, for all h ∈ C∞(S),
+22
+
+which implies that d(λi − iZiαi) = 0 for all i. Thus (uϕ, (dλi)) = (Tϕ ◦ Zr, (diZiαi)) is the
+infinitesimal generator at (ϕ, (αi)) for Z = (Zi) ∈ X(S; ι). This ensures that the kernel of Ω is
+spanned by the infinitesimal generators of the Diff(S; ι) action.
+As a consequence, Ω descends to a leafwise symplectic form ¯Ω on the associated bundle
+(Flagwt iso
+S,ι
+(M), ¯D) = (Embiso
+Sr (M) ×Diff(S,ι) Den×(S), D ×Diff(S,ι) ker ThS,ι).
+The restriction of ¯Ω to (Flagwt iso
+S,ι,µ (M), ¯D) is a leafwise symplectic form as well. Thus every leaf
+of ¯D is symplectic.
+Let F denote a leaf of the isodrastic distribution ¯D on Flagwt iso
+S,ι
+(M), equipped with the
+symplectic form ¯Ω. Restricting (18), we obtain a Ham(M) equivariant smooth map
+J : F → hamc(M)∗,
+⟨J(N, ν), Xf⟩ =
+r
+�
+i=1
+�
+Ni
+fνi,
+(63)
+where we identify hamc(M) = C∞
+0 (M) as in (52). This is a moment map for the (Hamilto-
+nian) action of Hamc(M) on (F, ¯Ω). Indeed, this follows readily by combining (56) with the
+expression for the moment map in (52). Moreover, J is injective according to Proposition 2.5.
+By Proposition 3.10, the Hamc(M) action on F is transitive. Using Proposition 3.8, we thus
+obtain the following generalization of Theorem 3.7.
+Theorem 3.14. The moment map J : (F, ¯Ω) → hamc(M)∗ in (63) is one-to-one onto a coad-
+joint orbit of the Hamiltonian group Hamc(M). The Kostant-Kirillov-Souriau symplectic form
+ωKKS on the coadjoint orbit satisfies J∗ωKKS = ¯Ω.
+Remark 3.15. In Section 3.3 we have seen that the inclusion
+Flagwt iso
+S,ι,µ (M) ⊆
+r
+�
+i=1
+Grwt iso
+Si,µi (M).
+is a splitting smooth submanifold.
+The symplectic leaf F described in Theorem 3.14 is a
+splitting symplectic submanifold of a product �r
+i=1 Gi of symplectic leaves described in Theo-
+rem 3.7. To show that this is indeed a splitting smooth submanifold, one can proceed as in the
+proof of Theorem 2.11(c), using the fact that each isodrastic leaf in Flagiso
+S,ι(M) is a splitting
+smooth submanifold in a product of isodrastic leaves in �r
+i=1 Griso
+Si (M). The latter can be show
+by induction on the depth of the flag using Lemma 3.4.
+Remark 3.16. The leafwise symplectic form ¯Ω on Flagwt iso
+S,ι
+(M) can be seen as a Poisson struc-
+ture whose symplectic leaves are the isodrasts F from the Theorem 3.14 (see Remark 3.4 in
+[26] about isodrasts in the nonlinear Grassmannian of weighted Lagrangian submanifolds).
+Restricting (18), we obtain a Poisson moment map
+J : Flagwt iso
+S,ι
+(M) → hamc(M)∗,
+⟨J(N, ν), Xf ⟩ =
+r
+�
+i=1
+�
+Ni
+fνi,
+for the Poisson action of Hamc(M) on weighted isotropic nonlinear flags, where we identify
+hamc(M) = C∞
+0 (M) as before.
+Example 3.17. Let M be a symplectic manifold that possesses isotropic submanifolds diffeo-
+morphic to the sphere Sr with r > 1. We use the setting of Example 2.19 to describe some
+coadjoint orbits of Hamc(M) consisting of nested weighted spheres.
+Since H1(Sr; R) = 0,
+23
+
+by Remark 3.9 each connected component of Flagwt iso
+S,ι,µ (M) is a coadjoint orbit of Hamc(M).
+Similarly to (44) we get
+Flagwt iso
+S,ι,µ (M) =
+�
+(N, ν) ∈ Flagwt iso
+S,ι
+(M)
+�����
+{
+�
+N+
+i νi,
+�
+N−
+i νi} = {a+
+i , a−
+i }, where N +
+i
+and N −
+i
+denote the connected components of Ni \ Ni−1
+�
+.
+Example 3.18. The coadjoint orbits of Hamc(R2) consisting of pointed vortex loops, studied
+in [3], are the lowest dimensional examples of coadjoint orbits of weighted nonlinear flags as
+described in Theorem 3.14.
+Their type is (S, ι), with ι : {1, . . . , k} → S1, ι(i) = ti consecutive points on the circle. We
+get the cohomology space
+H(S, ι) ∼= Rk × Rk,
+where the class [µ] ∈ H(S, ι) is identified with its integrals over connected components of
+{1, . . . , k} resp. S1 \ {t1, . . . , tk}. These are Γi =
+�
+{i} µ0 and wi =
+� ti+1
+ti
+µ1, for i = 1, . . . , k.
+The Diff(S, ι) action on H(S, ι) factorizes through an action of the dihedral group D2k on
+Rk ×Rk. More precisely, one let the dihedral group act on a regular k-gon, with Γi assigned to
+the vertex i and wi assigned to the edge [i, i + 1]. For generic density µ ∈ Den×(S), the orbit
+H(S, ι)[µ] consists of 2k elements. The orbit is a one-point set if and only if Γ1 = · · · = Γk and
+w1 = · · · = wk.
+Let us denote the weighted flags of type (S, ι) in R2 by (({x1, . . . , xk}, ν0), (C, ν1)). The
+area a enclosed by the curve C singles out one isodrast La ⊂ Griso
+S1(R2), as in Example 3.3.
+From Theorem 3.14 follows that each connected component of Flagwt iso
+S,ι,µ (R2)|La is a coadjoint
+orbit of Hamc(R2). Using the above description of Diff(S, ι) action on H(S, ι), we get
+Flagwt iso
+S,ι,µ (R2)|La =
+�
+(({x1, . . . , xk}, ν0), (C, ν1))
+����
+xi consecutive points in C ∈ La, ∃σ ∈ D2k
+s.t.
+�
+{xi} ν0 = σ(Γi),
+� xi+1
+xi
+ν1 = σ(wi)
+�
+.
+(64)
+In particular, the invariants are the area a enclosed by the loop C, the point vorticities Γi, and
+the net vorticities wi =
+� xi+1
+xi
+ν1 between two consecutive points on the loop.
+Example 3.19. Let L[a1,a2] be a union of isodrastic leaves of Lagrangian 2-tori in R4, as in
+Example 3.3, with [a1, a2] the GL(2, Z) orbit of the pair of action integrals (a1, a2) ∈ R2. Let
+S1 be a disjoint union of k circles, S2 = T2 the 2-torus, and ι : S1 → S2 the embedding that
+maps the i-th circle to the circle {ti} × S1 ⊆ T2, with consecutive points t1, . . . , tk ∈ S1. For
+fixed density µ ∈ Den×(S), we aim at describing the coadjoint orbits of Hamc(R4) that are
+connected components of Flagwt iso
+S,ι,µ (R4)|L[a1,a2]. To this end we need the isomorphism
+H(S, ι) ∼= Rk × Rk
+given by integration of µ1 over the components of S1 and of µ2 over the torus surface between
+two successive embedded circles. As in the previous example, the Diff(S, ι) action on H(S, ι)
+factorizes through the dihedral group. One can now express these coadjoint orbits of Hamc(R4)
+as in (64), with the help of the dihedral group. Thus, the invariants are: the GL(2, Z) orbit of
+the two action integrals for the embedded 2-torus, the total weights of the k isotopic loops on
+the torus, and the partial weights between two such consecutive loops on the torus.
+A
+Transitive actions on associated bundles
+The manifolds and Lie groups in this appendix may be infinite dimensional and are assumed
+to be modelled on convenient vector spaces as in [14].
+24
+
+Recall that a smooth G action on M is said to admit local smooth sections if every point
+x0 in M admits an open neighborhood U and a smooth map σ : U → G such that σ(x)x0 = x,
+for all x ∈ U. Clearly, such an action is locally and infinitesimally transitive. Due to the lack
+of a general implicit function theorem, one can not expect the converse implication to hold for
+general Fr´echet manifolds.
+Lemma A.1. Let P → B be a principal G-bundle endowed with the action of a Lie group
+H on P that commutes with the principal G action. Suppose the structure group G acts on
+another manifold Q, and consider the canonically induced H action on the associated bundle
+P ×G Q. If the H action on P and the G action on Q both admit local smooth sections, then
+the H action on P ×G Q admits local smooth sections too.
+Proof. Suppose ξ0 ∈ P ×G Q. As the canonical projection P × Q → P ×G Q is a locally
+trivial smooth bundle [14, Theorem 37.12], there exist an open neighborhood U of ξ0 as well
+as smooth maps π : U → P and ρ : U → Q such that for all ξ ∈ U we have
+[π(ξ), ρ(ξ)] = ξ.
+Put p0 := π(ξ0) ∈ P and q0 := ρ(ξ0) ∈ Q. As the H action on P admits local sections, there
+exist an open neighborhood V of p0 in P and a smooth map σ′ : V → H such that
+σ′(p0) = eH
+and
+σ′(p)p0 = p,
+for all p ∈ V . As the G action on Q admits local sections, there exist an open neighborhood
+W of q0 in Q and a smooth map σ′′ : W → G such that
+σ′′(q0) = eG
+and
+σ′′(q)q0 = q,
+for all q ∈ W. Possibly replacing U with a smaller neighborhood of ξ0, we may assume that
+for all ξ ∈ U we have ρ(ξ) ∈ W and π(ξ)σ′′(ρ(ξ)) ∈ V . Hence, σ : U → H,
+σ(ξ) := σ′�
+π(ξ)σ′′(ρ(ξ))
+�
+is a well defined smooth map. For ξ ∈ U we obtain
+σ(ξ)ξ0 = σ(ξ)[p0, q0] = [σ(ξ)p0, q0] = [σ′(π(ξ)σ′′(ρ(ξ)))p0, q0]
+= [π(ξ)σ′′(ρ(ξ)), q0] = [π(ξ), σ′′(ρ(ξ))q0] = [π(ξ), ρ(ξ)] = ξ.
+Hence, σ is the desired local smooth section of the H action on P ×G Q.
+We will denote the fundamental vector fields of a smooth G action on M by
+ζX(x) := ∂
+∂t
+��
+t=0exp(tX)x,
+where X ∈ g and x ∈ M.
+Lemma A.2. Let G be a regular Lie group acting on a smooth manifold M. Suppose every
+point x0 in M admits an open neighborhood U and a smooth map σ : TM|U → g such that
+ζσ(X)(x) = X,
+(65)
+for all x ∈ U and X ∈ TxM. Then the G action on M admits local smooth sections.
+25
+
+Proof. We will construct a local section using an argument due to Moser [21, Section 4].
+Suppose c : [0, 1] → U is a smooth curve. We seek a smooth curve g : [0, 1] → G such that
+c(t) = g(t)c(0).
+(66)
+Differentiating, we obtain
+c′(t) = ζ ˙g(t)(c(t)),
+(67)
+where ˙g(t) :=
+∂
+∂h
+��
+h=tg(h)g(t)−1 denotes the right logarithmic derivative of g.
+Since G is regular [14, Definition 38.4], there exists a unique smooth curve g = Evolr�
+σ◦c′�
+in G such that ˙g(t) = σ(c′(t)) and g(0) = e. Using (65), we see that (67) and, thus, (66) hold
+true. Evaluating at t = 1, we obtain a smooth map
+s : C∞([0, 1], U) → G,
+s(c) := g(1) = evolr�
+σ ◦ c′�
+.
+By construction, c(1) = s(c)c(0), for all smooth curves c : [0, 1] → U.
+To obtain a local smooth section for the G action on M, it suffices to compose s with a
+smooth map U → C∞([0, 1], U), x �→ cx satisfying cx(0) = x0 and cx(1) = x. The latter can
+readily be constructed using a chart for M centered at x0 = 0. Indeed, shrinking U so that it
+becomes star shaped with center x0 = 0 in such a chart, we may use cx(t) := tx.
+References
+[1] C. Balleier and T. Wurzbacher, On the geometry and quantization of symplectic Howe pairs, Math. Z.
+271 (2012), 577–591.
+[2] E. Binz and H.R. Fischer, The manifold of embeddings of a closed manifold. With an appendix by
+P. Michor. Lecture Notes in Phys. 139, Differential geometric methods in mathematical physics (Proc.
+Internat. Conf., Tech. Univ. Clausthal, Clausthal-Zellerfeld, 1978), pp. 310–329, Springer, Berlin-New
+York (1981).
+[3] I. Ciuclea and C. Vizman, Pointed weighted vortex loops in 2D ideal fluids, arXiv:2212.02612.
+[4] D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,
+Ann. of Math. 92 (1970), 102–163.
+[5] F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Global Anal. Geom. 41 (2012), 1–24.
+[6] F. Gay-Balmaz and C. Vizman, Principal bundles of embeddings, Ann. Global Anal. Geom. 46 (2014),
+293–312.
+[7] F. Gay-Balmaz and C. Vizman, Isotropic submanifolds and coadjoint orbits of the Hamiltonian group, J.
+Symp. Geom. 17 (2019), 663–702.
+[8] S. Haller and C. Vizman, Non-linear Grassmannians as coadjoint orbits, Math. Ann. 329 (2004), 771–785.
+[9] S. Haller and C. Vizman, Non-linear flag manifolds as coadjoint orbits, Ann. Global Anal. Geom. 58
+(2020), 385–413.
+[10] S. Haller and C. Vizman, A dual pair for the contact group, Math. Z. 301 (2022), 2937–2973.
+[11] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982),
+65–222.
+[12] M. W. Hirsch, Differential topology, Graduate Texts in Math. 33, Springer, 1976.
+[13] S. Jung, I. Dryden, and J.S. Marron, Analysis of principal nested spheres, Biometrika 99 (2012), 551–568.
+[14] A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and
+Monographs 53, American Mathematical Society, Providence, RI, 1997.
+[15] B. Lee, Geometric structures on spaces of weighted submanifolds, SIGMA 5 (2009), 099, 46 pages.
+[16] J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, Springer, 2003.
+26
+
+[17] Libermann, P. and C.-M. Marle [1987], Symplectic Geometry and Analytical Mechanics, D. Reidel Pub-
+lishing Company.
+[18] J.E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,
+Phys. D, 7 (1983), 305–323.
+[19] P. W. Michor, Manifolds of smooth maps. III. The principal bundle of embeddings of a noncompact
+smooth manifold. Cahiers Topologie G´eom. Diff´erentielle 21, 325–337 (1980)
+[20] P. W. Michor, Manifolds of differentiable mappings. Shiva Mathematics Series 3. Shiva Publishing Ltd.,
+Nantwich (1980)
+[21] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294.
+[22] C. Vizman, Induced differential forms on manifolds of functions, Archivum Mathematicum, 47 (2011),
+201–215.
+[23] A. Weinstein, Lectures on symplectic manifolds. Expository lectures from the CBMS Regional Conference
+held at the University of North Carolina, March 8–12, 1976. Regional Conference Series in Mathematics,
+No. 29. American Mathematical Society, Providence, R.I., 1977.
+[24] A. Weinstein, Neighborhood classification of isotropic embeddings, J. Diff. Geom. 16 (1981), 125–128.
+[25] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983), 523–557.
+[26] A. Weinstein, Connections of Berry and Hannay type for moving Lagrangian submanifolds, Adv. Math.
+82 (1990), 133–159.
+27
+
diff --git a/hdAyT4oBgHgl3EQfkPgm/content/tmp_files/load_file.txt b/hdAyT4oBgHgl3EQfkPgm/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..dfd5ab0779832a8823495f8c9df5ff16cea0ae10
--- /dev/null
+++ b/hdAyT4oBgHgl3EQfkPgm/content/tmp_files/load_file.txt
@@ -0,0 +1,1288 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf,len=1287
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='00428v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='DG]  1 Jan 2023  Weighted nonlinear flag manifolds as coadjoint orbits  Stefan Haller1 and Cornelia Vizman2  Abstract  A weighted nonlinear flag is a nested set of closed submanifolds, each submanifold  endowed with a volume density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We study the geometry of Fr´echet manifolds of weighted  nonlinear flags, in this way generalizing the weighted nonlinear Grassmannians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' When the  ambient manifold is symplectic, we use these nonlinear flags to describe a class of coadjoint  orbits of the group of Hamiltonian diffeomorphisms, orbits that consist of weighted isotropic  nonlinear flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  1  Introduction  In this article we study manifolds of weighted nonlinear flags, motivated by the fact that  one can use them to describe new coadjoint orbits of the Hamiltonian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This adds to  the already known coadjoint orbits described with configuration spaces of points, weighted  isotropic nonlinear Grassmannians [26, 15, 7], symplectic nonlinear Grassmannians [8], and  manifolds of symplectic nonlinear flags [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let M be a smooth manifold, and let S = (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Sr) be a collection of closed manifolds  of strictly increasing dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We consider the Fr´echet manifold FlagS(M) of nonlinear  flags of type S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=', sequences of nested embedded submanifolds N1 ⊆ · · · ⊆ Nr in M, with  Ni diffeomorphic to Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Considering submanifolds equipped with nowhere zero densities, one  obtains the manifold Flagwt  S (M) of weighted nonlinear flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We describe its Fr´echet manifold  structure in two ways: as a splitting smooth submanifold of the cartesian product of weighted  nonlinear Grassmannians of type Si in M, and as a locally trivial smooth fiber bundle over  FlagS(M) associated to the principal bundle of nonlinear frames of type S in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' To each  weighted nonlinear flag one associates a compactly supported distribution on M with controlled  singularities, by the Diff(M) equivariant inclusion  J : Flagwt  S (M) ֒→ C∞(M)∗,  �  J((N1, ν1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (Nr, νr)), f  �  :=  r  �  i=1  �  Ni  fνi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The Diff(M) orbits in Flagwt  S (M) are submanifolds of finite codimension, for which we give an  explicit homological description, up to connected components, using nonlinear flags decorated  with cohomology classes (see Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='11 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Inspired by the results in [26, 15, 7] on weighted isotropic nonlinear Grassmannians, we  consider the manifold of weighted isotropic nonlinear flags in a symplectic manifold (M, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The orbits for the natural action of the Hamiltonian group Hamc(M) are submanifolds of finite  1Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='haller@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='at  2Department of Mathematics, West University of Timi¸soara, Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Parvan 4, 300223 Timi¸soara, Romania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  cornelia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='vizman@e-uvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='ro  32020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' 58D10 (primary);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' 37K65, 53C30, 53D20, 58D05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  1  codimension, described as leaves of an isodrastic foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Each isodrastic leaf of weighted  nonlinear flags comes equipped with a canonical weakly non-degenerate symplectic form, and  the map J restricts to an equivariant moment map for the Hamc(M) action, thus identifying  the leaf with a coadjoint orbit of the Hamiltonian group (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover, this  coadjoint orbit is a splitting symplectic submanifold in a product of coadjoint orbits of weighted  submanifolds of type Si in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The lowest dimensional examples are the coadjoint orbits of Hamc(R2) consisting of pointed  weighted vortex loops, treated in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We give more examples with nested spheres or tori, and  provide explicit descriptions of the corresponding coadjoint orbits of the Hamiltonian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The first author would like to thank the West University of Timi¸soa-  ra for the warm hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' He gratefully acknowledges the support of the Austrian Science  Fund (FWF) grant P31663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The second author would like to thank the University of Vienna  for the warm hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' She was partially supported by CNCS UEFISCDI, project number  PN-III-P4-ID-PCE-2020-2888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  2  Manifolds of weighted nonlinear flags  A nonlinear flag is a sequence of nested closed submanifolds N1 ⊆ · · · ⊆ Nr in a smooth  manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' A weighted nonlinear flag is a nonlinear flag together with a volume density νi  on each submanifold Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Integrating against test functions f ∈ C∞(M), a weighted nonlinear  flag provides a compactly supported distribution on M with mild singularities, �r  i=1  �  Ni fνi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We will show that the space of all weighted nonlinear flags in M is a Fr´echet manifold  in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In fact, this is the total space of a locally trivial smooth bundle over the  manifold of nonlinear flags discussed in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The natural Diff(M) action on the base of this  bundle is locally transitive [9, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='9(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The main aim of this section is to describe  the Diff(M) orbits in the space of weighted nonlinear flags, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1  Weighted nonlinear Grassmannians  In this section we recall some basic facts about the manifolds of weighted submanifolds that  appear in [26, 15, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' These weighted nonlinear Grassmannians constitute a special case of the  weighted nonlinear flags to be introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We present them here in a manner  that readily generalizes to the setting of nonlinear flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let S be a closed manifold of dimension k, allowed to be nonconnected and nonorientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  For each manifold M, we let GrS(M) denote the nonlinear Grassmannian of type S in M,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=', the space of all smooth submanifolds in M that are diffeomorphic to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover, we let  EmbS(M) denote the space of all parametrized submanifolds of type S in M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=', the space of  all smooth embeddings of S into M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Both, EmbS(M) and GrS(M), are Fr´echet manifolds in  a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Furthermore, the Diff(M) equivariant map  EmbS(M) → GrS(M),  ϕ �→ ϕ(S),  (1)  is a smooth principal bundle with structure group Diff(S), a Fr´echet Lie group, see [2, 19, 20]  and [14, Theorem 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  For each closed k-dimensional manifold S, let  Den(S) := Γ∞(|Λ|S) = Ωk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS)  2  denote the space of all smooth densities on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Here OS denotes the orientation bundle of S  and |Λ|S = ΛkT ∗S ⊗OS, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Densities on S are the geometric quantities which can be  integrated over S in a coordinate independent way, without specifying an orientation or even  assuming orientability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We denote by Den×(S) the space of volume densities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=', the space of  nowhere vanishing densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Clearly, this is an open subset in the Fr´echet space Den(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We define the weighted nonlinear Grassmannian of type S in M by  Grwt  S (M) :=  �  (N, ν)  �� N ∈ GrS(M), ν ∈ Den×(N)  �  ,  (2)  that is, the space of all submanifolds of type S in M, decorated with a nowhere zero density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We equip this space with the structure of a Fr´echet manifold by declaring the natural bijection  Grwt  S (M) = EmbS(M) ×Diff(S) Den×(S),  �  ϕ(S), ϕ∗µ  �  ↔ [ϕ, µ],  (3)  to be a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Here the right hand side denotes the total space of the bundle asso-  ciated to the nonlinear frame bundle in (1) and the natural Diff(S) action on Den×(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In  particular, the canonical forgetful map  Grwt  S (M) → GrS(M),  (N, ν) �→ N,  (4)  becomes a locally trivial smooth bundle with typical fiber Den×(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Indeed, it corresponds to  the bundle projection of the associated bundle EmbS(M) ×Diff(S) Den×(S) → GrS(M) via the  identification in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  There is a canonical Diff(M) equivariant map  J : Grwt  S (M) → C∞(M)∗,  ⟨J(N, ν), f⟩ :=  �  N  fν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  This map is injective, and its image consists of compactly supported distributions with mild  singularities: J(N, ν) is supported on N, and its wave front set coincides with the conormal  bundle of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let µ ∈ Den×(S) be a volume density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The space  Grwt  S,µ(M) :=  �  (N, ν) ∈ Grwt  S (M)  �� (S, µ) ∼= (N, ν)  �  is called the nonlinear Grassmannian of weighted submanifolds of type (S, µ) in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' It consists  of all weighted submanifolds (N, ν) in M such that there exists a diffeomorphism S → N  taking µ to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Denoting the Diff(S) orbit of µ by Den(S)µ, the identification in (3) restricts  to a canonical bijection  Grwt  S,µ(M) = EmbS(M) ×Diff(S) Den(S)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (5)  It is well known [21] that the Diff(S)0 orbit of µ is a convex subset that consists of all volume  densities on S that represent the same cohomology class as µ in Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS), the de Rham  cohomology with coefficients in the orientation bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, the Diff(S) orbit of µ coincides  with the set of all volume densities on S that are in the preimage of Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS)[µ], the (finite)  Diff(S) orbit of [µ] in Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS), under the Diff(S) equivariant linear map  hS : Den(S) → Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS),  hS(α) = [α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (6)  More succinctly,  Den(S)µ = Den×(S) ∩ h−1  S  �  Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS)[µ]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (7)  3  Hence, Den(S)µ is an open subset in a finite union of parallel closed affine subspaces with finite  codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In particular, Den(S)µ is a splitting smooth submanifold in Den×(S) with finite  codimension dim Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS) and with tangent spaces  Tα Den(S)µ = ker hS = dΩk−1(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (8)  Using (5) we conclude that Grwt  S,µ(M) is a splitting smooth submanifold in Grwt  S (M) with finite  codimension dim Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover, the canonical forgetful map in (4) restricts to a locally  trivial smooth fiber bundle Grwt  S,µ(M) → GrS(M) with typical fiber Den(S)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The space of cohomologically weighted submanifolds of type S in M is defined as  Grhwt  S  (M) :=  �  (N, [ν]) : N ∈ GrS(M), [ν] ∈ Hk(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ON)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Using the canonical bijection  Grhwt  S  (M) = EmbS(M) ×Diff(S) Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS),  (9)  we turn Grhwt  S  (M) into a smooth vector bundle of finite rank dim Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS) over GrS(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The canonical Diff(M) equivariant map  hGrS(M) : Grwt  S (M) → Grhwt  S  (M),  (N, ν) �→ (N, [ν]),  is a smooth bundle map over GrS(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Indeed, via the diffeomorphisms in (5) and (9) it  corresponds to the map induced by (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The space of cohomologically weighted submanifolds of type (S, [µ]) in M is defined by  Grhwt  S,[µ](M) :=  �  (N, [ν]) ∈ Grhwt  S  (M) : (N, [ν]) ∼= (S, [µ])  �  and consists of all cohomologically weighted submanifolds (N, [ν]) such that there exists a  diffeomorphism S → N taking the cohomology class [µ] to [ν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' As (9) restricts to a bijection  Grhwt  S,[µ](M) = EmbS(M) ×Diff(S) Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS)[µ],  we see that Grhwt  S,[µ](M) is a finite covering of GrS(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Using (7) we conclude  Grwt  S,µ(M) = h−1  GrS(M)  �  Grhwt  S,[µ](M)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (10)  It is well known that the Diffc(M) action on EmbS(M) admits local smooth sections, see  for instance [9, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Furthermore, the (transitive) Diff(S) action on Den(S)µ also  admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The latter can be shown using Moser’s method of proof in [21,  Section 4], see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='12 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1 in the Appendix, we conclude that  the natural Diffc(M) action on Grwt  S,µ(M) admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In particular, this  action is locally transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Hence, each connected component of Grwt  S,µ(M) is a Diffc(M)0  orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Consequently, each Diffc(M) or Diff(M) orbit in Grwt  S,µ(M) is a union of connected  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Poincar´e duality provides a canonical Diff(S) equivariant isomorphism  Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS) = H0(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Hence, specifying a cohomology class [µ] ∈ Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS) amounts to specifying the total volume  of µ on each connected component of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  4  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' If S is connected, then Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS) = R and the Diff(S) action is trivial on this  cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, the orbit Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS)[µ] is a one-point set, and  Den(S)µ =  �  α ∈ Den×(S) :  �  S α =  �  S µ  �  (11)  is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Correspondingly,  Grwt  S,µ(M) =  �  (N, ν) ∈ Grwt  S (M) :  �  N ν =  �  S µ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  This is the case considered in [26, 15, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  If S is built out of two diffeomorphic connected components, then Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS) ∼= R2 and any  diffeomorphism swapping the two connected components acts nontrivially on this cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  If µ has equal total volume on the two connected components, then the orbit Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS)[µ] is  a one-point set and Den(S)µ is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Otherwise Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS)[µ] consists of two points and,  by (7), Den(S)µ has two connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Suppose µ ∈ Den×(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  It is well known that Diff(S, µ), the group of diffeo-  morphisms preserving µ, is a splitting Lie subgroup in Diff(S), see [11, Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='3  on page 203].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Moreover, the map provided by the action, Diff(S) → Den(S)µ, f �→ f∗µ,  is a smooth principal bundle with structure group Diff(S, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Via (5) this implies that the  surjective and Diff(M) equivariant map  EmbS(M) → Grwt  S,µ(M),  ϕ �→  �  ϕ(S), ϕ∗µ  �  ,  is smooth principal bundle with structure group Diff(S, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Suppose (N, ν) ∈ Grwt  S (M) and let Grwt  S (M)(N,ν) denote its Diffc(M) orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Combining the preceeding remark with the fact that the Diffc(M) action on EmbS(M) admits  local smooth sections [9, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1(c)], we see that the map provided by the action,  Diffc(M) → Grwt  S (M)(N,ν),  f �→  �  f(N), f∗ν  �  is a smooth principal bundle with structure group Diffc(M, N, ν), the group of diffeomorphisms  preserving N and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The latter is a splitting Lie subgroup in Diffc(M), for it coincides with  the preimage of Diff(N, ν) under the canonical bundle projection Diffc(M, N) → Diff(N), see  [9, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, each orbit may be regarded as a homogeneous space,  Grwt  S (M)(N,ν) = Diffc(M)/ Diffc(M, N, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='2  Weighted nonlinear flag manifolds  Fix natural numbers ki such that  0 ≤ k1 < k2 < · · · < kr  (12)  and let S = (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Sr) be a collection of closed smooth manifolds with dim Si = ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  For a smooth manifold M we let  FlagS(M) :=  �  �  N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Nr) ∈  r  �  i=1  GrSi(M)  �����∀i : Ni ⊆ Ni+1  �  denote the space of nonlinear flags of type S in M, and we write  FrS(M) :=  �  (ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ϕr) ∈  r  �  i=1  EmbSi(M)  �����∀i : ϕi(Si) ⊆ ϕi+1(Si+1)  �  5  for the space of the space of nonlinear frames of type S in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In [9, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='3] it has  been shown that FlagS(M) and FrS(M) are splitting smooth submanifolds of �r  i=1 GrSi(M)  and �r  i=1 EmbSi(M), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover, the canonical Diff(M) equivariant map  FrS(M) → FlagS(M),  (ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ϕr) �→  �  ϕ1(S1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ϕr(Sr)  �  (13)  is a smooth principal fiber bundle with structure group  Diff(S) :=  r  �  i=1  Diff(Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We denote the space of weighted nonlinear flags of type S in M by  Flagwt  S (M) :=  �  �  (N1, ν1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (Nr, νr)  �  ∈  r  �  i=1  Grwt  Si (M)  �����∀i : Ni ⊆ Ni+1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (14)  This is a splitting smooth submanifold in �r  i=1 Grwt  Si (M), for it coincides with the preimage  of the splitting smooth submanifold FlagS(M) under the bundle projection �r  i=1 Grwt  Si (M) →  �r  i=1 GrSi(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover, the canonical Diff(M) equivariant forgetful map  Flagwt  S (M) → FlagS(M),  �  (N1, ν1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (Nr, νr)  �  �→ (N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Nr),  (15)  is a smooth fiber bundle with typical fiber  Den×(S) :=  r  �  i=1  Den×(Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (16)  The latter is a Diff(S) invariant open subset in the Fr´echet space Den(S) := �r  i=1 Den(Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Furthermore, the canonical Diff(M) equivariant bijection  Flagwt  S (M) = FrS(M) ×Diff(S) Den×(S),  (17)  ��  ϕ1(S1), (ϕ1)∗µ1  �  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ,  �  ϕr(Sr), (ϕr)∗µr  ��  ↔  �  (ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ϕr), (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , µr)  �  ,  is a diffeomorphism between Flagwt  S (M) and the bundle associated to the nonlinear frame  bundle in (13) and the canonical Diff(S) action on Den×(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Indeed, this is just the bundle  diffeomorphism �r  i=1 Grwt  Si (M) = �r  i=1 EmbSi(M) ×Diff(Si) Den×(Si) obtained by taking the  product of the diffeomorphisms in (3), restricted over the submanifold FlagS(M) in its base  �r  i=1 GrSi(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We have a canonical Diff(M) equivariant map  J : Flagwt  S (M) → C∞(M)∗,  �  J((N1, ν1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (Nr, νr)), f  �  :=  r  �  i=1  �  Ni  fνi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (18)  The image of J consists of compactly supported distributions on M with mild singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  More precisely, the wave front set of J((N1, ν1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (Nr, νr)) coincides with the union of the  conormal bundles of N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The map in (18) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  6  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Suppose J((N1, ν1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (Nr, νr)) = J((N ′  1, ν′  1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (N ′  r, ν′  r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Proceeding by induction  on r, it suffices to show Nr = N ′  r and νr = ν′  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  To show Nr = N ′  r, we assume by contradiction that there exists x ∈ Nr with x /∈ N ′  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Using  (12), we see that Nr \\Nr−1 is dense in Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Thus, we may w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' assume x /∈ Nr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover,  νr(x) ̸= 0 as νr does not vanish on Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, if f is a smooth bump function supported on  a sufficiently small neighborhood of x, then ⟨J((N1, ν1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (Nr, νr)), f⟩ =  �  Nr fνr ̸= 0 and  ⟨J((N ′  1, ν′  1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (N ′  r, ν′  r)), f⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Since this contradicts our assumption, we conclude Nr = N ′  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  To show νr = ν′  r, we assume by contradiction that there exists x ∈ Nr = N ′  r with  νr(x) ̸= ν′  r(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  As before, we may w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' assume x /∈ Nr−1 and x /∈ N ′  r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Hence, if  f is a smooth bump function supported in a sufficiently small neighborhood of x, then  ⟨J((N1, ν1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (Nr, νr)), f⟩ =  �  Nr fνr ̸=  �  N′r fν′  r = ⟨J((N ′  1, ν′  1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (N ′  r, ν′  r)), f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Since this  contradicts our assumption, we conclude νr = ν′  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Suppose ω is a symplectic form on M, and let Flagsymp  S  (M) denote the manifold  of symplectic nonlinear flags of type S, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' [9, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Recall that this is the open subset  consisting of all flags (N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Nr) ∈ FlagS(M) such that ω restricts to a symplectic form on  each Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, ki must be even and ωki/2 pulls back to a volume form on Ni which in turn  gives rise to a volume density νi = |ι∗  Niωki/2| on Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Consequently, the symplectic form ω  provides a Symp(M, ω) equivariant injective smooth map (section)  Flagsymp  S  (M) → Flagwt  S (M)  (19)  which is right inverse to the restriction of the canonical bundle projection in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Composing  the map in (19) with J in (18), we obtain the moment map considered in [9, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' (38)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' A Riemannian metric g on M induces a volume density on every submanifold  of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, g provides a smooth section FlagS(M) → Flagwt  S (M) of the canonical bundle  projection in (15), which is Isom(M, g) equivariant, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' [26, Section 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='3  Reduction of structure group  It will be convenient to use a reduction of the structure group for the principal frame bundle  in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' To this end, we fix embeddings ιi : Si → Si+1 and put ι = (ι1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ιr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We begin by recalling some facts from [9, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The space of nonlinear flags  of type (S, ι) in M,  FlagS,ι(M) :=  �  (N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Nr) ∈ FlagS(M)  ����  �  S1  ι1  −→ S2  ι2  −→ · · · → Sr  �  ∼=  �  N1 ⊆ N2 ⊆ · · · ⊆ Nr  �  �  ,  consists of all nonlinear flags (N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Nr) in M such that there exist diffeomorphisms Si → Ni,  1 ≤ i ≤ r intertwining ιi with the canonical inclusion Ni ⊆ Ni+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This is a Diff(M) invariant  open and closed subset in FlagS(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The space of parametrized nonlinear flags (nonlinear  frames) of type (S, ι) in M,  FrS,ι(M) := {(ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ϕr) ∈ FrS(M)|∀i : ϕi = ϕi+1 ◦ ιi} ,  is a splitting smooth submanifold of FrS(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Moreover, the map FrS,ι(M) → FlagS,ι(M)  obtained by restriction of (13) is a smooth principal bundle with structure group  Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) :=  �  (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , gr) ∈  r  �  i=1  Diff(Si)  �����∀i : gi+1 ◦ ιi = ιi ◦ gi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (20)  7  The latter is a splitting Lie subgroup in Diff(S) with Lie algebra  X(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) =  �  (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Zr) ∈  r  �  i=1  X(Si)  �����∀i : Zi+1 ◦ ιi = Tιi ◦ Zi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (21)  We obtain a Diff(M) equivariant commutative diagram  FrS,ι(M)  Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='ι)  �  � �  � FrS(M)  Diff(S)  �  FlagS,ι(M)� �  � FlagS(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (22)  which may be regarded as a reduction of the structure group for (13) along the inclusion  Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) ⊆ Diff(S) over FlagS,ι(M), see [9, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The Diff(M) equivariant bijection  FrS,ι(M) = EmbSr(M),  (ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ϕr) �→ ϕr,  (23)  is a diffeomorphism [9, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Correspondingly, we have a group isomorphism  Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) = Diff(Sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Σ),  (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , gr) �→ gr,  (24)  where Diff(Sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Σ) denotes the subgroup of all diffeomorphisms of Sr preserving the nonlinear  flag Σ = (Σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Σr−1) in Sr, where Σi := (ιr−1 ◦ · · · ◦ ιi)(Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The latter is a splitting Lie  subgroup of Diff(Sr), see [9, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='9(b)], and (24) is a diffeomorphism of Lie groups  [9, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The Lie algebra of Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι), can be identified in a similar way with  X(Sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Σ), the Lie algebra of vector fields on Sr that are tangent to Σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Σr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We are interested in the reduction of structure group (22) because the Diffc(M) action on  FrS,ι(M) admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This follows from [9, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1(c)] and the diffeo-  morphism in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let Flagwt  S,ι(M) denote the preimage of FlagS,ι(M) under the bundle projection in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Restricting the diffeomorphism in (17) over FlagS,ι(M) and combining this with the diffeomor-  phism in (23), we obtain a Diff(M) equivariant diffeomorphism of bundles over FlagS,ι(M),  Flagwt  S,ι(M) = EmbSr(M) ×Diff(S,ι) Den×(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (25)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='4  The Diff(M) action on the space of weighted nonlinear flags  In this section we aim at describing the Diff(M) orbits in Flagwt  S (M), see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let ι = (ι1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ιr−1) be a collection of embeddings ιi : Si → Si+1 and suppose µ =  (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , µr) ∈ Den×(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We define the space of weighted flags of type (S, ι, µ) in M by  Flagwt  S,ι,µ(M) :=  ��  (N1, ν1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (Nr, νr)  �  ∈ Flagwt  S (M)  ����  �  S1  ι1  −→ S2  ι2  −→ · · · → Sr, µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , µr  �  ∼=  �  N1 ⊆ N2 ⊆ · · · ⊆ Nr, ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , νr  �  �  ,  that is, the space of all weighted flags  �  (N1, ν1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , (Nr, νr)  �  in M such that there exist  diffeomorphisms Si → Ni, 1 ≤ i ≤ r, intertwining ιi with the canonical inclusion Ni ⊆ Ni+1,  and taking µi to νi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  8  Denoting the Diff(S, ι) orbit of µ by Den(S)ι,µ, the diffeomorphism in (25) restricts to a  Diff(M) equivariant bijection  Flagwt  S,ι,µ(M) = EmbSr(M) ×Diff(S,ι) Den(S)ι,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (26)  Consider the finite dimensional vector space  H(S, ι) :=  r  �  i=1  Hki�  Si, ιi−1(Si−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OSi  �  =  r  �  i=1  H0  �  Si \\ ιi−1(Si−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (27)  Here the left hand side denotes relative de Rham cohomology with coefficients in the orientation  bundle, and we are using the convention S0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The Diff(S, ι) equivariant identification on the  right hand side indicates Poincar´e–Lefschetz duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We have a Diff(S, ι) equivariant linear  map  hS,ι : Den(S) → H(S, ι),  hS,ι(µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , µr) :=  �  [µ1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , [µr]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (28)  Pinning down the class [µ] := hS,ι(µ) thus amounts to specifying the integrals of µi over each  connected component of Si \\ ιi−1(Si−1) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='9 (Large codimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' If the codimensions dim(Si) − dim(Si−1) are all strictly  larger than one, then Hki�  Si, ιi−1(Si−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OSi  �  = Hki�  Si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OSi  �  = H0(Si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R), and  H(S, ι) =  r  �  i=1  Hki�  Si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OSi  �  =  r  �  i=1  H0  �  Si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Hence, in this case, the cohomology space H(S, ι) does not depend on the embeddings ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In this situation the following hold true:  (a) The Diff(S, ι)0 orbit of µ coincides with the convex set Den×(S) ∩ h−1  S,ι([µ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In particular,  this orbit is a splitting smooth submanifold in Den×(S) with finite codimension dim H(S, ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (b) The Diff(S, ι)0 action on Den×(S) ∩ h−1  S,ι([µ]) admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (c) Denoting the (finite) Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) orbit of [µ] by H(S, ι)[µ], the Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) orbit of µ is  Den(S)ι,µ = Den×(S) ∩ h−1  S,ι  �  H(S, ι)[µ]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (29)  In particular, Den(S)ι,µ is a splitting smooth submanifold in Den×(S) with finite codimen-  sion dim H(S, ι) and with tangent spaces  Tα Den(S)ι,µ = ker hS,ι =  �  (dγ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , dγr) : γi ∈ Ωki−1(Si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OSi), ι∗  i−1γi = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (30)  Moreover, the Diff(S, ι) action on Den(S)ι,µ admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (d) The canonical inclusion Den(S)ι,µ ⊆ �r  i=1 Den(Si)µi is a splitting smooth submanifold of  finite codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We postpone the proof of this proposition and proceed with the main result in this section:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In this situation the following hold true:  (a) The space Flagwt  S,ι,µ(M) is a splitting smooth submanifold in Flagwt  S,ι(M) with finite codi-  mension dim H(S, ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  9  (b) The canonical Diff(M) equivariant forgetful map Flagwt  S,ι,µ(M) → FlagS,ι(M) is a locally  trivial smooth fiber bundle with typical fiber Den(S)ι,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (c) The canonical inclusion Flagwt  S,ι,µ(M) ⊆ �r  i=1 Grwt  Si,µi(M) is a splitting smooth submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (d) The Diffc(M) action on Flagwt  S,ι,µ(M) admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In particular, each  connected component of Flagwt  S,ι,µ(M) is a Diffc(M)0 orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Furthermore, every Diff(M)  or Diffc(M) orbit in Flagwt  S,ι,µ(M) is a union of connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Parts (a) and (b) follow by combining (25) and (26) with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Part (c) follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10(d) and the reduction of structure groups in (22) via  the diffeomorphisms in (5) and (26), see also (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let us finally turn to part (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10(c), the (transitive) Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) action  on Den(S)ι,µ admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The Diffc(M) action on EmbSr(M) admits local  smooth sections too, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' [9, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1, we conclude that the Diffc(M)  action on Flagwt  S,ι,µ(M) admits local smooth sections, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We will prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10 by induction on the depth of the flags, using the following  crucial lemma whose proof we postpone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Let S be a closed submanifold of N such that dim(S) < dim(N) = n, and  consider the Diff(N, S) equivariant linear map  h : Den(N) → Hn(N, S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ON),  h(µ) = [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Then, for each κ ∈ Hn(N, S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ON), the natural Diff(N, S)0 action on  Diff(S) ×  �  Den×(N) ∩ h−1(κ)  �  (31)  admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We proceed by induction on r using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Let us denote  the truncated sequences by S′ := (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Sr−1), µ′ := (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , µr−1), and ι′ := (ι1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ιr−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  By induction, the Diff(S′, ι′)0 action on Den×(S′) ∩ h−1  S′,ι′([µ′]) admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Hence, there exist an open neighborhood U ′ of µ′ in Den×(S′) ∩ h−1  S′,ι′([µ′]) and a smooth map  Den×(S′) ∩ h−1  S′,ι′([µ′]) ⊇ U ′ σ′  −→ Diff(S′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι′)0,  such that for all ˜µ′ ∈ U ′ we have  �  σ′(˜µ′)  �  ∗µ′ = ˜µ′  and  σ′(µ′) = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (32)  Recall that Diff(S′, ι′) is a splitting Lie subgroup in Diff(Sr−1), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' [9, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='9(b)  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Using [9, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1(d)] this implies that Diff(S, ι) is a splitting Lie subgroup in  Diff(Sr, ιr−1(Sr−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, restricting a local smooth section as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='12, we see that  the Diff(S, ι)0 action on Diff(S′, ι′) ×  �  Den×(Sr) ∩ h−1([µr])  �  admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  In other words, there exist an open neighborhood V of the identity in Diff(S′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι′), an open  neighborhood U ′′ of µr in Den×(Sr) ∩ h−1([µr]), and a smooth map  Diff(S′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι′) ×  �  Den×(Sr) ∩ h−1([µr])  �  ⊇ V × U ′′ σ′′  −→ Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι)0,  10  such that for all g ∈ V and ˜µr ∈ U ′′ we have  σ′′(g, ˜µr) · (id, µr) = (g, ˜µr)  and  σ′′(id, µr) = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (33)  Hence, U := (σ′)−1(V ) × U ′′ is an open neighborhood of µ in Den×(S) ∩ h−1  S,ι([µ]), and  Den×(S) ∩ h−1  S,ι([µ]) ⊇ U  σ−→ Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι)0,  σ(˜µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ˜µr) := σ′′(σ′(˜µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ˜µr−1), ˜µr)  is a local smooth section for the Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι)0 action on Den×(S) ∩ h−1  S,ι([µ]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=',  σ(µ) = id  and  σ(˜µ)∗µ = ˜µ,  for all ˜µ ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' By convexity, Den×(S) ∩ h−1  S,ι([µ]) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Therefore, the Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι)0 action  is transitive on Den×(S) ∩ h−1  S,ι([µ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This shows (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Part (c) follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  To see (d), let A denote the preimage of �r  i=1 Hki(Si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OSi)[µi] under the canonical linear  surjection H(S, ι) → �r  i=1 Hki(Si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OSi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Hence, A is a finite union of affine subspaces in  H(S, ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In view of (7) we have �r  i=1 Den(Si)µi = Den×(S) ∩ h−1  S,ι(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Combining this with  (29), we conclude that Den(S)ι,µ is a splitting smooth submanifold in �r  i=1 Den(Si)µi with  finite codimension dim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let us next establish the following infinitesimal version of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='12:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Let S be a closed submanifold of N such that dim(S) < dim(N) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Suppose  µ ∈ Den×(N), γ ∈ Ωn−1(N, S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ON) := {α ∈ Ω(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ON) : ι∗  Sα = 0}, and Z ∈ X(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Then there  exists a vector field X ∈ X(N) such that LXµ = dγ and X|S = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Let ˜Z ∈ X(N) be any extension of Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ˜Z|S = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Note that i ˜Zµ ∈ Ωn−1(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ON)  vanishes when pulled back to S, hence the same holds for β := γ − i ˜Zµ ∈ Ωn−1(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ON).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let us first construct ˜γ ∈ Ωn−1(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ON) such that d˜γ = dβ and ˜γ|S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' To this end, we fix  a smooth homotopy h: N × [0, 1] → N such that h1 = idN, ht|S = idS, and such that h0 maps  a neighborhood of S into S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Consider the corresponding chain homotopy φ: Ω∗(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ON) →  Ω∗−1(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ON) defined by φ(α) :=  � 1  0 ι∗  t i∂th∗α dt, where ιt : N → N ×[0, 1] denotes the inclusion  at t, that is, ιt(x) := (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Then φ(α)|S = 0 and d(φ(α)) + φ(dα) = h∗  1α − h∗  0α, for all forms  α ∈ Ω∗(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ON).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In particular, dα = d  �  h∗  0α + φ(dα)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Defining ˜γ := h∗  0β + φ(dβ), we obtain  d˜γ = dβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover, h∗  0β|S = 0 because β vanishes when pulled back to S, hence ˜γ|S = 0 as  desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Defining a vector field Y ∈ X(N) by iY µ := ˜γ, we obtain Y |S = 0 and LY µ = diY µ = d˜γ =  d(γ − i ˜Zµ) = dγ − L ˜Zµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, the vector field X := ˜Z + Y has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Recall that the infinitesimal Diff(N, S) action on Diff(S) × Den(N) is  ζX(f, µ) =  �  RX|S(f), −LXµ  �  ,  where X ∈ X(N, S), f ∈ Diff(S), and µ ∈ Den(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Here, for Z ∈ Tid Diff(S) = X(S) we let  RZ(f) denote the right invariant vector field on Diff(S) such that RZ(id) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Note that the vector field ˜Z in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='13 can be chosen to depend smoothly  (and linearly) on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, the proof of said lemma actually provides a smooth map  ˜σ : Tid Diff(S) × T  �  Den×(N) ∩ h−1(κ)  �  → X(N, S)  11  such that ζ˜σ(Z,dγ)(id, µ) = (Z, dγ) for all Z ∈ X(S) = Tid Diff(S), µ ∈ Den×(N) ∩ h−1(κ), and  dγ ∈ dΩn−1(N, S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ON) = Tµ  �  Den×(N) ∩ h−1(κ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Combining this with the right trivialization  of T Diff(S), we obtain a smooth map  σ : T  �  Diff(S) ×  �  Den×(N) ∩ h−1(κ)  ��  → X(N, S)  such that ζσ(Z,ξ)(f, µ) = (Z, ξ) for all f ∈ Diff(S), Z ∈ Tf Diff(S), µ ∈ Den×(N) ∩ h−1(κ), and  ξ ∈ Tµ  �  Den×(N) ∩ h−1(κ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' As Diff(N, S) is a regular Lie group, we may apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='2  to conclude that the Diff(N, S)0 action on (31) admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  This completes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In view of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='3, we expect that the isotropy subgroup  Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι, µ) :=  �  (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , gr) ∈ Diff(S, ι)  �� ∀i : g∗  i µi = µi  �  (34)  is a splitting Lie subgroup of Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) with Lie algebra  X(S, ι, µ) =  �  (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Zr) ∈ X(S, ι)  �� ∀i : LZiµi = 0  �  ,  (35)  and the surjective map provided by the action, Diff(S, ι) → Den(S)ι,µ, is a locally trivial  smooth principal fiber bundle with structure group Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This would follow in a rather  straigthforward manner, via induction on the depth of the flags, if one could show that the  isotropy group {f ∈ Diff(N, S) : f|S = id, f ∗µ = µ} is a splitting Lie subgroup in Diff(N, S),  whenever S is a closed submanifold of N and µ is a volume density on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The proof in [11,  Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='3 on page 203] covers the case S = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' However, the adaptation of said proof  to nontrivial S is not entirely straigthforward, and we will not attempt to prove this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Note that via the diffeomorphism in (24) the group Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι, µ) corresponds to the subgroup  of Diff(Sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Σ) consisting of all diffeomorphisms that preserve µr and whose restriction to Σi  preserves (ιr−1 ◦ · · · ◦ ιi)∗µi, for 1 ≤ i ≤ r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Similarly, the Lie algebra X(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι, µ) can be  identified to the corresponding subalgebra of X(Sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  If the expectation formulated in the preceding paragraph were indeed true, then the sur-  jective and Diff(M) equivariant map  EmbSr(M) = FrS,ι(M) → Flagwt  S,ι,µ(M),  (36)  (ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ϕr) �→  ��  ϕ1(S1), (ϕ1)∗µ1  �  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ,  �  ϕr(Sr), (ϕr)∗µr  ��  ,  (37)  would be a locally trivial smooth principal fiber bundle with structure group Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι, µ),  see (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover, generalizing Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='4, the isotropy group of a weighted flag (N, ν),  Diffc(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' N, ν) :=  �  g ∈ Diffc(M)  �� ∀i : g(Ni) = Ni, g|∗  Niνi = νi  �  ,  would be a splitting Lie subgroup of Diffc(M), for it coincides with the preimage of Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι, µ)  under the bundle projection Diffc(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' N) → Diff(N, ιN ), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' [9, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1(d), Proposi-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='9(b), and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Furthermore, the orbit map Diffc(M) → Flagwt  S (M)(N ,ν)  provided by the action would be a locally trivial smooth principal bundle with structure group  Diffc(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' N, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, the Diffc(M) orbit of (N, ν) could be regarded as a homogeneous space  Flagwt  S (M)(N ,ν) = Diffc(M)/ Diffc(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' N, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='5  A homological description  In this section we give a more explicit description of the manifold Flagwt  S,ι,µ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  If N = (N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Nr) is a nonlinear flag of type S in M, we put  H(N) :=  r  �  i=1  Hki�  Ni, Ni−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ONi  �  =  r  �  i=1  H0  �  Ni \\ Ni−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R  �  ,  with the convention that N0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We define the space of homologically weighted flags of type S in M by  Flaghwt  S  (M) :=  �  (N, [ν]) : N ∈ Flagwt  S (M), [ν] ∈ H(N)  �  Note that we have a Diff(M) equivariant forgetful map  Flaghwt  S  (M) → FlagS(M),  (38)  as well as a Diff(M) equivariant map  hFlagS(M) : Flagwt  S (M) → Flaghwt  S  (M),  (N, ν) �→ (N, [ν]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (39)  Let Flaghwt  S,ι (M) denote the preimage of the open subset FlagS,ι(M) under the projection  in (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Using the canonical Diff(M) equivariant identifications  Flaghwt  S,ι (M) = EmbSr(M) ×Diff(S,ι) H(S, ι),  (40)  we equip Flaghwt  S  (M) with the structure of a smooth vector bundle of finite (possibly noncon-  stant) rank over FlagS(M) and with projection (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The map in (39) is a smooth bundle map  over FlagS(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Indeed, via the identifications in (25) and (40), the map hS,ι in (28) induces  a bundle map Flagwt  S,ι(M) → Flaghwt  S,ι (M) which coincides with the restriction of (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We define the space of homologically weighted flags of type (S, ι, [µ]) in M by  Flaghwt  S,ι,[µ](M) :=  �  (N, [ν]) ∈ Flaghwt  S  (M)  ����  �  S1  ι1  −→ S2  ι2  −→ · · · → Sr, [µ]  �  ∼=  �  N1 ⊆ N2 ⊆ · · · ⊆ Nr, [ν]  �  �  ,  that is, the space of all homologically weighted flags (N, [ν]) in M such that there exist dif-  feomorphisms Si → Ni, 1 ≤ i ≤ r, intertwining ιi with the canonical inclusion Ni ⊆ Ni+1 and  taking [µ] ∈ H(S, ι) to [ν] ∈ H(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The diffeomorphism in (40) restricts to a bijection  Flaghwt  S,ι,[µ](M) = EmbSr(M) ×Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='ι) H(S, ι)[µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (41)  Hence, since the orbit H(S, ι)[µ] is finite, Flaghwt  S,ι,[µ](M) is a Diff(M) invariant closed subman-  ifold in Flaghwt  S,ι (M) of codimension dim H(S, ι) by (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover, the forgetful map  Flaghwt  S,ι,[µ](M) → FlagS,ι(M)  (42)  is a Diff(M) equivariant covering map with finite fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We complement Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='11 with the following:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In this situation we have  Flagwt  S,ι,µ(M) = h−1  FlagS(M)  �  Flaghwt  S,ι,[µ](M)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This follows from the identity (29) in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10(c), using (26) and (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' If the action of Diff(S, ι) on H(S, ι) is trivial, then the Diff(S, ι)0 and Diff(S, ι)  orbits in Den×(S) coincide in view of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10, and the forgetful (covering) map in  (42) is a diffeomorphism, Flaghwt  S,ι,[µ](M) = FlagS,ι(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This happens in particular when all  Si \\ι(Si−1) are connected, as in Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='17 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='18 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Under the latter connectedness  assumption, H(S, ι) = Rr via the isomorphism [µ] �→  ��  S1 µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ,  �  Sr µr  �  and  Den(S)ι,µ =  �  α ∈ Den×(S) :  �  Si αi =  �  Si µi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Moreover, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='15 ensures that  Flagwt  S,ι,µ(M) =  �  (N, ν) ∈ Flagwt  S,ι(M) :  �  Ni νi =  �  Si µi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (43)  This also applies in the situation of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='9, provided each model manifold Si is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  A description for nested spheres in the same vein can be found in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='17 (Nested tori).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' If (S, ι) denotes the standard (meridional) embeddings between  tori, T0 ⊆ T1 ⊆ · · · ⊆ Tr, then H(S, ι) = Rr+1 via the isomorphism [µ] �→ (  �  Ti µi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='18 (Nested projective spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' If (S, ι) denotes the standard embeddings between  projective spaces, P0 ⊆ P1 ⊆ · · · ⊆ Pr, then H(S, ι) = Rr+1 via the isomorphism [µ] �→ (  �  Pi µi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='19 (Nested spheres [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' If (S, ι) denotes the standard equatorial embeddings be-  tween spheres, S0 ⊆ S1 ⊆ · · · ⊆ Sr, then H(S, ι) = R2(r+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The 2(r + 1) numbers assigned to  [µ] ∈ H(S, ι) by this isomorphism are  (a+  0 , a−  0 , a+  1 , a−  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , a+  r , a−  r ) =  ��  S0  + µ0,  �  S0  − µ0,  �  S1  + µ1,  �  S1  − µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ,  �  Sr  + µr,  �  Sr  − µr  �  ,  where Si  + and Si  − denote the northern and southern hemispheres of Si, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Consid-  ering reflections on hyperplanes, we see that for each 0 ≤ i ≤ r there exists a diffeomorphism  in Diff(S, ι) swapping Si  + with Si  −, but leaving all other hemispheres Sk  ± invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Such a  diffeomorphism interchanges a+  i with a−  i , but leaves all other numbers a±  k unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence,  the Diff(S, ι) orbit H(S, ι)[µ] has 2s elements, where s is the number of 0 ≤ i ≤ r with a+  i ̸= a−  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Actually the Diff(S, ι) action on H(S, ι) ∼= R2(r+1) factorizes through an (Z2)2(r+1) action by  switching or not the numbers a+  i and a−  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We obtain a description similar to the one in (43):  Flagwt  S,ι,µ(M) =  �  (N, ν) ∈ Flagwt  S,ι(M)  �����  {  �  N+  i νi,  �  N−  i νi} = {a+  i , a−  i }, where N +  i  and N −  i  denote the connected components of Ni \\ Ni−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (44)  3  Coadjoint orbits of the Hamiltonian group  Throughout this section (M, ω) denotes a symplectic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The nonlinear Grassmannian  of all isotropic submanifolds of type Sr in M, denoted by Griso  Sr (M), is a splitting smooth  submanifold in GrSr(M) which is invariant under the Hamiltonian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In fact the Ham(M)  orbits provide a smooth foliation of finite codimension in Griso  Sr (M) which is called the isodrastic  foliation [26, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Suppose L is an isodrastic leaf in Griso  Sr (M) and let Flagwt iso  S,ι,µ (M)|L denote the preimage  of L under the canonical bundle projection Flagwt  S,ι,µ(M) → GrSr(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We will show that the  14  natural Hamc(M) action on Flagwt iso  S,ι,µ (M)|L admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In particular, each  connected component of the latter space is an orbit of Hamc(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The space Flagwt iso  S,ι,µ (M)|L comes equipped with a canonical weakly non-degenerate sym-  plectic form, and the restriction of (18) provides an Ham(M) equivariant injective moment  map for the Hamc(M) action,  J : Flagwt iso  S,ι,µ (M)|L ֒→ hamc(M)∗,  ⟨J(N, ν), Xf⟩ =  r  �  i=1  �  Ni  fνi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  This moment map J maps each connected component of Flagwt iso  S,ι,µ (M)|L one-to-one onto the  corresponding coadjoint orbit, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Thereby, we identify coadjoint orbits of the  Hamiltonian group Hamc(M) that can be modeled on weighted nonlinear flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The material in this section is inspired by the results in [26, 15, 7] on weighted isotropic  nonlinear Grassmannians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1  Isodrasts as Ham(M) orbits  In view of the tubular neighborhood theorem for isotropic embeddings [23, 24], the space  Griso  S (M) of all isotropic submanifolds of type S in M is a splitting smooth submanifold of  GrS(M), see for instance [15, Section 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The tangent space at an isotropic submanifold N is  TN Griso  S (M) = {uN ∈ Γ(TN ⊥)|ι∗  NiuN ω ∈ Ω1(N) closed},  where TN ⊥ := TM|N/TN denotes the normal bundle to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Weinstein’s [26] isodrastic distribution D on Griso  S (M) is given by  DN := {uN ∈ Γ(TN ⊥)|ι∗  NiuN ω ∈ dC∞(N)}  (45)  and has finite codimension dim H1(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This is an integrable distribution [26, 15] whose  leaves are orbits of Ham(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' It gives rise to a smooth foliation of Griso  S (M) called the isodrastic  foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In particular, the Ham(M) orbits in Griso  S (M) are splitting smooth submanifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Restricting the fundamental frame bundle in (1), we obtain a principal Diff(S) bundle  Embiso  S (M) → Griso  S (M) with total space Embiso  S (M), the splitting smooth submanifold of  EmbS(M) consisting of all isotropic embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' On Embiso  S (M) we consider the pullback of  the isodrastic distribution D:  Dϕ :=  �  uϕ ∈ Γ(ϕ∗TM) : ϕ∗iuϕω ∈ dC∞(S)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (46)  This is an integrable distribution with the same codimension, dim H1(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R), and the leaves  of D are connected components in the preimage of a leaf of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' According to [26, 15], the  group Ham(M) acts transitively on the leaves of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We need the subsequent slightly stronger  statement in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The Lie algebra of compactly supported Hamiltonian vector fields will be denoted by  hamc(M) = {Xf : f ∈ C∞  c (M)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We let Hamc(M) denote the group of diffeomorphisms  obtained by integrating time dependent vector fields in hamc(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' For our purpose it will not  be necessary to consider Hamc(M) as an infinite dimensional Lie group, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' [14, Section 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  By a smooth map (section) into Hamc(M) we will simply mean a smooth map into Diffc(M)  that takes values in Hamc(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The Hamc(M) action on each leaf E ⊆ Embiso  S (M) of the isodrastic foliation  D admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  15  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' To show infinitesimal transitivity, suppose ϕ ∈ Embiso  S (M) and uϕ ∈ Dϕ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Hence, there exists ¯f ∈ C∞(S) such that d ¯f = ϕ∗iuϕω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We extend ¯f to a function f1 ∈ C∞  c (M)  such that ¯f = f1 ◦ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The 1-form β on M along S defined by  β = df1 ◦ ϕ − iuϕω ∈ Γ(ϕ∗T ∗M)  (47)  vanishes on vectors tangent to ϕ(S) ⊆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, β can be seen as a fiberwise linear function  on the normal bundle Tϕ(S)⊥ whose differential along the zero section ϕ(S) coincides with  β itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Thus, with the help of a tubular neighborhood of ϕ(S) in M and a suitable bump  function, we get f2 ∈ C∞  c (M) such that β = df2 ◦ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' It follows from (47) that df ◦ ϕ = iuϕω  for f = f1 − f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We conclude that uϕ = Xf ◦ ϕ is the infinitesimal generator at ϕ for the  Hamiltonian vector field Xf ∈ hamc(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Using tubular neighborhoods constructed with the help of a Riemannian metric, say, we see  that the function f may be chosen to depend smoothly on ϕ und uϕ, for ϕ in a sufficiently small  open neighborhood of a fixed isotropic embedding ϕ0 ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, we may apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='2  and conclude that the Hamc(M) action admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The Hamc(M) action on each leaf L ⊆ Griso  S (M) of the isodrastic foliation D  admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Every embedded closed curve in the plane is a Lagrangian submanifold of (R2, ω),  where ω is the canonical area form, thus an element of Griso  S1(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The isodrastic distribution  D has codimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The enclosed area a singles out one isodrast La ⊆ Griso  S1(R2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' one  orbit of Hamc(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  A similar phenomena happens for Lagrangian k-tori in R2k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' elements of Griso  Tk(R2k),  where Tk := (S1)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' To any ϕ ∈ Embiso  Tk(R2k) we assign the symplectic area ai of the surface  in R2k enclosed by the ith meridian ϕi(θ) = ϕ(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , θ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , 1) of the embedded k-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' These  numbers are independent of the choice of the meridian in its homotopy class and of the surface  having the meridian as boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The k-tuple (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ak) is an invariant under isodrastic  deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Actually ai is the action integral of the ith meridian, as defined in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let Ea1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=',ak ⊆ Embiso  Tk(R2k) be the space of all isotropic embeddings having symplectic areas  a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' It is a union of isodrastic leaves of Lagrangian embeddings, but it is not necessarily  Diff(Tk) saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The Diff(Tk) action on the k-tuples (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ak) factorizes through an  GL(k, Z) action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let [a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ak] denote the orbit of (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We define L[a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=',ak] ⊆  Griso  Tk(R2k) to be the image of Ea1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=',ak under the principal Diff(Tk) bundle projection (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Thus L[a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=',ak] is a union of isodrastic leaves of Lagrangian k-tori in R2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  There is also a direct description of L[a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=',ak].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Given a Lagrangian torus N ∈ GrS  Tk(R2k),  we choose a base {[γ1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , [γk]} of H1(N, Z), where γi are loops in N with action integrals ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We observe that the GL(k, Z) orbit [a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ak] is independent of the choices and L[a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=',ak] is  the space of all Lagrangian tori in GrS  Tk(R2k) such that the orbit of these action integrals is  [a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ak].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We will also need the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' If L is an isotropic submanifold in M, then the canonical inclusion GrS(L) ⊆  Griso  S (M) is a splitting smooth submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover, each connected component of GrS(L)  is a splitting smooth submanifold in an isodrastic leaf in Griso  S (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Suppose N ∈ GrS(L), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=', N ∼= S is a closed submanifold in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  By the tubular  neighborhood theorem, we may w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' assume that L is the total space of a vector bundle  p: L → N, the normal bundle of N in L, and identify N with the zero section in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We  16  have a canonical short exact sequence 0 → p∗L → TL → p∗TN → 0 of vector bundles  over L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Choosing a linear connection on L, we obtain a splitting of this sequence and thus an  isomorphism TL ∼= p∗TN ⊕p∗L of vector bundles over L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Dualizing, we obtain an isomorphism  T ∗L ∼= p∗T ∗N ⊕ p∗L∗ of vector bundles over L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We regard this as a diffeomorphism  T ∗L ∼= T ∗N ⊕ L ⊕ L∗  that maps the zero section L ⊆ T ∗L identically onto the summand L on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Via this isomorphism we have  θL = π∗  1θN + π∗  2κ,  where π1 and π2 denote the projections from T ∗N ⊕L⊕L∗ onto T ∗N and L⊕L∗, respectively,  θL ∈ Ω1(T ∗L) and θN ∈ Ω1(T ∗N) denote the tautological 1-forms, and κ ∈ Ω1(L ⊕ L∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Indeed, a straightforward computation yields κ(ξ) = ℓ′(C(Tq · ξ)) where x ∈ N, ℓ ∈ Lx,  ℓ′ ∈ L∗  x, ξ ∈ T(ℓ,ℓ′)(L ⊕ L∗), q: L ⊕ L∗ → L denotes the projection and C denotes the linear  connection on L, viewed as a fiberwise linear map C : TL → p∗L over L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  By the tubular neighborhood theorem for isotropic embeddings [23, 24], we may assume  M = T ∗L ⊕ p∗E and ω = ˜π∗  1dθL + ˜π∗  2ρ, where E is a vector bundle over N, ˜π1 and ˜π2 denote  the projections from T ∗L ⊕ p∗E onto T ∗L and p∗E, respectively, and ρ is a closed 2-form on  the total space of p∗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Combining this with the diffeomorphism in the previous paragraph,  we obtain a diffeomorphism  M = T ∗N ⊕ L ⊕ L∗ ⊕ E,  (48)  mapping L ⊆ M identically onto the summand L on the right hand side, and such that  ω = π∗  1dθN + π∗  2σ,  where π1 and π2 denote the projections from T ∗N ⊕ L ⊕ L∗ ⊕ E onto T ∗N and L ⊕ L∗ ⊕ E,  respectively, and σ is a closed 2-form on the total space of L ⊕ L∗ ⊕ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The diffeomorphism in (48) provides a (standard) chart for the smooth structure on GrS(M)  centered at N,  Γ(T ∗N ⊕ L ⊕ L∗ ⊕ E) → GrS(M),  φ �→ φ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  In this chart, the inclusions GrS(L) ⊆ Griso  S (M) ⊆ GrS(M) become  Γ(L) ⊆ {(α, ξ) ∈ Ω1(N) × Γ(L ⊕ L∗ ⊕ E) : dα + ξ∗σ = 0} ⊆ Ω1(N) × Γ(L ⊕ L∗ ⊕ E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (49)  As L is isotropic, σ vanishes when pulled back to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, by the Poincar´e lemma, σ = dβ for  a 1-form β on the total space of L ⊕ L∗ ⊕ E which vanishes along L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Via the diffeomorphism  of Ω1(N) × Γ(L ⊕ L∗ ⊕ E) given by (α, ξ) �→ (α − ξ∗β, ξ), the inclusions in (49) become linear  inclusions,  Γ(L) ⊆ Z1(N) × Γ(L ⊕ L∗ ⊕ E) ⊆ Ω1(N) × Γ(L ⊕ L∗ ⊕ E),  where Z1(N) denotes the space of closed 1-forms on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Clearly, both inclusions admit com-  plementary subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In particular, GrS(L) is a splitting smooth submanifold in Griso  S (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The second assertion follows from the fact that the isodrastic leaf through N corresponds to  the subspace B1(N) × Γ(L ⊕ L∗ ⊕ E), see [15, Section 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  17  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='2  Weighted isotropic nonlinear Grassmannians as coadjoint orbits  In this section we recall the results in [26, 15, 7] about coadjoint orbits of the Hamiltonian  group Hamc(M) modeled on weighted isotropic nonlinear Grassmannians of type S in M, and  extend them to a possibly nonconnected model manifold S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Here we present them in a manner  that readily generalizes to manifolds of weighted nonlinear flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let S be a closed k-dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The preimage of Griso  S (M) under the canonical  bundle projection Grwt  S (M) → GrS(M) is a splitting smooth submanifold in Grwt  S (M) which  will be denoted by Grwt iso  S  (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The diffeomorphism in (3) restricts to a diffeomorphism of  bundles over Griso  S (M),  Grwt iso  S  (M) = Embiso  S (M) ×Diff(S) Den×(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The preimage of Griso  S (M) under the canonical bundle projection Grwt  S,µ(M) → GrS(M) is  a splitting smooth submanifold in Grwt  S,µ(M) which will be denoted by Grwt iso  S,µ (M) and will be  referred to as the nonlinear Grassmannian of weighted isotropic submanifolds of type (S, µ) in  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The diffeomorphism in (5) restricts to a diffeomorphism of bundles over Griso  S (M),  Grwt iso  S,µ (M) = Embiso  S (M) ×Diff(S) Den(S)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We recall the Diff(S) equivariant linear map hS : Den(S) → Hk(S, OS), hS(α) = [α]  in (6), with kernel dΩk−1(S, OS), which we restrict to Den×(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The product of integrable  distributions D × ker ThS on Embiso  S (M) × Den×(S) descends to an integrable distribution  ¯D := D ×Diff(S) ker ThS  on Grwt iso  S  (M), of codimension dim H1(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R) + dim Hk(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The image of ¯D under the  forgetful map Grwt iso  S  (M) → Griso  S (M) is the isodrastic distribution D in (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In view of (7),  each leaf G of ¯D is a connected component in the preimage of an isodrast L in Griso  S (M) under  the bundle projection Grwt iso  S,µ (M) → Griso  S (M), for some volume density µ on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, each  leaf G of ¯D is a splitting smooth submanifold of codimension dim H1(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R) in Grwt iso  S,µ (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  According to [26, 15], the group Ham(M) acts transitively on the leaves of ¯D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We need the  following slightly stronger statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The Hamc(M) action on each leaf G ⊆ Grwt iso  S  (M) of the isodrastic foliation  ¯D admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Suppose L is an isodrastic leaf in Griso  S (M) and let Grwt iso  S,µ (M)|L denote its preimage  under the bundle projection Grwt iso  S,µ (M) → Griso  S (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' It suffices to show that the Hamc(M)  action on Grwt iso  S,µ (M)|L admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The diffeomorphism in (5) restricts to  a diffeomorphism  Grwt iso  S,µ (M)|L = Embiso  S (M)|L ×Diff(S) Den(S)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (50)  By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1, the Hamc(M) action on Embiso  S (M)|L admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Ac-  cording to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10(b), the Diff(S) action on Den(S)µ admits local smooth sections too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1 in the Appendix, we conclude that the Hamc(M) action on the associated  bundle in (50) admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='6 ([15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The leafwise differential 2-form on (Embiso  S (M) × Den×(S), D × ker ThS),  given by  Ω(ϕ,α)((uϕ, dλ), (vϕ, dγ)) :=  �  S  (ω(uϕ, vϕ)α − ϕ∗iuϕω ∧ γ + ϕ∗ivϕω ∧ λ),  (51)  18  is closed and Diff(S) invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover, its kernel is spanned by the infinitesimal generators  of the Diff(S) action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='6, the leafwise differential 2-form Ω in (51) descends to a leafwise symplectic  form ¯Ω on (Grwt iso  S  (M), ¯D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Thus every leaf G of the isodrastic distribution ¯D in Grwt iso  S  (M)  is endowed with a symplectic form, the restriction of ¯Ω, which we denote by the same letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='5, G is an orbit for the Hamc(M) action on the weighted isotropic nonlinear  Grassmannian Grwt iso  S  (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This action is Hamiltonian with injective and Ham(M) equivariant  moment map [15]  J : G ⊆ Grwt iso  S  (M) → hamc(M)∗,  ⟨J(N, ν), Xf ⟩ =  �  N  fν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (52)  Here we use the Symp(M) equivariant isomorphism hamc(M) = C∞  0 (M) where the latter  denotes the Lie algebra of all compactly supported functions on M for which the integral with  respect to the Liouville form vanishes on all closed connected components of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  For connected S, the subsequent theorem is due to Weinstein [26] in the Lagrangian case  and due to Lee [15] for isotropic submanifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='7 ([26, 15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The moment map J : (G, ¯Ω) → hamc(M)∗ in (52) is one-to-one onto  a coadjoint orbit of the Hamiltonian group Hamc(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The Kostant-Kirillov-Souriau symplectic  form ωKKS on the coadjoint orbit satisfies J∗ωKKS = ¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  In this generality the theorem follows from the discussion above and the following folklore  result (see for instance the Appendix in [9]):  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Suppose the action of G on (M, Ω) is transitive with injective equivariant  moment map J : M → g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Then J is one-to-one onto a coadjoint orbit of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover, it  pulls back the Kostant–Kirillov–Souriau symplectic form ωKKS on the coadjoint orbit to the  symplectic form Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The same coadjoint orbits of the Hamiltonian group, under the additional restriction  H1(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R) = 0 on the closed connected manifold S, can be obtained via symplectic reduction  in the Marsden-Weinstein ideal fluid dual pair, as shown in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='3  Manifolds of isotropic nonlinear flags  In this section we extend the constructions of the previous section to nonlinear flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  As in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='3, we let ι = (ι1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , ιr−1) denote a collection of fixed embeddings ιi : Si →  Si+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We let Flagiso  S,ι(M) denote the preimage of Griso  Sr (M) under the canonical bundle pro-  jection FlagS,ι(M) → GrSr(M), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' [9, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This is a splitting smooth submanifold  in FlagS,ι(M) called the manifold of isotropic nonlinear flags of type (S, ι) in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Using  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='4, and proceeding by induction on the depth of the flag, one readily shows that the  canonical inclusion  Flagiso  S,ι(M) ⊆  r  �  i=1  Griso  Si (M)  (53)  is a splitting smooth submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let Flagwt iso  S,ι  (M) denote the preimage of Flagiso  S,ι(M) under the canonical bundle projection  Flagwt  S,ι(M) → FlagS,ι(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This is a splitting smooth submanifold in Flagwt  S,ι(M) called the  19  manifold of weighted isotropic nonlinear flags of type (S, ι) in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The diffeomorphism in (25)  restricts to a diffeomorphism of bundles over Flagiso  S,ι(M),  Flagwt iso  S,ι  (M) = Embiso  Sr (M) ×Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='ι) Den×(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The canonical inclusion Flagwt iso  S,ι  (M) ⊆ �r  i=1 Grwt iso  Si  (M) is a splitting smooth submanifold in  view of (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  As in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='4, we fix µ = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , µr) ∈ Den×(S), and let Flagwt iso  S,ι,µ (M) denote the  preimage of Flagiso  S,ι(M) under the canonical bundle projection Flagwt  S,ι,µ(M) → FlagS,ι(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  This is a splitting smoth submanifold in Flagwt  S,ι,µ(M) called the manifold of weighted isotropic  nonlinear flags of type (S, ι, µ) in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The diffeomorphism in (26) restricts to a diffeomorphism  Flagwt iso  S,ι,µ (M) = Embiso  Sr (M) ×Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='ι) Den(S)ι,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (54)  Using (53) and proceeding as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='11(c), one readily shows that the  canonical inclusion Flagwt iso  S,ι,µ (M) ⊆ �r  i=1 Grwt iso  Si,µi (M) is a splitting smooth submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Recall the Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) equivariant linear map hS,ι : Den(S) → H(S, ι) in (28) with kernel  ker ThS,ι = {(dγ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , dγr) : γi ∈ Ωki−1(Si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OSi), ι∗  i−1γi = 0},  whose restriction to Den×(S) we also denote by hS,ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The product with the isodrastic dis-  tribution gives the integrable distribution D × ker ThS,ι on Embiso  Sr (M) × Den×(S), of finite  codimension dim H1(Sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R)+dim H(S, ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) invariant distribution descends to an  integrable distribution  ¯D = D ×Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='ι) ker ThS,ι  of the same codimension on Flagwt iso  S,ι  (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The image of ¯D under the forgetful map is an  integrable distribution on Flagiso  S,ι(M) of codimension dim H1(Sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R), which coincides with the  distribution that descends from the Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) invariant isodrastic distribution D on Embiso  Sr (M)  by the principal bundle projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10(c) and (54), we see that each leaf F  of ¯D is a connected component in the preimage of an isodrast L in Griso  Sr (M) under the bundle  projection Flagwt iso  S,ι,µ (M) → Griso  Sr (M), for some volume density µ on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, each leaf F of  ¯D is a splitting smooth submanifold of codimension dim H1(Sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R) in Flagwt iso  S,ι,µ (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' If H1(Sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R) = 0, then the leaves of the isodrastic foliation ¯D are the connected  components of Flagwt iso  S,ι,µ (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The Hamc(M) action on each leaf F ⊆ Flagwt iso  S,ι  (M) of the isodrastic  foliation ¯D admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Suppose L is an isodrastic leaf in Griso  Sr (M) and let Flagwt iso  S,ι,µ (M)|L denote its preimage  under the canonical projection Flagwt  S,ι,µ(M) → GrSr(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' It suffices to show that the Hamc(M)  action on Flagwt iso  S,ι,µ (M)|L admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The diffeomorphism in (54) restricts  to a diffeomorphism  Flagwt iso  S,ι,µ (M)|L = Embiso  Sr (M)|L ×Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='ι) Den(S)ι,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (55)  By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1, the Hamc(M) action on Embiso  Sr (M)|L admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Ac-  cording to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10(c), the Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) action on Den(S)ι,µ admits local smooth sections  too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' With the help of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1, we conclude that the Hamc(M) action on the associated  bundle (55) admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The Hamc(M) action on each leaf of the isodrastic foliation on Flagiso  S,ι(M)  admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='4  Weighted isotropic nonlinear flag manifolds as coadjoint orbits  In this section we describe coadjoint orbits of the Hamiltonian group consisting of weighted  isotropic nonlinear flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We aim at defining a leafwise symplectic form ¯Ω on (Flagwt iso  S,ι  (M), ¯D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We start with  leafwise differential 2-forms Ωi on (Embiso  Si (M) × Den×(Si), Di × ker ThSi), for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , r,  defined as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Let ji := ιr−1 ◦ · · · ◦ ιi ∈ Emb(Si, Sr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We consider  qi := j∗  i × pri : Embiso  Sr (M) × Den×(S) → Embiso  Si (M) × Den×(Si),  which maps the distribution D × ker ThS,ι to Di × ker ThSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Then  Ω :=  r  �  i=1  q∗  i Ωi  (56)  is a closed leafwise differential 2-form on (Embiso  Sr (M) × Den×(S), D × ker ThS,ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The image under Tqi of the distribution D × ker ThS,ι is, in general, strictly  included in the distribution Di × ker ThSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The reason is that the projection on the ith factor  pri : ker ThS,ι → ker ThSi is not surjective in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This happens for instance in the setting  of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='18, where ι : {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , k} → S1 with k ≥ 2: the projection on the second factor is  d{γ ∈ C∞(S1) : γ ◦ ι = 0}, strictly included in ker ThS1 = dC∞(S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) action on Embiso  Sr (M) × Den×(S),  g · (ϕ, (αi)) = (ϕ ◦ gr, (g∗  i αi)), for g = (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , gr) ∈ Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι),  (57)  has infinitesimal generators of the form  ζZ(ϕ, (αi)) = (Tϕ ◦ Zr, (diZiαi)), for Z = (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Zr) ∈ X(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (58)  They belong to the integrable distribution D×ker ThS,ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Notice that qi = j∗  i ×pri is equivariant  over the homomorphism g ∈ Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) �→ gi ∈ Diff(Si), because ji ◦ gi = gr ◦ ji for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Now,  from the Diff(Si) invariance of Ωi and (56), we deduce the Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) invariance of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The next lemma is a generalization of the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='6:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The kernel of the leafwise differential form Ω is spanned by the infinitesimal  generators of the Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) action (57) on Embiso  Sr (M) × Den×(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The formula for the leafwise 2-form in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='6 provides a similar formula for Ω:  Ω(ϕ,(αi))((uϕ, (dλi)), (vϕ, (dγi))) =  r  �  i=1  �  Si  j∗  i ω(uϕ, vϕ)αi  −  r  �  i=1  �  Si  j∗  i (ϕ∗iuϕω) ∧ γi +  r  �  i=1  �  Si  j∗  i (ϕ∗ivϕω) ∧ λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (59)  The contraction of Ω with an infinitesimal generator ζZ, given in (58), vanishes for all Z ∈  X(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This follows from the analogous statement for Ωi and the infinitesimal generators ζZi  on Embiso  Si (M) × Den×(Si), together with the fact that the infinitesimal generators ζZ and ζZi  are qi related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  For the converse statement, we start with an arbitrary tangent vector  (uϕ, (dλi)) ∈ ker Ω(ϕ,(αi)) ⊆ Dϕ × ker hS,ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  21  We need to find Z = (Zi) ∈ X(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) such that uϕ = Tϕ ◦ Zr and dλi = diZiαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In particular  the rth component Zr has to be tangent to the nonlinear flag Σ = (Σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Σr−1) in Sr, where  Σi := ji(Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The condition that (uϕ, (dλi)) lies in the kernel of Ω is equivalent to  r  �  i=1  �  Si  j∗  i (ϕ∗iuϕω) ∧ γi = 0, for all γi ∈ Ωki−1(Si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OSi), ι∗  i−1γi = 0,  (60)  and  r  �  i=1  �  Si  j∗  i ω(uϕ, vϕ)αi +  r  �  i=1  �  Si  j∗  i (ϕ∗ivϕω) ∧ λi = 0, for all vϕ ∈ Dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (61)  Choosing differential forms γi = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , r − 1 in (60), we get that  �  Sr ϕ∗iuϕω ∧ γr = 0,  for all γr ∈ Ωkr−1(Sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' OSr) with ι∗  r−1γr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This ensures that ϕ∗iuϕω = 0, meaning that uϕ  takes values in the symplectic orthogonal to ϕ(Sr) in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We assume by contradiction that uϕ ∈ Dϕ is not everywhere tangent to the isotropic  submanifold ϕ(Sr) ⊆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The subset Sr \\ Σr−1 is dense in Sr (because dim Sr−1 < dim Sr), so  there exists x ∈ Sr \\Σr−1 such that uϕ(x) is not tangent to ϕ(Sr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We choose another tangent  vector vϕ ∈ Dϕ taking values in the symplectic orthogonal to ϕ(Sr) in M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ϕ∗ivϕω = 0,  supported in an open neighborhood of x in Sr \\ Σr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The isotropic embedding theorem  of Weinstein [24] allows to choose vϕ such that ω(uϕ, vϕ) ∈ C∞(Sr) is a nonzero nowhere  negative function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Plugging vϕ in (61), all terms are zero except  �  Sr ω(uϕ, vϕ)αr > 0, leading  to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Thus there exists Zr ∈ X(Sr) with uϕ = Tϕ ◦ Zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The identity (61), rewritten with our uϕ = Tϕ ◦ Zr and arbitrary vϕ ∈ Dϕ, so ϕ∗ivϕω = dh  with arbitrary h ∈ C∞(S), becomes:  r  �  i=1  �  Si  j∗  i (iZrdh)αi =  r  �  i=1  �  Si  j∗  i (dh) ∧ λi, for all h ∈ C∞(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (62)  We need to show that Zr ∈ X(Sr) is tangent to all the submanifolds of the nonlinear flag  Σ1 ⊆ · · · ⊆ Σr−1 ⊆ Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We do it successively, starting with Σr−1 and ending with Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We assume by contradiction that Zr is not tangent to Σr−1 ⊆ Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Since dim Σr−2 <  dim Σr−1, we can find a point x ∈ Σr−1 \\ Σr−2 such that Zr(x) is not tangent to Σr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We  choose h ∈ C∞(S) supported in a tubular neighorhod of Σr−1 with the following properties:  h vanishes on Σr−1 (so that j∗  i dh = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , r − 1) and iZrdh restricted to Σr−1 is a  positive bump function at x that vanishes on Σr−2 (so that j∗  i (iZrdh) = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , r − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Inserting it in (62) only three terms survive, giving  �  Sr−1  j∗  r−1(iZrdh)αr−1 =  �  Sr  hd(iZrαr − λr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The integral over Sr−1 on the left doesn’t change when cutting h with a function supported  in a tubular ε-neighborhood of Σr−1 which is constant in an ε/2-neighborhood of Σr−1, while  the absolute value of the integral over Sr on the right can be made as small as we need by  making ε small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Thus Zr must be tangent to Σr−1, so  there exists Zr−1 ∈ X(Sr−1) which is jr−1 related to Zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  In a similar way, step by step, we address all the remaining submanifolds Σr−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , Σ1 of  the nonlinear flag Σ, finding Zi ∈ X(Si) that are ji related to Zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Knowing this, the identity  (62) becomes  r  �  i=1  �  Si  j∗  i dh ∧ (λi − iZiαi) = 0, for all h ∈ C∞(S),  22  which implies that d(λi − iZiαi) = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Thus (uϕ, (dλi)) = (Tϕ ◦ Zr, (diZiαi)) is the  infinitesimal generator at (ϕ, (αi)) for Z = (Zi) ∈ X(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This ensures that the kernel of Ω is  spanned by the infinitesimal generators of the Diff(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' ι) action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  As a consequence, Ω descends to a leafwise symplectic form ¯Ω on the associated bundle  (Flagwt iso  S,ι  (M), ¯D) = (Embiso  Sr (M) ×Diff(S,ι) Den×(S), D ×Diff(S,ι) ker ThS,ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The restriction of ¯Ω to (Flagwt iso  S,ι,µ (M), ¯D) is a leafwise symplectic form as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Thus every leaf  of ¯D is symplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let F denote a leaf of the isodrastic distribution ¯D on Flagwt iso  S,ι  (M), equipped with the  symplectic form ¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Restricting (18), we obtain a Ham(M) equivariant smooth map  J : F → hamc(M)∗,  ⟨J(N, ν), Xf⟩ =  r  �  i=1  �  Ni  fνi,  (63)  where we identify hamc(M) = C∞  0 (M) as in (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' This is a moment map for the (Hamilto-  nian) action of Hamc(M) on (F, ¯Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Indeed, this follows readily by combining (56) with the  expression for the moment map in (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Moreover, J is injective according to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='10, the Hamc(M) action on F is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='8, we thus  obtain the following generalization of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The moment map J : (F, ¯Ω) → hamc(M)∗ in (63) is one-to-one onto a coad-  joint orbit of the Hamiltonian group Hamc(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The Kostant-Kirillov-Souriau symplectic form  ωKKS on the coadjoint orbit satisfies J∗ωKKS = ¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='3 we have seen that the inclusion  Flagwt iso  S,ι,µ (M) ⊆  r  �  i=1  Grwt iso  Si,µi (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  is a splitting smooth submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The symplectic leaf F described in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='14 is a  splitting symplectic submanifold of a product �r  i=1 Gi of symplectic leaves described in Theo-  rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' To show that this is indeed a splitting smooth submanifold, one can proceed as in the  proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='11(c), using the fact that each isodrastic leaf in Flagiso  S,ι(M) is a splitting  smooth submanifold in a product of isodrastic leaves in �r  i=1 Griso  Si (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The latter can be show  by induction on the depth of the flag using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The leafwise symplectic form ¯Ω on Flagwt iso  S,ι  (M) can be seen as a Poisson struc-  ture whose symplectic leaves are the isodrasts F from the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='14 (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='4 in  [26] about isodrasts in the nonlinear Grassmannian of weighted Lagrangian submanifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Restricting (18), we obtain a Poisson moment map  J : Flagwt iso  S,ι  (M) → hamc(M)∗,  ⟨J(N, ν), Xf ⟩ =  r  �  i=1  �  Ni  fνi,  for the Poisson action of Hamc(M) on weighted isotropic nonlinear flags, where we identify  hamc(M) = C∞  0 (M) as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Let M be a symplectic manifold that possesses isotropic submanifolds diffeo-  morphic to the sphere Sr with r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We use the setting of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='19 to describe some  coadjoint orbits of Hamc(M) consisting of nested weighted spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Since H1(Sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' R) = 0,  23  by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='9 each connected component of Flagwt iso  S,ι,µ (M) is a coadjoint orbit of Hamc(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Similarly to (44) we get  Flagwt iso  S,ι,µ (M) =  �  (N, ν) ∈ Flagwt iso  S,ι  (M)  �����  {  �  N+  i νi,  �  N−  i νi} = {a+  i , a−  i }, where N +  i  and N −  i  denote the connected components of Ni \\ Ni−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The coadjoint orbits of Hamc(R2) consisting of pointed vortex loops, studied  in [3], are the lowest dimensional examples of coadjoint orbits of weighted nonlinear flags as  described in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Their type is (S, ι), with ι : {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , k} → S1, ι(i) = ti consecutive points on the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We  get the cohomology space  H(S, ι) ∼= Rk × Rk,  where the class [µ] ∈ H(S, ι) is identified with its integrals over connected components of  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , k} resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' S1 \\ {t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , tk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' These are Γi =  �  {i} µ0 and wi =  � ti+1  ti  µ1, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  The Diff(S, ι) action on H(S, ι) factorizes through an action of the dihedral group D2k on  Rk ×Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' More precisely, one let the dihedral group act on a regular k-gon, with Γi assigned to  the vertex i and wi assigned to the edge [i, i + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' For generic density µ ∈ Den×(S), the orbit  H(S, ι)[µ] consists of 2k elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The orbit is a one-point set if and only if Γ1 = · · · = Γk and  w1 = · · · = wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Let us denote the weighted flags of type (S, ι) in R2 by (({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , xk}, ν0), (C, ν1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The  area a enclosed by the curve C singles out one isodrast La ⊂ Griso  S1(R2), as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='14 follows that each connected component of Flagwt iso  S,ι,µ (R2)|La is a coadjoint  orbit of Hamc(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Using the above description of Diff(S, ι) action on H(S, ι), we get  Flagwt iso  S,ι,µ (R2)|La =  �  (({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , xk}, ν0), (C, ν1))  ����  xi consecutive points in C ∈ La, ∃σ ∈ D2k  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  �  {xi} ν0 = σ(Γi),  � xi+1  xi  ν1 = σ(wi)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (64)  In particular, the invariants are the area a enclosed by the loop C, the point vorticities Γi, and  the net vorticities wi =  � xi+1  xi  ν1 between two consecutive points on the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Let L[a1,a2] be a union of isodrastic leaves of Lagrangian 2-tori in R4, as in  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='3, with [a1, a2] the GL(2, Z) orbit of the pair of action integrals (a1, a2) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Let  S1 be a disjoint union of k circles, S2 = T2 the 2-torus, and ι : S1 → S2 the embedding that  maps the i-th circle to the circle {ti} × S1 ⊆ T2, with consecutive points t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' , tk ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' For  fixed density µ ∈ Den×(S), we aim at describing the coadjoint orbits of Hamc(R4) that are  connected components of Flagwt iso  S,ι,µ (R4)|L[a1,a2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' To this end we need the isomorphism  H(S, ι) ∼= Rk × Rk  given by integration of µ1 over the components of S1 and of µ2 over the torus surface between  two successive embedded circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' As in the previous example, the Diff(S, ι) action on H(S, ι)  factorizes through the dihedral group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' One can now express these coadjoint orbits of Hamc(R4)  as in (64), with the help of the dihedral group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Thus, the invariants are: the GL(2, Z) orbit of  the two action integrals for the embedded 2-torus, the total weights of the k isotopic loops on  the torus, and the partial weights between two such consecutive loops on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  A  Transitive actions on associated bundles  The manifolds and Lie groups in this appendix may be infinite dimensional and are assumed  to be modelled on convenient vector spaces as in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  24  Recall that a smooth G action on M is said to admit local smooth sections if every point  x0 in M admits an open neighborhood U and a smooth map σ : U → G such that σ(x)x0 = x,  for all x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Clearly, such an action is locally and infinitesimally transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Due to the lack  of a general implicit function theorem, one can not expect the converse implication to hold for  general Fr´echet manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Let P → B be a principal G-bundle endowed with the action of a Lie group  H on P that commutes with the principal G action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Suppose the structure group G acts on  another manifold Q, and consider the canonically induced H action on the associated bundle  P ×G Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' If the H action on P and the G action on Q both admit local smooth sections, then  the H action on P ×G Q admits local smooth sections too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Suppose ξ0 ∈ P ×G Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' As the canonical projection P × Q → P ×G Q is a locally  trivial smooth bundle [14, Theorem 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='12], there exist an open neighborhood U of ξ0 as well  as smooth maps π : U → P and ρ : U → Q such that for all ξ ∈ U we have  [π(ξ), ρ(ξ)] = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Put p0 := π(ξ0) ∈ P and q0 := ρ(ξ0) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' As the H action on P admits local sections, there  exist an open neighborhood V of p0 in P and a smooth map σ′ : V → H such that  σ′(p0) = eH  and  σ′(p)p0 = p,  for all p ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' As the G action on Q admits local sections, there exist an open neighborhood  W of q0 in Q and a smooth map σ′′ : W → G such that  σ′′(q0) = eG  and  σ′′(q)q0 = q,  for all q ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Possibly replacing U with a smaller neighborhood of ξ0, we may assume that  for all ξ ∈ U we have ρ(ξ) ∈ W and π(ξ)σ′′(ρ(ξ)) ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Hence, σ : U → H,  σ(ξ) := σ′�  π(ξ)σ′′(ρ(ξ))  �  is a well defined smooth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' For ξ ∈ U we obtain  σ(ξ)ξ0 = σ(ξ)[p0, q0] = [σ(ξ)p0, q0] = [σ′(π(ξ)σ′′(ρ(ξ)))p0, q0]  = [π(ξ)σ′′(ρ(ξ)), q0] = [π(ξ), σ′′(ρ(ξ))q0] = [π(ξ), ρ(ξ)] = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Hence, σ is the desired local smooth section of the H action on P ×G Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  We will denote the fundamental vector fields of a smooth G action on M by  ζX(x) := ∂  ∂t  ��  t=0exp(tX)x,  where X ∈ g and x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Let G be a regular Lie group acting on a smooth manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Suppose every  point x0 in M admits an open neighborhood U and a smooth map σ : TM|U → g such that  ζσ(X)(x) = X,  (65)  for all x ∈ U and X ∈ TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Then the G action on M admits local smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  25  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We will construct a local section using an argument due to Moser [21, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Suppose c : [0, 1] → U is a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' We seek a smooth curve g : [0, 1] → G such that  c(t) = g(t)c(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  (66)  Differentiating, we obtain  c′(t) = ζ ˙g(t)(c(t)),  (67)  where ˙g(t) :=  ∂  ∂h  ��  h=tg(h)g(t)−1 denotes the right logarithmic derivative of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  Since G is regular [14, Definition 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='4], there exists a unique smooth curve g = Evolr�  σ◦c′�  in G such that ˙g(t) = σ(c′(t)) and g(0) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Using (65), we see that (67) and, thus, (66) hold  true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Evaluating at t = 1, we obtain a smooth map  s : C∞([0, 1], U) → G,  s(c) := g(1) = evolr�  σ ◦ c′�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  By construction, c(1) = s(c)c(0), for all smooth curves c : [0, 1] → U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content='  To obtain a local smooth section for the G action on M, it suffices to compose s with a  smooth map U → C∞([0, 1], U), x �→ cx satisfying cx(0) = x0 and cx(1) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' The latter can  readily be constructed using a chart for M centered at x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
+page_content=' Indeed, shrinking U so that it  becomes star shaped with center x0 = 0 in such a chart, we may use cx(t) := tx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdAyT4oBgHgl3EQfkPgm/content/2301.00428v1.pdf'}
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diff --git a/hdE5T4oBgHgl3EQfFQ6l/content/tmp_files/2301.05421v1.pdf.txt b/hdE5T4oBgHgl3EQfFQ6l/content/tmp_files/2301.05421v1.pdf.txt
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+Analyzing and Improving the Pyramidal Predictive
+Network for Future Video Frame Prediction
+Chaofan Ling
+South China University of Technology
+wichaofan@mail.scut.edu.cn
+Weihua Li
+South China University of Technology
+whlee@scut.edu.cn
+Junpei Zhong
+The Hong Kong Polytechnic University
+joni.zhong@polyu.edu.hk
+Abstract—The pyramidal predictive network (PPNV1) pro-
+poses an interesting temporal pyramid architecture and yields
+promising results on the task of future video-frame prediction.
+We expose and analyze its signal dissemination and characteristic
+artifacts, and propose corresponding improvements in model
+architecture and training strategies to address them. Although
+the PPNV1 theoretically mimics the workings of human brain,
+its careless signal processing leads to aliasing in the network. We
+redesign the network architecture to solve the problems. In ad-
+dition to improving the unreasonable information dissemination,
+the new architecture also aims to solve the aliasing in neural
+networks. Different inputs are no longer simply concatenated,
+and the downsampling and upsampling components have also
+been redesigned to ensure that the network can more easily
+construct images from Fourier features of low-frequency inputs.
+Finally, we further improve the training strategies, to alleviate
+the problem of input inconsistency during training and testing.
+Overall, the improved model is more interpretable, stronger,
+and the quality of its predictions is better. Code is available
+at https://github.com/Ling-CF/PPNV2.
+Index Terms—video-frame prediction, pyramidal predictive
+network, aliasing
+I. INTRODUCTION
+Future video frame prediction has been rapidly developed
+recently [1]–[8]. This looking-ahead ability could be applied to
+various fields including autonomous driving [9], [10], robotic
+systems [11], [12], rainfall forecasting [13], [14] and so on.
+In particular, this self-supervised learning method for video
+representation can be also migrated to downstream tasks [15],
+[16]. Specifically, we can first perform self-supervised video
+prediction learning on the backbone network, which is then
+fixed and used in tasks such as video classification, to solve
+the difficulties of label acquisition in supervised learning tasks.
+Video frame prediction is a pixel-level task designed to use
+historical frames to predict future frames. One of the state-
+of-the-art video prediction methods is the Pyramid Prediction
+Network (PPNV1) [17], which is an interesting work and
+has achieved promising results. The distinguishing feature of
+PPNV1 is its unconventional temporal pyramid architecture.
+The total model is updated through a combination of bottom-
+up and top-down information flows, and uses prediction errors
+to achieve effective feedback connections, which is where
+predictive coding models differ from traditional generators
+[18]–[21]. In particular, the update frequency of neurons in this
+model decreases as the network level increases, so that neurons
+in higher layers can capture information over a longer time
+Fig. 1. The architecture of PPNV1. According to the definitions in PPNV1
+[17], ft
+l (green), P t
+l
+(orange) and Et
+l (red) represent the sensory input,
+prediction and prediction error at time-step t and level l, respectively. In the
+picture, the sensory input propagating up is lagged, in which ft+1
+l
+should be a
+higher-level representation of ft+1
+l−1 rather than ft
+l−1. In addition, since ft+1
+l
+characterizes Et+1
+l−1 at the same time, P t
+l can be considered to predict Et+1
+l−1 .
+However, Et+1
+l−1 is actually indirectly generated by P t
+l , which is contradictory
+and incomprehensible.
+range, and the computational cost is also reduced. However,
+the authors seem to prefer to build network models from
+theories of cognitive science or neuroscience, ignoring the
+importance of how computers process signals. Overall, the
+PPNV1 provides an effective predictive framework, but its
+signal processing is careless. Our main work is to fix its
+signal processing and characteristic artifacts to improve the
+final prediction quality.
+We first observed that there are inconsistencies in the way of
+temporal signal processing in PPNV1 (shown in Fig 1). Firstly,
+the sensory input (green) in the neural network is propagated
+up to higher level along with the prediction error (red), which
+has been proved to be necessary [17]. However, there is a prob-
+lem with the choice of sensory information. The model should
+have transmitted sensory input of next time-step (f t+1
+l−1 ) instead
+arXiv:2301.05421v1  [cs.CV]  13 Jan 2023
+
+ConvLSTM
+Conv+Pool
+D/+1
+E]+
+ConvLSTM
+ConvLSTM
+ft
+Conv+Pool
+Conv+Pool
+ConvLSTM
+Ei-1
+ConvLSTM
+ConvLSTM
+fi
+fi1Fig. 2. The complete architecture of the improved pyramidal predictive network (PPNV2). Similar to PPNV1, ft
+l (green), P t
+l (orange) and Et
+l (red) represent
+the sensory input, prediction and prediction error at time-step t and level l, respectively. We first made improvements to how sensory input and prediction errors
+are propagated and computed, making the model more interpretable (will be introduced in detail in Section 2). Second, we redesigned several characteristic
+artifacts to address the problem of information aliasing. Here “US” denotes the upsampling artifact, in which we replace the traditional bilinear or bicubic
+upsampling methods by interleaving zeros and then using depthwise separable convolutions to implement interpolation calculations (Section 3, C). A low-pass
+filter with a learnable cutoff frequency is used for anti-aliasing filtering. “DS” denotes the downsampling artifact, in which low-pass filtering is also introduced
+to prevent high-frequency information from being mixed into low-frequency information (Section 3, B). “ModError” and “ModPred” are special modulation
+modules designed to better handle different inputs and make gradient propagation easier and more stable (Section 3, A). Zoom in for a better view.
+of current time-step (f t
+l−1). Secondly, the sensory input and
+prediction error are simply concatenated for calculation, which
+will cause information aliasing. Moreover, the predictive unit
+at higher-level needs to simultaneously predict the prediction
+error from a lower level, which is contradictory and difficult.
+We will expose and analyze these issues in detail and redesign
+the signal processing specification in Section 2, to make the
+model stronger and more interpretable.
+The rough splicing of various signals is another problem
+that cannot be ignored in PPNV1 (Section 3). For instance,
+the prediction, prediction error and sensory information from
+different levels are directly spliced and input into the predictive
+unit (ConvLSTM) for calculation, which will cause serious
+information aliasing and huge computational overhead. We
+propose an alternative design to alleviate the problems — a
+modulation module is added before and after the predictive
+unit, and only the modulated information is sent to the
+predictive unit for calculation — this is a special network
+module designed to modulate the primary input with a scaling
+and deviation tensor computed from the auxiliary input. In
+addition, the downsampling and upsampling artifacts have
+also been redesigned to ensure that the network can more
+easily construct images from Fourier features of low-frequency
+inputs. The total improved network architecture is shown in
+Fig 2. This new design makes the model more efficient and
+interpretable, which also allows us to further improve the
+training strategy.
+In addition to network architecture, input inconsistency be-
+tween training and testing in temporal sequential tasks remains
+a challenging topic (Section 4). In general, the predicted
+frames needed to be served as new inputs to enable continuous
+prediction, thus causing inconsistency between training and
+testing. We observe that the improved pyramidal predictive
+network can alleviate this problem by computing losses at
+multiple levels. The new convolutional unit will learn to extract
+accurate and effective information from ”less perfect” data,
+that is, the overall model is more robust. Additionally, we
+propose to train the model with LPIPS loss [22]. In tasks
+of future video-frame prediction, LPIPS is usually used as
+an evaluation metric to evaluate the similarity of images, but
+there is almost no work uses it as a loss function. In this work,
+it will be combined with Euclidean distance loss to generate
+more accurate and clearer future video frames.
+
+P
+DS
+ : Downsampling
+US
+: Upsampling
+ ModPred
+Prediction of
+P
+local level
+Output
+Input
+ModPred
+Conv
+Pi+1
+ConvLSTM
+A
+Encoding
+Decoding
+Prediction from
+tanh
+ higher level
+ ModError
+Conv
+Conv
+Zero
+:.
+ Downsample
+interleaving
+Pt
+DS
+DS
+US
+ConvLSTM
+Lowpass
+Anti-aliasing
+filtering
+f out
+filter
+ModPred
+ModPred
+4
+4
+Convolution
+ConvLSTM
+ConvLSTM
+interpol ati on
+tanh
+4
+tanh
+ ModError
+ModError
++1-11
+ Convolutional L ayer
+h;
+ Feature
+extraction
+P
+DS
+ DS
+US
+DS
+DS
+US
+ ModError
+Low-pass
+ Anti-aliasing
+filtering
+ ModPred
+ModPred
+ModPred
+filter
+Error of 
+Convtanh
+Input
+ current level
+ ConvLSTM
+ConvLSTM
+ConvLSTM
+reconstruction
+E,
+X
+ModError
+Ei1
+ ModError
+ ModError
+Conv
+Error from
+Ei-1
+ Sensory input of
+Ji1
+lower level
+current ievel
+ndnoFig. 3.
+We first redesign the signal processing of pyramidal predictive network. (a)The original model (PPNV1). (b) The improved model (PPNV2). We
+make several changes to the original structure as shown in the red boxes in the figure: 1. We adjust the sensory input being propagated upward, where we
+use sensory input of next time-step (ft+1
+l−1 ) instead of the current one (ft
+l−1) for propagation, to solve the problem of lagging propagation. 2. We separate
+prediction error from sensory input and calculate them with novel downsampling artifacts respectively, to prevent the information aliasing and make the model
+more interpretable. 3. Instead of roughly feeding different signals directly into the ConvLSTM for computation, we use a network block to process them
+separately, which is made up of “ModError”, “ConvLSTM” and “ModPred” shown in Fig 2.
+II. IMPROVING SIGNAL PROCESSING
+We first observe that there are problems in the signal
+processing of the pyramidal predictive network (PPNV1). The
+authors seem to prefer to build network models from cognitive
+science or neuroscience theories, ignoring the importance of
+how convolutional units perform calculation. Simply treating
+the convolution units as brain neurons to build a brain-like
+model may cause problems. Specifically, we have identified
+two serious signal processing issues in PPNV1: 1) there is a
+lag in the upward propagation of sensory information. 2) the
+integrated calculation of sensory input and prediction error
+will cause the information aliasing and makes the prediction
+difficult and ambiguous. These problems make the entire
+PPNV1 incomprehensible, it works in a puzzling way. As
+shown in Fig 3, we propose corresponding solutions to the
+problems to address the disturbing ”black box” in pyramidal
+predictive network, the details will be described in the follow-
+ing subsection.
+A. Revisiting the processing of sensory input
+In the pyramidal predictive network (PPNV1), the sensory
+information is input to the predictive unit (ConvLSTM) for
+prediction generation, which is also propagated up to higher
+level along with the prediction error. The authors [17] have
+confirmed the necessity of this operation. However, in such
+a temporal sequential model, there is a lag in the sensory
+information used in the up-propagation process.
+As shown in Fig 3 (a), the sensory input f t+1
+l
+is a higher-
+level representation of the combination of f t
+l−1 and prediction
+error Et+1
+l−1, while P t
+l is considered to be a prediction of f t+1
+l
+.
+In other words, the P t
+l can be considered to predict f t
+l−1
+and Et+1
+l−1. However, the P t
+l will be eventually propagated
+downward and sent to the predictive unit together with f t
+l−1
+to make the prediction of lower-level, so what P l
+t predicts is
+lag and meaningless. In particular, this kind of delay becomes
+more severe as the network level increases. For instance, the
+sensory information contained in f t+1
+l+1 (the third level in Fig
+3 (a)) is actually derived from f t−1
+l−1 . Therefore, to solve this
+problem, we should propagate up the sensory input f t+1
+l−1 (Fig
+3 (b) 1.) of next time step instead of f t
+l−1, which matches in
+the time dimension.
+B. Revisiting the processing of prediction error
+Secondly, the prediction error and sensory input in PPNV1
+are simply concatenated, which is then computed with a con-
+volution unit to obtain the higher-level representation (f t+1
+l
+).
+Unfortunately, this kind of integrated calculation will lead to
+aliasing of information, in which the higher-level sensory input
+is no longer a simple representation of lower-level sensory
+input, but mixed with the information of prediction error.
+Moreover, predicting the prediction error is actually puzzling
+and difficult.
+According to what we mentioned in the previous subsection,
+the P t
+l is considered to predict the lower-level information
+
+(a)
+Pt
+(b)
+Pt
+onLSTM
+id
+pr+1
+E++
+E++1
+Pi
+Pi1
+Pl
+US
+Conv+Pool
+DS
+DS
+2.
+E
+ConvLSTM
+ConvLSTM
+nvLSTM
+NetBlock
+E
+fi1
+fi
+1.
+DS
+: Downsampling artifacts
+US
+: Upsampling artifacts
+ConvLSTM
+: Convolutional LSTM
+ NetBlock
+: Network blockFig. 4. We use a special modulation module to perform computation on different input signals x1 and x2 to further address the problems of signal aliasing.
+The auxiliary signal x2 is calculated by convolution and activated by the sigmoid and tanh functions to obtain scaling and shifting matrices, and then
+the main signal x1 is scaled and shifted with the matrices, which can prevent the signal of x2 from leaking into x1. Additionally, there is only pixel-level
+multiplication and addition on the x1 computational path (red), so the gradient flow is more stable. We make comparison with the traditional methods of
+“Concat” and “Add”, where “Concat” denotes concating two matrices together by channel and “Add” denotes adding two matrices together at the pixel-level.
+To make sure the same number of parameters and computational overhead, we add additional convolutional unit for “Add” method.
+of prediction error Et+1
+l−1. Paradoxically, the prediction error
+Et+1
+l−1 is calculated from P t
+l−1 and f t+1
+l−1 , and the generation
+of P t
+l−1 is inseparable from the higher-level prediction P t
+l . In
+short, P t
+l predicts the Et+1
+l−1, while the Et+1
+l−1 must be generated
+by P t
+l first, which is contradictory. To solve the problem, we
+propose to calculate the prediction error and sensory input
+separately in the new improved pyramidal predictive network
+(PPNV2). As shown in Fig 3 (b), the higher-level sensory input
+f t+1
+l
+only represents f t+1
+l−1 at lower-level, while the prediction
+error is excluded, thus avoiding the aliasing of information.
+Importantly, since the generation of f t+1
+l
+is independent of
+prediction error Et+1
+l−1, the aforementioned contradiction is
+resolved and the overall model becomes more interpretable. In
+particular, we use a novel downsampling artifact for upward
+propagation, which will be introduced in detail in the next
+section.
+III. ANTI-ALIASING DESIGN
+Another major problem with pyramidal predictive network
+(PPNV1) is that it handles different signals carelessly. As
+shown in Fig 3 (a), it simply concatenates different signals
+and feeds them into the ConvLSTM predictive unit for com-
+putation, which will inevitably lead to aliasing of information.
+We argue that different information is of different importance.
+For example, the sensory input which contains more low-
+frequency signals can provide more effective information than
+prediction errors, which deserves more attention. Additionally,
+the information contained in higher-level predictions (P t
+l ),
+prediction errors (Et
+l−1) and sensory input (f t
+l−1) are quite
+different, it is better to calculate them separately instead of
+splicing together. Although the convolution unit can be theoret-
+ically trained to distinguish them eventually, it is undoubtedly
+difficult. Repeated multiplication of huge weight matrices may
+cause the gradients vanishing or exploding.
+In particular, in view of the special calculation of ConvL-
+STM (the ConvLSTM needs to compute four outputs [14]),
+this concatenation operation will also lead to a substantial
+increase in the amount of parameters and calculation overhead.
+Therefore, to solve the problems, we propose to use a network
+module before and after the ConvLSTM predictive unit to
+process different inputs, and only propagate the necessary
+signals to the ConvLSTM for calculation. The purpose is to
+solve the problem of information aliasing and reduce network
+parameters and computational overhead as much as possible.
+Moreover, aliasing of different frequencies in a signal is also
+of concern. In the process of downsampling and upsampling,
+high-frequency signals are often mixed with low-frequency
+signals, which is easy to cause the final generated image to be
+incoherent and unnatural. To alleviate the problem, we propose
+to redesign the upsampling and downsampling artifacts with
+anti-aliasing low-pass filters to ensure that the model can more
+easily learn low-frequency Fourier features. The details will be
+described in the next several subsections.
+A. modulation module
+As shown in Fig 4, we add a modulation module before and
+after the ConvLSTM prediction unit for signal preprocessing
+and postprocessing. This is a special module designed to alle-
+viate the problems of signal aliasing and difficulties of gradient
+propagation. Different from the traditional concatenating or
+adding operation, the modulation module needs to distinguish
+and determine the main signal x1 and the auxiliary signal x2
+first. The design of the module is inspired by the attention
+mechanism — the brain responds more strongly when predic-
+tions are severely inconsistent with the environment [23]–[25].
+
+Modulate
+Concatenate
+Add
+US
+X, (b, c, h, w)
+X2 (b, c, h, w)
+X, (b, c, h, w)
+X, (b,c,h,w)
+X (b, c, h, w) X2 (b, c, h, w)
+ModPred
+Conv
+Conv
+concat
+add
+ConvL STM
+(b, c, h, w)
+Ei++
+ModError
+a
+tanh
+x
+(b, 2c, h, w)
+Conv
+(b, c, h, w)
+(b, c, h, w)
+(b, c, h, w)
+Conv
+Conv
+DS
+DS
+(b, c, h, w)
+y
+(b, c, h, w)
+y
+2
+(b, c, h, w)
+Et1For “ModError” (Fig 4), the operation of modulation can
+be seen as the process of attention attachment and transfer.
+The sensory input contains more information that effectively
+describes the current scene, while the prediction error consists
+of sparse error signals. We have pointed out that the neural net-
+work preferentially learns low-frequency features [26], [27], so
+it is better to take the sensory input as the main signal x1 and
+the prediction error as an auxiliary signal x2. According to
+Clark et al. [23], [28], [29], the larger the prediction error,
+the more attention is given. Therefore, we propose that the
+prediction error can be viewed as an attention matrix, and the
+attachment of attention is equivalent to the process of scaling
+and shifting the sensory input.
+The calculation is as follows: First, the auxiliary signal x2 is
+convolved separately, and then the sigmoid and tanh are used
+as activation functions to obtain the scaling and shift matrices.
+The calculation formulas are shown in Eq.1 and Eq.2:
+msc = sigmoid f(x2, θsc)
+(1)
+msf = tanh f(x2, θsf)
+(2)
+where f(·) represents a typical convolutional network and
+θ is the model parameter, msc and msf denote the scaling
+and shifting matrices respectively. The role of sigmoid and
+tanh is to constrain the matrices between (0, 1) and (-1, 1),
+respectively. The final output is shown in Eq.3, where α is a
+learnable scaling factor. The scaling matrix is limited to (0, 1)
+by simoid, which only plays the role of downscaling but not
+upscaling, so we recommend adding a scaling factor α and
+setting it as a learnable parameter.
+y = α · msc · x1 + msf
+(3)
+Similarly, in the post-processing stage, we also use the
+modulation module to process the prediction P t+1
+l+1 (Fig 4)
+from higher level and the output of ConvLSTM. In this work,
+we take the higher-level prediction as the main signal x1 and
+the output of the ConvLSTM as the auxiliary signal x2. The
+downward propagation of high-level prediction is actually the
+process of decoding, so we take the decoding feature as the
+main signal.
+B. Downsampling artifact
+The purpose of downsampling is to reduce memory and
+computation, prevent overfitting and increase the receptive
+field. In PPNV1, the conventional method of convolution and
+pooling is used for downsampling (Fig 3 (a) 2.). Related
+studies have shown that classifier networks focus on texture
+rather than shape [30], and one of the reasons is the use of
+pooling. Pooling, especially max pooling, is more sensitive
+to texture information and has better translation or rotation
+invariance. Therefore, no matter how the images are translated
+or rotated, the classifier network can still recognize them.
+However, continuous video frame sequences are semantically
+consistent, but different in pixel space [31]. The essence of
+Fig. 5.
+Simplified flowchart for downsampling (a) and upsampling (b).
+We introduce a low-pass filter for anti-aliasing filtering. Importantly, we set
+learnable cutoff frequencies for each feature map of each input. “ZeroPad”
+means that the input is interleaved by m−1 zeros in m-fold upsampling, and
+then interpolated using depthwise separable convolution.
+future video frame prediction is to learn changes by observing
+the differences between frames. As a result, this invariance
+brought by pooling can be disastrous for video prediction
+tasks, which motivates us to look for an alternative that
+preserves the spatial location information as much as possible
+while downsampling.
+The first thing that comes to mind is convolutional down-
+sampling, which preserves spatial details better than pooling,
+but can still be improved. For example, using a low-pass filter
+for downsampling first, followed by convolution calculation.
+Aliasing, a subtle and critical issue, has recently attracted
+attention [32]–[35]. In order to prevent the high frequency
+signal from being mixed into the low frequency signal during
+the downsampling process, it is better to perform anti-aliasing
+filtering first. As shown in Fig 5 (a), We first perform fea-
+ture extraction on the input signal using convolutional units,
+followed by low-pass filtering for anti-aliasing and downsam-
+pling. Finally, another convolution calculation is performed to
+encode the signal.
+Next we will focus on the design and implementation of
+low-pass filters. According to the Nyquist-Shannon theorem
+[36], in order to recover the signal without distortion, the
+sampling frequency should be greater than 2 times the highest
+frequency in the signal spectrum. Otherwise, the sampled
+frequencies will overlap. Therefore, in the case of a fixed sam-
+pling frequency, we assume that the signal whose frequencies
+are higher than half the sampling frequency are negligible,
+which is usually achieved with a low-pass filter.
+w(n) = 0.54 − 0.46 cos(2πn
+N ),
+0 ≤ n ≤ N − 1
+(4)
+
+y
+(a) i (b)
+y
+↑
+Encoding
+Convolution
+Convolution
+Decoding
+Anti-aliasing filtering
+Downsample
+ZeroPad + DSConv
+Low-pass filter
+Ups ample
+Low-pass filter
+Feature extraction
+Anti-aliasing filtering
+Convolution
+Convolution
+Reconstruction
+x
+x
+DSConv: Depthwise Separable Convolution
+ZeroPad: Zero padding (upsample with 0)λ(n) = f 2
+fs
+sinc(f 2
+fs
+(n − 0.5(N − 1)))
+(5)
+h(n) = λ(n) · w(n)
+(6)
+In this work, we prefer to use the Hamming Window [37]
+for the design of low-pass filter, which is described as Eq.4,
+where N denotes the length of filter. And the calculation
+of coefficients is shown in Eq.5, where f and fs represent
+the cut-off and sampling frequency respectively. As what we
+mentioned above, the cutoff frequency f should be between
+0 and half the sampling frequency fs/2, , and it needs to be
+limited to (0, 1) when calculating, which is achieved by 2f/fs.
+Finally we get the low-pass filter by multiplying the Hamming
+Window and the coefficient (Eq.6).
+We use a low-pass filter of length N = 25 and resize it to a
+5 × 5 convolution kernel in this paper. In particular, we define
+the cut-off frequency f as a learnable parameter, which is im-
+portant because the ratio of low-frequency and high-frequency
+signals may be different at different resolutions. In general,
+the higher the resolution, the higher the proportion of low
+frequency signals, so we can limit the bandwidth even lower.
+Taking a step further, we propose to define different learnable
+cut-off frequencies fi, 1 ≤ i ≤ C for each input feature map
+and use the sigmoid function to constrain it between (0, 1),
+where C denotes the number of channels. Finally, we can ob-
+tain a convolution kernel of shape (C, 1, 5, 5), which is depth-
+separable and will not fuse the features of different channels,
+only performs anti-aliasing filtering and downsampling (stride
+= 2). The traditional convolution calculation is placed after
+downsampling, so a larger receptive field is obtained.
+C. Upsampling artifact
+Inspired by styleGAN3 [35], ideal upsampling should not
+modify the continuous representation, its only purpose is to
+increase the output sampling rate. Similarly, the upsampling
+operation consists of two steps: we first increase the sampling
+rate to sout = sin · m by interleaving m − 1 zeros between
+each input sample (m is usually set to 2), which is the same as
+styleGAN3. The feature map after staggered filling with zeros
+is shown in Fig 6 (Zoom in for a better view), where x refers
+to the original input features. Next, we perform convolution
+processing on the result and then low-pass filtering.
+In this work, we replace the bilinear or bicubic 2× up-
+sampling filter with a 7 × 7 depthwise separable convolution
+approximation, and consider the four computational cases
+shown in Fig 6. In 2× upsampling, we focus on the calcu-
+lation at different positions in the 2 × 2 box ( 0 0
+0 x ) , which
+has different convolution parameters and input features (x)
+involved in the calculation. Taking the first calculation (Fig 6
+(a)) as an example, there are 4 × 4 features involved in the
+calculation, which can be equivalent to bicuxic interpolation
+in some cases. For the other three cases, they calculate fewer
+features, thus saving computational overhead compared to
+bicubic interpolation, but introducing additional parameters.
+Since the kernel is depth-separable, it is still tolerable.
+Fig. 6. The convolution interpolation calculation. We focus on the calculation
+at different positions in the 2 × 2 box
+� 0 0
+0 x
+�
+, which can be discussed in four
+different cases (a)-(d). Particularly, the convolutional kernel parameters used
+in each case are non-overlapping (the second row), and we can initialize the
+parameters for each case according to the distance between the parameters to
+the center (green). What the third row shows is a total initialized convolution
+kernel. Zoom in for a better view.
+In view of the special function of this convolution, the
+initialization of parameters is also important. We use the
+weighted average method to initialize the parameters in this
+paper, in which the weight of the feature (x) is inversely
+proportional to its distance. Specifically, taking the first calcu-
+lation (Fig 6 (a)) as an example again, we need to calculate
+the interpolation of 0 in the center (green). There are three
+kinds of features with different distances around the center
+0 , and we will assign different weights to these features.
+Firstly, the weight is proportional to the inverse of the distance,
+which is shown as Eq.7 and Eq. 8, where w1, w2, w3 represent
+the weights under the three kinds of distances (l1, l2, l3)
+respectively. Secondly, the sum of all weights should be 1
+(Eq.9), so that the three kinds of weights can be calculated
+(Eq.10). The parameters for cases of (b) and (c) are calculated
+in the same way (note that for different cases, what the wn
+refers to is different). In particular, for the last case (d), we
+expect to preserve the original feature x, so we set the central
+parameter to 1 and the others to 0. Finally, the complete
+initialized kernel is shown in the third row of Fig 6.
+w1 : w2 : w3 = 1
+l1
+: 1
+l2
+: 1
+l3
+(7)
+l1 : l2 : l3 =
+√
+2 :
+�
+32 + 1 :
+�
+32 + 32
+(8)
+4w1 + 8w2 + 4w3 = 1
+(9)
+w1 = 0.1122 ;
+w2 = 0.0502 ;
+w3 = 0.0374
+(10)
+
+(a)
+(b)
+(c)
+(d)
+x
+0
+1.
+0
+1
+0
+x
+0
+x
+0
+x
+0
+x
+0
+0
+0
+0
+r
+0
+r
+0
+x
+0
+r
+0
+x
+0
+x
+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
+0
+0
+x
+0
+r
+0
+x
+0
+x
+0
+x
+0
+x
+0
+x
+0
+x
+0
+x
+0
+x
+0
+x
+0
+x
+x
+0
+x
+0
+x
+0
+x
+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
+0
+0
+r
+0
+x
+0
+r
+0
+x
+0
+x
+0
+x
+0
+x
+0
+x
+0
+0
+x
+0
+x
+0
+x
+0
+.r
+0
+x
+0
+x
+0
+x
+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
+0
+0
+0
+x
+0
+x
+0
+0
+x
+0
+0
+x
+0
+0
+x
+0
+0
+x
+0
+x
+0
+x
+0
+0
+x
+0
+x
+0
+x
+0
+W2
+W2
+ta
+W3
+tan
+W4
+W2
+W2
+W4
+0
+0
+0
+W2
+Wi
+W2
+W2
+Wi
+W2
+W
+Wi
+Wi
+W3
+0
+1
+0
+W2
+Wi
+Wi
+W2
+W2
+Wi
+W2
+W4
+W2
+W2
+W4
+0
+0
+0
+W2
+W2
+W3
+ta
+W3
+0.0347
+0.0498
+0.0502
+0.0599
+0.0502
+0.0498
+0.0347
+0.0498
+0.
+0.0804
+0.
+0.0804
+0.
+0.0498
+0.0502
+0.0804
+0.1122
+0.1797
+0.1122
+0.0804
+0.0502
+0.0599
+0.
+0.1797
+1.
+0.1797
+0.
+0.0599
+0.0502
+0.0804
+0.1122
+0.1797
+0.1122
+0.0804
+0.0502
+0.0498
+0.
+0.0804
+0.
+0.0804
+0.
+0.0498
+0.0347
+0.0498
+0.0502
+0.0599
+0.0502
+0.0498
+0.0347The complete downward propagation process is shown in
+Fig 5 (b), the input y is first decoded with convolution
+calculation, which corresponds to the encoding process of
+up-propagation. Next, upsampling is performed using the
+method described above, followed by low-pass filtering to
+eliminate aliasing frequencies above s/2. Finally, we use
+another convolution calculation to reconstruct the result, which
+corresponding to the feature extraction process in the upward
+propagation
+IV. REVISITING TRAINING STRATEGIES
+PPNV1 is committed to constructing neural network models
+from predictive coding theories, while the training strategies
+are less explored. First, PPNV1 is a typical end-to-end training
+model, which can only utilize the bottom predictions to
+calculate the loss. The careless signal processing makes the
+whole network model difficult to interpret (section 2), so the
+training of higher levels in the model are uncontrollable. Even
+worse result may be obtained if the higher-level prediction
+errors are used to calculate losses. Additionally, inconsistency
+between training and testing in image time series tasks remains
+a challenging topic, which has not been addressed in PPNV1.
+In response to the above problems, we propose several im-
+provements. First, benefiting from the clear improved network
+architecture, the entire network becomes controllable, we can
+simultaneously calculate the training loss for higher levels.
+Second, we further optimize the calculation process of long-
+term prediction, in which the encoding loss and decoding loss
+are calculated simultaneously to enhance the anti-interference
+ability of neural units. The details will be introduced in the
+next subsections.
+A. Long-term prediction training strategy
+Long-term prediction is a very important task for future
+video frame prediction. Specifically, if we need to predict
+further future frames, we have to predict the next video frame
+by inputting the predicted frame instead of the real frame, thus
+causing the problem of mismatch between training and testing.
+The predicted frames are “imperfect”, and the predictions
+will become increasingly blurry with the accumulation of
+prediction errors. In order to alleviate this problem, we propose
+a tracking training method that takes the predicted frame as
+input and the real frame as target during training.
+First, we fixedly take T (T is usually set to 10) real sequence
+frames as inputs for calculation, and use f t
+l to denote the
+representation of the input frame at time-step t and level l.
+These inputs are real frames which can be directly used to
+compare predictions from the previous time step to calculate
+prediction error and loss. The calculation of the prediction
+error is the same as that of PPNV1, which is divided into
+positive and negative errors Et
+l = C(|f t
+l − P t−1
+l
+|, |P t−1
+l
+−
+f t
+l |), where C denotes “concatenate”, that is, concatenate two
+matrices together. The loss is obtained by calculating their
+squared Euclidean distance L = � (f t
+l − P t−1
+l
+)2, which will
+be introduced in the next subsection.
+Fig. 7. We improve the training strategy by calculating encoding and decoding
+losses. The decoding loss L1, which is the prediction loss at each level,
+is obtained by comparing the current prediction P t
+l with the real sensory
+representation ft+1
+l
+of the next time step. The encoding loss L2 is only
+calculated when the predicted frame is used as input. We need to maintain
+an additional encoding pathway of the real frame to obtain the target for
+calculating the loss, which shares the encoding pathway (DSf) of the
+predicted frame.
+Second, we take the last predicted frame P T as a new input
+to predict P T +1, and then continue to predict with P T +1, so
+as to achieve continuous prediction. In particular, in order to
+be distinguished from the representation of the real frame,
+we denote the representation of the predicted frame input at
+time-step t and level l by ˆf t
+l . Importantly, ˆf t
+l will be used
+to calculate the prediction error Et
+l , but not the loss, which
+is different from the case where the real frame is used as
+input. As what we mentioned above, the predicted frame is
+“imperfect”, and the loss calculated using predicted frame
+might be “false”. As a result, we must use the real frame
+to calculate the loss. Importantly, in order to calculate loss
+for higher level, we need to maintain an encoding pathway
+for real frames during training, which shares parameters with
+the encoding pathway of the predicted frames (Fig 7). The
+real frame is only used to provide the target required for
+loss calculation during training, which will not participate
+in the calculation of prediction generation, and this encoding
+pathway will also be closed during testing.
+Furthermore, we calculate not only the prediction loss (Fig
+7, L1) of each level, but also the loss of the encoding process
+of predicted frame (Fig 7, L2). The purpose is to enhance the
+anti-interference ability of the downsampling artifact. We hope
+that the convoolutional units are robust enough to obtain the
+same features from “imperfect” predicted frames as real ones,
+further mitigating the effects of the prediction error.
+
+Pi+1
+Pi
+US
+Ei
+NetBlock
+fi
+fi
+DSE
+DSf
+DSf
+DSf
+Ei
+fi1
+ : Parameter sharing
+L, : Decoding loss
+L, : Encoding lossB. Loss function
+In this subsection, we would like to introduce the details of
+loss function used in this work. We split the computation into
+two stages during training: first calculating with real frames
+as input, followed by predicted frames. We assume that the
+length of the input real frame sequence is T1, and the length
+of predicted frame sequence is T2, then the total input length
+is T = T1 + T2. Firstly, we need to compute the prediction
+loss (Fig 7, L1) of each level at each time-step throughout the
+whole calculation, which can be expressed as Eq.11:
+L1 =
+T
+�
+t=0
+L
+�
+l=0
+λt · λl · (f t
+l − P t−1
+l
+)2
+(11)
+where L denotes the number of network level, λt and λl
+represent the weighting coefficient at time-step t and level
+l respectively. Importantly, we use the squared Euclidean
+distance to calculate the loss instead of the traditional mean
+squared error (MSE). The difference is that MSE is divided
+by the total number of features on the basis of the Euclidean
+distance, which will cause the loss value to be too small, and
+the gradient of some parameters may be ignored. It will cause
+the training process to converge first and then diverge, which
+often occurs in low-precision training.
+L2 =
+T
+�
+t=T1
+L
+�
+l=1
+λt · λl · (f t
+l − ˆf t
+l )2
+(12)
+Secondly, we need to calculate the encoding loss (Fig 7, L2)
+simultaneously in the second stage, to make the convolution
+units robust enough to extract similar features from “imper-
+fect” predicted frames to real frames. The calculation of L2 is
+defined as Eq.12: where f t
+l and ˆf t
+l indicate the representation
+of the real frame and predicted frame at time-step t and level
+l, respectively.
+Llpips =
+T
+�
+t=0
+λt · flpips(f t, P t−1)
+(13)
+Ltotal = L1 + L2 + α · Llpips
+(14)
+In addition, we also use LPIPS [22] to calculate the training
+loss. Zhang et al. proposes to use deep features to measure
+the perceptual similarity between two images, which is imple-
+mented by a neural network. It is often used as an evaluation
+metric for video prediction, but no one seems to have tried
+to use it as a function to compute the training loss. In this
+work, we try to use the provided pretrained model (VGG as
+backbone) to compute the loss between the predicted frame
+and the target, which is very important for sharpening the
+generated images (Eq.13, where flpips denotes the LPIPS
+pretrained model). Finally, the total loss is calculated as the
+sum of the above three losses (Eq.14) — since the loss
+provided by LPIPS is very small, we need to additionally
+multiply by a coefficient α.
+TABLE I
+QUANTITATIVE EVALUATION ON THE KTH DATASET. THE METRICS ARE
+AVERAGED OVER THE PREDICTED FRAMES. RED AND BLUE INDICATE THE
+BEST AND SECOND BEST RESULTS, RESPECTIVELY.
+Methods
+10 → 20
+10 → 40
+SSIM ↑
+PSNR ↑
+LPIPS ↓
+SSIM ↑
+PSNR ↑
+LPIPS ↓
+MCNet [38]
+0.804
+25.95
+-
+0.73
+23.89
+-
+fRNN [39]
+0.771
+26.12
+-
+0.678
+23.77
+-
+PredRNN [40]
+0.839
+27.55
+-
+0.703
+24.16
+-
+PredRNN++ [41]
+0.865
+28.47
+-
+0.741
+25.21
+-
+VarNet [42]
+0.843
+28.48
+-
+0.739
+25.37
+-
+SAVP-VAE [43]
+0.852
+27.77
+8.36
+0.811
+26.18
+11.33
+E3D-LSTM [44]
+0.879
+29.31
+-
+0.810
+27.24
+-
+STMF [5]
+0.893
+29.85
+11.81
+0.851
+27.56
+14.13
+Conv-TT-LSTM [45]
+0.907
+28.36
+13.34
+0.882
+26.11
+19.12
+LMC-Memory [4]
+0.894
+28.61
+13.33
+0.879
+27.50
+15.98
+PPNet [17]
+0.886
+31.02
+13.12
+0.821
+28.37
+23.19
+MSPN [46]
+0.881
+31.87
+7.98
+0.831
+28.86
+14.04
+PPNV2 (Ours)
+0.893
+32.05
+4.76
+0.833
+28.97
+8.93
+Fig. 8.
+Visualization examples on the KTH datasets. We use 10 frames as
+input to predict next 30 frames. Zoom in for a better view.
+V. RESULTS
+A. Evaluation with SOTA
+Similar to PPNV1 and most video prediction works, we use
+three kinds of metrics including structural similarity (SSIM),
+peak signal noise ratio (PSNR) and LPIPS for quantitative
+evaluation. The calculation of SSIM and PSNR are shown
+in Eq.15 and Eq.16. The LPIPS score is calculated using
+the designated pre-trained model (Alex network as the back-
+bone). Among them, higher scores of SSIM and PSNR and
+lower score of LPIPS indicate better results. We validate the
+proposed model on KTH, Human3.6M, Caltech and KITTI
+datasets, and the data preprocessing method is similar to
+previous works [17], [21], [47], [48] to ensure the fairness of
+the evaluation. The specific experimental setup of this work is
+
+Inputs 0→10
+Ground truth or prediction 10 -→ 40
+PPNV2 (Ours)
+人人人人美美
+MSPN
+人久#人人人人
+PPNet
+E3D-LSTM
+Conv-TT-LSTM
+PPNV2 (Ours)
+永人夫
+MSPN
+PPNet
+人人
+E3D-LSTM
+人人
+Conv-TT-LSTMTABLE II
+QUANTITATIVE EVALUATION ON THE HUMAN3.6M DATASET. THE BEST
+RESULTS ARE MARKED IN BOLD.
+Methods
+Metric
+T=2
+T=4
+T=6
+T=8
+T=10
+fRNN [39]
+PSNR
+27.58
+26.10
+25.06
+24.26
+23.66
+SSIM
+0.9000
+0.8885
+0.8799
+0.8729
+0.8675
+LPIPS
+0.0515
+0.0530
+0.0540
+0.0539
+0.0542
+MAFENet [48]
+PSNR
+31.36
+28.38
+26.61
+25.47
+24.61
+SSIM
+0.9663
+0.9528
+0.9414
+0.9326
+0.9235
+LPIPS
+0.0151
+0.0219
+0.0287
+0.0339
+0.0419
+MSPN [46]
+PSNR
+31.95
+29.19
+27.46
+26.44
+25.52
+SSIM
+0.9687
+0.9577
+0.9478
+0.9382
+0.9293
+LPIPS
+0.0146
+0.0271
+0.0384
+0.0480
+0.0571
+PPNV2 (Ours)
+PSNR
+32.07
+30.08
+28.81
+28.12
+27.55
+SSIM
+0.9645
+0.9566
+0.9510
+0.9461
+0.9421
+LPIPS
+0.0169
+0.0239
+0.0288
+0.0337
+0.0381
+TABLE III
+QUANTITATIVE EVALUATION ON THE CALTECH AND KITTI DATASETS.
+THE BEST RESULTS ARE MARKED IN BOLD.
+Methods
+Caltech (10 → 15)
+KITTI (10 → 15)
+SSIM
+PSNR
+LPIPS
+SSIM
+PSNR
+LPIPS
+MCNet [38]
+0.705
+-
+37.34
+0.555
+-
+37.39
+PredNet [21]
+0.753
+-
+36.03
+0.475
+-
+62.95
+Voxel Flow [49]
+0.711
+-
+28.79
+0.426
+-
+41.59
+Vid2vid [50]
+0.751
+-
+20.14
+-
+-
+-
+FVSOMP [51]
+0.756
+-
+16.50
+0.608
+-
+30.49
+PPNet [17]
+0.812
+21.3
+14.83
+0.617
+18.24
+31.07
+MSPN [46]
+0.818
+23.88
+10.98
+0.629
+19.44
+32.10
+PPNV2 (Ours)
+0.865
+25.44
+5.287
+0.621
+19.32
+15.45
+as follows: We use PyTorch as the platform and Adam as the
+optimizer to realize the calculation of the above model. The
+settings of the hyper-parameters defined in Eq.12 are shown
+in Eq.17, where L denotes the network level. The value of
+weighting factor α defined in Eq.14 is set to the total number
+of pixels in the input image (α = C·H·W), where the purpose
+is to ensure that the dimension of LPIPS loss is consistent with
+L1 and L2 losses.
+SSIM(x, y) =
+(2µxµy + c1)(2σxy + c2)
+(µ2x + µ2y + c1)(σ2x + σ2y + c2)
+(15)
+PSNR = 10 · log10(MAX2
+I
+MSE )
+(16)
+λt =
+�
+0.2,
+t = 0
+1.0,
+t > 0
+λl =
+�
+1.0,
+l = 0
+1.0/L,
+l > 0
+(17)
+Table I and Figure 8 give the quantitative results and visu-
+alization examples on the KTH dataset, respectively. It can be
+observed that the proposed method has a huge breakthrough in
+LPIPS evaluation while maintaining a high SSIM score, which
+Fig. 9. Visualization examples on the Human3.6M dataset. We use 10 frames
+as input to predict the next 5 frames. The other results are obtained from [46].
+Zoom in for a better view.
+Fig. 10. Visualization examples on the Caltech dataset. We use 10 frames as
+input to predict the next frame. Zoom in for a better view.
+means that it can generate sufficiently clear and sharp images
+while maintaining high pixel accuracy. Stochastic video pre-
+diction methods (such as SVAP-VAE [43]) can generate clearer
+future frames, so their LPIPS evaluation results are better, but
+they may be more biased towards unconditional generation,
+so the pixel accuracy is low. On the contrary, some methods
+improve pixel accuracy through averaging, but the generated
+images are blurry (such as Conv-TT-LSTM [45])
+Table II and Figure 9 give the results on the Human3.6M
+dataset. It can be observed from Table II that the proposed
+method performs much better than other works in terms
+of long-term prediction, although the performance in early
+
+Ground Truth
+PPNV2 (Ours)
+MSPN
+CrevNet
+ContextVPT+1
+T+2
+T+3
+T+4
+T+5
+fRNN
+MAFENet
+MSPN
+PPNV2 (Ours)
+Ground TruthFig. 11. Visualization examples on the KITTI dataset. We use 10 frames as
+input to predict the next 5 frames. In each group, the first row indicate the
+ground truth while the second row is prediction.
+TABLE IV
+ABLATIONS AND COMPARISONS ON THE KTH DATASET. THE
+QUANTITATIVE EVALUATION RESULTS.
+Ablation
+10 → 20
+10 → 40
+SSIM ↑
+PSNR ↑
+LPIPS ↓
+SSIM ↑
+PSNR ↑
+LPIPS ↓
+Default
+0.893
+32.05
+4.76
+0.833
+28.97
+8.93
+Add
+0.886
+31.87
+5.08
+0.817
+28.46
+9.75
+Concat
+0.888
+31.92
+5.16
+0.820
+28.71
+9.56
+NoFilter
+0.882
+31.70
+5.21
+0.808
+28.39
+9.71
+Bilinear
+0.887
+31.85
+5.10
+0.816
+28.38
+9.37
+NoLPIPS
+0.889
+31.99
+11.45
+0.824
+28.79
+20.41
+stage is slightly worse. The reconstruction of the character’s
+silhouette and motion is the difficulty of prediction on this
+dataset. Since most of the image area is a static background,
+the model only needs to keep the background unchanged to
+obtain high pixel accuracy, thus diluting the generation of the
+character in the image. Nevertheless, according to Fig. 9, the
+proposed method can better solve the above problem.
+Table III shows the quantitative evaluation results on Caltech
+and KITTI datasets. It can be observed that the proposed
+method achieves a great improvement on the Caltech dataset. It
+can also be seen from Figure 10 that our method do generates
+finer images and recovers more details. Although the pixel
+accuracy (SSIM and PSNR scores) on the KITTI dataset is not
+satisfactory, we obtain good LPIPS evaluation results, which
+means better visual performance (Figure 11).
+B. Ablations and Comparisons
+We perform several ablation studies to estimate the effects
+of each artifact or method proposed above. Each ablation
+experiment is performed on the basis of the default method
+with corresponding artifacts removed or replaced. Table IV
+and Figure 12 give the quantitative and qualitative results
+on the KTH dataset, respectively. Firstly, we highlight the
+superiority of “Modulate” model by comparing with the “Add”
+Fig. 12.
+Ablations and comparisons on the KTH dataset. The qualitative
+evaluation results. Shown in the figure are visualization examples of the
+predicted future 30 frames (10 → 40).
+and “Concat” methods (Section III-A). We have previously
+proposed a new module to better handle the two different input
+signals. It can be observed that the “Modulate” method out-
+performs the traditional “Add” and “Concat” methods both in
+quantitative and qualitative evaluation. In addition, the “Add”
+and “Concat” methods both experienced crashes (sudden drop
+in accuracy) during training, while the “Modulate” method did
+not, indicating that the gradient propagation of this method is
+indeed more stable.
+Whether the low-pass filter (Section III-B) is used or not has
+a greater impact on the prediction effect. It can be observed
+from Table IV and Figure 12 that in the absence of a low-pass
+filter to remove the aliasing signal (“NoFilter”), the long-term
+prediction performance is worse and the characterization of
+the human profile is indeed worse. In addition, we remove
+the interleaved upsampling method proposed in Section III-C
+by padding with zeros, and revert to the traditional bilinear
+upsampling. It can be observed that the gain of the pro-
+posed method is not very large, and the traditional bilinear
+upsampling method can also obtain good results. However,
+the proposed method is more coherent in depicting the color
+of the character without gradually fading, which is also in
+line with our original intention of proposing such a method.
+Finally, we also remove the LPIPS loss and only use Euclidean
+distance to train the model. It can be observed from the results
+that although its pixel accuracy (SSIM and PSNR) is high, its
+
+T+1
+T+2
+T+3
+T+4
+T+5#人人人人人
+G.T.
+Default
+Add
+Concat
+NoFilter
+人人人人
+Bilinear
+NoLPIPS
+G.T.
+Default
+Add
+Concat
+NoFilter
+Bilinear
+NoLPIPSvisual performance is worse, and the LPIPS evaluation result
+is also the worst. This shows the huge role of LPIPS loss in
+image sharpening.
+VI. LIMITATIONS AND FUTURE WORK
+In this work, we modified several artifacts in the pyramidal
+predictive network as well as the signal processing of the
+overall architecture from a signal processing perspective. The
+improved model is more interpretable and more powerful. In
+addition, we also improved the training strategy to obtain
+accurate and sharp predicted frames. The experimental results
+have shown that our method does achieve very good perfor-
+mance. Ablation study proves the rationality and validity of
+our proposed improvements.
+Nevertheless, the computational efficiency of the proposed
+model still needs to be improved. The overall model is still
+based on the LSTM recurrent neural network architecture,
+which cannot be well parallelized. The introduction of modules
+such as low-pass filters also increases the difficulty for parallel
+computing, resulting in a decrease in computing efficiency. In
+the future, we will focus on solving the above problems. At
+the same time, we will refer to the diffusion calculation and
+transformer attention calculation methods to further improve
+the performance of the model.
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+page_content='Analyzing and Improving the Pyramidal Predictive  Network for Future Video Frame Prediction  Chaofan Ling  South China University of Technology  wichaofan@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='scut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='cn  Weihua Li  South China University of Technology  whlee@scut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='cn  Junpei Zhong  The Hong Kong Polytechnic University  joni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='zhong@polyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='hk  Abstract—The pyramidal predictive network (PPNV1) pro-  poses an interesting temporal pyramid architecture and yields  promising results on the task of future video-frame prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  We expose and analyze its signal dissemination and characteristic  artifacts, and propose corresponding improvements in model  architecture and training strategies to address them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Although  the PPNV1 theoretically mimics the workings of human brain,  its careless signal processing leads to aliasing in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We  redesign the network architecture to solve the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In ad-  dition to improving the unreasonable information dissemination,  the new architecture also aims to solve the aliasing in neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Different inputs are no longer simply concatenated,  and the downsampling and upsampling components have also  been redesigned to ensure that the network can more easily  construct images from Fourier features of low-frequency inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Finally, we further improve the training strategies, to alleviate  the problem of input inconsistency during training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Overall, the improved model is more interpretable, stronger,  and the quality of its predictions is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Code is available  at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='com/Ling-CF/PPNV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Index Terms—video-frame prediction, pyramidal predictive  network, aliasing  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' INTRODUCTION  Future video frame prediction has been rapidly developed  recently [1]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' This looking-ahead ability could be applied to  various fields including autonomous driving [9], [10], robotic  systems [11], [12], rainfall forecasting [13], [14] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  In particular, this self-supervised learning method for video  representation can be also migrated to downstream tasks [15],  [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Specifically, we can first perform self-supervised video  prediction learning on the backbone network, which is then  fixed and used in tasks such as video classification, to solve  the difficulties of label acquisition in supervised learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Video frame prediction is a pixel-level task designed to use  historical frames to predict future frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' One of the state-  of-the-art video prediction methods is the Pyramid Prediction  Network (PPNV1) [17], which is an interesting work and  has achieved promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The distinguishing feature of  PPNV1 is its unconventional temporal pyramid architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  The total model is updated through a combination of bottom-  up and top-down information flows, and uses prediction errors  to achieve effective feedback connections, which is where  predictive coding models differ from traditional generators  [18]–[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In particular, the update frequency of neurons in this  model decreases as the network level increases, so that neurons  in higher layers can capture information over a longer time  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The architecture of PPNV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' According to the definitions in PPNV1  [17], ft  l (green), P t  l  (orange) and Et  l (red) represent the sensory input,  prediction and prediction error at time-step t and level l, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In the  picture, the sensory input propagating up is lagged, in which ft+1  l  should be a  higher-level representation of ft+1  l−1 rather than ft  l−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In addition, since ft+1  l  characterizes Et+1  l−1 at the same time, P t  l can be considered to predict Et+1  l−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  However, Et+1  l−1 is actually indirectly generated by P t  l , which is contradictory  and incomprehensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  range, and the computational cost is also reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' However,  the authors seem to prefer to build network models from  theories of cognitive science or neuroscience, ignoring the  importance of how computers process signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Overall, the  PPNV1 provides an effective predictive framework, but its  signal processing is careless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Our main work is to fix its  signal processing and characteristic artifacts to improve the  final prediction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  We first observed that there are inconsistencies in the way of  temporal signal processing in PPNV1 (shown in Fig 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Firstly,  the sensory input (green) in the neural network is propagated  up to higher level along with the prediction error (red), which  has been proved to be necessary [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' However, there is a prob-  lem with the choice of sensory information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The model should  have transmitted sensory input of next time-step (f t+1  l−1 ) instead  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='05421v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='CV]  13 Jan 2023  ConvLSTM  Conv+Pool  D/+1  E]+  ConvLSTM  ConvLSTM  ft  Conv+Pool  Conv+Pool  ConvLSTM  Ei-1  ConvLSTM  ConvLSTM  fi  fi1Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The complete architecture of the improved pyramidal predictive network (PPNV2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Similar to PPNV1, ft  l (green), P t  l (orange) and Et  l (red) represent  the sensory input, prediction and prediction error at time-step t and level l, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We first made improvements to how sensory input and prediction errors  are propagated and computed, making the model more interpretable (will be introduced in detail in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Second, we redesigned several characteristic  artifacts to address the problem of information aliasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Here “US” denotes the upsampling artifact, in which we replace the traditional bilinear or bicubic  upsampling methods by interleaving zeros and then using depthwise separable convolutions to implement interpolation calculations (Section 3, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' A low-pass  filter with a learnable cutoff frequency is used for anti-aliasing filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' “DS” denotes the downsampling artifact, in which low-pass filtering is also introduced  to prevent high-frequency information from being mixed into low-frequency information (Section 3, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' “ModError” and “ModPred” are special modulation  modules designed to better handle different inputs and make gradient propagation easier and more stable (Section 3, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Zoom in for a better view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  of current time-step (f t  l−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Secondly, the sensory input and  prediction error are simply concatenated for calculation, which  will cause information aliasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Moreover, the predictive unit  at higher-level needs to simultaneously predict the prediction  error from a lower level, which is contradictory and difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  We will expose and analyze these issues in detail and redesign  the signal processing specification in Section 2, to make the  model stronger and more interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  The rough splicing of various signals is another problem  that cannot be ignored in PPNV1 (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' For instance,  the prediction, prediction error and sensory information from  different levels are directly spliced and input into the predictive  unit (ConvLSTM) for calculation, which will cause serious  information aliasing and huge computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We  propose an alternative design to alleviate the problems — a  modulation module is added before and after the predictive  unit, and only the modulated information is sent to the  predictive unit for calculation — this is a special network  module designed to modulate the primary input with a scaling  and deviation tensor computed from the auxiliary input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In  addition, the downsampling and upsampling artifacts have  also been redesigned to ensure that the network can more  easily construct images from Fourier features of low-frequency  inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The total improved network architecture is shown in  Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' This new design makes the model more efficient and  interpretable, which also allows us to further improve the  training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  In addition to network architecture, input inconsistency be-  tween training and testing in temporal sequential tasks remains  a challenging topic (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In general, the predicted  frames needed to be served as new inputs to enable continuous  prediction, thus causing inconsistency between training and  testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We observe that the improved pyramidal predictive  network can alleviate this problem by computing losses at  multiple levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The new convolutional unit will learn to extract  accurate and effective information from ”less perfect” data,  that is, the overall model is more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Additionally, we  propose to train the model with LPIPS loss [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In tasks  of future video-frame prediction, LPIPS is usually used as  an evaluation metric to evaluate the similarity of images, but  there is almost no work uses it as a loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In this work,  it will be combined with Euclidean distance loss to generate  more accurate and clearer future video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  P  DS  : Downsampling  US  : Upsampling  ModPred  Prediction of  P  local level  Output  Input  ModPred  Conv  Pi+1  ConvLSTM  A  Encoding  Decoding  Prediction from  tanh  higher level  ModError  Conv  Conv  Zero  :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Downsample  interleaving  Pt  DS  DS  US  ConvLSTM  Lowpass  Anti-aliasing  filtering  f out  filter  ModPred  ModPred  4  4  Convolution  ConvLSTM  ConvLSTM  interpol ati on  tanh  4  tanh  ModError  ModError  +1-11  Convolutional L ayer  h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Feature  extraction  P  DS  DS  US  DS  DS  US  ModError  Low-pass  Anti-aliasing  filtering  ModPred  ModPred  ModPred  filter  Error of  Convtanh  Input  current level  ConvLSTM  ConvLSTM  ConvLSTM  reconstruction  E,  X  ModError  Ei1  ModError  ModError  Conv  Error from  Ei-1  Sensory input of  Ji1  lower level  current ievel  ndnoFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  We first redesign the signal processing of pyramidal predictive network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' (a)The original model (PPNV1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' (b) The improved model (PPNV2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We  make several changes to the original structure as shown in the red boxes in the figure: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We adjust the sensory input being propagated upward, where we  use sensory input of next time-step (ft+1  l−1 ) instead of the current one (ft  l−1) for propagation, to solve the problem of lagging propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We separate  prediction error from sensory input and calculate them with novel downsampling artifacts respectively, to prevent the information aliasing and make the model  more interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Instead of roughly feeding different signals directly into the ConvLSTM for computation, we use a network block to process them  separately, which is made up of “ModError”, “ConvLSTM” and “ModPred” shown in Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' IMPROVING SIGNAL PROCESSING  We first observe that there are problems in the signal  processing of the pyramidal predictive network (PPNV1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The  authors seem to prefer to build network models from cognitive  science or neuroscience theories, ignoring the importance of  how convolutional units perform calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Simply treating  the convolution units as brain neurons to build a brain-like  model may cause problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Specifically, we have identified  two serious signal processing issues in PPNV1: 1) there is a  lag in the upward propagation of sensory information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 2) the  integrated calculation of sensory input and prediction error  will cause the information aliasing and makes the prediction  difficult and ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' These problems make the entire  PPNV1 incomprehensible, it works in a puzzling way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' As  shown in Fig 3, we propose corresponding solutions to the  problems to address the disturbing ”black box” in pyramidal  predictive network, the details will be described in the follow-  ing subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Revisiting the processing of sensory input  In the pyramidal predictive network (PPNV1), the sensory  information is input to the predictive unit (ConvLSTM) for  prediction generation, which is also propagated up to higher  level along with the prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The authors [17] have  confirmed the necessity of this operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' However, in such  a temporal sequential model, there is a lag in the sensory  information used in the up-propagation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  As shown in Fig 3 (a), the sensory input f t+1  l  is a higher-  level representation of the combination of f t  l−1 and prediction  error Et+1  l−1, while P t  l is considered to be a prediction of f t+1  l  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  In other words, the P t  l can be considered to predict f t  l−1  and Et+1  l−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' However, the P t  l will be eventually propagated  downward and sent to the predictive unit together with f t  l−1  to make the prediction of lower-level, so what P l  t predicts is  lag and meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In particular, this kind of delay becomes  more severe as the network level increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' For instance, the  sensory information contained in f t+1  l+1 (the third level in Fig  3 (a)) is actually derived from f t−1  l−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Therefore, to solve this  problem, we should propagate up the sensory input f t+1  l−1 (Fig  3 (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=') of next time step instead of f t  l−1, which matches in  the time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Revisiting the processing of prediction error  Secondly, the prediction error and sensory input in PPNV1  are simply concatenated, which is then computed with a con-  volution unit to obtain the higher-level representation (f t+1  l  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Unfortunately, this kind of integrated calculation will lead to  aliasing of information, in which the higher-level sensory input  is no longer a simple representation of lower-level sensory  input, but mixed with the information of prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Moreover, predicting the prediction error is actually puzzling  and difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  According to what we mentioned in the previous subsection,  the P t  l is considered to predict the lower-level information  (a)  Pt  (b)  Pt  onLSTM  id  pr+1  E++  E++1  Pi  Pi1  Pl  US  Conv+Pool  DS  DS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  E  ConvLSTM  ConvLSTM  nvLSTM  NetBlock  E  fi1  fi  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  DS  : Downsampling artifacts  US  : Upsampling artifacts  ConvLSTM  : Convolutional LSTM  NetBlock  : Network blockFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We use a special modulation module to perform computation on different input signals x1 and x2 to further address the problems of signal aliasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  The auxiliary signal x2 is calculated by convolution and activated by the sigmoid and tanh functions to obtain scaling and shifting matrices, and then  the main signal x1 is scaled and shifted with the matrices, which can prevent the signal of x2 from leaking into x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Additionally, there is only pixel-level  multiplication and addition on the x1 computational path (red), so the gradient flow is more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We make comparison with the traditional methods of  “Concat” and “Add”, where “Concat” denotes concating two matrices together by channel and “Add” denotes adding two matrices together at the pixel-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  To make sure the same number of parameters and computational overhead, we add additional convolutional unit for “Add” method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  of prediction error Et+1  l−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Paradoxically, the prediction error  Et+1  l−1 is calculated from P t  l−1 and f t+1  l−1 , and the generation  of P t  l−1 is inseparable from the higher-level prediction P t  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In  short, P t  l predicts the Et+1  l−1, while the Et+1  l−1 must be generated  by P t  l first, which is contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' To solve the problem, we  propose to calculate the prediction error and sensory input  separately in the new improved pyramidal predictive network  (PPNV2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' As shown in Fig 3 (b), the higher-level sensory input  f t+1  l  only represents f t+1  l−1 at lower-level, while the prediction  error is excluded, thus avoiding the aliasing of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Importantly, since the generation of f t+1  l  is independent of  prediction error Et+1  l−1, the aforementioned contradiction is  resolved and the overall model becomes more interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In  particular, we use a novel downsampling artifact for upward  propagation, which will be introduced in detail in the next  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' ANTI-ALIASING DESIGN  Another major problem with pyramidal predictive network  (PPNV1) is that it handles different signals carelessly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' As  shown in Fig 3 (a), it simply concatenates different signals  and feeds them into the ConvLSTM predictive unit for com-  putation, which will inevitably lead to aliasing of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  We argue that different information is of different importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  For example, the sensory input which contains more low-  frequency signals can provide more effective information than  prediction errors, which deserves more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Additionally,  the information contained in higher-level predictions (P t  l ),  prediction errors (Et  l−1) and sensory input (f t  l−1) are quite  different, it is better to calculate them separately instead of  splicing together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Although the convolution unit can be theoret-  ically trained to distinguish them eventually, it is undoubtedly  difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Repeated multiplication of huge weight matrices may  cause the gradients vanishing or exploding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  In particular, in view of the special calculation of ConvL-  STM (the ConvLSTM needs to compute four outputs [14]),  this concatenation operation will also lead to a substantial  increase in the amount of parameters and calculation overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Therefore, to solve the problems, we propose to use a network  module before and after the ConvLSTM predictive unit to  process different inputs, and only propagate the necessary  signals to the ConvLSTM for calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The purpose is to  solve the problem of information aliasing and reduce network  parameters and computational overhead as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Moreover, aliasing of different frequencies in a signal is also  of concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In the process of downsampling and upsampling,  high-frequency signals are often mixed with low-frequency  signals, which is easy to cause the final generated image to be  incoherent and unnatural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' To alleviate the problem, we propose  to redesign the upsampling and downsampling artifacts with  anti-aliasing low-pass filters to ensure that the model can more  easily learn low-frequency Fourier features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The details will be  described in the next several subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' modulation module  As shown in Fig 4, we add a modulation module before and  after the ConvLSTM prediction unit for signal preprocessing  and postprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' This is a special module designed to alle-  viate the problems of signal aliasing and difficulties of gradient  propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Different from the traditional concatenating or  adding operation, the modulation module needs to distinguish  and determine the main signal x1 and the auxiliary signal x2  first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The design of the module is inspired by the attention  mechanism — the brain responds more strongly when predic-  tions are severely inconsistent with the environment [23]–[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Modulate  Concatenate  Add  US  X, (b, c, h, w)  X2 (b, c, h, w)  X, (b, c, h, w)  X, (b,c,h,w)  X (b, c, h, w) X2 (b, c, h, w)  ModPred  Conv  Conv  concat  add  ConvL STM  (b, c, h, w)  Ei++  ModError  a  tanh  x  (b, 2c, h, w)  Conv  (b, c, h, w)  (b, c, h, w)  (b, c, h, w)  Conv  Conv  DS  DS  (b, c, h, w)  y  (b, c, h, w)  y  2  (b, c, h, w)  Et1For “ModError” (Fig 4), the operation of modulation can  be seen as the process of attention attachment and transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  The sensory input contains more information that effectively  describes the current scene, while the prediction error consists  of sparse error signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We have pointed out that the neural net-  work preferentially learns low-frequency features [26], [27], so  it is better to take the sensory input as the main signal x1 and  the prediction error as an auxiliary signal x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' According to  Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' [23], [28], [29], the larger the prediction error,  the more attention is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Therefore, we propose that the  prediction error can be viewed as an attention matrix, and the  attachment of attention is equivalent to the process of scaling  and shifting the sensory input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  The calculation is as follows: First, the auxiliary signal x2 is  convolved separately, and then the sigmoid and tanh are used  as activation functions to obtain the scaling and shift matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  The calculation formulas are shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='2:  msc = sigmoid f(x2, θsc)  (1)  msf = tanh f(x2, θsf)  (2)  where f(·) represents a typical convolutional network and  θ is the model parameter, msc and msf denote the scaling  and shifting matrices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The role of sigmoid and  tanh is to constrain the matrices between (0, 1) and (-1, 1),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The final output is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='3, where α is a  learnable scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The scaling matrix is limited to (0, 1)  by simoid, which only plays the role of downscaling but not  upscaling, so we recommend adding a scaling factor α and  setting it as a learnable parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  y = α · msc · x1 + msf  (3)  Similarly, in the post-processing stage, we also use the  modulation module to process the prediction P t+1  l+1 (Fig 4)  from higher level and the output of ConvLSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In this work,  we take the higher-level prediction as the main signal x1 and  the output of the ConvLSTM as the auxiliary signal x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The  downward propagation of high-level prediction is actually the  process of decoding, so we take the decoding feature as the  main signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Downsampling artifact  The purpose of downsampling is to reduce memory and  computation, prevent overfitting and increase the receptive  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In PPNV1, the conventional method of convolution and  pooling is used for downsampling (Fig 3 (a) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Related  studies have shown that classifier networks focus on texture  rather than shape [30], and one of the reasons is the use of  pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Pooling, especially max pooling, is more sensitive  to texture information and has better translation or rotation  invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Therefore, no matter how the images are translated  or rotated, the classifier network can still recognize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  However, continuous video frame sequences are semantically  consistent, but different in pixel space [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The essence of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Simplified flowchart for downsampling (a) and upsampling (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  We introduce a low-pass filter for anti-aliasing filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Importantly, we set  learnable cutoff frequencies for each feature map of each input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' “ZeroPad”  means that the input is interleaved by m−1 zeros in m-fold upsampling, and  then interpolated using depthwise separable convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  future video frame prediction is to learn changes by observing  the differences between frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' As a result, this invariance  brought by pooling can be disastrous for video prediction  tasks, which motivates us to look for an alternative that  preserves the spatial location information as much as possible  while downsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  The first thing that comes to mind is convolutional down-  sampling, which preserves spatial details better than pooling,  but can still be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' For example, using a low-pass filter  for downsampling first, followed by convolution calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Aliasing, a subtle and critical issue, has recently attracted  attention [32]–[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In order to prevent the high frequency  signal from being mixed into the low frequency signal during  the downsampling process, it is better to perform anti-aliasing  filtering first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' As shown in Fig 5 (a), We first perform fea-  ture extraction on the input signal using convolutional units,  followed by low-pass filtering for anti-aliasing and downsam-  pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Finally, another convolution calculation is performed to  encode the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Next we will focus on the design and implementation of  low-pass filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' According to the Nyquist-Shannon theorem  [36], in order to recover the signal without distortion, the  sampling frequency should be greater than 2 times the highest  frequency in the signal spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Otherwise, the sampled  frequencies will overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Therefore, in the case of a fixed sam-  pling frequency, we assume that the signal whose frequencies  are higher than half the sampling frequency are negligible,  which is usually achieved with a low-pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  w(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='54 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='46 cos(2πn  N ),  0 ≤ n ≤ N − 1  (4)  y  (a) i (b)  y  ↑  Encoding  Convolution  Convolution  Decoding  Anti-aliasing filtering  Downsample  ZeroPad + DSConv  Low-pass filter  Ups ample  Low-pass filter  Feature extraction  Anti-aliasing filtering  Convolution  Convolution  Reconstruction  x  x  DSConv: Depthwise Separable Convolution  ZeroPad: Zero padding (upsample with 0)λ(n) = f 2  fs  sinc(f 2  fs  (n − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='5(N − 1)))  (5)  h(n) = λ(n) · w(n)  (6)  In this work, we prefer to use the Hamming Window [37]  for the design of low-pass filter, which is described as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='4,  where N denotes the length of filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' And the calculation  of coefficients is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='5, where f and fs represent  the cut-off and sampling frequency respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' As what we  mentioned above, the cutoff frequency f should be between  0 and half the sampling frequency fs/2, , and it needs to be  limited to (0, 1) when calculating, which is achieved by 2f/fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Finally we get the low-pass filter by multiplying the Hamming  Window and the coefficient (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  We use a low-pass filter of length N = 25 and resize it to a  5 × 5 convolution kernel in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In particular, we define  the cut-off frequency f as a learnable parameter, which is im-  portant because the ratio of low-frequency and high-frequency  signals may be different at different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In general,  the higher the resolution, the higher the proportion of low  frequency signals, so we can limit the bandwidth even lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Taking a step further, we propose to define different learnable  cut-off frequencies fi, 1 ≤ i ≤ C for each input feature map  and use the sigmoid function to constrain it between (0, 1),  where C denotes the number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Finally, we can ob-  tain a convolution kernel of shape (C, 1, 5, 5), which is depth-  separable and will not fuse the features of different channels,  only performs anti-aliasing filtering and downsampling (stride  = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The traditional convolution calculation is placed after  downsampling, so a larger receptive field is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Upsampling artifact  Inspired by styleGAN3 [35], ideal upsampling should not  modify the continuous representation, its only purpose is to  increase the output sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Similarly, the upsampling  operation consists of two steps: we first increase the sampling  rate to sout = sin · m by interleaving m − 1 zeros between  each input sample (m is usually set to 2), which is the same as  styleGAN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The feature map after staggered filling with zeros  is shown in Fig 6 (Zoom in for a better view), where x refers  to the original input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Next, we perform convolution  processing on the result and then low-pass filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  In this work, we replace the bilinear or bicubic 2× up-  sampling filter with a 7 × 7 depthwise separable convolution  approximation, and consider the four computational cases  shown in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In 2× upsampling, we focus on the calcu-  lation at different positions in the 2 × 2 box ( 0 0  0 x ) , which  has different convolution parameters and input features (x)  involved in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Taking the first calculation (Fig 6  (a)) as an example, there are 4 × 4 features involved in the  calculation, which can be equivalent to bicuxic interpolation  in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' For the other three cases, they calculate fewer  features, thus saving computational overhead compared to  bicubic interpolation, but introducing additional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Since the kernel is depth-separable, it is still tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The convolution interpolation calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We focus on the calculation  at different positions in the 2 × 2 box  � 0 0  0 x  �  , which can be discussed in four  different cases (a)-(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Particularly, the convolutional kernel parameters used  in each case are non-overlapping (the second row), and we can initialize the  parameters for each case according to the distance between the parameters to  the center (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' What the third row shows is a total initialized convolution  kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Zoom in for a better view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  In view of the special function of this convolution, the  initialization of parameters is also important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We use the  weighted average method to initialize the parameters in this  paper, in which the weight of the feature (x) is inversely  proportional to its distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Specifically, taking the first calcu-  lation (Fig 6 (a)) as an example again, we need to calculate  the interpolation of 0 in the center (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' There are three  kinds of features with different distances around the center  0 , and we will assign different weights to these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Firstly, the weight is proportional to the inverse of the distance,  which is shown as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='7 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 8, where w1, w2, w3 represent  the weights under the three kinds of distances (l1, l2, l3)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Secondly, the sum of all weights should be 1  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='9), so that the three kinds of weights can be calculated  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The parameters for cases of (b) and (c) are calculated  in the same way (note that for different cases, what the wn  refers to is different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In particular, for the last case (d), we  expect to preserve the original feature x, so we set the central  parameter to 1 and the others to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Finally, the complete  initialized kernel is shown in the third row of Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  w1 : w2 : w3 = 1  l1  : 1  l2  : 1  l3  (7)  l1 : l2 : l3 =  √  2 :  �  32 + 1 :  �  32 + 32  (8)  4w1 + 8w2 + 4w3 = 1  (9)  w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='1122 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  w2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='0502 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  w3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='0374  (10)  (a)  (b)  (c)  (d)  x  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
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+page_content='0347The complete downward propagation process is shown in  Fig 5 (b), the input y is first decoded with convolution  calculation, which corresponds to the encoding process of  up-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Next, upsampling is performed using the  method described above, followed by low-pass filtering to  eliminate aliasing frequencies above s/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Finally, we use  another convolution calculation to reconstruct the result, which  corresponding to the feature extraction process in the upward  propagation  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' REVISITING TRAINING STRATEGIES  PPNV1 is committed to constructing neural network models  from predictive coding theories, while the training strategies  are less explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' First, PPNV1 is a typical end-to-end training  model, which can only utilize the bottom predictions to  calculate the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The careless signal processing makes the  whole network model difficult to interpret (section 2), so the  training of higher levels in the model are uncontrollable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Even  worse result may be obtained if the higher-level prediction  errors are used to calculate losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Additionally, inconsistency  between training and testing in image time series tasks remains  a challenging topic, which has not been addressed in PPNV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  In response to the above problems, we propose several im-  provements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' First, benefiting from the clear improved network  architecture, the entire network becomes controllable, we can  simultaneously calculate the training loss for higher levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Second, we further optimize the calculation process of long-  term prediction, in which the encoding loss and decoding loss  are calculated simultaneously to enhance the anti-interference  ability of neural units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The details will be introduced in the  next subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Long-term prediction training strategy  Long-term prediction is a very important task for future  video frame prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Specifically, if we need to predict  further future frames, we have to predict the next video frame  by inputting the predicted frame instead of the real frame, thus  causing the problem of mismatch between training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  The predicted frames are “imperfect”, and the predictions  will become increasingly blurry with the accumulation of  prediction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In order to alleviate this problem, we propose  a tracking training method that takes the predicted frame as  input and the real frame as target during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  First, we fixedly take T (T is usually set to 10) real sequence  frames as inputs for calculation, and use f t  l to denote the  representation of the input frame at time-step t and level l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  These inputs are real frames which can be directly used to  compare predictions from the previous time step to calculate  prediction error and loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The calculation of the prediction  error is the same as that of PPNV1, which is divided into  positive and negative errors Et  l = C(|f t  l − P t−1  l  |, |P t−1  l  −  f t  l |), where C denotes “concatenate”, that is, concatenate two  matrices together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The loss is obtained by calculating their  squared Euclidean distance L = � (f t  l − P t−1  l  )2, which will  be introduced in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We improve the training strategy by calculating encoding and decoding  losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The decoding loss L1, which is the prediction loss at each level,  is obtained by comparing the current prediction P t  l with the real sensory  representation ft+1  l  of the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The encoding loss L2 is only  calculated when the predicted frame is used as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We need to maintain  an additional encoding pathway of the real frame to obtain the target for  calculating the loss, which shares the encoding pathway (DSf) of the  predicted frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Second, we take the last predicted frame P T as a new input  to predict P T +1, and then continue to predict with P T +1, so  as to achieve continuous prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In particular, in order to  be distinguished from the representation of the real frame,  we denote the representation of the predicted frame input at  time-step t and level l by ˆf t  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Importantly, ˆf t  l will be used  to calculate the prediction error Et  l , but not the loss, which  is different from the case where the real frame is used as  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' As what we mentioned above, the predicted frame is  “imperfect”, and the loss calculated using predicted frame  might be “false”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' As a result, we must use the real frame  to calculate the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Importantly, in order to calculate loss  for higher level, we need to maintain an encoding pathway  for real frames during training, which shares parameters with  the encoding pathway of the predicted frames (Fig 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The  real frame is only used to provide the target required for  loss calculation during training, which will not participate  in the calculation of prediction generation, and this encoding  pathway will also be closed during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Furthermore, we calculate not only the prediction loss (Fig  7, L1) of each level, but also the loss of the encoding process  of predicted frame (Fig 7, L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The purpose is to enhance the  anti-interference ability of the downsampling artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We hope  that the convoolutional units are robust enough to obtain the  same features from “imperfect” predicted frames as real ones,  further mitigating the effects of the prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Pi+1  Pi  US  Ei  NetBlock  fi  fi  DSE  DSf  DSf  DSf  Ei  fi1  : Parameter sharing  L, : Decoding loss  L, : Encoding lossB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Loss function  In this subsection, we would like to introduce the details of  loss function used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We split the computation into  two stages during training: first calculating with real frames  as input, followed by predicted frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We assume that the  length of the input real frame sequence is T1, and the length  of predicted frame sequence is T2, then the total input length  is T = T1 + T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Firstly, we need to compute the prediction  loss (Fig 7, L1) of each level at each time-step throughout the  whole calculation, which can be expressed as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='11:  L1 =  T  �  t=0  L  �  l=0  λt · λl · (f t  l − P t−1  l  )2  (11)  where L denotes the number of network level, λt and λl  represent the weighting coefficient at time-step t and level  l respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Importantly, we use the squared Euclidean  distance to calculate the loss instead of the traditional mean  squared error (MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The difference is that MSE is divided  by the total number of features on the basis of the Euclidean  distance, which will cause the loss value to be too small, and  the gradient of some parameters may be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' It will cause  the training process to converge first and then diverge, which  often occurs in low-precision training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  L2 =  T  �  t=T1  L  �  l=1  λt · λl · (f t  l − ˆf t  l )2  (12)  Secondly, we need to calculate the encoding loss (Fig 7, L2)  simultaneously in the second stage, to make the convolution  units robust enough to extract similar features from “imper-  fect” predicted frames to real frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The calculation of L2 is  defined as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='12: where f t  l and ˆf t  l indicate the representation  of the real frame and predicted frame at time-step t and level  l, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Llpips =  T  �  t=0  λt · flpips(f t, P t−1)  (13)  Ltotal = L1 + L2 + α · Llpips  (14)  In addition, we also use LPIPS [22] to calculate the training  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' proposes to use deep features to measure  the perceptual similarity between two images, which is imple-  mented by a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' It is often used as an evaluation  metric for video prediction, but no one seems to have tried  to use it as a function to compute the training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In this  work, we try to use the provided pretrained model (VGG as  backbone) to compute the loss between the predicted frame  and the target, which is very important for sharpening the  generated images (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='13, where flpips denotes the LPIPS  pretrained model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Finally, the total loss is calculated as the  sum of the above three losses (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='14) — since the loss  provided by LPIPS is very small, we need to additionally  multiply by a coefficient α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  TABLE I  QUANTITATIVE EVALUATION ON THE KTH DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' THE METRICS ARE  AVERAGED OVER THE PREDICTED FRAMES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' RED AND BLUE INDICATE THE  BEST AND SECOND BEST RESULTS, RESPECTIVELY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Methods  10 → 20  10 → 40  SSIM ↑  PSNR ↑  LPIPS ↓  SSIM ↑  PSNR ↑  LPIPS ↓  MCNet [38]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='804  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='73  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='89 fRNN [39]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='771  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='678  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='77 PredRNN [40]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='839  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
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+page_content='16 PredRNN++ [41]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
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+page_content='93  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Visualization examples on the KTH datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We use 10 frames as  input to predict next 30 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Zoom in for a better view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Evaluation with SOTA  Similar to PPNV1 and most video prediction works, we use  three kinds of metrics including structural similarity (SSIM),  peak signal noise ratio (PSNR) and LPIPS for quantitative  evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The calculation of SSIM and PSNR are shown  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='15 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The LPIPS score is calculated using  the designated pre-trained model (Alex network as the back-  bone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Among them, higher scores of SSIM and PSNR and  lower score of LPIPS indicate better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We validate the  proposed model on KTH, Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='6M, Caltech and KITTI  datasets, and the data preprocessing method is similar to  previous works [17], [21], [47], [48] to ensure the fairness of  the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The specific experimental setup of this work is  Inputs 0→10  Ground truth or prediction 10 -→ 40  PPNV2 (Ours)  人人人人美美  MSPN  人久#人人人人  PPNet  E3D-LSTM  Conv-TT-LSTM  PPNV2 (Ours)  永人夫  MSPN  PPNet  人人  E3D-LSTM  人人  Conv-TT-LSTMTABLE II  QUANTITATIVE EVALUATION ON THE HUMAN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='6M DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' THE BEST  RESULTS ARE MARKED IN BOLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Methods  Metric  T=2  T=4  T=6  T=8  T=10  fRNN [39]  PSNR  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
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+page_content='0381  TABLE III  QUANTITATIVE EVALUATION ON THE CALTECH AND KITTI DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  THE BEST RESULTS ARE MARKED IN BOLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Methods  Caltech (10 → 15)  KITTI (10 → 15)  SSIM  PSNR  LPIPS  SSIM  PSNR  LPIPS  MCNet [38]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
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+page_content='32  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='45  as follows: We use PyTorch as the platform and Adam as the  optimizer to realize the calculation of the above model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The  settings of the hyper-parameters defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='12 are shown  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='17, where L denotes the network level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The value of  weighting factor α defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='14 is set to the total number  of pixels in the input image (α = C·H·W), where the purpose  is to ensure that the dimension of LPIPS loss is consistent with  L1 and L2 losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  SSIM(x, y) =  (2µxµy + c1)(2σxy + c2)  (µ2x + µ2y + c1)(σ2x + σ2y + c2)  (15)  PSNR = 10 · log10(MAX2  I  MSE )  (16)  λt =  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='2,  t = 0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='0,  t > 0  λl =  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='0,  l = 0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='0/L,  l > 0  (17)  Table I and Figure 8 give the quantitative results and visu-  alization examples on the KTH dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' It can be  observed that the proposed method has a huge breakthrough in  LPIPS evaluation while maintaining a high SSIM score, which  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Visualization examples on the Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='6M dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We use 10 frames  as input to predict the next 5 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The other results are obtained from [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Zoom in for a better view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Visualization examples on the Caltech dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We use 10 frames as  input to predict the next frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Zoom in for a better view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  means that it can generate sufficiently clear and sharp images  while maintaining high pixel accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Stochastic video pre-  diction methods (such as SVAP-VAE [43]) can generate clearer  future frames, so their LPIPS evaluation results are better, but  they may be more biased towards unconditional generation,  so the pixel accuracy is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' On the contrary, some methods  improve pixel accuracy through averaging, but the generated  images are blurry (such as Conv-TT-LSTM [45])  Table II and Figure 9 give the results on the Human3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='6M  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' It can be observed from Table II that the proposed  method performs much better than other works in terms  of long-term prediction, although the performance in early  Ground Truth  PPNV2 (Ours)  MSPN  CrevNet  ContextVPT+1  T+2  T+3  T+4  T+5  fRNN  MAFENet  MSPN  PPNV2 (Ours)  Ground TruthFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Visualization examples on the KITTI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We use 10 frames as  input to predict the next 5 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In each group, the first row indicate the  ground truth while the second row is prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  TABLE IV  ABLATIONS AND COMPARISONS ON THE KTH DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' THE  QUANTITATIVE EVALUATION RESULTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Ablation  10 → 20  10 → 40  SSIM ↑  PSNR ↑  LPIPS ↓  SSIM ↑  PSNR ↑  LPIPS ↓  Default  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='893  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='05  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='833  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='97  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='93  Add  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
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+page_content='75  Concat  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
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+page_content='56  NoFilter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
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+page_content='39  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='71  Bilinear  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='887  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='85  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='816  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='38  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='37  NoLPIPS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='889  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='99  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='824  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='79  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='41  stage is slightly worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The reconstruction of the character’s  silhouette and motion is the difficulty of prediction on this  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Since most of the image area is a static background,  the model only needs to keep the background unchanged to  obtain high pixel accuracy, thus diluting the generation of the  character in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Nevertheless, according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 9, the  proposed method can better solve the above problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Table III shows the quantitative evaluation results on Caltech  and KITTI datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' It can be observed that the proposed  method achieves a great improvement on the Caltech dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' It  can also be seen from Figure 10 that our method do generates  finer images and recovers more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Although the pixel  accuracy (SSIM and PSNR scores) on the KITTI dataset is not  satisfactory, we obtain good LPIPS evaluation results, which  means better visual performance (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Ablations and Comparisons  We perform several ablation studies to estimate the effects  of each artifact or method proposed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Each ablation  experiment is performed on the basis of the default method  with corresponding artifacts removed or replaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Table IV  and Figure 12 give the quantitative and qualitative results  on the KTH dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Firstly, we highlight the  superiority of “Modulate” model by comparing with the “Add”  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Ablations and comparisons on the KTH dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The qualitative  evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Shown in the figure are visualization examples of the  predicted future 30 frames (10 → 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  and “Concat” methods (Section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' We have previously  proposed a new module to better handle the two different input  signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' It can be observed that the “Modulate” method out-  performs the traditional “Add” and “Concat” methods both in  quantitative and qualitative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In addition, the “Add”  and “Concat” methods both experienced crashes (sudden drop  in accuracy) during training, while the “Modulate” method did  not, indicating that the gradient propagation of this method is  indeed more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Whether the low-pass filter (Section III-B) is used or not has  a greater impact on the prediction effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' It can be observed  from Table IV and Figure 12 that in the absence of a low-pass  filter to remove the aliasing signal (“NoFilter”), the long-term  prediction performance is worse and the characterization of  the human profile is indeed worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In addition, we remove  the interleaved upsampling method proposed in Section III-C  by padding with zeros, and revert to the traditional bilinear  upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' It can be observed that the gain of the pro-  posed method is not very large, and the traditional bilinear  upsampling method can also obtain good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' However,  the proposed method is more coherent in depicting the color  of the character without gradually fading, which is also in  line with our original intention of proposing such a method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Finally, we also remove the LPIPS loss and only use Euclidean  distance to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' It can be observed from the results  that although its pixel accuracy (SSIM and PSNR) is high, its  T+1  T+2  T+3  T+4  T+5#人人人人人  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Default  Add  Concat  NoFilter  人人人人  Bilinear  NoLPIPS  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Default  Add  Concat  NoFilter  Bilinear  NoLPIPSvisual performance is worse, and the LPIPS evaluation result  is also the worst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' This shows the huge role of LPIPS loss in  image sharpening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' LIMITATIONS AND FUTURE WORK  In this work, we modified several artifacts in the pyramidal  predictive network as well as the signal processing of the  overall architecture from a signal processing perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The  improved model is more interpretable and more powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In  addition, we also improved the training strategy to obtain  accurate and sharp predicted frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The experimental results  have shown that our method does achieve very good perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' Ablation study proves the rationality and validity of  our proposed improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content='  Nevertheless, the computational efficiency of the proposed  model still needs to be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The overall model is still  based on the LSTM recurrent neural network architecture,  which cannot be well parallelized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' The introduction of modules  such as low-pass filters also increases the difficulty for parallel  computing, resulting in a decrease in computing efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' In  the future, we will focus on solving the above problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
+page_content=' At  the same time, we will refer to the diffusion calculation and  transformer attention calculation methods to further improve  the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf'}
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+Entangled Phase of Simultaneous Fermion and Exciton Condensations Realized
+LeeAnn M. Sager and David A. Mazziotti∗
+Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, IL 60637
+(Dated: Submitted August 8, 2021; Revised December 30, 2021)
+Fermion-exciton condensates (FECs)—computationally- and theoretically-predicted states that
+simultaneously exhibit character of superconducting states and exciton condensates—are novel quan-
+tum states whose properties may involve a hybridization of superconductivity and the dissipationless
+flow of energy. Here, we exploit prior investigations of superconducting states and exciton conden-
+sates on quantum devices to identify a tuneable quantum state preparation entangling the wavefunc-
+tions of the individual condensate states. Utilizing this state preparation, we prepare a variety of
+fermion-exciton condensate states on quantum computers—realizing strongly-correlated FEC states
+on current, noisy intermediate-scale quantum devices—and verify the presence of the dual conden-
+sate via post-measurement analysis. This confirmation of the previously-predicted condensate state
+on quantum devices as well as the form of its wavefunction motivates further theoretical and exper-
+imental exploration of the properties, applications, and stability of fermion-exciton condensates.
+I.
+INTRODUCTION
+It may be possible to create materials that con-
+duct both electric current and exciton excitation en-
+ergy through the realization a single quantum state that
+simultaneously demonstrates properties of two differ-
+ent condensates—one composed of “Cooper” (particle-
+particle) pairs and the other composed of excitons
+(particle-hole pairs) [1]. Bose-Einstein condensation al-
+lows for multiple bosons aggregating in a single quantum
+state [2, 3] at sufficiently low temperatures, resulting in
+the superfluidity of the constituent bosons [4, 5]. A su-
+perconducting quantum phase is created upon the con-
+densation of pairs of fermions into a single quantum state,
+which results in the frictionless flow of the constituent
+particle-particle pairs [6, 7]. Significant theoretical and
+experimental investigation [6–28] has centered on super-
+conductors in an effort to determine a commercially-
+viable material supporting superconductivity at suffi-
+ciently high temperatures. However, the relatively low
+energy of the Cooper pairs [6, 7] cause them to become
+unstable, reverting to traditional conductors above a crit-
+ical temperature too low for commercial applications.
+One avenue towards higher-temperature condensate
+phases has been an examination of condensations com-
+posed of particle-hole pairs (excitons) in a single quantum
+state, which can carry exciton excitation energy without
+resistance [16, 29]. Excitons are more-tightly bound than
+Cooper pairs, meaning that the condensation of excitons
+can persist at higher temperatures than those at which
+superconductors form, although the natural recombina-
+tion of particles and holes is a cause of experimental diffi-
+culties in creating stable, high-temperature, ground-state
+exciton condensates. As such, much recent literature has
+explored the characteristics of exciton condensation as
+well as established various methodologies for overcom-
+ing the problem of annihilation upon recombination [15–
+19, 30–33]. Specifically, exciton condensates have been
+∗ damazz@uchicago.edu
+observed in optical traps with polaritons [34–37], the elec-
+tronic double layers of semiconductors [15, 30, 38–40] and
+graphene [19, 41, 42], and in systems composed of tran-
+sition metal chalcogenides [18, 43–47].
+FIG. 1: A figure of the condensate phase diagram in
+the phase space of the signatures of particle-particle
+condensation, λD, and exciton condensation, λG,—
+previously presented as Fig. 1 in Ref. 1—is shown.
+Here, we present an entangled quantum phase of mat-
+ter in which a superconductor and an exciton conden-
+sate coexist in a single quantum state—a fermion-exciton
+condensate (FEC). We leverage the ability of quantum
+computation to explore strongly correlated phenomena
+[48–53] as well as prior investigation [28, 33] to prepare
+a variety of fermion-exciton condensate states on quan-
+tum devices via a tuneable quantum state preparation—
+verifying the presence of the condensate state through
+probing the signatures of particle-particle (λD) [54, 55]
+and exciton (λG) [17, 56] condensations. Our results not
+only confirm the existence of a new class of condensates,
+they verify the theoretical prediction of the form of the
+wavefunction of the FEC as well as the phase diagram of
+the states (see Fig. 1). These results suggest that such
+condensates can potentially be prepared in physical sys-
+tems such as twisted graphene bilayers in which forces
+favoring exciton condensation and superconducting, re-
+spectively, are in fierce competition.
+arXiv:2301.01626v1  [quant-ph]  29 Dec 2022
+
+Superconducting
+Coexistence
+Condensate
+1
+入D
+Exciton
+No Condensation
+Condensation
+Phenomenon
+1 入G2
+Note that while both particle-particle and exciton con-
+densates are known to exist in systems designed to use
+exciton condensates to mediate the creation of Cooper
+pairs at higher temperatures [57, 58], this coexistence of
+fermion pair and excitonic condensation occurs in two
+adjacent systems that interact with one another [58] in-
+stead of existing in a joint FEC state.
+II.
+THEORY
+A.
+Signatures of Condensation
+Condensation occurs when multiple bosons aggregate
+into a single quantum state [2, 3] at temperatures below
+a certain critical temperature, resulting in the superflu-
+idity of the constituent bosons [4, 5]. In a condensation
+of particle-particle pairs, pairs of fermions condense into
+a single geminal, a two-fermion function directly analo-
+gous to a one-fermion orbital [54, 55, 59–62], resulting in
+the frictionless flow of the particle-particle pairs [6, 7].
+As established by Yang [54] and Sasaki [55], a computa-
+tional signature of such superconducting states—denoted
+as λD—is a large eigenvalue (λD > 1) of the particle-
+particle reduced density matrix (2-RDM) with elements
+given by
+2Di,j
+k,l = ⟨Ψ|ˆa†
+iˆa†
+jˆalˆak|Ψ⟩
+(1)
+where |Ψ⟩ is an N-fermion wavefunction and where ˆa†
+i
+and ˆai are fermionic creation and annihilation operators
+for orbital i, respectively. This signature directly probes
+the presence and extent of non-classical (off-diagonal)
+long-range order [60].
+Exciton condensation, similarly, results from the con-
+densation of particle-hole pairs (excitons) condensing
+into a single quantum state [16, 29]. A computational
+signature of exciton condensation—denoted as λG—is a
+large eigenvalue (λG > 1) of a modified version of the
+particle-hole reduced density matrix [17, 56, 63], with el-
+ements given by
+2 ˜Gi,j
+k,l = 2Gi,j
+k,l − 1Di
+j
+1Dl
+k
+= ⟨Ψ|ˆa†
+iˆajˆa†
+l ˆak|Ψ⟩ − ⟨Ψ|ˆa†
+iˆaj|Ψ⟩⟨Ψ|ˆa†
+l ˆak|Ψ⟩
+(2)
+where 1D is the one-fermion reduced density matrix (1-
+RDM). Note that this modification removes the extrane-
+ous large eigenvalue from a ground-state-to-ground-state
+transition.
+See the Methods section of the Supplemental Mate-
+rial [64] for specifics of how the signatures of supercon-
+ductivity (λD) and exciton condensation (λG) are ob-
+tained from the result data of a given quantum state
+preparation.
+B.
+Fermion-Exciton Condensate
+A fermion-exciton condensate is a quantum state in
+which character of both particle-particle condensation
+and exciton condensation coexist [1]; thus, a fermion-
+exciton condensate exhibits simultaneous large eigenval-
+ues of the particle-particle and modified particle-hole
+RDMs—i.e., λD, λG > 1.
+As we have previously the-
+oretically established in the thermodynamic limit [1], a
+fermion-exciton condensate should result from the entan-
+glement of a wavefunction exhibiting superconductivity,
+|ΨD⟩, with a wavefunction exhibiting exciton condensa-
+tion, |ΨG⟩), mathematically represented as
+|ΨF EC⟩ =
+1
+�
+2 − |∆|
+(|ΨD⟩ − sgn(∆)|ΨG⟩) ,
+(3)
+where ∆ = 2⟨ΨD|ΨG⟩ [1].
+From our previous work [1], we note that a fermion-
+exciton condensate state is accessible in systems as small
+as four fermions (N = 4) in eight orbitals (r = 8),
+and from our investigations of condensate behavior on
+quantum devices, wavefunctions demonstrating maximal
+particle-particle condensation [28] and maximal exciton
+condensation [33], individually, have been identified, pre-
+pared, and probed on quantum devices for N = 4, r = 8
+systems. Using the forms of these wavefunctions, we con-
+struct a state preparation that allows for the entangle-
+ment of the non-zero elements of the separate conden-
+sates, which is shown in Fig. 2. The input angles (θ1, θ2)
+are then optimized to generate a fermion-exciton conden-
+sate with character of each (i.e., a dual maximization of
+λD and λG).
+See the Methods section of the Supplemental Material
+for details of state preparations using both the bosonic
+representation of a qubit—in which each qubit is inter-
+preted as a two-fermion geminal—and the fermionic rep-
+resentation of a qubit—in which each qubit is interpreted
+as a one-fermion orbital—as well as the optimization pro-
+cedure for the input angles. Also note that—as in Ref.
+28—in our fermionic preparation, the pairing of qubits
+causes the usual difference between fermion and qubit
+statistics to disappear and hence allows for the direct rep-
+resentation of electron pairs on the quantum computer.
+III.
+RESULTS
+Using the bosonic state preparations with input angles
+that span the region exhibiting dual condensate charac-
+ter, we prepare a fermion-exciton condensate on a five-
+qubit quantum device [65]. As can be seen in Fig. 3a,
+with the blue x’s representing device data before any er-
+ror mitigation, various input angles yield quantum states
+with the signatures of both superconductor (λD) and ex-
+citon condensate (λG) character simultaneously exceed-
+ing the Pauli-like bound of one. Moreover, as statistical
+analysis was conducted with the average and standard
+
+3
+FIG. 2: A schematic demonstrating the fermionic quantum state preparation that yields an entanglement of the
+non-zero elements of the separate particle-particle and particle-hole condensates [28, 33], where Ry and Rz repre-
+sent rotations about the y and z axes of the Bloch sphere and where two-qubit gates are represented such that the
+control qubit is specified by a dot connected to the target qubit, which is specified by the appropriate gate. The
+wavefunction that results from this state preparation is given in the Supplemental Material. Note that the conden-
+sate character—and hence the signatures of condensation λG, λD—are varied by scanning over input angles (θ1, θ2).
+deviation determined from a sample size of ten trials per
+state preparation, these large eigenvalues are not spuri-
+ous; rather, they are statistically-significant within one
+standard deviation. Further, for several of these unmiti-
+gated, experimental fermion-exciton condensates, the sig-
+nature of condensation persists within two standard de-
+viations. (See the Supplemental Material for the stan-
+dard deviation ranges of the signatures of condensation.)
+Note that divergence from the maximal dual condensate
+character predicted for these input angles (i.e., the degree
+to which the reported data deviates from points along
+the elliptical fit) is likely due to preparation and readout
+errors on this noisy intermediate-scale quantum (NISQ)
+computer [66–69].
+(See the Supplemental Material for
+device error specifications.)
+As is shown in Fig. 3b, using the fermionic state prepa-
+ration on a noisier, fifteen-qubit quantum device [70] fails
+to realize a fermion-exciton condensate before error mit-
+igation. Further, as shown in Table I which presents the
+λD and λG values for a range of preparations simulated
+using four noise models that simulate errors consistent
+with real-world quantum device backends, this fermionic
+preparation would likely not yield fermion-exciton con-
+densates on even the newer and less-error-prone Mon-
+treal and Mumbai quantum computers. Likely, the four
+additional two-qubit, CNOT gates introduced into the
+fermionic state preparation—relative to the bosonic state
+preparation—introduce sufficient error to the quantum
+state such that the degree that condensate character is
+decreased or lost altogether. This is further evidenced by
+both simulated Montreal and Mumbai being capable of
+demonstrating dual condensate behavior indicative of a
+fermion-exciton condensate for the bosonic state prepara-
+tion and by simulated Melbourne demonstrating higher
+signatures of condensation for the bosonic preparation
+relative to the fermionic preparation.
+In order to use NISQ devices to better-model these
+fermion-exciton condensate phases, we introduce an er-
+ror mitigation scheme. As can be seen in the Methods
+section of the Supplemental Material, the state prepara-
+tions should yield quantum states with only six of the
+qubit basis states contributing to the overall wavefunc-
+tion. Any contribution from states other than these six
+basis states are unexpected and are directly caused by er-
+ror on the quantum devices (or simulators) employed. As
+such, we perform an error mitigation technique in which
+we project contributions from the qubit basis states that
+are not expected to contribute to zero and renormalize
+the resultant wavefunction. As can be seen in Fig. 3 and
+Table I, using this error mitigation technique improves re-
+sults from both the fermionic and bosonic preparations.
+Specifically—as can be readily observed from the green
+x’s representing the mitigated, projected results in Fig.
+3—, this projection technique leads to values approxi-
+mating the ideal dual existence of excitonic and fermionic
+behavior along the elliptical fit, allowing us to prepare
+and probe ideal fermion-exciton condensates despite sig-
+nificant amounts of error on the NISQ quantum devices.
+One interesting aside is that—for both the raw and
+projected data—the trade-off between character of a su-
+perconductor and that of an exciton condensate first
+noted in Ref. 1 is also observed here. This trade-off ap-
+pears to be elliptic in nature—consistent with the convex
+nature of the 2-RDMs when projected onto two dimen-
+sions [71–73]—even for the noisy, non-mitigated Santiago
+results, and nearly the exact elliptic fit established in Ref.
+1 is observed when the contributions from the compo-
+nents that should not contribute are projected to zero.
+
+Ry()
+q0
+q1
+Ry(01)
+q2
+Ry(-2
+Ry(02
+q3
+q4
+q5
+q6
+2b4
+Computer
+Quantum
+Full
+Projected
+Volume
+λG
+λD
+λG
+λD
+Fermionic Preparation
+Melbourne
+8
+0.804 0.994 1.191 1.352
+0.921 0.597 1.387 1.303
+0.934 0.581 1.440 1.271
+0.910 0.567 1.461 1.261
+Montreal
+128
+0.918 1.134 1.169 1.329
+1.193 0.868 1.472 1.255
+1.241 0.849 1.527 1.215
+1.267 0.824 1.580 1.176
+Mumbai
+128
+0.866 1.051 1.217 1.334
+0.928 0.697 1.360 1.319
+0.925 0.636 1.435 1.278
+1.113 0.721 1.537 1.212
+Bosonized Preparation
+Melbourne
+8
+0.875 1.256 1.130 1.377
+1.013 0.919 1.328 1.334
+1.077 0.906 1.409 1.289
+1.094 0.914 1.422 1.281
+Montreal
+128
+1.126 1.281 1.244 1.310
+1.306 1.107 1.454 1.265
+1.400 1.066 1.547 1.205
+1.406 1.041 1.567 1.188
+Mumbai
+128
+0.934 1.295 1.138 1.364
+1.160 1.022 1.408 1.293
+1.080 0.965 1.376 1.311
+1.233 0.990 1.497 1.238
+Santiago
+32
+1.169 1.278 1.264 1.304
+1.344 1.130 1.465 1.260
+1.390 1.100 1.517 1.225
+1.460 1.060 1.589 1.174
+TABLE I: Table of eigenvalues for the 2 ˜G (λG) and 2D
+(λD) matrices obtained from noise model simulating er-
+rors from real-world quantum computers both before
+(full) and after (projected) error mitigation via projec-
+tion of appropriate components to zero.
+(See Ref. 1 for additional details.) This trade-off is signif-
+icant as it precludes a fermion-exciton condensate with
+maximal condensate character of both particle-particle
+and exciton condensations. However, as the trade-off is
+elliptic in nature and as the maximal λD and λD values
+increase with system size (N), the lengths of the major
+and minor axes of the elliptical fit will increase as the
+size of the system is increased, causing the trade-off to
+become less and less stark.
+IV.
+CONCLUSIONS
+In
+this
+paper,
+we
+prepare
+a
+fermion-exciton
+condensate—a
+single
+quantum
+state
+demonstrating
+both superconductivity and exciton condensation—on a
+quantum device. This both realizes a highly-correlated
+state of matter on a noisy intermediate-scale quantum
+device and verifies the theoretical hypothesis from Ref.
+1 that such a state can be generated by entangling
+wavefunctions that separately exhibit particle-particle
+and exciton condensation. Further, the error mitigation
+technique introduced leads to signatures of fermion-pair
+(λD) and exciton condensation (λG) approaching the
+ideal dual existence of excitonic and fermionic behav-
+ior along the elliptical fit on NISQ devices, allowing
+for better-modelling of these highly-correlated dual
+condensate phases on even extremely-noisy devices.
+An experimental system that may result in such an
+entanglement may be a bilayer system—as bilayers are
+often known to exhibit exciton condensation—in which
+the geometric orientation of the layers such as the twist
+angles are optimized to generate competition between
+forces favoring an exciton condensate and a supercon-
+ductor. It may also be possible to consider bilayers in
+which each layer is composed of a traditional supercon-
+ductor, which can demonstrate particle-particle conden-
+sation. These systems should be studied both compu-
+tationally and experimentally as there are many open
+questions regarding the formation, properties, applica-
+tion, and stability of fermion-exciton condensates. The
+possibility of a hybridization of the properties of super-
+conductors with those of an exciton condensate definitely
+motivate further examination of this new state of matter.
+ACKNOWLEDGMENTS
+Acknowledgments:
+D.A.M. gratefully acknowledges
+the Department of Energy, Office of Basic Energy Sci-
+ences, Grant DE-SC0019215 and the U.S. National Sci-
+ence Foundation Grants No. CHE-2035876, No. DMR-
+2037783, No. CHE-1565638, and DGE-1746045.
+Data availability. Data will be made available upon
+reasonable request.
+Code availability. Code will be made available on a
+public Github repository upon publication.
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diff --git a/m9AzT4oBgHgl3EQfqP2M/content/tmp_files/load_file.txt b/m9AzT4oBgHgl3EQfqP2M/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
+++ b/m9AzT4oBgHgl3EQfqP2M/content/tmp_files/load_file.txt
@@ -0,0 +1,894 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf,len=893
+page_content='Entangled Phase of Simultaneous Fermion and Exciton Condensations Realized  LeeAnn M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Sager and David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Mazziotti∗  Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, IL 60637  (Dated: Submitted August 8, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Revised December 30, 2021)  Fermion-exciton condensates (FECs)—computationally- and theoretically-predicted states that  simultaneously exhibit character of superconducting states and exciton condensates—are novel quan-  tum states whose properties may involve a hybridization of superconductivity and the dissipationless  flow of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Here, we exploit prior investigations of superconducting states and exciton conden-  sates on quantum devices to identify a tuneable quantum state preparation entangling the wavefunc-  tions of the individual condensate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Utilizing this state preparation, we prepare a variety of  fermion-exciton condensate states on quantum computers—realizing strongly-correlated FEC states  on current, noisy intermediate-scale quantum devices—and verify the presence of the dual conden-  sate via post-measurement analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' This confirmation of the previously-predicted condensate state  on quantum devices as well as the form of its wavefunction motivates further theoretical and exper-  imental exploration of the properties, applications, and stability of fermion-exciton condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  INTRODUCTION  It may be possible to create materials that con-  duct both electric current and exciton excitation en-  ergy through the realization a single quantum state that  simultaneously demonstrates properties of two differ-  ent condensates—one composed of “Cooper” (particle-  particle) pairs and the other composed of excitons  (particle-hole pairs) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Bose-Einstein condensation al-  lows for multiple bosons aggregating in a single quantum  state [2, 3] at sufficiently low temperatures, resulting in  the superfluidity of the constituent bosons [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' A su-  perconducting quantum phase is created upon the con-  densation of pairs of fermions into a single quantum state,  which results in the frictionless flow of the constituent  particle-particle pairs [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Significant theoretical and  experimental investigation [6–28] has centered on super-  conductors in an effort to determine a commercially-  viable material supporting superconductivity at suffi-  ciently high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' However, the relatively low  energy of the Cooper pairs [6, 7] cause them to become  unstable, reverting to traditional conductors above a crit-  ical temperature too low for commercial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  One avenue towards higher-temperature condensate  phases has been an examination of condensations com-  posed of particle-hole pairs (excitons) in a single quantum  state, which can carry exciton excitation energy without  resistance [16, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Excitons are more-tightly bound than  Cooper pairs, meaning that the condensation of excitons  can persist at higher temperatures than those at which  superconductors form, although the natural recombina-  tion of particles and holes is a cause of experimental diffi-  culties in creating stable, high-temperature, ground-state  exciton condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' As such, much recent literature has  explored the characteristics of exciton condensation as  well as established various methodologies for overcom-  ing the problem of annihilation upon recombination [15–  19, 30–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Specifically, exciton condensates have been  ∗ damazz@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='edu  observed in optical traps with polaritons [34–37], the elec-  tronic double layers of semiconductors [15, 30, 38–40] and  graphene [19, 41, 42], and in systems composed of tran-  sition metal chalcogenides [18, 43–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' 1: A figure of the condensate phase diagram in  the phase space of the signatures of particle-particle  condensation, λD, and exciton condensation, λG,—  previously presented as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' 1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' 1—is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  Here, we present an entangled quantum phase of mat-  ter in which a superconductor and an exciton conden-  sate coexist in a single quantum state—a fermion-exciton  condensate (FEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' We leverage the ability of quantum  computation to explore strongly correlated phenomena  [48–53] as well as prior investigation [28, 33] to prepare  a variety of fermion-exciton condensate states on quan-  tum devices via a tuneable quantum state preparation—  verifying the presence of the condensate state through  probing the signatures of particle-particle (λD) [54, 55]  and exciton (λG) [17, 56] condensations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Our results not  only confirm the existence of a new class of condensates,  they verify the theoretical prediction of the form of the  wavefunction of the FEC as well as the phase diagram of  the states (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' These results suggest that such  condensates can potentially be prepared in physical sys-  tems such as twisted graphene bilayers in which forces  favoring exciton condensation and superconducting, re-  spectively, are in fierce competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='01626v1  [quant-ph]  29 Dec 2022  Superconducting  Coexistence  Condensate  1  入D  Exciton  No Condensation  Condensation  Phenomenon  1 入G2  Note that while both particle-particle and exciton con-  densates are known to exist in systems designed to use  exciton condensates to mediate the creation of Cooper  pairs at higher temperatures [57,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' 58],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' this coexistence of  fermion pair and excitonic condensation occurs in two  adjacent systems that interact with one another [58] in-  stead of existing in a joint FEC state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  THEORY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  Signatures of Condensation  Condensation occurs when multiple bosons aggregate  into a single quantum state [2, 3] at temperatures below  a certain critical temperature, resulting in the superflu-  idity of the constituent bosons [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' In a condensation  of particle-particle pairs, pairs of fermions condense into  a single geminal, a two-fermion function directly analo-  gous to a one-fermion orbital [54, 55, 59–62], resulting in  the frictionless flow of the particle-particle pairs [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  As established by Yang [54] and Sasaki [55], a computa-  tional signature of such superconducting states—denoted  as λD—is a large eigenvalue (λD > 1) of the particle-  particle reduced density matrix (2-RDM) with elements  given by  2Di,j  k,l = ⟨Ψ|ˆa†  iˆa†  jˆalˆak|Ψ⟩  (1)  where |Ψ⟩ is an N-fermion wavefunction and where ˆa†  i  and ˆai are fermionic creation and annihilation operators  for orbital i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' This signature directly probes  the presence and extent of non-classical (off-diagonal)  long-range order [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  Exciton condensation, similarly, results from the con-  densation of particle-hole pairs (excitons) condensing  into a single quantum state [16, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' A computational  signature of exciton condensation—denoted as λG—is a  large eigenvalue (λG > 1) of a modified version of the  particle-hole reduced density matrix [17, 56, 63], with el-  ements given by  2 ˜Gi,j  k,l = 2Gi,j  k,l − 1Di  j  1Dl  k  = ⟨Ψ|ˆa†  iˆajˆa†  l ˆak|Ψ⟩ − ⟨Ψ|ˆa†  iˆaj|Ψ⟩⟨Ψ|ˆa†  l ˆak|Ψ⟩  (2)  where 1D is the one-fermion reduced density matrix (1-  RDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Note that this modification removes the extrane-  ous large eigenvalue from a ground-state-to-ground-state  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  See the Methods section of the Supplemental Mate-  rial [64] for specifics of how the signatures of supercon-  ductivity (λD) and exciton condensation (λG) are ob-  tained from the result data of a given quantum state  preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  Fermion-Exciton Condensate  A fermion-exciton condensate is a quantum state in  which character of both particle-particle condensation  and exciton condensation coexist [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' thus, a fermion-  exciton condensate exhibits simultaneous large eigenval-  ues of the particle-particle and modified particle-hole  RDMs—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=', λD, λG > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  As we have previously the-  oretically established in the thermodynamic limit [1], a  fermion-exciton condensate should result from the entan-  glement of a wavefunction exhibiting superconductivity,  |ΨD⟩, with a wavefunction exhibiting exciton condensa-  tion, |ΨG⟩), mathematically represented as  |ΨF EC⟩ =  1  �  2 − |∆|  (|ΨD⟩ − sgn(∆)|ΨG⟩) ,  (3)  where ∆ = 2⟨ΨD|ΨG⟩ [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  From our previous work [1], we note that a fermion-  exciton condensate state is accessible in systems as small  as four fermions (N = 4) in eight orbitals (r = 8),  and from our investigations of condensate behavior on  quantum devices, wavefunctions demonstrating maximal  particle-particle condensation [28] and maximal exciton  condensation [33], individually, have been identified, pre-  pared, and probed on quantum devices for N = 4, r = 8  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Using the forms of these wavefunctions, we con-  struct a state preparation that allows for the entangle-  ment of the non-zero elements of the separate conden-  sates, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' The input angles (θ1, θ2)  are then optimized to generate a fermion-exciton conden-  sate with character of each (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=', a dual maximization of  λD and λG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  See the Methods section of the Supplemental Material  for details of state preparations using both the bosonic  representation of a qubit—in which each qubit is inter-  preted as a two-fermion geminal—and the fermionic rep-  resentation of a qubit—in which each qubit is interpreted  as a one-fermion orbital—as well as the optimization pro-  cedure for the input angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Also note that—as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  28—in our fermionic preparation, the pairing of qubits  causes the usual difference between fermion and qubit  statistics to disappear and hence allows for the direct rep-  resentation of electron pairs on the quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  RESULTS  Using the bosonic state preparations with input angles  that span the region exhibiting dual condensate charac-  ter, we prepare a fermion-exciton condensate on a five-  qubit quantum device [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' 3a,  with the blue x’s representing device data before any er-  ror mitigation, various input angles yield quantum states  with the signatures of both superconductor (λD) and ex-  citon condensate (λG) character simultaneously exceed-  ing the Pauli-like bound of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Moreover, as statistical  analysis was conducted with the average and standard  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' 2: A schematic demonstrating the fermionic quantum state preparation that yields an entanglement of the  non-zero elements of the separate particle-particle and particle-hole condensates [28, 33], where Ry and Rz repre-  sent rotations about the y and z axes of the Bloch sphere and where two-qubit gates are represented such that the  control qubit is specified by a dot connected to the target qubit, which is specified by the appropriate gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' The  wavefunction that results from this state preparation is given in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Note that the conden-  sate character—and hence the signatures of condensation λG, λD—are varied by scanning over input angles (θ1, θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  deviation determined from a sample size of ten trials per  state preparation, these large eigenvalues are not spuri-  ous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' rather, they are statistically-significant within one  standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Further, for several of these unmiti-  gated, experimental fermion-exciton condensates, the sig-  nature of condensation persists within two standard de-  viations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' (See the Supplemental Material for the stan-  dard deviation ranges of the signatures of condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=')  Note that divergence from the maximal dual condensate  character predicted for these input angles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=', the degree  to which the reported data deviates from points along  the elliptical fit) is likely due to preparation and readout  errors on this noisy intermediate-scale quantum (NISQ)  computer [66–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  (See the Supplemental Material for  device error specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=')  As is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' 3b, using the fermionic state prepa-  ration on a noisier, fifteen-qubit quantum device [70] fails  to realize a fermion-exciton condensate before error mit-  igation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Further, as shown in Table I which presents the  λD and λG values for a range of preparations simulated  using four noise models that simulate errors consistent  with real-world quantum device backends, this fermionic  preparation would likely not yield fermion-exciton con-  densates on even the newer and less-error-prone Mon-  treal and Mumbai quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Likely, the four  additional two-qubit, CNOT gates introduced into the  fermionic state preparation—relative to the bosonic state  preparation—introduce sufficient error to the quantum  state such that the degree that condensate character is  decreased or lost altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' This is further evidenced by  both simulated Montreal and Mumbai being capable of  demonstrating dual condensate behavior indicative of a  fermion-exciton condensate for the bosonic state prepara-  tion and by simulated Melbourne demonstrating higher  signatures of condensation for the bosonic preparation  relative to the fermionic preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  In order to use NISQ devices to better-model these  fermion-exciton condensate phases, we introduce an er-  ror mitigation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' As can be seen in the Methods  section of the Supplemental Material, the state prepara-  tions should yield quantum states with only six of the  qubit basis states contributing to the overall wavefunc-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Any contribution from states other than these six  basis states are unexpected and are directly caused by er-  ror on the quantum devices (or simulators) employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' As  such, we perform an error mitigation technique in which  we project contributions from the qubit basis states that  are not expected to contribute to zero and renormalize  the resultant wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' 3 and  Table I, using this error mitigation technique improves re-  sults from both the fermionic and bosonic preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  Specifically—as can be readily observed from the green  x’s representing the mitigated, projected results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  3—, this projection technique leads to values approxi-  mating the ideal dual existence of excitonic and fermionic  behavior along the elliptical fit, allowing us to prepare  and probe ideal fermion-exciton condensates despite sig-  nificant amounts of error on the NISQ quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  One interesting aside is that—for both the raw and  projected data—the trade-off between character of a su-  perconductor and that of an exciton condensate first  noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' 1 is also observed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' This trade-off ap-  pears to be elliptic in nature—consistent with the convex  nature of the 2-RDMs when projected onto two dimen-  sions [71–73]—even for the noisy, non-mitigated Santiago  results, and nearly the exact elliptic fit established in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  1 is observed when the contributions from the compo-  nents that should not contribute are projected to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  Ry()  q0  q1  Ry(01)  q2  Ry(-2  Ry(02  q3  q4  q5  q6  2b4  Computer  Quantum  Full  Projected  Volume  λG  λD  λG  λD  Fermionic Preparation  Melbourne  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
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+page_content='174  TABLE I: Table of eigenvalues for the 2 ˜G (λG) and 2D  (λD) matrices obtained from noise model simulating er-  rors from real-world quantum computers both before  (full) and after (projected) error mitigation via projec-  tion of appropriate components to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  (See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' 1 for additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=') This trade-off is signif-  icant as it precludes a fermion-exciton condensate with  maximal condensate character of both particle-particle  and exciton condensations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' However, as the trade-off is  elliptic in nature and as the maximal λD and λD values  increase with system size (N), the lengths of the major  and minor axes of the elliptical fit will increase as the  size of the system is increased, causing the trade-off to  become less and less stark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  CONCLUSIONS  In  this  paper,  we  prepare  a  fermion-exciton  condensate—a  single  quantum  state  demonstrating  both superconductivity and exciton condensation—on a  quantum device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' This both realizes a highly-correlated  state of matter on a noisy intermediate-scale quantum  device and verifies the theoretical hypothesis from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  1 that such a state can be generated by entangling  wavefunctions that separately exhibit particle-particle  and exciton condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Further, the error mitigation  technique introduced leads to signatures of fermion-pair  (λD) and exciton condensation (λG) approaching the  ideal dual existence of excitonic and fermionic behav-  ior along the elliptical fit on NISQ devices, allowing  for better-modelling of these highly-correlated dual  condensate phases on even extremely-noisy devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  An experimental system that may result in such an  entanglement may be a bilayer system—as bilayers are  often known to exhibit exciton condensation—in which  the geometric orientation of the layers such as the twist  angles are optimized to generate competition between  forces favoring an exciton condensate and a supercon-  ductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' It may also be possible to consider bilayers in  which each layer is composed of a traditional supercon-  ductor, which can demonstrate particle-particle conden-  sation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' These systems should be studied both compu-  tationally and experimentally as there are many open  questions regarding the formation, properties, applica-  tion, and stability of fermion-exciton condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' The  possibility of a hybridization of the properties of super-  conductors with those of an exciton condensate definitely  motivate further examination of this new state of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  Acknowledgments:  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' gratefully acknowledges  the Department of Energy, Office of Basic Energy Sci-  ences, Grant DE-SC0019215 and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' National Sci-  ence Foundation Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' CHE-2035876, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' DMR-  2037783, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' CHE-1565638, and DGE-1746045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Data will be made available upon  reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  Code availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Code will be made available on a  public Github repository upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Sager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Safaei, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Mazziotti, Potential  coexistence of exciton and fermion-pair condensations,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' B 101, 081107 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Bose and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Einstein, Planck’s law and light quan-  tum hypothesis, Zeitscrift f¨ur Physik 26, 178 (1924).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' Einstein, Quantentheorie des einatomigen idealen  gases, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
+page_content=' , 261–267 (1924).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfqP2M/content/2301.01626v1.pdf'}
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+Highly uniform and efficient, broadband meta-
+beam-splitter/combiner 
+Saeed Hemayat, Liyi Hsu, Jeongho Ha, and Abdoulaye Ndao* 
+Department of Electrical and Computer Engineering & Photonics Center, Boston University, 
+8 Saint Mary’s Street, Boston, MA 02215, USA 
+*andao@bu.edu 
+Abstract: Subwavelength planar structured interfaces, also known as metasurfaces, are ultra-
+thin optical elements modulating the amplitude, phase, and polarization of incident light using 
+nanostructures called meta-atoms. The optical properties of such metasurfaces can be 
+controlled across wavelengths by selecting geometries and materials of the meta-atoms. Given 
+recent technological developments in optical device miniaturization, components for beam 
+splitting and beam combining are sought for use within these devices as two quintessential 
+components of every optical setup. However, realizing such devices using metasurfaces 
+typically leads to poor uniformity of diffraction orders and narrow-band operation. Using a 
+modified version of particle swarm optimization, we propose and numerically demonstrate a 
+broadband, reciprocal metasurface beam combiner/splitter with uniformity>97% and 
+diffraction efficiency>90% in the continuous band from λ=1525 nm to λ=1575 nm. The 
+proposed approach significantly extends the current state of the art of metasurfaces design in 
+terms of uniformity, bandwidth, and efficiency, and opens the door for devices requiring high 
+power or near-unit uniformity. 
+1. Introduction 
+Since the first demonstration of laser by Maiman [1], the demand for high-power lasers has 
+increased remarkably; such lasers are of great importance in industrial applications such as 
+material processing [2], observation and analysis of ultrafast electronic processes [3,4], X-Ray 
+lasers [5], surface hardening [6], quantum electrodynamics [7], and white light generation [8]. 
+Unfortunately, conventional single-channel high-power emitters suffer from various physical 
+limitations like nonlinear and thermo-optical effects and to mode instabilities [9–13]. An 
+alternate approach is using beam combiners [10,14–17] to combine multiple lasers in an array-
+like fashion to obtain a high-power beam, avoiding the difficulties above associated with single 
+emitters. 
+
+In their simplest form, beam combiners, can be considered beam splitters used in reverse 
+with the phase and amplitude of the beams adjusted [10]. This can also be explained from the 
+antenna’s perspective through the reciprocity theorem, as an antenna (and also an array of 
+antennas) transmits and receives with the same radiation pattern (antenna gain is the same 
+whether it operates in transmitter or receiver mode) [18]. As a result, if a beam splitter generates 
+highly uniform diffraction orders in the farfield, one can expect the same device to act as a 
+beam combiner with high diffraction efficiency if used with reverse-propagating fields. 
+Dammann gratings, which are binary phase holograms, have been used over decades as a 
+trustworthy candidate for generating uniform diffraction orders in the far-field [19,20] and have 
+been utilized to realize beam splitter/combiners, multi-imaging systems, laser beam shaping, 
+and laser parallel micro-processing [21–29]. Conventionally, Dammann gratings were 
+fabricated by controlling the etch depths to vary the phase, but this approach gives rise to 
+fabrication difficulties as it requires multiple lithographic procedures [30]. Therefore, an 
+alternative method is highly desirable for implementing Dammann gratings to achieve near-
+perfect uniformity and diffraction efficiencies in beam splitters/combiners. 
+As current technology rapidly advances toward the complete miniaturization of optical 
+components, a fully integrable beam splitter/combiner is highly demanding. Metasurfaces 
+emerge as an excellent platform to achieve the miniaturization of optical components due to 
+their localized phase control. In recent years both plasmonic and dielectric metasurfaces have 
+been used to achieve various optical functionalities such as metalenses [31–34], Bessel beam 
+generation and super-resolution focusing [35,36], power-limiters [37], large field of view 
+structured light projection [38], carpet cloaking [39,40], holograms [41], sensing [42–46], 
+anisotropic metasurfaces [47–50], and beam splitters and combiners [51–56]. However, 
+because of the phase discretization on the metasurfaces, which gives rise to unwanted high 
+order components (diffraction orders in the far-field), most existing metasurface-based beam 
+splitters/combiners suffer from low uniformity, particularly in the form of stronger zeroth-order 
+diffraction with respect to other diffraction orders, or low diffraction efficiency, which highly 
+affects the diffraction orders in the far-field [53–55,57–61]. On the other hand, it is challenging 
+to correct this phase discretization in a theoretical framework due to a large number of variables 
+(rotation of each meta-atoms) in geometrical metasurfaces. 
+Here we present a broadband, metasurface beam splitter/combiner with diffraction 
+efficiencies of 93% (2D), uniformity of 98.6% as a beam splitter, as well as diffraction 
+efficiencies 88.6% as a beam combiner at the central wavelength of λ=1550 nm. Our design is 
+based on a Dammann grating with the primary goal of obtaining highly uniform diffraction 
+orders in the far-field. In addition to the commonly adapted geometric phase, we further propose 
+
+and adapt a modified version of particle swarm optimization (PSO) as a global optimization 
+strategy to compensate for the phase discretization effects which in turn gives rise to undesired 
+higher diffraction orders. Our adaption jointly maximizes both diffraction efficiency and 
+uniformity of diffraction orders, and successfully finds an optimum solution in huge parameter 
+space (729 meta-atoms with rotation angles between 0 and 180 degrees). Numerical results 
+show a 16.5% and 15.5% increase in diffraction efficiency and uniformity of the 2D beam 
+splitter and a 12.2% increase in the diffraction efficiency of the 2D beam combiner following 
+the optimization stage. Our approach avoids the complex structures resulting from counterpart 
+methods such as inverse design because it is based on the optimization of rotations of meta-
+atoms, thus lowering the fabrication complexity. This generic method is highly scalable and 
+can be easily applied to achieve a higher number of uniform diffraction orders in the far-field. 
+Furthermore, the proposed device is broadband with uniformity>97% and diffraction 
+efficiency>90% (diffraction efficiency>87% for the beam combiner) over the continuous band 
+from λ=1525 nm to λ=1575 nm. The proposed design and optimization procedure paves the 
+way for miniaturized beam splitting and combining, making it a potential centerpiece of many 
+applications which require high uniformity, like, quantum-photonics, depth sensing and facial 
+recognition, or applications that require high power, avoiding the nonlinear effects and mode 
+instabilities in conventional high-power lasers. 
+2. Structure design and operation principles 
+As previously elucidated, a beam combiner can be designed by optimizing its beam-splitting 
+capabilities. Figure 1 shows an exemplary 3×3 beam splitting/combining application using our 
+proposed metasurface. As the incident beam goes toward the metasurface along the +z 
+direction, as shown in Fig. 1(a), the metasurface splits the beam into 3×3 diffraction orders. 
+Conversely, if nine laser beams are incident at the exact diffraction angles but in the reverse 
+direction, as depicted in Fig. 1(b), the metasurface will combine the nine beams into one high-
+power beam. 
+ 
+Fig. 1. Concept figure showing the operation principle of the reciprocal beam splitter/combiner. 
+(a) beam splitter; (b) beam combiner. 
+
+(a)
+(b)
+BeamSplitter
+BeamCombiner
+laser
+Side view
+Side viewGiven the binarized nature of Dammann gratings, as proposed in [19], the only free 
+parameter in the design of Dammann gratings are transition points where the phase changes 
+from 0 to π or vice-versa. Here, we aim to design a 1D (1×7) and a 2D (3×3) beam splitter, and 
+consequently, there exist three and one transition points (Nd=2N+1, where Nd is the total number 
+of desired diffraction orders, and N is the number of transition points) for the 1×7 and the 3×3 
+case, respectively. The diffraction orders in the far-field can be obtained by taking the Fourier 
+transform of the device’s phase profile. As a result, one can write the diffraction orders obtained 
+using Dammann grating as the following [19,55]: 
+ 
+0
+1
+1
+2
+( 1) (
+),
+N
+i
+i
+i
+i
+A
+
+ +
+=
+=
+−
+−
+
+ 
+(1) 
+ 
+1
+1
+1
+( 1) (sin2
+sin2
+).
+N
+i
+n
+i
+i
+i
+A
+n
+n
+n
+
+
+
++
+=
+=
+−
+−
+
+ 
+(2) 
+Here A0 and An are amplitudes of the zeroth-order and nth-order diffraction spots, 
+respectively, and ξi are the transition points. To achieve uniform 1×7 (N=3) diffraction orders 
+in the far-field, the three transition points along the x-axis in 0≤x≤1/2 range are ξ1 = 0.2405, ξ2 
+= 0.3655, and ξ3 = 0.4380 as shown in Fig. 2(a). To construct a 2D theoretical phase on the 
+metasurface capable of generating uniform 3×3 diffraction orders, first, we design the phase 
+vector on the metasurface to obtain uniform 1×3 diffraction orders in the far-field, where the 
+only transition point while 0≤x≤1/2 is ξ1= 0.18905. Consequently, the 2-dimensional theoretical 
+phase to achieve 3×3 diffraction orders can be obtained by multiplying the 1D phase vector (for 
+1×3 splitter) by its transpose vector, which will result in a phase profile shown in Fig. 2(b). 
+To achieve such a binarized phase profile with a metasurface, we use Pancharatnam-Berry 
+(P.B.) phase to design the proposed geometrical metasurface; the phase shifts between adjacent 
+meta-atoms are induced via rotation of meta-atoms [35,62,63]. None of the other geometrical 
+parameters (e.g., height, R1, R2, or period as shown in Fig. 2(c)) change throughout the entire 
+design procedure. This feature prohibits generating entirely random structures that are 
+challenging to fabricate, which occurs in most of the structures generated by inverse-design. 
+To perform the numerical simulation, we used FDTD module of ANSYS LUMERICAL for 
+both unit cell and full-wave simulations of the whole structure throughout the optimization 
+procedure. For optimization procedures, LUMERICAL was linked to MATLAB by introducing 
+MATLAB API to LUMERICAL, which enabled us to gain full control of LUMERICAL from 
+inside MATLAB. In each iteration, a set of rotation angles are generated for each of 729 meta-
+atoms, evaluating the cost function. 
+
+To find the optimum values of the fixed geometrical parameters for the unit cell, a 
+comprehensive sweep of different geometrical parameters is done using the Finite-difference 
+Time-domain (FDTD) method to achieve the maximum transmission and the maximum 
+conversion efficiency between the incident circularly polarized (C.P.) beam and the orthogonal 
+output C.P. beam. The metasurface is designed for the center wavelength of λ=1550 nm using 
+amorphous silicon (α-Si). Each unit cell comprises an α-Si elliptical cylinder on a SiO2 substrate 
+(Optical constants of silicon and SiO2 are extracted from Palik [64]). Since each unit cell is a 
+Pancharatnam-Berry Optical Element (PBOE), in the ideal case, PBOEs should act as a half-
+waveplate and transform the incident LCP/RCP beam to its orthogonal output 
+polarization [36,65,66]. The sweep of different geometrical parameters, the maximum 
+transmission and conversion efficiency is achieved for a unit cell with P=750 nm, R1=215 nm, 
+R2=140 nm, and H=850 nm. The unit cell of the metasurface and the corresponding electric 
+field is shown in Fig. 2(c) and Fig. 2(d), respectively. Intuitively a smaller period gives rise to 
+more accurate results, as the number of sampling points in the phase space is increased and 
+hence leads to a less discretized phase response; However, to mitigate the fabrication 
+complexities, the minimum distance between two adjacent meta-atoms must be chosen 
+carefully (typically more than 50 nm). The analytical equation for the transmission of a rotated 
+unit cell can be derived using T(θ)=R(-θ)T1R(θ), where R(θ) is the rotation matrix [63]: 
+ 
+( )
+0
+cos
+sin
+cos
+sin
+.
+0
+sin
+cos
+sin
+cos
+o
+e
+T
+T
+T
+
+
+
+
+
+
+
+
+
+−
+
+
+
+
+
+
+=
+
+
+
+
+
+
+−
+
+
+
+
+
+
+ 
+(3) 
+Here, To and Te represent the respective complex transmission coefficients when the 
+polarization of incoming light is aligned toward the principal axes of the meta-atom, and θ is 
+the rotation angle. Eventually, if the incident light is circularly polarized, the expression for the 
+transmitted electric field is given as [35,63]: 
+ 
+(2 )
+/
+1
+1
+.
+(
+)
+.
+(
+).
+2
+2
+2
+2
+t
+i
+o
+e
+o
+e
+x
+y
+x
+y
+L R
+T
+T
+T
+T
+E
+e
+ie
+e
+e
+ie
+
++
+−
+=
+
++
+
+ 
+(4) 
+where
+xe and
+ye are electric field components along x and y directions. The second term in 
+the equation above shows that one can induce a phase equal to two times the rotation of each 
+element by rotating each unit cell while using the C.P. light. Rotation of each meta-atoms can 
+be obtained by dividing the desired phase on the metasurface by two, i.e., θ=φ(x,y)/2 where φ 
+is the phase profile of the metasurface and θ is the rotation of each meta-atoms. The Dammann 
+
+metasurface beam splitter capable of generating 3×3 uniform diffraction orders in the far-field 
+is shown in Fig. 2(b). 
+ 
+Fig. 2. (a) Phase profile of a unit cell of 1×7 Dammann beam splitter spanned from 0 to +1/2, 1 
+and -1 on the vertical axis correspond to the relative phases of π and 0, (b) phase profile of a unit 
+cell of 3×3 Dammann beam splitter with white and black regions corresponding to π and 0 phase, 
+respectively, (c) a unit cell of the designed metasurface, (d) electric field intensity in one unit 
+cell at λ=1550 nm. 
+To quantitatively assess the performance of the device, two criteria have been selected: 
+diffraction efficiency, given by total power concentrated in the desired diffraction orders, and 
+uniformity U, defined as: 
+ 
+max( )
+min( )
+1
+,
+max( )
+min( )
+g
+g
+U
+g
+g
+−
+= −
++
+ 
+(5) 
+where vector g = [g1, g2, g3, g4, g5, g6, g7, g8, g9], in which each g1-9 represents the fraction of 
+power concentrated in (-1,-1), (-1,0), (-1,+1), (0,-1), (0,0), (0,+1), (+1,-1), (+1,0), and (+1,+1) 
+diffraction orders, respectively. As a result, uniformity can be obtained by finding the minimum 
+
+(a)
+(b)
+Y
+12
++1
++1+
+0-
+x
+3-1-
+-1
+5+1
+5+2 5+3 :
+1-2
+5-1 0 专+1
+1-2
+X
+2
+(c)
+(d)
+max
+R
+850
+H
+wu
+Si
+425
+N
+SiO2
+0
+min
+-375
+0
+375
+x (nm)and maximum of vector g and using Eq. (5). The total diffraction efficiency of the metasurface, 
+defined as the sum of all the power concentrated in the desired orders is defined as G. The 
+results obtained directly from theoretical formulations are presented in Fig. 3. The diffraction 
+efficiency for the beam combiner is defined as the fraction of total power concentrated in the 
+zeroth-order diffraction order. The diffraction efficiency (G) of the 1D beam splitter and beam 
+combiner is 78.5% (and 77.9% uniformity) and 81.2%, respectively. For the 2D designs, the 
+diffraction efficiencies for the beam splitter and beam combiner are respectively 77% (and 83% 
+uniformity) and 76.4%. The low values of uniformity and diffraction efficiency for the 
+theoretical metasurface design are attributed to discretization effects and sidelobes, which arise 
+due to the interference between the radiation patterns of each array element (meta-atoms). 
+Consequently, the results obtained from the theory of Dammann gratings are far from ideal if 
+realized using metasurfaces due to the phase discretization in the regions where there should 
+be no transition points, and hence no phase changes are allowed. 
+ 
+Fig. 3. Diffraction orders obtained from theoretical designs (following P.B. phase and before 
+optimization) at λ=1550 nm. (a) The 1D beam splitter with 77.9% uniformity and 78.5% 
+diffraction efficiency, (b) 1D beam combiner with 81.2% diffraction efficiency, (c) 2D beam 
+splitter uniformity=83% and 77% diffraction efficiency, and (d) 2D beam combiner with 76.4% 
+diffraction efficiency. 
+3. Optimization procedure 
+One of the most successful categories of global optimization is evolutionary algorithms [67], 
+such as genetic algorithms [68], PSO [69], ant colony [70], artificial bee colony [71], and many 
+more that have been used with great success in a wide variety of applications like image 
+
+1DBeamsplitter(P.B.)
+1DBeamcombiner(P.B.
+0.13
+(a)
+0.713
+(b)
+0.089
+0.113
+0.124
+0.130
+0.126
+0.116
+0.087
+0.004
+0.009
+0.033
+0.713
+0.035
+0.012
+0.004
+0.0
+-3
++3
+0.0
+-2
+-1
+0
++1
++2
+-3
+-2
+-1
+0
++1
++2
++3
+Diffractionorders
+Diffraction orders
+(c)
+2DBeamsplitter(P.B.)
+(d)
+2DBeamcombiner(P.B.)
+0.110
+0.708
+0.074
+0.078
+0.109
+0.003
+0.002
+0.003
++1
++1
+Diffractionorders
+Diffraction orders
+0.084
+0.080
+0.082
+0.002
+0.708
+0.004
+0
+0.055
+0
+0.354
+0.110
+0.098
+0.061
+0.007
+0.003
+0.003
+-1
+-1
+0.0
+0.0
+-1
+0
++1
+-1
+0
++1
+Diffraction orders
+Diffractionordersprocessing and enhancement [72–74], industry [75], and medical applications [76]. The 
+significant advantage of PSO over other evolutionary algorithms, such as genetic algorithm, 
+besides its ease of implementation and faster convergence speed, lies in the memory of the 
+particles and the information flow between them. Each particle (agent) in PSO has memory and 
+shares information with all other particles while obeying some general rules (self-organization). 
+However, as the size of the current problem is extremely large (729 meta-atoms, each accepting 
+a rotation value between 0 and 180), one must carefully modify the conventional PSO to prevent 
+premature convergence. PSO can be summarized in three steps: 1. slight movement of each 
+particle along the same direction as its previous velocity vector, 2. slight movement of each 
+particle toward its own best cost, and 3. slight movement of each particle toward the global best 
+of all particles. The result would be a vector summation of all the vectors obtained from the 
+three steps mentioned above. These steps can be mathematically written as the following: 
+ 
+1 1
+2 2
+(
+1)
+( )
+[
+( )
+( )]
+[ ( )
+( )],
+i
+i
+i
+i
+i
+i
+v t
+wv t
+rc p t
+x t
+r c g t
+x t
++
+=
++
+−
++
+−
+ 
+(6) 
+where, vi(t+1) is the new velocity vector of the ith particle, w is the inertia coefficient 
+(typically less than one and generally takes values from 0.4 to 0.9) which determines how much 
+the particle tends to hold its current movement direction, vi(t) is the current velocity vector of 
+the particle, r1 and r2 are random numbers, c1 and c2 are personal and global learning 
+coefficients, respectively. The first term on the right-hand side of Eq. (6) represents the 
+tendency of the particle to move along the same direction as its current velocity vector. The 
+second term determines the movement along the best personal experience of the present 
+particle, and the third term illustrates the movement along the global best experience of all 
+particles. The new position of the particle can be obtained as follows: 
+ 
+(
+1)
+( )
+(
+1),
+i
+i
+i
+x t
+x t
+v t
++
+=
++
++
+ 
+(7) 
+where xi(t+1) is the new position of the particle. To ensure the best balance between 
+exploration and exploitation capabilities of the algorithm, constriction coefficients are used for 
+the inertia, personal, and global learning coefficients [77]. However, as discussed above, due 
+to the large number of variables that can take value in an uncountable set, the optimization 
+algorithm is highly likely to get stuck in local minima. 
+To prevent this, we have modified the conventional PSO by adding a layer of conservative 
+local search. This is implemented by defining a saturation threshold for the number of function 
+evaluations. We define saturation as the cases in which the cost function does not change after 
+a certain number of function evaluations. First, the conventional PSO runs, and if it reaches the 
+
+saturation threshold, instead of terminating, random noises within a specific range are added to 
+both positions and velocities of the particles, and then again, the algorithm enters a new PSO 
+procedure, this time with the updated noisy positions and velocities. This noise distribution is 
+generated using Latin hypercube sampling (LHS) of a multivariate uniform distribution [78] to 
+ensure that the noise is distributed uniformly all over the local search space, where a cumulative 
+density function (CDF) is divided into non-overlapping regions. One value is randomly chosen 
+from each region, which inherently ensures that at least one value is sampled from every region 
+(as opposed to simple random sampling) [79].  
+The noise injection phase compensates for the fact that in conventional PSO, the velocities 
+gradually decrease in each run to help with the exploitation. However, this increases the 
+probability of premature convergence, i.e., if the initial population was not covering the whole 
+space and the particles were gathered in clusters, they would converge toward a local minimum. 
+Due to the degraded velocity vector, the particles will never escape that local minimum. As a 
+result, the algorithm shows no exploration capabilities. The range for these random noises must 
+be chosen carefully, as we know that the optimum solution lies in the neighboring theoretical 
+design. We started the optimization procedure to optimize the results obtained from the theory. 
+Consequently, there is no need to search regions far from the current global best achieved with 
+the conventional PSO occurring before noise injection. It is worth noting that the position and 
+velocity of particles lie in different spaces, and as a result, the noise and its range are different 
+for positions and velocities. After the noise injection phase, 1% of all particles are selected 
+randomly in each saturation threshold step. Their position will be mirrored spatially to help 
+with the local exploration (apart from the principal coordinate axis of the problem, new local 
+coordinate axes are defined in each saturation threshold step and their centers are set at the 
+center of each particle cluster (local minima) just before the noise injection phase). As a result 
+of increased local exploration and exploitation capabilities of the algorithm, they act 
+synergistically to get out of a local minimum. To maximize the sum of desired diffraction orders 
+and minimize the difference between diffraction orders, the cost function of the problem is 
+defined below: 
+ 
+2
+2
+2
+2
+1
+1
+1
+1
+,
+(
+)
+n
+n
+n
+i
+j
+k
+i
+j
+k
+i
+j
+g
+g
+Cost Function
+g
+g
+g
+=
+=
+=
+
+
+−
+=
++ 
+
++
+
+
+
+
+
+
+ 
+(8) 
+where, gi,j,k are diffraction orders. The pseudocode of the modified PSO is presented below: 
+ 
+
+ALGORITHM 1. PSEUDOCODE FOR THE PROPOSED MODIFIED PSO 
+ 
+Begin procedure; 
+ 
+ 
+Pseudorandom generation of particles; 
+ 
+ 
+for (particle i∈N): 
+ 
+ 
+ 
+Calculate Cost_function(i); 
+ 
+ 
+ 
+Initialize inertia_coeff, constriction_coeffs, rand_vel_noise, rand_pos_noise, and 
+random_noise_coeff; 
+ 
+ 
+ 
+for each particle: 
+ 
+ 
+ 
+ 
+pBest(i) ← particle(i).BestPosition; 
+ 
+ 
+ 
+ 
+if (Cost_function(i)<pBest): 
+ 
+ 
+ 
+ 
+ 
+pBest(i) ← Cost_function(i); 
+ 
+ 
+ 
+ 
+end 
+ 
+ 
+ 
+end 
+ 
+ 
+ 
+gBest ← best cost of all particles; 
+ 
+ 
+ 
+for each particle: 
+ 
+ 
+ 
+ 
+Calculate particle(i).velocity using Eq. (6); 
+ 
+ 
+ 
+ 
+Update the particle(i).position using Eq. (7); 
+ 
+ 
+ 
+end 
+ 
+ 
+ 
+Update inertia_coeff; 
+ 
+ 
+ 
+if (saturation_threshold=true): /*Saturation threshold layer*/ 
+ 
+ 
+ 
+ 
+for (particle j∈N) 
+ 
+ 
+ 
+ 
+ 
+rand_pos_noise ← min(particle.pos(j)) + [(randperm(num_particles)'-… 
+ 
+ 
+ 
+ 
+ 
+rand(:,j))/(num_particles.* (max(particle.pos(j))-min(particle.pos(j)))]; 
+ 
+ 
+ 
+ 
+ 
+/*rand() generates random dist*/ 
+ 
+ 
+ 
+ 
+ 
+/*randperm() generates random perm. of integers*/ 
+ 
+ 
+ 
+ 
+ 
+rand_vel_noise ← min(particle.vel(j)) + [(randperm(num_particles)'-… 
+ 
+ 
+ 
+ 
+ 
+rand(:,j))/(num_particles.* (max(particle.vel(j))-min(particle.vel(j)))]; 
+ 
+ 
+ 
+ 
+end 
+ 
+ 
+ 
+ 
+rand_pos_noise ← rand_pos_noise* rand_noise_coeff; 
+ 
+ 
+ 
+ 
+rand_vel_noise ← rand_vel_noise* rand_noise_coeff; 
+ 
+ 
+ 
+ 
+add rand_pos_noise to particle(i).pos; 
+ 
+ 
+ 
+ 
+add rand_vel_noise to particle(i).vel; 
+ 
+ 
+ 
+ 
+local_center ←Find average x and y coordinates of particle clusters; 
+ 
+ 
+ 
+ 
+selected_mirror_particles ←Select 1% of each cluster; 
+ 
+ 
+ 
+ 
+for each selected_mirror_particles 
+ 
+ 
+ 
+ 
+ 
+selected_mirror_particles.x← -(selected_mirror_particles.x); 
+ 
+ 
+ 
+ 
+ 
+selected_mirror_particles.y← -(selected_mirror_particles.y); 
+ 
+ 
+ 
+ 
+end 
+ 
+ 
+ 
+ 
+Update rand_noise_coeff; 
+ 
+ 
+ 
+ 
+Run PSO’s main loop; /*with updated positions and velocities*/ 
+ 
+ 
+ 
+end 
+ 
+ 
+ 
+Check the termination criteria; /*if gBest is less than termination threshold*/ 
+ 
+ 
+end 
+ 
+END 
+gBest in the above pseudocode is a global property, and no index has been used to define it 
+since it is not assigned to a particular particle. The normalized cost function for the 1D and 2D 
+beam splitter/combiner is plotted with respect to the number of function evaluations. As shown 
+in Fig. 4(a), the algorithm starts to converge for the 1D design after approximately 4500 
+function evaluations, and the operation is terminated early (NFE<10000 for 1D case); as 
+uniformity reaches 100%. For the 2D case, however, the convergence is achieved after 
+approximately 14000 function evaluations, which is expected due to the larger problem size 
+and more significant local minima. In the 2D case, however, the algorithm faces multiple local 
+minima, and the saturation-threshold layer runs numerous times. 
+
+ 
+Fig. 4. Cost function values for 1D and 2D metasurface Dammann gratings, plotted with respect 
+to the number of function evaluations. (a) 1D and (b) 2D. 
+Since the current design is reciprocal, the optimization procedure must be done only once 
+for one of the cases (either for the beam splitter or the beam combiner). This is because the 
+device can be viewed as a 2D phased array of antenna elements (meta-atoms) arranged adjacent 
+to each other with different phases. Therefore, according to the reciprocity theorem, the antenna 
+gain is the same in transmission and receive modes [18]. 
+4. Optimized metasurface beam splitter/combiner 
+This section presents the results obtained for the optimized 1D and 2D beam splitter/combiner 
+(Fig. (5)) at λ=1550 nm. For the optimized design, one can observe the improvement in 
+diffraction efficiency and uniformity of the diffraction orders as compared to the designs 
+obtained directly from theory. The diffraction orders for the optimized 1D beam splitter are 
+perfectly uniform, and 100% uniformity is achieved (22.1% increase compared to the 
+unoptimized structure), as shown in Fig. 5(a). Also, 96.6% of the total diffracted power is 
+concentrated in these seven diffraction orders (diffraction efficiency=96.6%). In the case of the 
+1D beam combiner (shown in Fig. 5(b)), 92% diffraction efficiency is achieved, which is 10.8% 
+higher than the unoptimized design. For the 2D beam splitter, 98.5% uniformity and 93.5% 
+diffraction efficiency are reached (Fig. 5(c) and (e)), a 15.5% increase in uniformity and 16.5% 
+increase in diffraction efficiency with respect to the unoptimized design, while an 88.6% 
+diffraction efficiency corresponds to a 12.2% improvement over the unoptimized case for the 
+beam combiner (Fig. 5(d) and (f)). Both the theoretical and optimized results are summarized 
+in Table 1. It is noteworthy that the sum of all diffraction orders equals one. However, the sum 
+is less than one in Fig. (3) and Fig. (5), as in these figures only the desired orders are shown. 
+The orders in the extended region are not shown (i.e., if we extend the whole area in Fig. 3(c) 
+and (d) or Fig. 5(c-f) to ±3 diffraction orders, then very dim diffraction orders will appear to be 
+
+(a)
+1D
+(b)
+2D
+100
+100
+Normalized cost function
+Normalized costfunction
+2
+10
+10~6
+10-1
+0
+2000
+4000
+6000
+0
+5000
+10000
+15000
+20000
+NFE
+NFEvisible where the remaining power is concentrated in these orders. The same argument also 
+applies to the 1D cases). 
+ 
+Fig. 5. Diffraction orders at λ=1550 nm after optimization. (a) 1D beam splitter with 100% 
+uniformity and 96.6% diffraction efficiency, (b) 1D beam combiner with 92% diffraction 
+efficiency, (c) 2D beam splitter with 98.5% uniformity and 93.5% diffraction efficiency, (d) 2D 
+beam combiner with 88.6% diffraction efficiency. For better clarification of the concept, a 3D 
+representation of the diffraction orders for the 2D beam splitter and beam combiner are shown 
+in (e) and (f), respectively. 
+Table 1. Diffraction efficiency and uniformity of diffraction orders in the far-field for 1D and 2D beam 
+splitter/combiner. 
+Name 
+Diffraction efficiency 
+Uniformity 
+1D BS, Theory 
+78.5% 
+96.6% 
+81.2% 
+92% 
+77% 
+77.9% 
+100% 
+- 
+- 
+83% 
+1D BS, Optimized 
+1D BC, Theory 
+1D BC, Optimized 
+2D BS, Theory 
+2D BS, Optimized 
+93.5% 
+98.5% 
+2D BC, Theory 
+76.4% 
+- 
+2D BC, Optimized 
+88.6% 
+- 
+
+Beamsplitter(PSO)
+Beamcombiner(PSO)
+0.138
+0.92
+(a)
+(b)
+0.138
+0.138
+0.138
+0.138
+0.138
+0.138
+0.138
+0.92
+31
+6
++2
+0.00
+0.00
+-3
+-2
+-1
+0
++1
++3
+-3
+2
+0
++1
++2
++3
+Diffractionorders
+Diffractionorders
+(c)
+(d)
+0.106
+0.88
+0.106
+0.103
+0.104
++1
++1
+Diffraction orders
+Diffraction orders
+0.104
+0.103
+0.104
+0.88
+0
+0.053
+0
+0.44
+0.103
+0.103
+0.105
+-1
+-1
+0.0
+0.0
+-1
+0
++1
+-1
+0
++1
+Diffractionorders
+Diffractionorders
+(e)
+(f)
+0.106
+0.88
++2
++2
+Diffra
++1
++1
+ction
+0
+0
+0
+-1
+-1
++2
+2
+0
++1
++2
+-1
+-1
+0
+-2
+Diffraction orders
+-2
+DiffractionordersTo analyze the broadband characteristics of the optimized design, we define the operating 
+bandwidth as the spectral region where diffraction efficiency is greater than 90% and 
+uniformity is greater than 97%. The design is considered broadband only if it meets these two 
+criteria simultaneously. The proposed broadband design characteristics are illustrated in Fig. 
+(6). Even though the broadband requirements are chosen conservatively, the proposed design 
+satisfies both criteria over a 50 nm bandwidth. A consistent response has been achieved over 
+this bandwidth range, except for slight variations in the intensity of diffraction orders at each 
+wavelength and a slight variation in the angles of the diffraction orders. It is worth noting that 
+if we decrease the broadband limits to 85% and 92% for diffraction efficiency and uniformity, 
+a bandwidth of almost 100 nm is also achievable. This broadband behavior is attributed to the 
+metasurface's design based on dielectric waveguides. The broadband characteristics of the 
+proposed design are summarized in Table 2. 
+ 
+Fig. 6. Broadband characteristics of the proposed beam splitter/combiner. (a) Diffraction 
+efficiency and (b) uniformity of the diffraction orders with respect to changing the wavelength. 
+(c)-(e) diffraction orders at λ=1525 nm, λ=1550 nm, and λ=1575 nm, respectively, while 
+operating as a beam splitter, (f)-(h) diffraction orders at λ=1525 nm, λ=1550 nm, and λ=1575 
+nm, respectively, while operating as a beam combiner. 
+
+(a)
+(b)
+2100
+100
+90
+Uniformity (%)
+90
+80
+80
+BeamSplitter
+BeamCombiner
+70
+70
+60
+60
+50
+50
+1525
+1550
+1575
+1525
+1550
+1575
+Wavelength (nm)
+Wavelength (nm)
+(c)
+入=1525nm
+(d)
+入=1550nm
+(e)
+入=1575nm
+0.105
+0.106
+0.106
+0.099
+0.100
+0.102
+0.106
+0.103
+0.104
+0.106
+0.101
+0.101
++1
++1
++1
+Diffraction orders
+orders
+Diffraction orders
+0.101
+0.105
+0.100
+Diffraction
+0.104
+0.103
+0.104
+0.101
+0.101
+0.101
+0
+0
+0
+0
+0.100
+0.101
+0.099
+0.103
+0.103
+0.105
+0.101
+0.101
+0.101
+-1
+-1
+-1
+0
+0
+0
+0.89
+0.89
+0.89
+(f)
+(g)
+(h)
+Diffraction orders
++1
++1
+Diffraction orders
++1
+orders
+0.872
+Diffraction
+0.886
+0.881
+0
+0
+0
+-1
+-1
+-1
+0
+0
++1
+-1
+0
++1
+0
++1
+Diffraction orders
+Diffraction orders
+DiffractionordersTable 2. Broadband characteristics of the proposed 2D beam splitter/combiner 
+Name 
+λ=1525 nm 
+λ=1550 nm 
+λ=1575 nm 
+2D BS, Diffraction efficiency 
+90.4% 
+93.5% 
+91.4% 
+2D BS, Uniformity 
+97.1% 
+98.5% 
+97.5% 
+2D BC, Diffraction efficiency 
+87.2% 
+88.6% 
+88.1% 
+5. Conclusions 
+We proposed and numerically demonstrated a metasurface beam combiner/splitter generating 
+high diffraction efficiency (diffraction efficiency>90% for beam splitter and diffraction 
+efficiency>87% for beam combiner) and near unit uniformity (uniformity= 97%) in the 
+continuous wavelength range from 1525 nm to 1575 nm. Such performances are achieved using 
+a modified version of particle swarm optimization. To the best of our knowledge, this is the 
+highest uniformity, and highest diffraction efficiency reported to date. The proposed approach 
+significantly extends the current state of the art of metasurfaces design in terms of uniformity, 
+bandwidth, and efficiency, and opens the door for devices requiring high power or near unit 
+uniformity. 
+References 
+1.  T. H. Maiman, "Stimulated Optical Radiation in Ruby," Nature 187, 493–494 (1960). 
+2.  S. Nolte, G. Kamlage, F. Korte, T. Bauer, T. Wagner, A. Ostendorf, C. Fallnich, and H. Welling, 
+"Microstructuring with femtosecond lasers," ADV ENG MATER 2, 23–27 (2000). 
+3.  R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, 
+Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient 
+recorder," Nature 427, 817–821 (2004). 
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+
diff --git a/nNAzT4oBgHgl3EQfN_vv/content/tmp_files/load_file.txt b/nNAzT4oBgHgl3EQfN_vv/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf,len=1187
+page_content='Highly uniform and efficient, broadband meta- beam-splitter/combiner Saeed Hemayat, Liyi Hsu, Jeongho Ha, and Abdoulaye Ndao* Department of Electrical and Computer Engineering & Photonics Center, Boston University, 8 Saint Mary’s Street, Boston, MA 02215, USA *andao@bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='edu Abstract: Subwavelength planar structured interfaces, also known as metasurfaces, are ultra- thin optical elements modulating the amplitude, phase, and polarization of incident light using nanostructures called meta-atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The optical properties of such metasurfaces can be controlled across wavelengths by selecting geometries and materials of the meta-atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Given recent technological developments in optical device miniaturization, components for beam splitting and beam combining are sought for use within these devices as two quintessential components of every optical setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' However, realizing such devices using metasurfaces typically leads to poor uniformity of diffraction orders and narrow-band operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Using a modified version of particle swarm optimization, we propose and numerically demonstrate a broadband, reciprocal metasurface beam combiner/splitter with uniformity>97% and diffraction efficiency>90% in the continuous band from λ=1525 nm to λ=1575 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The proposed approach significantly extends the current state of the art of metasurfaces design in terms of uniformity, bandwidth, and efficiency, and opens the door for devices requiring high power or near-unit uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Introduction Since the first demonstration of laser by Maiman [1], the demand for high-power lasers has increased remarkably;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' such lasers are of great importance in industrial applications such as material processing [2], observation and analysis of ultrafast electronic processes [3,4], X-Ray lasers [5], surface hardening [6], quantum electrodynamics [7], and white light generation [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Unfortunately, conventional single-channel high-power emitters suffer from various physical limitations like nonlinear and thermo-optical effects and to mode instabilities [9–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' An alternate approach is using beam combiners [10,14–17] to combine multiple lasers in an array- like fashion to obtain a high-power beam, avoiding the difficulties above associated with single emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  In their simplest form, beam combiners, can be considered beam splitters used in reverse with the phase and amplitude of the beams adjusted [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' This can also be explained from the antenna’s perspective through the reciprocity theorem, as an antenna (and also an array of antennas) transmits and receives with the same radiation pattern (antenna gain is the same whether it operates in transmitter or receiver mode) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' As a result, if a beam splitter generates highly uniform diffraction orders in the farfield, one can expect the same device to act as a beam combiner with high diffraction efficiency if used with reverse-propagating fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Dammann gratings, which are binary phase holograms, have been used over decades as a trustworthy candidate for generating uniform diffraction orders in the far-field [19,20] and have been utilized to realize beam splitter/combiners, multi-imaging systems, laser beam shaping, and laser parallel micro-processing [21–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Conventionally, Dammann gratings were fabricated by controlling the etch depths to vary the phase, but this approach gives rise to fabrication difficulties as it requires multiple lithographic procedures [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Therefore, an alternative method is highly desirable for implementing Dammann gratings to achieve near- perfect uniformity and diffraction efficiencies in beam splitters/combiners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  As current technology rapidly advances toward the complete miniaturization of optical components, a fully integrable beam splitter/combiner is highly demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Metasurfaces emerge as an excellent platform to achieve the miniaturization of optical components due to their localized phase control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' In recent years both plasmonic and dielectric metasurfaces have been used to achieve various optical functionalities such as metalenses [31–34], Bessel beam generation and super-resolution focusing [35,36], power-limiters [37], large field of view structured light projection [38], carpet cloaking [39,40], holograms [41], sensing [42–46], anisotropic metasurfaces [47–50], and beam splitters and combiners [51–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' However, because of the phase discretization on the metasurfaces, which gives rise to unwanted high order components (diffraction orders in the far-field), most existing metasurface-based beam splitters/combiners suffer from low uniformity, particularly in the form of stronger zeroth-order diffraction with respect to other diffraction orders, or low diffraction efficiency, which highly affects the diffraction orders in the far-field [53–55,57–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' On the other hand, it is challenging to correct this phase discretization in a theoretical framework due to a large number of variables (rotation of each meta-atoms) in geometrical metasurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Here we present a broadband, metasurface beam splitter/combiner with diffraction efficiencies of 93% (2D), uniformity of 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='6% as a beam splitter, as well as diffraction efficiencies 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='6% as a beam combiner at the central wavelength of λ=1550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Our design is based on a Dammann grating with the primary goal of obtaining highly uniform diffraction orders in the far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' In addition to the commonly adapted geometric phase, we further propose  and adapt a modified version of particle swarm optimization (PSO) as a global optimization strategy to compensate for the phase discretization effects which in turn gives rise to undesired higher diffraction orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Our adaption jointly maximizes both diffraction efficiency and uniformity of diffraction orders, and successfully finds an optimum solution in huge parameter space (729 meta-atoms with rotation angles between 0 and 180 degrees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Numerical results show a 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% increase in diffraction efficiency and uniformity of the 2D beam splitter and a 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='2% increase in the diffraction efficiency of the 2D beam combiner following the optimization stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Our approach avoids the complex structures resulting from counterpart methods such as inverse design because it is based on the optimization of rotations of meta- atoms, thus lowering the fabrication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' This generic method is highly scalable and can be easily applied to achieve a higher number of uniform diffraction orders in the far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Furthermore, the proposed device is broadband with uniformity>97% and diffraction efficiency>90% (diffraction efficiency>87% for the beam combiner) over the continuous band from λ=1525 nm to λ=1575 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  The proposed design and optimization procedure paves the way for miniaturized beam splitting and combining, making it a potential centerpiece of many applications which require high uniformity, like, quantum-photonics, depth sensing and facial recognition, or applications that require high power, avoiding the nonlinear effects and mode instabilities in conventional high-power lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Structure design and operation principles As previously elucidated, a beam combiner can be designed by optimizing its beam-splitting capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Figure 1 shows an exemplary 3×3 beam splitting/combining application using our proposed metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' As the incident beam goes toward the metasurface along the +z direction, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 1(a), the metasurface splits the beam into 3×3 diffraction orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Conversely, if nine laser beams are incident at the exact diffraction angles but in the reverse direction, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 1(b), the metasurface will combine the nine beams into one high- power beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Concept figure showing the operation principle of the reciprocal beam splitter/combiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (a) beam splitter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (b) beam combiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  (a) (b) BeamSplitter BeamCombiner laser Side view Side viewGiven the binarized nature of Dammann gratings, as proposed in [19], the only free parameter in the design of Dammann gratings are transition points where the phase changes from 0 to π or vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Here, we aim to design a 1D (1×7) and a 2D (3×3) beam splitter, and consequently, there exist three and one transition points (Nd=2N+1, where Nd is the total number of desired diffraction orders, and N is the number of transition points) for the 1×7 and the 3×3 case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The diffraction orders in the far-field can be obtained by taking the Fourier transform of the device’s phase profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' As a result, one can write the diffraction orders obtained using Dammann grating as the following [19,55]:  0  1  1  2  ( 1) (  ),  N  i  i  i  i  A  \uf078  \uf078 +  =  =  −  −  \uf0e5  (1)  1  1  1  ( 1) (sin2  sin2  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  N  i  n  i  i  i  A  n  n  n  \uf070\uf078  \uf070\uf078  \uf070  +  =  =  −  −  \uf0e5  (2) Here A0 and An are amplitudes of the zeroth-order and nth-order diffraction spots, respectively, and ξi are the transition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' To achieve uniform 1×7 (N=3) diffraction orders in the far-field, the three transition points along the x-axis in 0≤x≤1/2 range are ξ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='2405, ξ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='3655, and ξ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='4380 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' To construct a 2D theoretical phase on the metasurface capable of generating uniform 3×3 diffraction orders, first, we design the phase vector on the metasurface to obtain uniform 1×3 diffraction orders in the far-field, where the only transition point while 0≤x≤1/2 is ξ1= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='18905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Consequently, the 2-dimensional theoretical phase to achieve 3×3 diffraction orders can be obtained by multiplying the 1D phase vector (for 1×3 splitter) by its transpose vector, which will result in a phase profile shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' To achieve such a binarized phase profile with a metasurface, we use Pancharatnam-Berry (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=') phase to design the proposed geometrical metasurface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' the phase shifts between adjacent meta-atoms are induced via rotation of meta-atoms [35,62,63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' None of the other geometrical parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=', height, R1, R2, or period as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 2(c)) change throughout the entire design procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' This feature prohibits generating entirely random structures that are challenging to fabricate, which occurs in most of the structures generated by inverse-design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  To perform the numerical simulation, we used FDTD module of ANSYS LUMERICAL for both unit cell and full-wave simulations of the whole structure throughout the optimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' For optimization procedures, LUMERICAL was linked to MATLAB by introducing MATLAB API to LUMERICAL, which enabled us to gain full control of LUMERICAL from inside MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' In each iteration, a set of rotation angles are generated for each of 729 meta- atoms, evaluating the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  To find the optimum values of the fixed geometrical parameters for the unit cell, a comprehensive sweep of different geometrical parameters is done using the Finite-difference Time-domain (FDTD) method to achieve the maximum transmission and the maximum conversion efficiency between the incident circularly polarized (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=') beam and the orthogonal output C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The metasurface is designed for the center wavelength of λ=1550 nm using amorphous silicon (α-Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Each unit cell comprises an α-Si elliptical cylinder on a SiO2 substrate (Optical constants of silicon and SiO2 are extracted from Palik [64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Since each unit cell is a Pancharatnam-Berry Optical Element (PBOE), in the ideal case, PBOEs should act as a half- waveplate and transform the incident LCP/RCP beam to its orthogonal output polarization [36,65,66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The sweep of different geometrical parameters, the maximum transmission and conversion efficiency is achieved for a unit cell with P=750 nm, R1=215 nm, R2=140 nm, and H=850 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The unit cell of the metasurface and the corresponding electric field is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 2(c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 2(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Intuitively a smaller period gives rise to more accurate results, as the number of sampling points in the phase space is increased and hence leads to a less discretized phase response;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' However, to mitigate the fabrication complexities, the minimum distance between two adjacent meta-atoms must be chosen carefully (typically more than 50 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  The analytical equation for the transmission of a rotated unit cell can be derived using T(θ)=R(-θ)T1R(θ), where R(θ) is the rotation matrix [63]:  ( )  0  cos  sin  cos  sin  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  0  sin  cos  sin  cos  o  e  T  T  T  \uf071  \uf071  \uf071  \uf071  \uf071  \uf071  \uf071  \uf071  \uf071  −  \uf0e6  \uf0f6  \uf0e6  \uf0f6  \uf0e6  \uf0f6  =  \uf0e7  \uf0f7  \uf0e7  \uf0f7  \uf0e7  \uf0f7  −  \uf0e8  \uf0f8  \uf0e8  \uf0f8  \uf0e8  \uf0f8  (3) Here, To and Te represent the respective complex transmission coefficients when the polarization of incoming light is aligned toward the principal axes of the meta-atom, and θ is the rotation angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Eventually, if the incident light is circularly polarized, the expression for the transmitted electric field is given as [35,63]:  (2 )  /  1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  (  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  (  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  2  2  2  2  t  i  o  e  o  e  x  y  x  y  L R  T  T  T  T  E  e  ie  e  e  ie  \uf071  +  −  =  \uf0b1  +  \uf0b1  (4) where xe and ye are electric field components along x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The second term in the equation above shows that one can induce a phase equal to two times the rotation of each element by rotating each unit cell while using the C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Rotation of each meta-atoms can be obtained by dividing the desired phase on the metasurface by two, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=', θ=φ(x,y)/2 where φ is the phase profile of the metasurface and θ is the rotation of each meta-atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The Dammann  metasurface beam splitter capable of generating 3×3 uniform diffraction orders in the far-field is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (a) Phase profile of a unit cell of 1×7 Dammann beam splitter spanned from 0 to +1/2, 1 and -1 on the vertical axis correspond to the relative phases of π and 0, (b) phase profile of a unit cell of 3×3 Dammann beam splitter with white and black regions corresponding to π and 0 phase, respectively, (c) a unit cell of the designed metasurface, (d) electric field intensity in one unit cell at λ=1550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' To quantitatively assess the performance of the device,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' two criteria have been selected: diffraction efficiency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' given by total power concentrated in the desired diffraction orders,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' and uniformity U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' defined as:  max( )  min( )  1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  max( )  min( )  g  g  U  g  g  −  = −  +  (5) where vector g = [g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' g5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' g6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' g7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' g8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' g9],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' in which each g1-9 represents the fraction of power concentrated in (-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='-1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+page_content='+1) diffraction orders,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' As a result, uniformity can be obtained by finding the minimum  (a) (b) Y 12 +1 +1+ 0- x 3-1- -1 5+1 5+2 5+3 : 1-2 5-1 0 专+1 1-2 X 2 (c) (d) max R 850 H wu Si 425 N SiO2 0 min -375 0 375 x (nm)and maximum of vector g and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The total diffraction efficiency of the metasurface, defined as the sum of all the power concentrated in the desired orders is defined as G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The results obtained directly from theoretical formulations are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The diffraction efficiency for the beam combiner is defined as the fraction of total power concentrated in the zeroth-order diffraction order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The diffraction efficiency (G) of the 1D beam splitter and beam combiner is 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% (and 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='9% uniformity) and 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='2%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' For the 2D designs, the diffraction efficiencies for the beam splitter and beam combiner are respectively 77% (and 83% uniformity) and 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The low values of uniformity and diffraction efficiency for the theoretical metasurface design are attributed to discretization effects and sidelobes, which arise due to the interference between the radiation patterns of each array element (meta-atoms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Consequently, the results obtained from the theory of Dammann gratings are far from ideal if realized using metasurfaces due to the phase discretization in the regions where there should be no transition points, and hence no phase changes are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Diffraction orders obtained from theoretical designs (following P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' phase and before optimization) at λ=1550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (a) The 1D beam splitter with 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='9% uniformity and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% diffraction efficiency, (b) 1D beam combiner with 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='2% diffraction efficiency, (c) 2D beam splitter uniformity=83% and 77% diffraction efficiency, and (d) 2D beam combiner with 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='4% diffraction efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Optimization procedure One of the most successful categories of global optimization is evolutionary algorithms [67], such as genetic algorithms [68], PSO [69], ant colony [70], artificial bee colony [71], and many more that have been used with great success in a wide variety of applications like image  1DBeamsplitter(P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+page_content='003 -1 -1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+page_content='0 -1 0 +1 -1 0 +1 Diffraction orders Diffractionordersprocessing and enhancement [72–74], industry [75], and medical applications [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The significant advantage of PSO over other evolutionary algorithms, such as genetic algorithm, besides its ease of implementation and faster convergence speed, lies in the memory of the particles and the information flow between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Each particle (agent) in PSO has memory and shares information with all other particles while obeying some general rules (self-organization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' However, as the size of the current problem is extremely large (729 meta-atoms, each accepting a rotation value between 0 and 180), one must carefully modify the conventional PSO to prevent premature convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' PSO can be summarized in three steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' slight movement of each particle along the same direction as its previous velocity vector, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' slight movement of each particle toward its own best cost, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' slight movement of each particle toward the global best of all particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  The result would be a vector summation of all the vectors obtained from the three steps mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' These steps can be mathematically written as the following:  1 1  2 2  (  1)  ( )  [  ( )  ( )]  [ ( )  ( )],  i  i  i  i  i  i  v t  wv t  rc p t  x t  r c g t  x t  +  =  +  −  +  −  (6) where, vi(t+1) is the new velocity vector of the ith particle, w is the inertia coefficient (typically less than one and generally takes values from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='4 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='9) which determines how much the particle tends to hold its current movement direction, vi(t) is the current velocity vector of the particle, r1 and r2 are random numbers, c1 and c2 are personal and global learning coefficients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The first term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (6) represents the tendency of the particle to move along the same direction as its current velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The second term determines the movement along the best personal experience of the present particle, and the third term illustrates the movement along the global best experience of all particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The new position of the particle can be obtained as follows:  (  1)  ( )  (  1),  i  i  i  x t  x t  v t  +  =  +  +  (7) where xi(t+1) is the new position of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' To ensure the best balance between exploration and exploitation capabilities of the algorithm, constriction coefficients are used for the inertia, personal, and global learning coefficients [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' However, as discussed above, due to the large number of variables that can take value in an uncountable set, the optimization algorithm is highly likely to get stuck in local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' To prevent this, we have modified the conventional PSO by adding a layer of conservative local search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' This is implemented by defining a saturation threshold for the number of function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' We define saturation as the cases in which the cost function does not change after a certain number of function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' First, the conventional PSO runs, and if it reaches the  saturation threshold, instead of terminating, random noises within a specific range are added to both positions and velocities of the particles, and then again, the algorithm enters a new PSO procedure, this time with the updated noisy positions and velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' This noise distribution is generated using Latin hypercube sampling (LHS) of a multivariate uniform distribution [78] to ensure that the noise is distributed uniformly all over the local search space, where a cumulative density function (CDF) is divided into non-overlapping regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' One value is randomly chosen from each region, which inherently ensures that at least one value is sampled from every region (as opposed to simple random sampling) [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The noise injection phase compensates for the fact that in conventional PSO, the velocities gradually decrease in each run to help with the exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' However, this increases the probability of premature convergence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=', if the initial population was not covering the whole space and the particles were gathered in clusters, they would converge toward a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Due to the degraded velocity vector, the particles will never escape that local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' As a result, the algorithm shows no exploration capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The range for these random noises must be chosen carefully, as we know that the optimum solution lies in the neighboring theoretical design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' We started the optimization procedure to optimize the results obtained from the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Consequently, there is no need to search regions far from the current global best achieved with the conventional PSO occurring before noise injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' It is worth noting that the position and velocity of particles lie in different spaces, and as a result, the noise and its range are different for positions and velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' After the noise injection phase, 1% of all particles are selected randomly in each saturation threshold step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Their position will be mirrored spatially to help with the local exploration (apart from the principal coordinate axis of the problem, new local coordinate axes are defined in each saturation threshold step and their centers are set at the center of each particle cluster (local minima) just before the noise injection phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' As a result of increased local exploration and exploitation capabilities of the algorithm, they act synergistically to get out of a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' To maximize the sum of desired diffraction orders and minimize the difference between diffraction orders, the cost function of the problem is defined below:  2  2  2  2  1  1  1  1  ,  (  )  n  n  n  i  j  k  i  j  k  i  j  g  g  Cost Function  g  g  g  =  =  =  \uf0e9  \uf0f9  −  =  + \uf0ea  \uf0fa  +  \uf0ea  \uf0fa  \uf0eb  \uf0fb  \uf0e5  \uf0e5\uf0e5  (8) where, gi,j,k are diffraction orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The pseudocode of the modified PSO is presented below:  ALGORITHM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' PSEUDOCODE FOR THE PROPOSED MODIFIED PSO  Begin procedure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Pseudorandom generation of particles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  for (particle i∈N):  Calculate Cost_function(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Initialize inertia_coeff, constriction_coeffs, rand_vel_noise, rand_pos_noise, and random_noise_coeff;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  for each particle:  pBest(i) ← particle(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='BestPosition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  if (Cost_function(i)<pBest):  pBest(i) ← Cost_function(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  end  end  gBest ← best cost of all particles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  for each particle:  Calculate particle(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='velocity using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Update the particle(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='position using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  end  Update inertia_coeff;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  if (saturation_threshold=true): / Saturation threshold layer /  for (particle j∈N)  rand_pos_noise ← min(particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content="pos(j)) + [(randperm(num_particles)'-…  rand(:,j))/(num_particles." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (max(particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='pos(j)) min(particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='pos(j)))];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  / rand() generates random dist /  /*randperm() generates random perm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' of integers*/  rand_vel_noise ← min(particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content="vel(j)) + [(randperm(num_particles)'-…  rand(:,j))/(num_particles." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (max(particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='vel(j)) min(particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='vel(j)))];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  end  rand_pos_noise ← rand_pos_noise rand_noise_coeff;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  rand_vel_noise ← rand_vel_noise rand_noise_coeff;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  add rand_pos_noise to particle(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='pos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  add rand_vel_noise to particle(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='vel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  local_center ←Find average x and y coordinates of particle clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  selected_mirror_particles ←Select 1% of each cluster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  for each selected_mirror_particles  selected_mirror_particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='x← (selected_mirror_particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  selected_mirror_particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='y← (selected_mirror_particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  end  Update rand_noise_coeff;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Run PSO’s main loop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' /*with updated positions and velocities*/  end  Check the termination criteria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' /*if gBest is less than termination threshold*/  end  END gBest in the above pseudocode is a global property, and no index has been used to define it since it is not assigned to a particular particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The normalized cost function for the 1D and 2D beam splitter/combiner is plotted with respect to the number of function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 4(a), the algorithm starts to converge for the 1D design after approximately 4500 function evaluations, and the operation is terminated early (NFE<10000 for 1D case);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' as uniformity reaches 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' For the 2D case, however, the convergence is achieved after approximately 14000 function evaluations, which is expected due to the larger problem size and more significant local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' In the 2D case, however, the algorithm faces multiple local minima, and the saturation-threshold layer runs numerous times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Cost function values for 1D and 2D metasurface Dammann gratings, plotted with respect to the number of function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (a) 1D and (b) 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Since the current design is reciprocal, the optimization procedure must be done only once for one of the cases (either for the beam splitter or the beam combiner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' This is because the device can be viewed as a 2D phased array of antenna elements (meta-atoms) arranged adjacent to each other with different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Therefore, according to the reciprocity theorem, the antenna gain is the same in transmission and receive modes [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Optimized metasurface beam splitter/combiner This section presents the results obtained for the optimized 1D and 2D beam splitter/combiner (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (5)) at λ=1550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' For the optimized design, one can observe the improvement in diffraction efficiency and uniformity of the diffraction orders as compared to the designs obtained directly from theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The diffraction orders for the optimized 1D beam splitter are perfectly uniform, and 100% uniformity is achieved (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='1% increase compared to the unoptimized structure), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Also, 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='6% of the total diffracted power is concentrated in these seven diffraction orders (diffraction efficiency=96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' In the case of the 1D beam combiner (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 5(b)), 92% diffraction efficiency is achieved, which is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='8% higher than the unoptimized design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' For the 2D beam splitter, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% uniformity and 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% diffraction efficiency are reached (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  5(c) and (e)), a 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% increase in uniformity and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% increase in diffraction efficiency with respect to the unoptimized design, while an 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='6% diffraction efficiency corresponds to a 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='2% improvement over the unoptimized case for the beam combiner (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 5(d) and (f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Both the theoretical and optimized results are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' It is noteworthy that the sum of all diffraction orders equals one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' However, the sum is less than one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (3) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (5), as in these figures only the desired orders are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The orders in the extended region are not shown (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=', if we extend the whole area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 3(c) and (d) or Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 5(c-f) to ±3 diffraction orders, then very dim diffraction orders will appear to be  (a) 1D (b) 2D 100 100 Normalized cost function Normalized costfunction 2 10 10~6 10-1 0 2000 4000 6000 0 5000 10000 15000 20000 NFE NFEvisible where the remaining power is concentrated in these orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The same argument also applies to the 1D cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Diffraction orders at λ=1550 nm after optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (a) 1D beam splitter with 100% uniformity and 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='6% diffraction efficiency, (b) 1D beam combiner with 92% diffraction efficiency, (c) 2D beam splitter with 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% uniformity and 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% diffraction efficiency, (d) 2D beam combiner with 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='6% diffraction efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' For better clarification of the concept, a 3D representation of the diffraction orders for the 2D beam splitter and beam combiner are shown in (e) and (f), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Diffraction efficiency and uniformity of diffraction orders in the far-field for 1D and 2D beam splitter/combiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Name Diffraction efficiency Uniformity 1D BS, Theory 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='6% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='2% 92% 77% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='9% 100% - - 83% 1D BS, Optimized 1D BC, Theory 1D BC, Optimized 2D BS, Theory 2D BS, Optimized 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% 2D BC, Theory 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='4% - 2D BC, Optimized 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='6% -  Beamsplitter(PSO) Beamcombiner(PSO) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='92 (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+page_content='92 31 6 +2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='00 -3 -2 -1 0 +1 +3 -3 2 0 +1 +2 +3 Diffractionorders Diffractionorders (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+page_content='104 +1 +1 Diffraction orders Diffraction orders 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+page_content='105 -1 -1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+page_content='0 -1 0 +1 -1 0 +1 Diffractionorders Diffractionorders (e) (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+page_content='88 +2 +2 Diffra +1 +1 ction 0 0 0 -1 -1 +2 2 0 +1 +2 -1 -1 0 -2 Diffraction orders -2 DiffractionordersTo analyze the broadband characteristics of the optimized design, we define the operating bandwidth as the spectral region where diffraction efficiency is greater than 90% and uniformity is greater than 97%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The design is considered broadband only if it meets these two criteria simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' The proposed broadband design characteristics are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Even though the broadband requirements are chosen conservatively, the proposed design satisfies both criteria over a 50 nm bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' A consistent response has been achieved over this bandwidth range, except for slight variations in the intensity of diffraction orders at each wavelength and a slight variation in the angles of the diffraction orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' It is worth noting that if we decrease the broadband limits to 85% and 92% for diffraction efficiency and uniformity, a bandwidth of almost 100 nm is also achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=" This broadband behavior is attributed to the metasurface's design based on dielectric waveguides." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  The broadband characteristics of the proposed design are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Broadband characteristics of the proposed beam splitter/combiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (a) Diffraction efficiency and (b) uniformity of the diffraction orders with respect to changing the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' (c)-(e) diffraction orders at λ=1525 nm, λ=1550 nm, and λ=1575 nm, respectively, while operating as a beam splitter, (f)-(h) diffraction orders at λ=1525 nm, λ=1550 nm, and λ=1575 nm, respectively, while operating as a beam combiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  (a) (b) 2100 100 90 Uniformity (%) 90 80 80 BeamSplitter BeamCombiner 70 70 60 60 50 50 1525 1550 1575 1525 1550 1575 Wavelength (nm) Wavelength (nm) (c) 入=1525nm (d) 入=1550nm (e) 入=1575nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+page_content='89 (f) (g) (h) Diffraction orders +1 +1 Diffraction orders +1 orders 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='872 Diffraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+page_content='881 0 0 0 -1 -1 -1 0 0 +1 -1 0 +1 0 +1 Diffraction orders Diffraction orders DiffractionordersTable 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Broadband characteristics of the proposed 2D beam splitter/combiner Name λ=1525 nm λ=1550 nm λ=1575 nm 2D BS, Diffraction efficiency 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='4% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='4% 2D BS, Uniformity 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='1% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='5% 2D BC, Diffraction efficiency 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='2% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='6% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='1% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Conclusions We proposed and numerically demonstrated a metasurface beam combiner/splitter generating high diffraction efficiency (diffraction efficiency>90% for beam splitter and diffraction efficiency>87% for beam combiner) and near unit uniformity (uniformity= 97%) in the continuous wavelength range from 1525 nm to 1575 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' Such performances are achieved using a modified version of particle swarm optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content=' To the best of our knowledge, this is the highest uniformity, and highest diffraction efficiency reported to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
+page_content='  The proposed approach significantly extends the current state of the art of metasurfaces design in terms of uniformity, bandwidth, and efficiency, and opens the door for devices requiring high power or near unit uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNAzT4oBgHgl3EQfN_vv/content/2301.01160v1.pdf'}
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+Five Common Misconceptions About
+Privacy-Preserving Internet of Things
+Mohammad Abu Alsheikh, Senior Member, IEEE
+Abstract—Billions of devices in the Internet of Things (IoT)
+collect sensitive data about people, creating data privacy risks and
+breach vulnerabilities. Accordingly, data privacy preservation is
+vital for sustaining the proliferation of IoT services. In particular,
+privacy-preserving IoT connects devices embedded with sensors
+and maintains the data privacy of people. However, common
+misconceptions exist among IoT researchers, service providers,
+and users about privacy-preserving IoT.
+This article refutes five common misconceptions about privacy-
+preserving IoT concerning data sensing and innovation, regula-
+tions, and privacy safeguards. For example, IoT users have a com-
+mon misconception that no data collection is permitted in data
+privacy regulations. On the other hand, IoT service providers
+often think data privacy impedes IoT sensing and innovation.
+Addressing these misconceptions is essential for making progress
+in privacy-preserving IoT. This article refutes such common
+misconceptions using real-world experiments and online survey
+research. First, the experiments indicate that data privacy should
+not be perceived as an impediment in IoT but as an opportunity to
+increase customer retention and trust. Second, privacy-preserving
+IoT is not exclusively a regulatory problem but also a functional
+necessity that must be incorporated in the early stages of any
+IoT design. Third, people do not trust services that lack sufficient
+privacy measures. Fourth, conventional data security principles
+do not guarantee data privacy protection, and data privacy can be
+exposed even if data is securely stored. Fifth, IoT decentralization
+does not attain absolute privacy preservation.
+Index Terms—Internet of things, data privacy.
+INTRODUCTION
+R
+ECENT years have witnessed a proliferation of Internet
+of Things (IoT) services in home automation, retail,
+telehealth, manufacturing, autonomous vehicles, and preci-
+sion agriculture. Cisco Systems estimates that 29.3 billion
+networked devices, 5.7 billion mobile subscribers with an
+average cellular speed of 43.9 Mbps, and 1.4 billion 5G-
+enabled devices will exist in 2023 [1]. This proliferation in
+ubiquitous computing and high-speed mobile networks enables
+service providers to collect valuable IoT datasets. In particular,
+IoT data is fundamental for creating new IoT services tailored
+to people’s needs.
+IoT data can contain sensitive data about people, creating
+genuine data privacy concerns for users. For example, more
+than 1,272 major data breaches happened in 2019, exposing
+163 million records (or 128,171 exposed records per data
+breach) [1]. Data breaches severely impact victims, including
+financial loss, psychological trauma, criminal impersonation,
+and online fraud. Accordingly, privacy-preserving IoT applies
+This work was supported by the Australian Research Council (ARC) under
+Grant DE200100863. The data collected in this project has been approved by
+the Human Research Ethics Committee at the University of Canberra under
+the application “4522 - Privacy Coupling: When Your Personal Devices Betray
+You”.
+strict measures to maintain people’s data privacy. Specifically,
+privacy-preserving IoT is a fully-operational IoT network that
+meets the relevant data privacy regulations, such as the General
+Data Protection Regulation (GDPR) [2] and the California
+Consumer Privacy Act (CCPA) [3].
+Nonetheless, privacy-preserving IoT has a few commonly
+misunderstood aspects, as IoT is a technology for pervasive
+sensing about people and their surroundings. Specifically,
+common misconceptions exist among IoT service providers
+and users about the possibility of maintaining data pri-
+vacy in pervasive IoT sensing. This article refutes com-
+mon misconceptions, which are organized into three cate-
+gories—misconceptions about IoT innovation, data regula-
+tions, and privacy safeguards. First, IoT is widely motivated as
+an efficient and cheap method for data collection about people
+and their surroundings. Thus, data privacy may be wrongly
+perceived as an impediment to IoT innovation and pervasive
+sensing. Second, privacy-preserving IoT is often recognized as
+an exclusive regulatory problem while ignoring its financial
+benefits and technical requirements. Third, misconceptions
+exist about the expected benefits of IoT privacy safeguards
+and decentralization.
+The author identifies five common misconceptions about
+privacy-preserving IoT that appear widely in the literature
+and recurred in discussions with students, researchers, IoT
+users, and industry professionals. This article refutes the mis-
+conceptions and provides corrections by applying quantitative
+research methodologies of experimental analysis on real-world
+datasets and survey research with statistical analysis. First,
+data privacy does not impede IoT innovation, and a trade-off
+exists between service accuracy and privacy budget (Miscon-
+ception 1). Second, data privacy is not exclusively a regulatory
+problem (Misconception 2). Third, users distrust services that
+do not make sufficient efforts to protect users’ privacy (Mis-
+conception 3). Fourth, basic data security principles, i.e., the
+confidentiality, integrity, and availability (CIA) conditions, do
+not guarantee privacy preservation (Misconception 4). For ex-
+ample, privacy attacks can be exploited to estimate users’ faces
+in exposed facial recognition systems even when the original
+data is securely stored. Fifth, decentralized IoT (DeIoT) does
+not provide absolute privacy preservation (Misconception 5).
+The rest of this article is organized as follows. First,
+an overview of privacy-preserving IoT and its main entities
+is presented. Then, misconceptions about IoT data sensing
+are discussed. After that, misconceptions about data privacy
+regulations in privacy-preserving IoT are debated. Common
+misconceptions about privacy safeguard tools are also de-
+bunked. Finally, critical questions for future research directions
+are discussed, and the article is concluded.
+arXiv:2301.00920v1  [cs.CR]  3 Jan 2023
+
+PRIVACY-PRESERVING IOT: OVERVIEW
+Connecting to IoT services is indispensable and makes
+people’s life more convenient. IoT services are characterized
+by their ability to interconnect people and physical objects
+embedded with sensors, actuators, and network connectiv-
+ity. Real-world examples of IoT services include CropX
+(www.cropx.com) for real-time crop monitoring in precision
+agriculture, HealthMap (www.healthmap.org) for crowdsens-
+ing healthcare and disease outbreak monitoring, and Fitbits
+(www.fitbit.com) for fitness tracking.
+Privacy-preserving IoT
+Privacy-preserving IoT refers to any IoT service, i.e., any
+network of objects embedded with sensors and connection
+links, that functions while maintaining the privacy rights of
+users. Figure 1 shows the four main entities of privacy-
+preserving IoT.
+• People (service users): People hold the ownership of their
+data in privacy-preserving IoT. Each user will possess
+3.6 devices in 2023 [1]. Accordingly, IoT is a cheap and
+efficient method for large-scale data sensing about people
+and their surroundings.
+• Service provider and business stakeholders: Service
+providers transmit IoT data to backend servers through
+high-speed networks. Service providers apply ambient
+intelligence, including data analysis and machine learn-
+ing, to create new IoT services and attain personalized
+user experiences. Business stakeholders apply revenue
+generation strategies and offer IoT services to interested
+customers subject to a subscription fee.
+• Adversary: An adversary is any third-party entity that ini-
+tiates privacy attacks to partially or fully attain user data.
+Many data privacy attacks exist and target all segments
+of the IoT network, including man-in-the-middle, data
+poising, membership, and model inversion attacks.
+• Data privacy analyst (regulatory officer): A data privacy
+analyst oversees the compliance of service providers
+with the relevant data privacy regulations. A data pri-
+vacy analyst produces the privacy impact assessment
+(PIA), which specifies all data flows in a privacy-
+preserving IoT network and their corresponding data
+privacy risks. Accordingly, data privacy safeguards, such
+as data anonymization, perturbation, tokenization, and
+encryption, are implemented to mitigate the identified
+privacy risks.
+Data privacy rights
+Recent years have witnessed the introduction of strict data
+privacy regulations that govern data operations in IoT ser-
+vices. For example, the General Data Protection Regulation
+(GDPR) [2] is a European Union law that defines eight data
+privacy rights of people—the right to be informed of all data
+operations, the right to review and access copies of personal
+data, the right to rectify incorrect data, the right to object data
+processing, the right to restrict data processing, the right of
+data portability to third parties, the right to be forgotten if
+personal data is no longer needed for the original purpose,
+and the right not to be a subject of automation and profiling.
+Likewise, the California Consumer Privacy Act (CCPA) [3]
+defines equivalent user rights for residents of California.
+MISCONCEPTIONS ABOUT IOT DATA SENSING AND
+INNOVATIONS
+Data privacy is widely deemed a fundamental human right.
+At the same time, IoT is motivated as an affordable and
+scalable technology for data sensing about people, their sur-
+roundings, and everyday activities. This section debunks the
+misconception that pervasive IoT sensing and data privacy
+cannot co-exist.
+Misconception 1: Data privacy impedes IoT innovation and
+implies that IoT data cannot be collected
+IoT data is collected to create new IoT services and tai-
+lor existing ones for user personalization through ambient
+intelligence. Therefore, IoT service providers and stakeholders
+commonly perceive privacy-preserving IoT as an impediment
+to innovation as data sensing about people is heavily regulated.
+On the other hand, some service users assume that service
+providers should not collect any data. These misconceptions
+arise from an inaccurate understanding of the concept of data
+privacy.
+Correction: The rights and responsibilities of each entity in
+privacy-preserving IoT are well-depicted. Specifically, privacy-
+preserving IoT is mainly about providing users with control
+over their data while promoting safeguarded IoT sensing and
+innovation. Therefore, the service provider must incorporate
+various privacy safeguards, including explicit consent, rec-
+tification forms, and meeting the differential privacy mea-
+surements when serving user requests. Privacy-preserving IoT
+is generally devised to meet the differential privacy require-
+ments [4] by adding noise to the input data, the parameters of
+ambient intelligence models, or the output results. Dwork and
+Roth [4] describe differential privacy as a guarantee provided
+by a service provider to users that they will not be affected by
+sharing their data, regardless of the availability of other infor-
+mation sources or personal data about them, i.e., the privacy
+guarantee is satisfied even when prior knowledge is available
+about a user. Therefore, this article chooses differential privacy
+as a state-of-the-art measure for privacy preservation and data
+leakage in IoT. Moreover, service providers generally apply
+cryptographic algorithms, e.g., tokenization, homomorphic en-
+cryption, and secure multi-party computation, to hide sensitive
+data during computation and data analysis.
+Data privacy is not absolute in privacy-preserving IoT.
+Instead, a trade-off exists between service accuracy and pri-
+vacy preservation. Figure 2 shows the accuracy performance
+of privacy-preserving and exposed IoT services for human
+activity recognition. Privacy-preserving and exposed services
+are created using logistic regression and Gaussian naive Bayes
+trained on a real-world activity prediction dataset [5]. The real-
+world dataset contains 1,098,207 accelerometer data points of
+36 subjects performing everyday activities, including walking,
+
+base station
+Internet  
+backbone
+people & physical devices 
+embedded with sensors
+wireless 
+access point
+packet data 
+network gateway
+serving 
+gateway
+service 
+provider
+data privacy analyst
+data analysis
+adversary
+attacks
+modem
+vehicles
+mobile 
+devices
+public 
+transport
+warehouse
+surveillance
+banking
+agriculture
+R&D
+smart homes
+shipping 
+& logistics
+privacy impact 
+assessment (PIA)
+Fig. 1. Main entities of privacy-preserving IoT.
+0
+5
+10
+15
+20
+25
+30
+Privacy budget
+20
+30
+40
+50
+60
+70
+80
+90
+Accuracy
+rapid accuracy 
+improvement
+marginal accuracy 
+improvement
+Logistic regression (privacy preserving)
+Logistic regression (exposed)
+Gaussian naive bayes (privacy preserving)
+Gaussian naive bayes (exposed)
+Fig. 2. Accuracy of privacy-preserving and exposed IoT services.
+jogging, climbing stairs, sitting, standing, and lying down. In-
+deed, accelerometer sensors are widely utilized in IoT gadgets
+and wearables for detecting users’ activity in fitness and e-
+health applications. The privacy budget in differential privacy
+is defined as the probability of accidental data leakage by an
+adversary. A small privacy budget requires adding significant
+noise to the machine learning parameters, resulting in high
+privacy preservation. Cross-validation of 5 folds was used in
+each experiment, and each experiment was repeated 20 times.
+Several important results can be noted from Figure 2. First,
+when the privacy budget increases, the service accuracy will
+increase. This relationship is expected because satisfying the
+tight privacy budgets needs adding more noise; thus, the ser-
+vice accuracy will decrease. Second, when the privacy budget
+is small, i.e., less than 5, significant accuracy improvement
+can be achieved for small increases in the privacy budget
+requirements. Third, there is a marginal gain in the service
+accuracy for increasing the privacy budget at high values.
+Fourth, different algorithms may have different accuracy ranks
+when changing the privacy budget. For example, Gaussian
+naive Bayes has higher accuracy than logistic regression when
+the privacy budget is less than 6.8. Then, logistic regression
+reports higher accuracy values when the privacy budget ex-
+ceeds 6.8. Finally, the exposed services retain higher accuracy
+values than the privacy-preserving ones, but that accuracy gain
+comes at the cost of risking users’ privacy.
+MISCONCEPTIONS ABOUT DATA PRIVACY REGULATIONS
+IoT standards, e.g., IEEE 2413-2019 [6], include data
+privacy as a functional requirement of IoT architectures.
+This section discusses two misconceptions about data privacy
+regulations and IoT. The first misconception emphasizes that
+privacy-preserving IoT is an exclusive regulatory problem.
+The second misconception undervalues the benefits of privacy-
+preserving IoT in assembling trust bridges with users and
+improving customer retention.
+Misconception 2: Privacy-preserving IoT is exclusively a
+regulatory problem
+The privacy paradox is a widely used concept in the litera-
+ture to describe the discrepancy between how people insist on
+the importance of their privacy and how they compromise their
+privacy in reality. For example, many users still provide their
+names and emails in marketing campaigns to receive discounts
+or free product samples. Accordingly, data privacy has been
+
+portrayed as an exclusive regulatory problem, i.e., people
+are wrongly perceived as incompetent in protecting their
+privacy. Therefore, Solove [7] argues for regulating service
+architectures and against privacy self-management, describing
+it as a complex task for users.
+Correction: Privacy-preserving IoT is not an exclusive
+regulatory problem. Two main issues regarding restricting
+user-level data control can be underlined. First, data privacy
+is individual-level ownership rather than a societal right.
+Accordingly, people should be able to provide consent to
+service providers for data collection and selling. Existing
+privacy regulation, such as the California Consumer Privacy
+Act (CCPA) [3], underlines that users may be offered discounts
+and financial incentives for data collection. This arrangement
+provides flexibility to both users and service providers. Sec-
+ond, a single government body cannot check the compliance
+of every service provider with the privacy regulations. The
+centralized privacy authority creates an unnecessary bottleneck
+in privacy-preserving IoT. Third, many sensing technologies
+exist in IoT systems, and it would be unattainable for a single
+entity to assess all possible privacy risks.
+Data privacy must be incorporated in the early design cycle
+of IoT. Internal and external policies must be drafted early
+in the design cycle. In addition, a data privacy analyst must
+be recruited to devise privacy impact assessments and breach
+response plans to comply with the data privacy regulations.
+Misconception 3: Privacy-preserving IoT is exclusively re-
+quired to comply with data privacy regulations
+A common misconception among service providers is per-
+ceiving data privacy in IoT as an obligation that does not retain
+direct financial benefits. Therefore, service providers adhere
+to the data privacy regulations as a compliance action, and
+IoT data privacy is not perceived as a functional requirement.
+Notably, this misconception underestimates the significance of
+privacy preservation in customer satisfaction, retention, and
+trust.
+Correction: Privacy-preserving IoT has many benefits for
+building trust bridges with users; hence, it boosts user retention
+and satisfaction. An online survey study was conducted to
+understand how people perceive their data privacy in exposed
+systems. The survey was created using the Qualtrics platform
+(www.qualtrics.com). The survey responses were collected
+from 200 participants recruited using Amazon Mechanical
+Turk (www.mturk.com) for crowdsourcing task assignments.
+• People will not use exposed services: Table I shows the
+response percents and 95% confidence intervals of the
+various actions that users would take if a company does
+not make sufficient efforts to protect their data privacy.
+A 95% confidence interval indicates the range of likely
+values of the responses for which the survey results are
+accurate with a 95% confidence level, i.e., 19 times out
+of 20 repetitions. The respondents indicated they would
+take all possible measures to protect their privacy. For
+example, 56.2% suggested that they would stop using the
+company’s services, and 67.2% said they would close the
+service accounts. Only 5.0% of the respondents would
+TABLE I
+USERS TAKE VARIOUS ACTIONS IF A COMPANY DOES NOT MAKE
+SUFFICIENT EFFORTS TO PROTECT THEIR ONLINE DATA PRIVACY.
+user action
+(a user can select multiple actions)
+percentage &
+95% confidence
+interval (%)
+Stop using the company’s services
+56.2 [49.3, 62.9]
+Close service accounts
+67.2 [60.4, 73.3]
+Request deleting data
+66.7 [59.9, 72.8]
+Report the company to cybersecurity services and
+government agencies, such as the National
+Cybersecurity Center
+41.3 [34.7, 48.2]
+Share the information with family and friends so
+they can avoid using the company’s services
+37.8 [31.4, 44.7]
+Unsubscribe from the company’s email list
+48.3 [41.4, 55.1]
+None
+5.0 [2.7, 8.9]
+not take any action. The 95% confidence intervals show
+that the survey results are accurate and represent the
+general population with high confidence. For example,
+the probability of a user to ”stop using the company’s
+services” in the confidence interval [49.3% to 62.9%] is
+accurate under the confidence level of 95%. The survey
+responses show that exposed services will lose many
+users due to the lack of proper privacy measures.
+• 92.5% of people are genuinely concerned about their
+data privacy and how service providers use their online
+data: 42.5% and 50.0% of the respondents reported that
+they are “strongly concerned” and “somewhat concerned”
+about their data privacy and how service providers use
+their online data. The “strongly concerned” and “some-
+what concerned” responses have 95% confidence inter-
+vals of [35.9% to 49.4%] and [43.1% to 56.9%], respec-
+tively. Only 2.5% of the respondents are unconcerned
+about their privacy. 5.0% of the respondents are neither
+concerned nor unconcerned about their data privacy.
+• 73% of people do not trust companies that do not make
+sufficient efforts to protect their data privacy: The survey
+results show that data privacy is essential for gaining
+users’ trust. Notably, 29.5% and 43.5% of the respondents
+indicated that they “strongly distrust” and “somewhat
+distrust” companies that do not make sufficient efforts to
+protect their data privacy with 95% confidence intervals
+of [23.6% to 36.2%] and [36.8% to 50.4%], respectively.
+Only 8.5% of the respondents trust exposed companies.
+In addition, 18.5% of the respondents do not have strong
+opinions.
+The survey research’s results indicate the importance of data
+privacy in improving user retention and overall satisfaction.
+Thus, data privacy should not be perceived as a compliance
+problem but rather as a business opportunity with financial
+yields.
+MISCONCEPTIONS ABOUT PRIVACY SAFEGUARDS
+There has been significant progress in data security tech-
+nologies and privacy safeguards. However, a common mis-
+understanding exists regarding the relationship between data
+
+people & physical devices 
+embedded with sensors
+service 
+provider
+extended data privacy layer 
+(intervenability, transparency, and 
+unlinkability)
+data 
+analysis
+user 
+data
+adversary
+attacks
+data security layer 
+(confidentiality, integrity, and availability)
+Fig. 3.
+Data privacy extends the data security conditions, providing users
+with control over their data and preventing data violations and misuse.
+security and data privacy in IoT. In particular, fulfilling
+the security principles, i.e., the confidentiality, integrity, and
+availability (CIA) conditions, does not mean data privacy is
+protected. This section debunks two myths about technical
+solutions for meeting the data privacy requirements in privacy-
+preserving IoT. First, this section shows that data privacy
+may not be met even if IoT data is securely stored. Then,
+it shows that IoT decentralization does not guarantee absolute
+data privacy for users.
+Misconception 4: Data privacy is fully preserved if IoT data
+is securely stored
+A widespread fallacy, even among cybersecurity practition-
+ers, is claiming data privacy preservation by applying data
+security measures, such as network security, access control,
+backups, authorization, firewalls, and intrusion detectors. In
+particular, such security methods are implemented to adhere to
+the confidentiality, integrity, and availability (CIA) principles.
+Confidentiality protocols, e.g., access control and authoriza-
+tion, aim to protect the data from unauthorized disclosure.
+Integrity, e.g., digital signatures and logging, aims to maintain
+the accuracy and completeness of data. Finally, availability,
+e.g., backups and firewalls, aims to promptly supply resource
+access to users when requested.
+Correction: Data security, defined in the confidentiality,
+integrity, and availability (CIA) triad, does not guarantee
+users’ data privacy. In particular, data security protects users
+from unauthorized data access or modification. On the other
+hand, data privacy protects users from violations and misuse,
+including how service providers use and process user data.
+Data privacy is a superset of data security and requires stricter
+conditions to comply with the privacy laws on how user data is
+collected, transmitted, stored, and processed, e.g., the data pri-
+vacy rights of users as depicted in the General Data Protection
+Original images
+Reconstructed images (using model)
+Fig. 4. Privacy attacks on an exposed IoT service that uses face recognition.
+Regulation (GDPR) [2] and the California Consumer Privacy
+Act (CCPA) [3]. For example, the confidentiality, integrity,
+and availability (CIA) conditions do not cover the rights to be
+informed or forgotten, which are fundamental privacy rights of
+users. Therefore, Hansen et al. [8] extend the confidentiality,
+integrity, and availability (CIA) triad to support data privacy
+by including intervenability, transparency, and unlinkability
+conditions. Intervenability enables the users to intervene and
+provoke their data rights, such as consent withdrawal and
+data rectification and erasure. Then, transparency ensures that
+users can understand and verify all data operations with
+reasonable effort, which enables users to provide informed
+consent for data operations. Finally, the unlinkability condition
+controls linking user data with information from other sources,
+preventing the risk of user re-identification and automated
+profiling. Figure 3 depicts how data privacy extends data
+security, providing users with control over their data and
+preventing data violations and misuse.
+Data privacy may not be met even when original data
+is securely stored. This can be demonstrated using a facial
+recognition system that is widely used for easy sign-in to IoT
+services. Figure 4 depicts how an adversary can reconstruct
+people’s faces in exposed IoT facial recognition systems. The
+facial recognition service is built by training a deep learning
+model on the CelebFaces dataset [9], which contains 202,599
+images of 10,177 identities. Even though the original face
+images are securely kept, the adversary can reconstruct an
+accurate estimation of people’s faces using the deep learning
+model, i.e., the original training images are not used in
+producing the reconstructed images. Such a privacy attack is an
+example of model inversion attacks [10] that produce sensitive
+data using outputs of a model. Figure 4 demonstrates how data
+privacy is not fully preserved even when the confidentiality,
+integrity, and availability (CIA) conditions are satisfied. The
+attack arises in many real-world IoT systems that ship trained
+deep learning models with IoT consumer products, e.g., for
+running face recognition with edge computing at IoT devices.
+Accordingly, an adversary can effortlessly obtain a copy of the
+trained deep learning model and apply model inversion attacks.
+Therefore, service providers should utilize privacy-preserving
+learning that adds reasonable noise to the modeling parameters
+during model training according to the differential privacy
+conditions [4]. Moreover, learning from encrypted data using
+
+000
+5
+10
+15
+20
+25
+30
+35
+40
+Number of data sensing
+0
+1
+2
+3
+4
+5
+Total privacy cost
+growing privacy 
+leakage gap
+3.7x privacy cost 
+of Service 1
+3x privacy cost 
+of Service 1
+Service 1 (privacy budget=0.1)
+Service 2 (privacy budget=0.15)
+Service 3 (privacy budget=0.3)
+Fig. 5. Total privacy cost of repeated data sensing at various privacy budgets.
+homomorphic encryption and secure multi-party computation
+must be utilized to thwart model inversion attacks and the
+reconstruction of user data.
+Misconception 5: Decentralized IoT (DeIoT) solves the pri-
+vacy problem and provides absolute data privacy preservation
+Decentralized IoT (DeIoT) is an emerging user-centered
+ecosystem that distributes IoT control functions and delegates
+operations to users without including a central authority. Edge
+computing, blockchain ledgers, and federated learning are the
+most promising technologies for DeIoT. For example, smart
+contracts and blockchain ledgers provide decentralized digital
+identities that are shared with all participating devices [11]. In
+addition, federated learning and edge computing can optimize
+a master ambient intelligence model without sharing users’
+original data with a central server [12]. DeIoT is widely
+suggested as a method for improving data privacy, security,
+transparency, and scalability using token-based operations and
+decentralization. As a result, DeIoT allows people to gain
+additional control over their data collection and processing.
+Correction: Unfortunately, DeIoT does not provide absolute
+data privacy preservation. First, DeIoT is challenging to regu-
+late, given the lack of a central moderation entity. Second,
+adversaries can impersonate legitimate users, and harmful
+content can be injected into DeIoT networks. Third, DeIoT
+introduces new privacy and security challenges. For example,
+blockchains and federated learning are two prominent DeIoT-
+enabling technologies that raise new privacy risks in IoT
+services. Blockchain uses asymmetric cryptography, i.e., pairs
+of private and public keys, to securely store transactions
+on a public ledger that all users can view. Therefore, user
+identities in DeIoT can be impersonated if the user’s private
+key, which is unique for each user, is stolen or hacked
+by an adversary. Then, blockchain-based DeIoT guarantees
+immutability [11], indicating that IoT data cannot be altered or
+deleted once a blockchain record is verified. Nonetheless, the
+immutability of DeIoT violates the user rights in data erasure
+and rectification as depicted in the General Data Protection
+Regulation (GDPR) [2] and the California Consumer Privacy
+Act (CCPA) [3]. Also, adversaries can participate in federated
+learning to access the final trained model, creating privacy
+risks that can be exploited using model inversion attacks [10].
+When the privacy budget equals zero, absolute data privacy
+is achieved, and differential privacy guarantees that an adver-
+sary cannot identify data about individual users. As discussed
+above, DeIoT does not fulfill the requirements of absolute data
+privacy. Assume that three services (Services 1-3) are built
+using blockchain ledgers. The privacy budget of a single data
+sensing is set at 0.1, 0.15, and 0.3 in Services 1-3, respectively.
+Figure 5 shows the total privacy costs of accessing Services 1-
+3 at different privacy budgets. The total privacy cost indicates
+the cumulative privacy budget of differential privacy for se-
+quential sensing events in IoT, i.e., overall privacy leakage as
+a function of the number of data sensing events. The total
+privacy cost is calculated using the composition theorem of
+privacy-preserving mechanisms [4]. Two substantial results
+can be underlined. First, the total privacy cost increases over
+repeated sensing in the three services. Therefore, although the
+privacy budget is small for a single sensing event, data privacy
+preservation in DeIoT is not absolute. Second, the difference
+in the privacy cost of users magnifies over time. For example,
+Service 3 has 3.7x the privacy cost of Service 1 after 38 data
+sensing attempts, even though the privacy budget of a single
+sensing attempt in Service 3 is 3x that of Service 1.
+CRITICAL QUESTIONS FOR FUTURE RESEARCH
+This section presents critical questions for future research
+in privacy-preserving IoT.
+Data privacy and criminal justice
+A
+widespread
+argument
+for
+supporting
+dataveillance,
+i.e., monitoring and profiling people’s data, is for criminal
+justice, law enforcement, and fraud prevention. For example,
+all United States privacy laws exclude law enforcement from
+information disclosure even though the Fourth Amendment
+protects individuals from unreasonable searches by the gov-
+ernment without a reasonable basis [13]. Furthermore, privacy-
+preserving IoT might be seen as a tool for boosting on-
+line incivility and unfriendly behavior. Such arguments for
+dataveillance undervalue the benefits of data privacy. First, data
+privacy is a powerful mechanism for preventing cybercrimes
+and online incivility. In particular, when personal data is
+privately reserved, people will be immune to adversaries and
+disruptive behaviors. Second, data privacy is a crucial enabler
+for the freedom of speech and civil liberty. Third, organized
+crime has the resources and motives to create custom en-
+crypted communication channels; thus, exposed services will
+have the most impact on regular users.
+Social benefits do not wipe out the personal benefits of data
+privacy. Several critical questions need further research. What
+is the proper procedure for requesting data disclosure for crim-
+inal justice? How can protected data be accessed for criminal
+justice without establishing an encryption backdoor? How can
+people oversee the levels of dataveillance by organizations and
+governments?
+
+User-in-the-loop (UIL) privacy-preserving IoT
+IoT standards, e.g., IEEE 2413-2019 [6], emphasize that
+user privacy and trust are essential components of any IoT
+design. However, people, i.e., data owners, are still not well-
+engaged in their privacy preservation. As discussed in Miscon-
+ception 2, the privacy paradox suggests doubt about the ability
+of users to protect their privacy. Moreover, users generally
+cannot verify the privacy measures taken by service providers
+due to the lack of transparency in the implemented privacy
+safeguards [14].
+User-in-the-loop (UIL) data privacy engages users in their
+privacy preservation. UIL is an emerging design concept in IoT
+that improves performance, e.g., reducing overall energy con-
+sumption, by engaging people in fundamental IoT operations
+beyond traffic consumption and generation [15]. UIL privacy-
+preserving IoT has garnered limited scholarly attention. Thus,
+a few critical questions required further exploration. How can
+user awareness of data privacy issues be increased? How can
+service providers provide people with data privacy measure-
+ments? How can users be incentivized to contribute to their
+data protection efforts?
+CONCLUSIONS
+Privacy-preserving IoT is essential given the substantial
+data privacy risks and their severe impacts on users. This
+article presented five common misconceptions about privacy-
+preserving IoT. Several results were presented using exper-
+iments on real-world datasets and survey research. First, a
+trade-off exists between privacy preservation and service ac-
+curacy in privacy-preserving IoT. Second, privacy-preserving
+IoT is beneficial for user retention and overall satisfaction.
+Third, users are concerned about their privacy and do not
+trust services that lack sufficient privacy procedures. Fourth,
+privacy attacks may occur even when user data is securely
+stored. Fifth, IoT decentralization does guarantee absolute
+data privacy. Finally, critical questions were raised for future
+research in privacy-preserving IoT.
+BIOGRAPHIES
+REFERENCES
+[1] Cisco Systems, Inc., “Cisco annual Internet report (2018–2023)
+white
+paper,”
+https://www.cisco.com/c/en/us/solutions/collateral/
+executive-perspectives/annual-internet-report/white-paper-c11-741490.
+html, 2020, online; accessed 16 January 2022.
+[2] European Parliament and Council of the European Union, “General
+data protection regulation (GDPR),” https://gdpr-info.eu, 2016, online;
+accessed 27 February 2022.
+[3] California Civil Code, “California consumer privacy act (CCPA),” https:
+//oag.ca.gov/privacy/ccpa, 2018, online; accessed 27 February 2022.
+[4] C. Dwork and A. Roth, “The algorithmic foundations of differential
+privacy,” Foundations and Trends in Theoretical Computer Science,
+vol. 9, no. 3-4, pp. 211–407, 2014.
+[5] J. R. Kwapisz, G. M. Weiss, and S. A. Moore, “Activity recognition us-
+ing cell phone accelerometers,” ACM SigKDD Explorations Newsletter,
+vol. 12, no. 2, pp. 74–82, March 2011.
+[6] IEEE SA Board of Governors/Corporate Advisory Group (BoG/CAG),
+“IEEE standard for an architectural framework for the Internet of things
+(IoT),” IEEE Std 2413-2019, pp. 1–269, March 2020.
+[7] D. J. Solove, “The myth of the privacy paradox,” George Washington
+Law Review, vol. 89, pp. 1–51, January 2021.
+[8] M. Hansen, M. Jensen, and M. Rost, “Protection goals for privacy engi-
+neering,” in Proceedings of the IEEE Security and Privacy Workshops.
+IEEE, 2015, pp. 159–166.
+[9] Z. Liu, P. Luo, X. Wang, and X. Tang, “Deep learning face attributes
+in the wild,” in Proceedings of the IEEE International Conference on
+Computer Vision, 2015, pp. 3730–3738.
+[10] Y. He, G. Meng, K. Chen, X. Hu, and J. He, “Towards security threats
+of deep learning systems: A survey,” IEEE Transactions on Software
+Engineering, pp. 1–28, November 2020.
+[11] L. D. Nguyen, A. E. Kalor, I. Leyva-Mayorga, and P. Popovski,
+“Trusted wireless monitoring based on distributed ledgers over NB-IoT
+connectivity,” IEEE Communications Magazine, vol. 58, no. 6, pp. 77–
+83, July 2020.
+[12] F. Malandrino and C. F. Chiasserini, “Federated learning at the network
+edge: When not all nodes are created equal,” IEEE Communications
+Magazine, vol. 59, no. 7, pp. 68–73, July 2021.
+[13] E. Murphy, “The politics of privacy in the criminal justice system:
+Information disclosure, the fourth amendment, and statutory law enforce-
+ment exemptions,” Michigan Law Review, vol. 111, no. 4, pp. 485–546,
+February 2013.
+[14] S. Zheng, N. Apthorpe, M. Chetty, and N. Feamster, “User perceptions of
+smart home IoT privacy,” Proceedings of the ACM on Human-Computer
+Interaction, vol. 2, no. CSCW, pp. 1–20, November 2018.
+[15] V. Petrov, K. Mikhaylov, D. Moltchanov, S. Andreev, G. Fodor,
+J. Torsner, H. Yanikomeroglu, M. Juntti, and Y. Koucheryavy, “When
+IoT keeps people in the loop: A path towards a new global utility,” IEEE
+Communications Magazine, vol. 57, no. 1, pp. 114–121, December 2018.
+Mohammad Abu Alsheikh [S’14, M’17, SM’22] is an Associate Professor
+and ARC DECRA Fellow at the University of Canberra, Australia.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf,len=480
+page_content='Five Common Misconceptions About  Privacy-Preserving Internet of Things  Mohammad Abu Alsheikh, Senior Member, IEEE  Abstract—Billions of devices in the Internet of Things (IoT)  collect sensitive data about people, creating data privacy risks and  breach vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Accordingly, data privacy preservation is  vital for sustaining the proliferation of IoT services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' In particular,  privacy-preserving IoT connects devices embedded with sensors  and maintains the data privacy of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' However, common  misconceptions exist among IoT researchers, service providers,  and users about privacy-preserving IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  This article refutes five common misconceptions about privacy-  preserving IoT concerning data sensing and innovation, regula-  tions, and privacy safeguards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For example, IoT users have a com-  mon misconception that no data collection is permitted in data  privacy regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' On the other hand, IoT service providers  often think data privacy impedes IoT sensing and innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Addressing these misconceptions is essential for making progress  in privacy-preserving IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' This article refutes such common  misconceptions using real-world experiments and online survey  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' First, the experiments indicate that data privacy should  not be perceived as an impediment in IoT but as an opportunity to  increase customer retention and trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Second, privacy-preserving  IoT is not exclusively a regulatory problem but also a functional  necessity that must be incorporated in the early stages of any  IoT design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Third, people do not trust services that lack sufficient  privacy measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Fourth, conventional data security principles  do not guarantee data privacy protection, and data privacy can be  exposed even if data is securely stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Fifth, IoT decentralization  does not attain absolute privacy preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Index Terms—Internet of things, data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  INTRODUCTION  R  ECENT years have witnessed a proliferation of Internet  of Things (IoT) services in home automation, retail,  telehealth, manufacturing, autonomous vehicles, and preci-  sion agriculture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Cisco Systems estimates that 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='3 billion  networked devices, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='7 billion mobile subscribers with an  average cellular speed of 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='9 Mbps, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='4 billion 5G-  enabled devices will exist in 2023 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' This proliferation in  ubiquitous computing and high-speed mobile networks enables  service providers to collect valuable IoT datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' In particular,  IoT data is fundamental for creating new IoT services tailored  to people’s needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  IoT data can contain sensitive data about people, creating  genuine data privacy concerns for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For example, more  than 1,272 major data breaches happened in 2019, exposing  163 million records (or 128,171 exposed records per data  breach) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Data breaches severely impact victims, including  financial loss, psychological trauma, criminal impersonation,  and online fraud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Accordingly, privacy-preserving IoT applies  This work was supported by the Australian Research Council (ARC) under  Grant DE200100863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The data collected in this project has been approved by  the Human Research Ethics Committee at the University of Canberra under  the application “4522 - Privacy Coupling: When Your Personal Devices Betray  You”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  strict measures to maintain people’s data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Specifically,  privacy-preserving IoT is a fully-operational IoT network that  meets the relevant data privacy regulations, such as the General  Data Protection Regulation (GDPR) [2] and the California  Consumer Privacy Act (CCPA) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Nonetheless, privacy-preserving IoT has a few commonly  misunderstood aspects, as IoT is a technology for pervasive  sensing about people and their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Specifically,  common misconceptions exist among IoT service providers  and users about the possibility of maintaining data pri-  vacy in pervasive IoT sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' This article refutes com-  mon misconceptions, which are organized into three cate-  gories—misconceptions about IoT innovation, data regula-  tions, and privacy safeguards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' First, IoT is widely motivated as  an efficient and cheap method for data collection about people  and their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Thus, data privacy may be wrongly  perceived as an impediment to IoT innovation and pervasive  sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Second, privacy-preserving IoT is often recognized as  an exclusive regulatory problem while ignoring its financial  benefits and technical requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Third, misconceptions  exist about the expected benefits of IoT privacy safeguards  and decentralization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  The author identifies five common misconceptions about  privacy-preserving IoT that appear widely in the literature  and recurred in discussions with students, researchers, IoT  users, and industry professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' This article refutes the mis-  conceptions and provides corrections by applying quantitative  research methodologies of experimental analysis on real-world  datasets and survey research with statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' First,  data privacy does not impede IoT innovation, and a trade-off  exists between service accuracy and privacy budget (Miscon-  ception 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Second, data privacy is not exclusively a regulatory  problem (Misconception 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Third, users distrust services that  do not make sufficient efforts to protect users’ privacy (Mis-  conception 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Fourth, basic data security principles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', the  confidentiality, integrity, and availability (CIA) conditions, do  not guarantee privacy preservation (Misconception 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For ex-  ample, privacy attacks can be exploited to estimate users’ faces  in exposed facial recognition systems even when the original  data is securely stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Fifth, decentralized IoT (DeIoT) does  not provide absolute privacy preservation (Misconception 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  The rest of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' First,  an overview of privacy-preserving IoT and its main entities  is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Then, misconceptions about IoT data sensing  are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' After that, misconceptions about data privacy  regulations in privacy-preserving IoT are debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Common  misconceptions about privacy safeguard tools are also de-  bunked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Finally, critical questions for future research directions  are discussed, and the article is concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='00920v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='CR]  3 Jan 2023  PRIVACY-PRESERVING IOT: OVERVIEW  Connecting to IoT services is indispensable and makes  people’s life more convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' IoT services are characterized  by their ability to interconnect people and physical objects  embedded with sensors, actuators, and network connectiv-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Real-world examples of IoT services include CropX  (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='cropx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='com) for real-time crop monitoring in precision  agriculture, HealthMap (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='healthmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='org) for crowdsens-  ing healthcare and disease outbreak monitoring, and Fitbits  (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='fitbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='com) for fitness tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Privacy-preserving IoT  Privacy-preserving IoT refers to any IoT service, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', any  network of objects embedded with sensors and connection  links, that functions while maintaining the privacy rights of  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Figure 1 shows the four main entities of privacy-  preserving IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  People (service users): People hold the ownership of their  data in privacy-preserving IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Each user will possess  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='6 devices in 2023 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Accordingly, IoT is a cheap and  efficient method for large-scale data sensing about people  and their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Service provider and business stakeholders: Service  providers transmit IoT data to backend servers through  high-speed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Service providers apply ambient  intelligence, including data analysis and machine learn-  ing, to create new IoT services and attain personalized  user experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Business stakeholders apply revenue  generation strategies and offer IoT services to interested  customers subject to a subscription fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Adversary: An adversary is any third-party entity that ini-  tiates privacy attacks to partially or fully attain user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Many data privacy attacks exist and target all segments  of the IoT network, including man-in-the-middle, data  poising, membership, and model inversion attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Data privacy analyst (regulatory officer): A data privacy  analyst oversees the compliance of service providers  with the relevant data privacy regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' A data pri-  vacy analyst produces the privacy impact assessment  (PIA), which specifies all data flows in a privacy-  preserving IoT network and their corresponding data  privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Accordingly, data privacy safeguards, such  as data anonymization, perturbation, tokenization, and  encryption, are implemented to mitigate the identified  privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Data privacy rights  Recent years have witnessed the introduction of strict data  privacy regulations that govern data operations in IoT ser-  vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' the General Data Protection Regulation  (GDPR) [2] is a European Union law that defines eight data  privacy rights of people—the right to be informed of all data  operations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' the right to review and access copies of personal  data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' the right to rectify incorrect data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' the right to object data  processing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' the right to restrict data processing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' the right of  data portability to third parties,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' the right to be forgotten if  personal data is no longer needed for the original purpose,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  and the right not to be a subject of automation and profiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Likewise, the California Consumer Privacy Act (CCPA) [3]  defines equivalent user rights for residents of California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  MISCONCEPTIONS ABOUT IOT DATA SENSING AND  INNOVATIONS  Data privacy is widely deemed a fundamental human right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  At the same time, IoT is motivated as an affordable and  scalable technology for data sensing about people, their sur-  roundings, and everyday activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' This section debunks the  misconception that pervasive IoT sensing and data privacy  cannot co-exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Misconception 1: Data privacy impedes IoT innovation and  implies that IoT data cannot be collected  IoT data is collected to create new IoT services and tai-  lor existing ones for user personalization through ambient  intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Therefore, IoT service providers and stakeholders  commonly perceive privacy-preserving IoT as an impediment  to innovation as data sensing about people is heavily regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  On the other hand, some service users assume that service  providers should not collect any data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' These misconceptions  arise from an inaccurate understanding of the concept of data  privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Correction: The rights and responsibilities of each entity in  privacy-preserving IoT are well-depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Specifically, privacy-  preserving IoT is mainly about providing users with control  over their data while promoting safeguarded IoT sensing and  innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Therefore, the service provider must incorporate  various privacy safeguards, including explicit consent, rec-  tification forms, and meeting the differential privacy mea-  surements when serving user requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Privacy-preserving IoT  is generally devised to meet the differential privacy require-  ments [4] by adding noise to the input data, the parameters of  ambient intelligence models, or the output results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Dwork and  Roth [4] describe differential privacy as a guarantee provided  by a service provider to users that they will not be affected by  sharing their data, regardless of the availability of other infor-  mation sources or personal data about them, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', the privacy  guarantee is satisfied even when prior knowledge is available  about a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Therefore, this article chooses differential privacy  as a state-of-the-art measure for privacy preservation and data  leakage in IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Moreover, service providers generally apply  cryptographic algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', tokenization, homomorphic en-  cryption, and secure multi-party computation, to hide sensitive  data during computation and data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Data privacy is not absolute in privacy-preserving IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Instead, a trade-off exists between service accuracy and pri-  vacy preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Figure 2 shows the accuracy performance  of privacy-preserving and exposed IoT services for human  activity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Privacy-preserving and exposed services  are created using logistic regression and Gaussian naive Bayes  trained on a real-world activity prediction dataset [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The real-  world dataset contains 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='098,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='207 accelerometer data points of  36 subjects performing everyday activities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' including walking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  base station  Internet  backbone  people & physical devices  embedded with sensors  wireless  access point  packet data  network gateway  serving  gateway  service  provider  data privacy analyst  data analysis  adversary  attacks  modem  vehicles  mobile  devices  public  transport  warehouse  surveillance  banking  agriculture  R&D  smart homes  shipping  & logistics  privacy impact  assessment (PIA)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Main entities of privacy-preserving IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  0  5  10  15  20  25  30  Privacy budget  20  30  40  50  60  70  80  90  Accuracy  rapid accuracy  improvement  marginal accuracy  improvement  Logistic regression (privacy preserving)  Logistic regression (exposed)  Gaussian naive bayes (privacy preserving)  Gaussian naive bayes (exposed)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Accuracy of privacy-preserving and exposed IoT services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  jogging, climbing stairs, sitting, standing, and lying down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' In-  deed, accelerometer sensors are widely utilized in IoT gadgets  and wearables for detecting users’ activity in fitness and e-  health applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The privacy budget in differential privacy  is defined as the probability of accidental data leakage by an  adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' A small privacy budget requires adding significant  noise to the machine learning parameters, resulting in high  privacy preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Cross-validation of 5 folds was used in  each experiment, and each experiment was repeated 20 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Several important results can be noted from Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' First,  when the privacy budget increases, the service accuracy will  increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' This relationship is expected because satisfying the  tight privacy budgets needs adding more noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' thus, the ser-  vice accuracy will decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Second, when the privacy budget  is small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', less than 5, significant accuracy improvement  can be achieved for small increases in the privacy budget  requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Third, there is a marginal gain in the service  accuracy for increasing the privacy budget at high values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Fourth, different algorithms may have different accuracy ranks  when changing the privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For example, Gaussian  naive Bayes has higher accuracy than logistic regression when  the privacy budget is less than 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Then, logistic regression  reports higher accuracy values when the privacy budget ex-  ceeds 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Finally, the exposed services retain higher accuracy  values than the privacy-preserving ones, but that accuracy gain  comes at the cost of risking users’ privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  MISCONCEPTIONS ABOUT DATA PRIVACY REGULATIONS  IoT standards, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', IEEE 2413-2019 [6], include data  privacy as a functional requirement of IoT architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  This section discusses two misconceptions about data privacy  regulations and IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The first misconception emphasizes that  privacy-preserving IoT is an exclusive regulatory problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  The second misconception undervalues the benefits of privacy-  preserving IoT in assembling trust bridges with users and  improving customer retention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Misconception 2: Privacy-preserving IoT is exclusively a  regulatory problem  The privacy paradox is a widely used concept in the litera-  ture to describe the discrepancy between how people insist on  the importance of their privacy and how they compromise their  privacy in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For example, many users still provide their  names and emails in marketing campaigns to receive discounts  or free product samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Accordingly, data privacy has been  portrayed as an exclusive regulatory problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', people  are wrongly perceived as incompetent in protecting their  privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Therefore, Solove [7] argues for regulating service  architectures and against privacy self-management, describing  it as a complex task for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Correction: Privacy-preserving IoT is not an exclusive  regulatory problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Two main issues regarding restricting  user-level data control can be underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' First, data privacy  is individual-level ownership rather than a societal right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Accordingly, people should be able to provide consent to  service providers for data collection and selling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Existing  privacy regulation, such as the California Consumer Privacy  Act (CCPA) [3], underlines that users may be offered discounts  and financial incentives for data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' This arrangement  provides flexibility to both users and service providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Sec-  ond, a single government body cannot check the compliance  of every service provider with the privacy regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The  centralized privacy authority creates an unnecessary bottleneck  in privacy-preserving IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Third, many sensing technologies  exist in IoT systems, and it would be unattainable for a single  entity to assess all possible privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Data privacy must be incorporated in the early design cycle  of IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Internal and external policies must be drafted early  in the design cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' In addition, a data privacy analyst must  be recruited to devise privacy impact assessments and breach  response plans to comply with the data privacy regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Misconception 3: Privacy-preserving IoT is exclusively re-  quired to comply with data privacy regulations  A common misconception among service providers is per-  ceiving data privacy in IoT as an obligation that does not retain  direct financial benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Therefore, service providers adhere  to the data privacy regulations as a compliance action, and  IoT data privacy is not perceived as a functional requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Notably, this misconception underestimates the significance of  privacy preservation in customer satisfaction, retention, and  trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Correction: Privacy-preserving IoT has many benefits for  building trust bridges with users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' hence, it boosts user retention  and satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' An online survey study was conducted to  understand how people perceive their data privacy in exposed  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The survey was created using the Qualtrics platform  (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='qualtrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The survey responses were collected  from 200 participants recruited using Amazon Mechanical  Turk (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='mturk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='com) for crowdsourcing task assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  People will not use exposed services: Table I shows the  response percents and 95% confidence intervals of the  various actions that users would take if a company does  not make sufficient efforts to protect their data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  A 95% confidence interval indicates the range of likely  values of the responses for which the survey results are  accurate with a 95% confidence level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', 19 times out  of 20 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The respondents indicated they would  take all possible measures to protect their privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For  example, 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='2% suggested that they would stop using the  company’s services, and 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='2% said they would close the  service accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Only 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='0% of the respondents would  TABLE I  USERS TAKE VARIOUS ACTIONS IF A COMPANY DOES NOT MAKE  SUFFICIENT EFFORTS TO PROTECT THEIR ONLINE DATA PRIVACY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  user action  (a user can select multiple actions)  percentage &  95% confidence  interval (%)  Stop using the company’s services  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='2 [49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='3, 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='9]  Close service accounts  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='2 [60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='4, 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='3]  Request deleting data  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='7 [59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='9, 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='8]  Report the company to cybersecurity services and  government agencies, such as the National  Cybersecurity Center  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='3 [34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='7, 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='2]  Share the information with family and friends so  they can avoid using the company’s services  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='8 [31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='4, 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='7]  Unsubscribe from the company’s email list  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='3 [41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='4, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='1]  None  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='0 [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='7, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='9]  not take any action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The 95% confidence intervals show  that the survey results are accurate and represent the  general population with high confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For example,  the probability of a user to ”stop using the company’s  services” in the confidence interval [49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='3% to 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='9%] is  accurate under the confidence level of 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The survey  responses show that exposed services will lose many  users due to the lack of proper privacy measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='5% of people are genuinely concerned about their  data privacy and how service providers use their online  data: 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='5% and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='0% of the respondents reported that  they are “strongly concerned” and “somewhat concerned”  about their data privacy and how service providers use  their online data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The “strongly concerned” and “some-  what concerned” responses have 95% confidence inter-  vals of [35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='9% to 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='4%] and [43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='1% to 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='9%], respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='5% of the respondents are unconcerned  about their privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='0% of the respondents are neither  concerned nor unconcerned about their data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  73% of people do not trust companies that do not make  sufficient efforts to protect their data privacy: The survey  results show that data privacy is essential for gaining  users’ trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Notably, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='5% and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='5% of the respondents  indicated that they “strongly distrust” and “somewhat  distrust” companies that do not make sufficient efforts to  protect their data privacy with 95% confidence intervals  of [23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='6% to 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='2%] and [36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='8% to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='4%], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Only 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='5% of the respondents trust exposed companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  In addition, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='5% of the respondents do not have strong  opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  The survey research’s results indicate the importance of data  privacy in improving user retention and overall satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Thus, data privacy should not be perceived as a compliance  problem but rather as a business opportunity with financial  yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  MISCONCEPTIONS ABOUT PRIVACY SAFEGUARDS  There has been significant progress in data security tech-  nologies and privacy safeguards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' However, a common mis-  understanding exists regarding the relationship between data  people & physical devices  embedded with sensors  service  provider  extended data privacy layer  (intervenability, transparency, and  unlinkability)  data  analysis  user  data  adversary  attacks  data security layer  (confidentiality, integrity, and availability)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Data privacy extends the data security conditions, providing users  with control over their data and preventing data violations and misuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  security and data privacy in IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' In particular, fulfilling  the security principles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', the confidentiality, integrity, and  availability (CIA) conditions, does not mean data privacy is  protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' This section debunks two myths about technical  solutions for meeting the data privacy requirements in privacy-  preserving IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' First, this section shows that data privacy  may not be met even if IoT data is securely stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Then,  it shows that IoT decentralization does not guarantee absolute  data privacy for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Misconception 4: Data privacy is fully preserved if IoT data  is securely stored  A widespread fallacy, even among cybersecurity practition-  ers, is claiming data privacy preservation by applying data  security measures, such as network security, access control,  backups, authorization, firewalls, and intrusion detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' In  particular, such security methods are implemented to adhere to  the confidentiality, integrity, and availability (CIA) principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Confidentiality protocols, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', access control and authoriza-  tion, aim to protect the data from unauthorized disclosure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Integrity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', digital signatures and logging, aims to maintain  the accuracy and completeness of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Finally, availability,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', backups and firewalls, aims to promptly supply resource  access to users when requested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Correction: Data security, defined in the confidentiality,  integrity, and availability (CIA) triad, does not guarantee  users’ data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' In particular, data security protects users  from unauthorized data access or modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' On the other  hand, data privacy protects users from violations and misuse,  including how service providers use and process user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Data privacy is a superset of data security and requires stricter  conditions to comply with the privacy laws on how user data is  collected, transmitted, stored, and processed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', the data pri-  vacy rights of users as depicted in the General Data Protection  Original images  Reconstructed images (using model)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Privacy attacks on an exposed IoT service that uses face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Regulation (GDPR) [2] and the California Consumer Privacy  Act (CCPA) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For example, the confidentiality, integrity,  and availability (CIA) conditions do not cover the rights to be  informed or forgotten, which are fundamental privacy rights of  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Therefore, Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' [8] extend the confidentiality,  integrity, and availability (CIA) triad to support data privacy  by including intervenability, transparency, and unlinkability  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Intervenability enables the users to intervene and  provoke their data rights, such as consent withdrawal and  data rectification and erasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Then, transparency ensures that  users can understand and verify all data operations with  reasonable effort, which enables users to provide informed  consent for data operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Finally, the unlinkability condition  controls linking user data with information from other sources,  preventing the risk of user re-identification and automated  profiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Figure 3 depicts how data privacy extends data  security, providing users with control over their data and  preventing data violations and misuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Data privacy may not be met even when original data  is securely stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' This can be demonstrated using a facial  recognition system that is widely used for easy sign-in to IoT  services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Figure 4 depicts how an adversary can reconstruct  people’s faces in exposed IoT facial recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The  facial recognition service is built by training a deep learning  model on the CelebFaces dataset [9], which contains 202,599  images of 10,177 identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Even though the original face  images are securely kept, the adversary can reconstruct an  accurate estimation of people’s faces using the deep learning  model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', the original training images are not used in  producing the reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Such a privacy attack is an  example of model inversion attacks [10] that produce sensitive  data using outputs of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Figure 4 demonstrates how data  privacy is not fully preserved even when the confidentiality,  integrity, and availability (CIA) conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The  attack arises in many real-world IoT systems that ship trained  deep learning models with IoT consumer products, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', for  running face recognition with edge computing at IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Accordingly, an adversary can effortlessly obtain a copy of the  trained deep learning model and apply model inversion attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Therefore, service providers should utilize privacy-preserving  learning that adds reasonable noise to the modeling parameters  during model training according to the differential privacy  conditions [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Moreover, learning from encrypted data using  000  5  10  15  20  25  30  35  40  Number of data sensing  0  1  2  3  4  5  Total privacy cost  growing privacy  leakage gap  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='7x privacy cost  of Service 1  3x privacy cost  of Service 1  Service 1 (privacy budget=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='1)  Service 2 (privacy budget=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='15)  Service 3 (privacy budget=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='3)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Total privacy cost of repeated data sensing at various privacy budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  homomorphic encryption and secure multi-party computation  must be utilized to thwart model inversion attacks and the  reconstruction of user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Misconception 5: Decentralized IoT (DeIoT) solves the pri-  vacy problem and provides absolute data privacy preservation  Decentralized IoT (DeIoT) is an emerging user-centered  ecosystem that distributes IoT control functions and delegates  operations to users without including a central authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Edge  computing, blockchain ledgers, and federated learning are the  most promising technologies for DeIoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For example, smart  contracts and blockchain ledgers provide decentralized digital  identities that are shared with all participating devices [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' In  addition, federated learning and edge computing can optimize  a master ambient intelligence model without sharing users’  original data with a central server [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' DeIoT is widely  suggested as a method for improving data privacy, security,  transparency, and scalability using token-based operations and  decentralization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' As a result, DeIoT allows people to gain  additional control over their data collection and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Correction: Unfortunately, DeIoT does not provide absolute  data privacy preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' First, DeIoT is challenging to regu-  late, given the lack of a central moderation entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Second,  adversaries can impersonate legitimate users, and harmful  content can be injected into DeIoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Third, DeIoT  introduces new privacy and security challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For example,  blockchains and federated learning are two prominent DeIoT-  enabling technologies that raise new privacy risks in IoT  services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Blockchain uses asymmetric cryptography, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', pairs  of private and public keys, to securely store transactions  on a public ledger that all users can view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Therefore, user  identities in DeIoT can be impersonated if the user’s private  key, which is unique for each user, is stolen or hacked  by an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Then, blockchain-based DeIoT guarantees  immutability [11], indicating that IoT data cannot be altered or  deleted once a blockchain record is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Nonetheless, the  immutability of DeIoT violates the user rights in data erasure  and rectification as depicted in the General Data Protection  Regulation (GDPR) [2] and the California Consumer Privacy  Act (CCPA) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Also, adversaries can participate in federated  learning to access the final trained model, creating privacy  risks that can be exploited using model inversion attacks [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  When the privacy budget equals zero, absolute data privacy  is achieved, and differential privacy guarantees that an adver-  sary cannot identify data about individual users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' As discussed  above, DeIoT does not fulfill the requirements of absolute data  privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Assume that three services (Services 1-3) are built  using blockchain ledgers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The privacy budget of a single data  sensing is set at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='15, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='3 in Services 1-3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Figure 5 shows the total privacy costs of accessing Services 1-  3 at different privacy budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The total privacy cost indicates  the cumulative privacy budget of differential privacy for se-  quential sensing events in IoT, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', overall privacy leakage as  a function of the number of data sensing events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' The total  privacy cost is calculated using the composition theorem of  privacy-preserving mechanisms [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Two substantial results  can be underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' First, the total privacy cost increases over  repeated sensing in the three services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Therefore, although the  privacy budget is small for a single sensing event, data privacy  preservation in DeIoT is not absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Second, the difference  in the privacy cost of users magnifies over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For example,  Service 3 has 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='7x the privacy cost of Service 1 after 38 data  sensing attempts, even though the privacy budget of a single  sensing attempt in Service 3 is 3x that of Service 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  CRITICAL QUESTIONS FOR FUTURE RESEARCH  This section presents critical questions for future research  in privacy-preserving IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Data privacy and criminal justice  A  widespread  argument  for  supporting  dataveillance,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', monitoring and profiling people’s data, is for criminal  justice, law enforcement, and fraud prevention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' For example,  all United States privacy laws exclude law enforcement from  information disclosure even though the Fourth Amendment  protects individuals from unreasonable searches by the gov-  ernment without a reasonable basis [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Furthermore, privacy-  preserving IoT might be seen as a tool for boosting on-  line incivility and unfriendly behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Such arguments for  dataveillance undervalue the benefits of data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' First, data  privacy is a powerful mechanism for preventing cybercrimes  and online incivility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' In particular, when personal data is  privately reserved, people will be immune to adversaries and  disruptive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Second, data privacy is a crucial enabler  for the freedom of speech and civil liberty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Third, organized  crime has the resources and motives to create custom en-  crypted communication channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' thus, exposed services will  have the most impact on regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Social benefits do not wipe out the personal benefits of data  privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Several critical questions need further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' What  is the proper procedure for requesting data disclosure for crim-  inal justice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' How can protected data be accessed for criminal  justice without establishing an encryption backdoor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' How can  people oversee the levels of dataveillance by organizations and  governments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  User-in-the-loop (UIL) privacy-preserving IoT  IoT standards, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', IEEE 2413-2019 [6], emphasize that  user privacy and trust are essential components of any IoT  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' However, people, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', data owners, are still not well-  engaged in their privacy preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' As discussed in Miscon-  ception 2, the privacy paradox suggests doubt about the ability  of users to protect their privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Moreover, users generally  cannot verify the privacy measures taken by service providers  due to the lack of transparency in the implemented privacy  safeguards [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  User-in-the-loop (UIL) data privacy engages users in their  privacy preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' UIL is an emerging design concept in IoT  that improves performance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=', reducing overall energy con-  sumption, by engaging people in fundamental IoT operations  beyond traffic consumption and generation [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' UIL privacy-  preserving IoT has garnered limited scholarly attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Thus,  a few critical questions required further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' How can  user awareness of data privacy issues be increased?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' How can  service providers provide people with data privacy measure-  ments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' How can users be incentivized to contribute to their  data protection efforts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  CONCLUSIONS  Privacy-preserving IoT is essential given the substantial  data privacy risks and their severe impacts on users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' This  article presented five common misconceptions about privacy-  preserving IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Several results were presented using exper-  iments on real-world datasets and survey research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' First, a  trade-off exists between privacy preservation and service ac-  curacy in privacy-preserving IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Second, privacy-preserving  IoT is beneficial for user retention and overall satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Third, users are concerned about their privacy and do not  trust services that lack sufficient privacy procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Fourth,  privacy attacks may occur even when user data is securely  stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Fifth, IoT decentralization does guarantee absolute  data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' Finally, critical questions were raised for future  research in privacy-preserving IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
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+page_content=' 57, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content=' 114–121, December 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
+page_content='  Mohammad Abu Alsheikh [S’14, M’17, SM’22] is an Associate Professor  and ARC DECRA Fellow at the University of Canberra, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndAyT4oBgHgl3EQf__rr/content/2301.00920v1.pdf'}
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+arXiv:2301.01171v1  [math.AP]  3 Jan 2023
+BMO–REGULARITY FOR A DEGENERATE
+TRANSMISSION PROBLEM
+VINCENZO BIANCA, EDGARD A. PIMENTEL, AND JOS´E MIGUEL URBANO
+Abstract. We examine a transmission problem driven by a degenerate
+nonlinear operator with a natural interface condition. Two aspects of
+the problem entail genuine difficulties in the analysis: the absence of
+representation formulas for the operator and the degenerate nature of
+the diffusion process. Our arguments circumvent these difficulties and
+lead to new regularity estimates. We prove the local boundedness of
+weak solutions and establish an estimate for their gradient in BMO–
+spaces. The latter implies solutions are of class C0,Log−Lip across the
+interface.
+1. Introduction
+Transmission problems describe diffusive processes within heterogeneous
+media that change abruptly across certain interfaces.
+They find applica-
+tion, for example, in the study of electromagnetic conductivity and compos-
+ite materials, and their mathematical formulation involves a domain split
+into sub-regions, where partial differential equations (PDEs) are prescribed.
+Since the PDEs vary from region to region, the problem may have disconti-
+nuities across the interfaces. Consequently, the geometry of these interfaces
+(which, in contrast to free boundary problems, are fixed and given a priori)
+and the structure of the underlying equations play a crucial role in analysing
+transmission problems.
+This class of problems first appeared circa 1950, in the work of Mauro
+Picone [16], as an attempt to address heterogeneous materials in elasticity
+theory. Several subsequent works developed the basics of the theory and
+generalised it in various directions [4, 8, 9, 10, 12, 17, 19, 21]. We refer the
+interested reader to [5] for a comprehensive account of this literature.
+Date: January 4, 2023.
+2020 Mathematics Subject Classification. 35B65; 35J92; 35Q74.
+Key words and phrases. Transmission problems; p-Laplace operator; local bounded-
+ness; BMO gradient estimates; Log-Lipschitz regularity.
+1
+
+2
+V. BIANCA, E. A. PIMENTEL, AND J.M. URBANO
+Developments concerning the regularity of the solutions to transmission
+problems are much more recent. In [14], the authors study a class of ellip-
+tic equations in divergence form, with discontinuous coefficients, modelling
+composite materials with closely spaced interfacial boundaries, such as fibre-
+reinforced structures.
+The main result in that paper is the local H¨older
+continuity for the gradient of the solutions, with estimates. The findings
+in [14] are relevant from the applied perspective since the gradient of a so-
+lution accounts for the stress of the material, and estimating it shows the
+stresses remain uniformly bounded, even when fibres are arbitrarily close to
+each other. The vectorial counterpart of the results in [14] appeared in [13],
+where regularity estimates for higher-order derivatives of the solutions are
+obtained. See also the developments reported in [3].
+A further layer of analysis concerns the proximity of sub-regions in lim-
+iting scenarios. In [1], the authors examine a domain containing two sub-
+regions, which are ε-apart, for some ε > 0. Within each sub-region, the
+diffusion process is given by a divergence-form equation with a diffusivity
+coefficient A ̸= 1. In the remainder of the domain, the diffusivity is also
+constant but equal to 1.
+By setting A = +∞, the authors examine the
+case of perfect conductivity. The remarkable fact about this model is that
+estimates on the gradient of the solutions deteriorate as the two regions ap-
+proach each other. In [1], the authors obtain blow-up rates for the gradient
+norm in terms of ε → 0. We also notice the findings reported in [2] extend
+those results to the context of multiple inclusions and also treat the case of
+perfect insulation A = 0. We also refer the reader to [6].
+More recently, the analysis of transmission problems focused on the ge-
+ometry of the interface. The minimum requirements on the transmission
+interface yielding regularity properties for the solutions are particularly in-
+teresting. In [7], the authors consider a domain split into two sub-regions.
+Inside each sub-region, the solution of the problem is required to be a har-
+monic function, and a flux condition is prescribed along the interface sepa-
+rating the sub-regions. By resorting to a representation formula for harmonic
+functions, the authors establish the existence of solutions to the problem and
+prove that solutions are of class C0,Log−Lip across the interface. In addition,
+under the assumption that the interface is locally of class C1,α, they prove
+the solutions are of class C1,α within each sub-region, up to the transmission
+interface. This fact follows from a new stability result allowing the argument
+
+A DEGENERATE TRANSMISSION PROBLEM
+3
+to import information from the case of flat interfaces. In [20], the authors
+extend the analysis in [7] to the context of fully nonlinear elliptic operators.
+Under the assumption that the interface is of class C1,α, they prove that
+solutions are of class C0,α across, and C1,α up to the interface. Further-
+more, if the interface is of class C2,α, then solutions became C2,α-regular,
+also up to the interface. The findings in [20] rely on a new Aleksandrov-
+Bakelman-Pucci estimate and variants of the maximum principle and the
+Harnack inequality.
+Our gist in this paper is to extend the results of [7] to the nonlinear de-
+generate case of p-harmonic functions in each sub-region. We first prove
+that weak solutions, properly defined and whose existence follows from well-
+known methods, are locally bounded. The proof combines delicate inequali-
+ties with the careful choice of auxiliary test functions and a cut-off argument
+to produce a variant of the weak Harnack inequality. Working under a C1
+interface geometry, our main contribution is an integral estimate for the
+gradient, leading to regularity in BMO–spaces. As a corollary, we infer that
+solutions are of class C0,Log−Lip across the fixed interface. This transmis-
+sion problem driven by the degenerate p-Laplace operator presents genuine
+difficulties compared to the Laplacian’s linear case. Firstly, the operator
+lacks representation formulas, and the strategy developed in [7] is no longer
+available. Secondly, the degenerate nature of the problem rules out the ap-
+proach put forward in [20]. Consequently, one must develop new machinery
+to examine the regularity of the solutions.
+Another fundamental question concerns the optimal regularity up to the
+interface. As mentioned before, results of this type appear in the recent
+works [7] and [20]; see also [11]. In the context of the p-Laplacian opera-
+tor, the problem remains open. We believe the analysis of the boundary
+behaviour of p-harmonic functions may yield helpful information in this di-
+rection.
+The remainder of this article is organised as follows. Section 2 contains the
+precise formulation of the problem, details the existence of a unique solution
+and gathers basic material used in the paper. In Section 3, we put forward
+the proof of the local boundedness. The proof of the BMO–regularity and
+its consequences is the object of Section 4.
+
+4
+V. BIANCA, E. A. PIMENTEL, AND J.M. URBANO
+2. Setting of the problem and auxiliary results
+In this section, we precisely state our transmission problem, introduce the
+notion of a weak solution and prove its existence and uniqueness. We then
+collect several auxiliary results.
+Let Ω ⊂ Rd be a bounded domain and fix Ω1 ⋐ Ω. Define Ω2 := Ω \ Ω1
+and consider the interface Γ := ∂Ω1, which we assume is a (d − 1)-surface
+of class C1. For a function u : Ω → R, we set
+u1 := u
+��
+Ω1
+and
+u2 := u
+��
+Ω2.
+Note that we necessarily have u1 = u2 on Γ.
+Denoting with ν the unit
+normal vector to Γ pointing inwards to Ω1, we write
+(ui)ν = ∂ui
+∂ν = Dui · ν,
+i = 1, 2.
+For p > 2 and g ∈ L∞(Γ), we consider the problem of finding a function
+u : Ω → R satisfying
+(2.1)
+
+
+
+div(|Du1|p−2Du1) = 0
+in
+Ω1
+div(|Du2|p−2Du2) = 0
+in
+Ω2,
+and the additional conditions
+(2.2)
+
+
+
+u = 0
+on
+∂Ω
+|Du2|p−2(u2)ν − |Du1|p−2(u1)ν = g
+on
+Γ.
+The precise definition of solution we have in mind is the object of the
+following definition.
+Definition 1. A function u ∈ W 1,p
+0 (Ω) is a weak solution of (2.1)-(2.2) if
+(2.3)
+�
+Ω
+|Du|p−2Du · Dv dx =
+�
+Γ
+gv dHd−1,
+∀ v ∈ W 1,p
+0 (Ω).
+We use the Hausdorff measure Hd−1 in the surface integral to emphasise
+that ∆pu is a measure supported along the interface, and we write
+−∆pu = g dHd−1��
+Γ.
+To justify the former definition, we multiply both equations in (2.1) by a
+test function ϕ ∈ C∞
+c (Ω), and formally integrate by parts to get
+�
+Ω1
+|Du1|p−2Du1 · Dϕ dx = −
+�
+Γ
+�
+|Du1|p−2Du1 · ν
+�
+ϕ dHd−1
+
+A DEGENERATE TRANSMISSION PROBLEM
+5
+and
+�
+Ω2
+|Du2|p−2Du2 · Dϕ dx =
+�
+Γ
+�
+|Du2|p−2Du2 · ν
+�
+ϕ dHd−1.
+Adding and using (2.2), we obtain, by density,
+�
+Ω
+|Du|p−2Du · Dϕ dx =
+�
+Γ
+gϕ dHd−1,
+∀ ϕ ∈ W 1,p
+0 (Ω).
+2.1. Existence and uniqueness of a weak solution. To prove the exis-
+tence of a unique weak solution to (2.1)-(2.2), we resort to standard methods
+in the literature. Indeed, the operator A : W 1,p
+0 (Ω) → W −1,p′(Ω) defined by
+⟨Au, v⟩ :=
+�
+Ω
+|Du|p−2Du · Dv dx
+is bounded, hemicontinuous, strictly monotone and coercive, and hence it
+is bijective. Since, due to the trace theorem and Poincar´e’s inequality, the
+right-hand side in (2.3) defines an element in W −1,p′(Ω), we obtain the result.
+Additionally, we remark that the weak solution is the global minimiser of
+the functional I : W 1,p
+0 (Ω) → R defined by
+(2.4)
+I(u) = 1
+p
+�
+Ω
+|Du|p dx −
+�
+Γ
+gu dHd−1,
+whose Euler-Lagrange equation, in its weak formulation, is precisely (2.3).
+2.2. Auxiliary results. We now collect some auxiliary material which will
+be instrumental in the proofs of the main results. We start with a technical
+inequality (c.f. [18, Lemma 2]).
+Lemma 1. Let p > 0, and N ∈ N. Let also a1, . . . , aN, q1, . . . , qN be real
+numbers such that 0 < ai < ∞ and 0 ≤ qi < p, for every i = 1, . . . , N.
+Suppose that z, z1, . . . , zN are positive real numbers satisfying
+zp ≤
+N
+�
+i=1
+aizqi
+i .
+Then there exists C > 0 such that
+z ≤ C
+N
+�
+i=1
+aγi
+i
+where γi = (p − qi)−1, for i = 1, . . . , N. Finally, C = C(N, p, q1, . . . , qN).
+Although standard in the field, the following result lacks detailed proof
+in the literature. We include it here for completeness and future reference.
+
+6
+V. BIANCA, E. A. PIMENTEL, AND J.M. URBANO
+Lemma 2. Fix R0 > 0 and let φ : [0, R0] → [0, ∞) be a non-decreasing
+function. Suppose there exist constants C1, α, β > 0, and C2, µ ≥ 0, with
+β < α, satisfying
+φ(r) ≤ C1
+�� r
+R
+�α
++ µ
+�
+φ(R) + C2Rβ,
+for every 0 < r ≤ R ≤ R0.
+Then, for every σ ≤ β, there exists µ0 =
+µ0(C1, α, β, σ) such that, if µ < µ0, for every 0 < r ≤ R ≤ R0, we have
+φ(r) ≤ C3
+� r
+R
+�σ�
+φ(R) + C2Rσ�
+,
+where C3 = C3(C1, α, β, σ) > 0. Moreover,
+φ(r) ≤ C4rσ,
+where C4 = C4(C2, C3, R0, φ(R0), σ).
+Proof. For clarity, we split the proof into two steps.
+First, an induction
+argument leads to an inequality at discrete scales; then, we pass to the
+continuous case and conclude the argument.
+Step 1 - We want to verify that
+(2.5)
+φ(θn+1R) ≤ θ(n+1)δφ(R) + C2θnβRβ
+n
+�
+j=0
+θj(δ−β),
+for every n ∈ N. We notice it suffices to prove the estimate for σ = β and
+work in this setting. For 0 < θ < 1 and 0 < R ≤ R0 the assumption of the
+lemma yields
+φ(θR) ≤ C1
+��θR
+R
+�α
++ µ
+�
+φ(R) + C2Rβ = θαC1(1 + µθ−α)φ(R) + C2Rβ.
+Choose θ ∈ (0, 1) such that 2C1θα = θδ with β < δ < α. Notice that θ
+depends only on C1, α, δ. Take µ0 > 0 such that µ0θ−α < 1. For every
+R ≤ R0 we then have
+(2.6)
+φ(θR) ≤ θδφ(R) + C2Rβ
+and the base case follows. Suppose the statement has already been verified
+for some k ∈ N, k ≥ 2; then
+φ(θkR) ≤ θkδφ(R) + C2θ(k−1)βRβ
+k−1
+�
+j=0
+θj(δ−β).
+
+A DEGENERATE TRANSMISSION PROBLEM
+7
+Thanks to (2.6), we have
+φ(θk+1R) = φ
+�
+θk(θR)
+�
+≤ θkδφ(θR) + C2θ(k−1)β(θR)β
+k−1
+�
+j=0
+θj(δ−β)
+≤ θkδ�
+θδφ(R) + C2Rβ�
++ C2θkβRβ
+k−1
+�
+j=0
+θj(δ−β)
+= θ(k+1)δφ(R) + C2θkδRβ + C2θkβRβ
+k−1
+�
+j=0
+θj(δ−β)
+= θ(k+1)δφ(R) + C2θkβRβ
+k
+�
+j=0
+θj(δ−β).
+Hence, (2.5) holds for every k ∈ N, and the induction argument is complete.
+Step 2 - Next, we pass from the discrete to the continuous case. In partic-
+ular, we claim that
+φ(r) ≤ C3
+� r
+R
+�β�
+φ(R) + C2Rβ�
+,
+for every 0 < r ≤ R ≤ R0.
+Indeed,
+φ(θk+1R) ≤θ(k+1)δφ(R) + C2θkβRβ
+1
+1 − θδ−β
+=θ(k+1)δφ(R) + C2Rβ θ(k+1)β
+θβ − θδ
+≤C3θ(k+1)β�
+φ(R) + C2Rβ�
+,
+for every k ∈ N.
+Taking k ∈ N such that θk+2R ≤ r < θk+1R, up to
+relabeling the constant C3, we get
+φ(r) ≤φ(θk+1R) ≤ C3θ(k+1)β�
+φ(R) + C2Rβ�
+=C3θ(k+2)βθ−β�
+φ(R) + C2Rβ�
+≤C3
+� r
+R
+�β�
+φ(R) + C2Rβ�
+.
+Finally, one notices
+φ(r) ≤ C3
+1
+Rβ
+0
+�
+φ(R0) + C2Rβ
+0
+�
+rβ =: C4rβ,
+and the proof is complete.
+□
+
+8
+V. BIANCA, E. A. PIMENTEL, AND J.M. URBANO
+3. Local boundedness
+In this section, we prove the local boundedness for the weak solutions to
+the problem. Our argument is inspired by the one put forward in [18].
+Theorem 1 (Local Boundedness). Let u ∈ W 1,p
+0 (Ω) be the weak solu-
+tion to (2.1)-(2.2). Then for any BR := BR(x0) ⋐ Ω, there exists C =
+C
+�
+d, p, R, ∥g∥L∞(Γ)
+�
+> 0 such that
+∥u∥L∞(BR/2) ≤ CR− d
+p �
+∥u∥Lp(BR) + R
+d
+p +1∥g∥L∞(Γ)
+�
+and
+∥Du∥Lp(BR/2) ≤ CR−1�
+∥u∥Lp(BR) + R
+d
+p +1∥g∥L∞(Γ)
+�
+.
+Proof. Fix R > 0 such that BR ⋐ Ω and set k := R∥g∥L∞(Γ).
+Define
+u : Ω → R as
+u(x) := |u(x)| + k
+for all x ∈ Ω. Fix q ≥ 1 and ℓ > k. For t ∈ R, denote t := |t| + k. To ease
+the presentation, we split the remainder of the proof into four steps.
+Step 1 - Let F : [k, ∞) → R be defined as
+F(s) :=
+
+
+
+sq
+if
+k ≤ s ≤ ℓ
+qℓq−1s − (q − 1)ℓq
+if
+ℓ < s.
+Then F ∈ C1�
+[k, ∞)
+�
+and F ∈ C∞�
+[k, ∞) \ {ℓ}
+�
+. Let G : R → R be defined
+as
+G(t) := sgn(t)
+�
+F(t)F ′(t)p−1 − qp−1kβ�
+,
+∀t ∈ R,
+where β = p(q − 1) + 1 > 1. A simple computation yields
+G′(t) =
+
+
+
+q−1βF ′(t)p
+if
+|t| < ℓ − k
+F ′(t)p
+if
+|t| > ℓ − k.
+Notice that
+|G(u)| ≤ F(u)F ′(u)p−1
+and
+uF ′(u) ≤ qF(u).
+Step 2 - In this step, we introduce auxiliary test functions, which build upon
+the former inequalities. Fix 0 < r < R. Let η ∈ C∞
+c (BR), 0 ≤ η ≤ 1, η = 1
+in Br, |Dη| ≤ (R − r)−1. Let v = ηpG(u). Since G ∈ C1�
+R \ {±(ℓ − k)}
+�
+is
+
+A DEGENERATE TRANSMISSION PROBLEM
+9
+continuous, with bounded derivative, it follows that G(u) ∈ W 1,p(Ω). Hence
+v is an admissible test function. We have
+Dv =
+
+
+
+pηp−1G(u)Dη + ηpG′(u)Du
+if
+u ̸= ±(ℓ − k)
+pηp−1G(u)Dη
+if
+u = ±(ℓ − k).
+Set w(x) = F
+�
+u(x)
+�
+. Notice that q−1β ≥ 1; hence G′(u) ≤ q−1βF ′(u)p.
+Notice also that |Du| = |Du|.
+Using the trace theorem and the Poincar´e inequality, we get
+(3.1)
+�
+Ω
+|Du|p−2Du · Dv dx ≤ ∥g∥L∞(Γ)
+�
+Γ
+|v| dHd−1 ≤ C
+�
+Ω
+|Dv| dx.
+Now we estimate the left-hand side of (3.1) from below. We get
+�
+B1
+|Du|p−2Du · Dv dx =
+�
+B1
+|Du|p−2Du ·
+�
+pηp−1G(u)Dη + ηpG′(u)Du
+�
+dx
+=p
+�
+B1
+ηp−1G(u)|Du|p−2Du · Dη dx
++
+�
+B1
+ηpG′(u)|Du|p dx
+≥ − p
+�
+B1
+ηp−1F(u)F ′(u)p−1|Du|p−1|Dη| dx
++
+�
+B1
+ηpF ′(u)p|Du|p dx
+= − p
+�
+B1
+ηp−1w|Dw|p−1|Dη| dx
++
+�
+B1
+ηp|Dw|p dx
+≥ − p∥wDη∥Lp(B1)∥ηDw∥p−1
+Lp(B1) + ∥ηDw∥p
+Lp(B1).
+(3.2)
+We also control the right-hand side of (3.1) by computing
+C
+�
+B1
+|Dv| dx =C
+�
+B1
+up−1
+up−1 |pηp−1G(u)Dη + ηpG′(u)Du| dx
+≤Ck1−pp
+�
+B1
+up−1ηp−1|G(u)Dη| dx
++ Ck1−p
+�
+B1
+up−1ηpG′(u)|Du| dx
+≤C
+�
+B1
+up−1ηp−1F(u)F ′(u)p−1|Dη| dx
+
+10
+V. BIANCA, E. A. PIMENTEL, AND J.M. URBANO
++ Cq−1β
+�
+B1
+up−1ηpF ′(u)p|Du| dx
+≤C
+�
+B1
+ηp−1qp−1F(u)p−1F(u)|Dη| dx
++ Cq−1β
+�
+B1
+qp−1F(u)p−1ηpF ′(u)|Du| dx
+=Cqp−1
+�
+B1
+(ηw)p−1w|Dη| dx
++ Cqp−2β
+�
+B1
+(ηw)p−1η|Dw| dx
+≤Cqp−1∥ηw∥p−1
+Lp(B1)∥wDη∥Lp(B1)
++ Cqp−2β∥ηw∥p−1
+Lp(B1)∥ηDw∥Lp(B1).
+(3.3)
+From (3.1), combining (3.3) with (3.2), we get
+∥ηDw∥p
+Lp(Ω) ≤p∥wDη∥Lp(Ω)∥ηDw∥p−1
+Lp(Ω)
++ Cqp−1∥ηw∥p−1
+Lp(Ω)∥wDη∥Lp(Ω)
++ Cqp−1∥ηw∥p−1
+Lp(Ω)∥ηDw∥Lp(Ω),
+(3.4)
+where we have used
+β = pq − p + 1 ≤ pq − p + q ≤ pq + q = (p + 1)q.
+Step 3 - Set
+z = ∥ηDw∥Lp(Ω)
+∥wDη∥Lp(Ω)
+,
+ζ =
+∥ηw∥Lp(Ω)
+∥wDη∥Lp(Ω)
+.
+By dividing (3.4) for ∥wDη∥p
+Lp(Ω), we have
+zp ≤pzp−1 + Cqp−1 ∥ηw∥p−1
+Lp(Ω)
+∥wDη∥p−1
+Lp(Ω)
++ Cqp−1 ∥ηw∥p−1
+Lp(Ω)
+∥wDη∥p−1
+Lp(Ω)
+∥ηDw∥Lp(Ω)
+∥wDη∥Lp(Ω)
+=pzp−1 + Cqp−1ζp−1 + Cqp−1ζp−1z.
+An application of Lemma 1, implies
+z ≤ C
+�
+p + q
+p−1
+p ζ
+p−1
+p
++ qζ
+�
+≤ Cq(1 + ζ),
+giving
+(3.5)
+∥ηDw∥Lp(Ω) ≤ Cq
+�
+∥ηw∥Lp(Ω) + ∥wDη∥Lp(Ω)
+�
+.
+
+A DEGENERATE TRANSMISSION PROBLEM
+11
+Using the Sobolev inequality, we get
+∥ηw∥Lp∗ (Ω) ≤C∥D(ηw)∥Lp(Ω)
+≤C
+�
+∥wDη∥Lp(Ω) + ∥ηDw∥Lp(Ω)
+�
+≤C
+�
+∥wDη∥Lp(Ω) + Cq
+�
+∥ηw∥Lp(Ω) + ∥wDη∥Lp(Ω)
+��
+and so
+(3.6)
+∥ηw∥Lp∗(Ω) ≤ Cq
+�
+∥ηw∥Lp(Ω) + ∥wDη∥Lp(Ω)
+�
+.
+Recall that η = 1 in Br and |Dη| ≤ (R − r)−1. Hence, (3.5) becomes
+∥Dw∥Lp(Br) ≤Cq
+�� �
+BR
+wp dx
+� 1
+p
++
+1
+R − r
+� �
+BR
+wp dx
+� 1
+p
+�
+=Cq∥w∥Lp(BR)
+�
+1 +
+1
+R − r
+�
+=CqR − r + 1
+R − r
+∥w∥Lp(BR)
+≤Cqdiam(B1) + 1
+R − r
+∥w∥Lp(BR)
+≤Cq
+1
+R − r∥w∥Lp(BR).
+Similarly, (3.6) becomes
+(3.7)
+∥w∥Lp∗(Br) ≤ Cq
+1
+R − r∥w∥Lp(BR).
+We claim that Fℓ ≤ Fℓ+1, for every ℓ ∈ N, ℓ > k. The only non-trivial case
+is when ℓ < t ≤ ℓ + 1. In this case, we have
+Fℓ(t) = qℓq−1t − (q − 1)ℓq
+and
+Fℓ+1(t) = tq.
+Let f : (ℓ, ℓ + 1] → R be defined by
+f(t) = tq − qℓq−1t + (q − 1)ℓq.
+We have f ′(t) = qtq−1 − qℓq−1 > 0, for every t ∈ (ℓ, ℓ + 1], and hence f is an
+increasing function. Since limt→ℓ f(t) = 0, we have f ≥ 0 in (ℓ, ℓ + 1], and
+so Fℓ ≤ Fℓ+1. Letting ℓ → ∞ in (3.7), since 0 ≤ Fℓ ≤ Fℓ+1 for every ℓ ∈ N,
+
+12
+V. BIANCA, E. A. PIMENTEL, AND J.M. URBANO
+ℓ > k, by the Monotone Convergence Theorem, we obtain
+� �
+Br
+uqp∗ dx
+� 1
+p∗
+≤ Cq
+1
+R − r
+� �
+BR
+uqp dx
+� 1
+p
+.
+Set
+s := qp
+and
+γ := p∗/p = d/(d − p);
+then
+� �
+Br
+usγ dx
+� 1
+pγ
+≤ Cq
+1
+R − r
+� �
+BR
+us dx
+� 1
+p
+.
+Raising both sides of the previous inequality to p/s, one gets
+(3.8)
+� �
+Br
+usγ dx
+� 1
+sγ
+≤ C
+p
+s
+�s
+p
+� p
+s �
+1
+R − r
+� p
+s
+� �
+BR
+us dx
+� 1
+s
+.
+Set sj = sγj and rj = r + 2−j(R − r), for every j ∈ N0. Iterating (3.8),
+which holds for every s ≥ p, we have
+� �
+Brj+1
+usjγ dx
+�
+1
+sjγ
+≤C
+p
+sj
+�sj
+p
+� p
+sj 2
+p
+sj (j+1)�
+1
+R − r
+� p
+sj
+� �
+Brj
+usj dx
+� 1
+sj
+=C
+p
+sj−1γ
+�sj−1γ
+p
+�
+p
+sj−1γ
+2
+p
+sj−1γ (j+1)�
+1
+R − r
+�
+p
+sj−1γ
+×
+� �
+Brj
+usj−1γ dx
+�
+1
+sj−1γ
+≤C(j, p, s, d)
+�
+1
+R − r
+� p
+s
+�j
+k=0 γ−k� �
+BR
+us dx
+� 1
+s
+,
+where
+C(j, p, s, d) := C
+p
+s
+�j
+k=0 γ−k�s
+p
+� p
+s
+�j
+k=0 γ−k
+γ
+p
+s
+�j
+k=0 kγ−k2
+p
+s
+�j
+k=0(k+1)γ−k.
+Notice that r < rj, for every j ∈ N0, the series are convergent and in
+particular �∞
+k=0 γ−k = d/p. By letting j → ∞, we get
+(3.9)
+sup
+Br
+u ≤ C
+�
+1
+(R − r)d
+�
+BR
+us dx
+� 1
+s
+.
+Step 4 - Now, we can choose some parameters in the former inequalities to
+complete the proof. By choosing q = 1, setting r := R/2, and recalling that
+u = |u| + k, we get
+∥u∥L∞(BR/2) ≤∥u∥L∞(BR/2) ≤ CR− d
+p �
+∥u∥Lp(BR) + R
+d
+p k
+�
+.
+
+A DEGENERATE TRANSMISSION PROBLEM
+13
+The second inequality in the theorem follows by setting q = 1 and r := R/2
+in (3.7), obtaining
+∥Du∥Lp(BR/2) =∥Du∥Lp(BR/2)
+(3.10)
+≤CR−1∥u∥Lp(BR)
+≤CR−1�
+∥u∥Lp(BR) + ∥k∥Lp(BR)
+�
+≤CR−1�
+∥u∥Lp(BR) + R
+d
+p k
+�
+.
+□
+4. Gradient regularity estimates in BMO–spaces
+In this section, we prove the main result of this paper. We start with
+a lemma, where we denote by W 1,p
+f
+(Ω) the affine space of functions w ∈
+W 1,p(Ω) such that
+w − f ∈ W 1,p
+0 (Ω).
+Lemma 3. Let w ∈ W 1,p(BR). Let also h ∈ W 1,p
+w (BR) be such that ∆ph = 0
+in BR, in the weak sense. Then there exists C = C(d, p) > 0 such that
+�
+BR
+|Dw|p − |Dh|p dx ≥ C
+�
+BR
+|D(w − h)|p dx.
+Proof. Fix τ ∈ [0, 1] and define vτ := τw + (1 − τ)h. We have
+�
+BR
+|Dw|p − |Dh|p dx =
+� 1
+0
+d
+dτ
+� �
+BR
+|Dvτ|p dx
+�
+dτ
+=p
+� 1
+0
+�
+BR
+|Dvτ|p−2Dvτ · D(w − h) dx dτ.
+(4.1)
+Since h ∈ W 1,p
+w (BR) and ∆ph = 0 in BR, we also get
+�
+BR
+|Dh|p−2Dh · D(w − h) dx = 0.
+Hence,
+p
+� 1
+0
+�
+BR
+|Dvτ|p−2Dvτ · D(w − h) dx dτ
+=p
+� 1
+0
+�
+BR
+(|Dvτ|p−2Dvτ − |Dh|p−2Dh) · D(w − h) dx dτ
+=p
+� 1
+0
+1
+τ
+�
+BR
+(|Dvτ|p−2Dvτ − |Dh|p−2Dh) · D(vτ − h) dx dτ,
+(4.2)
+
+14
+V. BIANCA, E. A. PIMENTEL, AND J.M. URBANO
+where the last equality relies on the fact that vτ −h = τ(w −h). Combining
+(4.1) and (4.2), and using a standard monotonicity property (recalling p >
+2), we obtain
+�
+BR
+|Dw|p − |Dh|p dx ≥C
+� 1
+0
+1
+τ
+�
+BR
+|D(vτ − h)|p dx dτ
+=C
+� 1
+0
+τ p−1 dτ
+�
+BR
+|D(w − h)|p dx
+=C
+�
+BR
+|D(w − h)|p dx,
+and the proof is complete.
+□
+The following result concerns the decay of the excess of the gradient of
+p-harmonic functions with respect to its average. Its proof can be found in
+[15, Lemma 5.1].
+Proposition 1. Let h ∈ W 1,p(BR) be a weak solution of the p-Laplace
+equation in BR. Then there exist constants C = C(d, p) > 0 and α ∈ (0, 1)
+such that, for every r ∈ (0, R], we have
+�
+Br
+|Dh − (Dh)r|p dx ≤ C
+� r
+R
+�d+pα �
+BR
+|Dh − (Dh)R|p dx.
+The next proposition provides a control on the decay of the excess for
+arbitrary Sobolev functions.
+Proposition 2. Let w ∈ W 1,p(BR). Let also h ∈ W 1,p
+w (BR) satisfy ∆ph = 0
+in BR, in the weak sense. Then there exists C = C(d, p) > 0 such that, for
+every 0 < r ≤ R, we have
+�
+Br
+|Dw − (Dw)r|p dx ≤C
+� r
+R
+�d+pα �
+BR
+|Dw − (Dw)R|p dx
++ C
+�
+BR
+|Dw − Dh|p dx,
+where α ∈ (0, 1) is the same as in Proposition 1.
+Proof. Let r ∈ (0, R]. We have
+�
+Br
+|Dw − (Dw)r|p dx ≤2p−1
+�
+Br
+|Dw − (Dh)r|p dx
++ 2p−1
+�
+Br
+|(Dw)r − (Dh)r|p dx.
+(4.3)
+
+A DEGENERATE TRANSMISSION PROBLEM
+15
+Similarly,
+�
+Br
+|Dw − (Dh)r|p dx ≤2p−1
+�
+Br
+|Dw − Dh|p dx
++ 2p−1
+�
+Br
+|Dh − (Dh)r|p dx.
+(4.4)
+Applying H¨older’s inequality, we get
+�
+Br
+|(Dw)r − (Dh)r|p dx =|Br|
+����
+1
+|Br|
+�
+Br
+Dw − Dh dx
+����
+p
+≤|Br|1−p
+�
+|Br|
+p−1
+p
+� �
+Br
+|Dw − Dh|p dx
+� 1
+p
+�p
+=
+�
+Br
+|Dw − Dh|p dx.
+(4.5)
+Combining (4.3), (4.4) and (4.5), we obtain
+�
+Br
+|Dw − (Dw)r|p dx ≤C
+�
+Br
+|Dh − (Dh)r|p dx
++ C
+�
+Br
+|Dw − Dh|p dx.
+Changing the roles of w and h, and integrating in the ball BR, we get
+�
+BR
+|Dh − (Dh)R|p dx ≤C
+�
+BR
+|Dw − (Dw)R|p dx
++ C
+�
+BR
+|Dw − Dh|p dx.
+(4.6)
+Now, Proposition 1 implies
+�
+Br
+|Dw − (Dw)r|p dx ≤C
+� r
+R
+�d+pα �
+BR
+|Dh − (Dh)R|p dx
++ C
+�
+BR
+|Dw − Dh|p dx.
+(4.7)
+Combining (4.6) with (4.7), we finally get
+�
+Br
+|Dw − (Dw)r|p dx ≤C
+� r
+R
+�d+pα �
+BR
+|Dw − (Dw)R|p dx
++ C
+� r
+R
+�d+pα �
+BR
+|Dw − Dh|p dx
++ C
+�
+BR
+|Dw − Dh|p dx
+
+16
+V. BIANCA, E. A. PIMENTEL, AND J.M. URBANO
+≤C
+� r
+R
+�d+pα �
+BR
+|Dw − (Dw)R|p dx
++ C
+�
+BR
+|Dw − Dh|p dx
+and the proof is complete.
+□
+We now state and prove our main theorem.
+Theorem 2 (Gradient regularity in BMO–spaces). Let u ∈ W 1,p
+0 (Ω) be the
+weak solution to (2.1)-(2.2). Then Du ∈ BMOloc(Ω). Moreover, for every
+Ω′ ⋐ Ω,
+∥Du∥BMO(Ω′) ≤ C,
+where C = C(p, d, ∥g∥L∞(Γ), diam(Ω), dist(Ω′, ∂Ω)) > 0.
+Proof. Let x0 ∈ Γ, and let R > 0 be such that BR := BR(x0) ⋐ Ω. Let
+h ∈ W 1,p
+u (BR) be the weak solution of ∆ph = 0 in BR. Since h = u on ∂BR
+in the trace sense, we can extend h in Ω \ BR such that h = u in Ω \ BR.
+This implies that h ∈ W 1,p
+0 (Ω) and hence, since u is the global minimizer of
+(2.4), we have
+1
+p
+�
+Ω
+|Du|p dx −
+�
+Γ
+gu dHd−1 ≤ 1
+p
+�
+Ω
+|Dh|p dx −
+�
+Γ
+gh dHd−1.
+(4.8)
+Set ΓR = BR ∩ Γ. Since h = u in Ω \ BR, (4.8) becomes
+1
+p
+�
+BR
+|Du|p dx −
+�
+ΓR
+gu dHd−1 ≤ 1
+p
+�
+BR
+|Dh|p dx −
+�
+ΓR
+gh dHd−1
+from which, applying the Trace Theorem, H¨older’s inequality and Poincar´e’s
+inequality, follows
+1
+p
+�
+BR
+|Du|p dx − 1
+p
+�
+BR
+|Dh|p dx ≤
+�
+ΓR
+gu dHd−1 −
+�
+ΓR
+gh dHd−1
+≤∥g∥L∞(Γ)
+�
+ΓR
+|u − h| dHd−1
+≤C
+�
+BR
+|u − h| dx + C
+�
+BR
+|D(u − h)| dx
+≤CRd p−1
+p ∥u − h∥Lp(BR)
++ CRd p−1
+p ∥D(u − h)∥Lp(BR)
+≤CRd p−1
+p ∥D(u − h)∥Lp(BR).
+(4.9)
+
+A DEGENERATE TRANSMISSION PROBLEM
+17
+Let us consider the left-hand side of (4.9). Using Lemma 3, we get
+1
+p
+�
+BR
+|Du|p dx − 1
+p
+�
+BR
+|Dh|p dx ≥C(d, p)
+�
+BR
+|Du − Dh|p dx
+=C∥D(u − h)∥p
+Lp(BR).
+(4.10)
+Combining now (4.9) with (4.10), we get
+∥D(u − h)∥p−1
+Lp(BR) ≤ CRd p−1
+p
+and, raising both sides to p/(p − 1), we obtain
+�
+BR
+|D(u − h)|p dx ≤ CRd.
+From Proposition 2, we get
+�
+Br
+|Du−(Du)r|p dx ≤ C
+� r
+R
+�d+pα �
+BR
+|Du−(Du)R|p dx+CRd, ∀r ∈ (0, R]
+and, applying Lemma 2, we reach
+�
+Br
+|Du − (Du)r|p dx ≤ Crd,
+∀r ∈ (0, R].
+Hence, Du ∈ BMOloc(Ω) and the proof is complete.
+□
+As a corollary to Theorem 2, we obtain a modulus of continuity for the
+solution u in C0,Log−Lip-spaces.
+Corollary 1 (Log-Lipschitz continuity estimates). Let u ∈ W 1,p
+0 (Ω) be the
+weak solution to problem (2.1)-(2.2). Then u ∈ C0,Log−Lip
+loc
+(Ω). Moreover,
+for every Ω′ ⋐ Ω,
+∥u∥C0,Log−Lip(Ω′) ≤ C
+�
+∥u∥L∞(Ω) + ∥g∥L∞(Γ)
+�
+,
+where C = C(p, d, diam(Ω), dist(Ω′, ∂Ω)) > 0.
+Indeed, a function whose partial derivatives are in BMO belongs to the
+Zygmund class (cf. [22]). Because functions in the latter have a C0,Log−Lip
+modulus of continuity, the corollary follows.
+Acknowledgments. VB supported by the Centre for Mathematics of the Univer-
+sity of Coimbra (UIDB/00324/2020, funded by the Portuguese Government through
+FCT/MCTES).
+
+18
+V. BIANCA, E. A. PIMENTEL, AND J.M. URBANO
+EP partially supported by the Centre for Mathematics of the University of Coimbra
+(UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES)
+and by FAPERJ (grants E26/200.002/2018 and E26/201.390/2021).
+JMU partially supported by the King Abdullah University of Science and Tech-
+nology (KAUST) and by the Centre for Mathematics of the University of Coimbra
+(UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES).
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+lineari del tipo dell’elasticit`a. I. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 13:223–
+258, 1959.
+[10] Sergio Campanato. Sui problemi al contorno per sistemi di equazioni differenziali
+lineari del tipo dell’elasticit`a. II. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 13:275–
+302, 1959.
+[11] Hongjie Dong. A simple proof of regularity for C1,α interface transmission problems.
+arXiv Preprint, arxiv.2004.09365, 2020.
+[12] Vladimir A. Il’in and Il’ya A. ˇSiˇsmarev. The method of potentials for the problems
+of Dirichlet and Neumann in the case of equations with discontinuous coefficients.
+Sibirsk. Mat. ˇZ., pages 46–58, 1961.
+
+A DEGENERATE TRANSMISSION PROBLEM
+19
+[13] YanYan Li and Louis Nirenberg. Estimates for elliptic systems from composite ma-
+terial. Comm. Pure Appl. Math., 56(7):892–925, 2003. Dedicated to the memory of
+J¨urgen K. Moser.
+[14] YanYan Li and Michael Vogelius. Gradient estimates for solutions to divergence
+form elliptic equations with discontinuous coefficients. Arch. Ration. Mech. Anal.,
+153(2):91–151, 2000.
+[15] Gary Lieberman. The natural generalisation of the natural conditions of Ladyzhen-
+skaya and Ural’tseva for elliptic equations. Comm. in Partial Differential Equations,
+16(2-3):311–361, 1991.
+[16] Mauro Picone. Sur un probl`eme nouveau pour l’´equation lin´eaire aux d´eriv´ees
+partielles de la th´eorie math´ematique classique de l’´elasticit´e. In Colloque sur les
+´equations aux d´eriv´ees partielles, CBRM, Bruxelles, pages 9–11, 1954.
+[17] Martin Schechter. A generalization of the problem of transmission. Ann. Scuola Norm.
+Sup. Pisa Cl. Sci. (3), 14:207–236, 1960.
+[18] James Serrin. Local behaviour of solutions of quasi-linear equations. Acta Math.,
+111:247–302, 1964.
+[19] Zinovi G. ˇSeftel’. Estimates in Lp of solutions of elliptic equations with discontinuous
+coefficients and satisfying general boundary conditions and conjugacy conditions.
+Soviet Math. Dokl., 4:321–324, 1963.
+[20] Mar´ıa Soria-Carro and Pablo R. Stinga, Regularity of viscosity solutions to fully
+nonlinear elliptic transmission problems. arXiv Preprint, arxiv.org/abs/2207.13772,
+2022.
+[21] Guido Stampacchia. Su un problema relativo alle equazioni di tipo ellittico del secondo
+ordine. Ricerche Mat., 5:3–24, 1956.
+[22] Antoni Zygmund. Trigonometric series. Vol. I, II. Cambridge University Press, Cam-
+bridge, 2002.
+University of Coimbra, CMUC, Department of Mathematics, 3001-501 Coim-
+bra, Portugal
+Email address: vincenzo@mat.uc.pt
+University of Coimbra, CMUC, Department of Mathematics, 3001-501 Coim-
+bra, Portugal and Pontifical Catholic University of Rio de Janeiro – PUC-
+Rio, 22451-900 G´avea, Rio de Janeiro-RJ, Brazil
+Email address: edgard.pimentel@mat.uc.pt
+King Abdullah University of Science and Technology (KAUST), Computer,
+Electrical and Mathematical Sciences and Engineering Division (CEMSE),
+Thuwal 23955-6900, Saudi Arabia and University of Coimbra, CMUC, Depart-
+ment of Mathematics, 3001-501 Coimbra, Portugal
+Email address: miguel.urbano@kaust.edu.sa
+
diff --git a/p9AzT4oBgHgl3EQfOvtl/content/tmp_files/load_file.txt b/p9AzT4oBgHgl3EQfOvtl/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf,len=527
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='01171v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='AP]  3 Jan 2023  BMO–REGULARITY FOR A DEGENERATE  TRANSMISSION PROBLEM  VINCENZO BIANCA, EDGARD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' PIMENTEL, AND JOS´E MIGUEL URBANO  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We examine a transmission problem driven by a degenerate  nonlinear operator with a natural interface condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Two aspects of  the problem entail genuine difficulties in the analysis: the absence of  representation formulas for the operator and the degenerate nature of  the diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Our arguments circumvent these difficulties and  lead to new regularity estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We prove the local boundedness of  weak solutions and establish an estimate for their gradient in BMO–  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' The latter implies solutions are of class C0,Log−Lip across the  interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Introduction  Transmission problems describe diffusive processes within heterogeneous  media that change abruptly across certain interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  They find applica-  tion, for example, in the study of electromagnetic conductivity and compos-  ite materials, and their mathematical formulation involves a domain split  into sub-regions, where partial differential equations (PDEs) are prescribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Since the PDEs vary from region to region, the problem may have disconti-  nuities across the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Consequently, the geometry of these interfaces  (which, in contrast to free boundary problems, are fixed and given a priori)  and the structure of the underlying equations play a crucial role in analysing  transmission problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  This class of problems first appeared circa 1950, in the work of Mauro  Picone [16], as an attempt to address heterogeneous materials in elasticity  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Several subsequent works developed the basics of the theory and  generalised it in various directions [4, 8, 9, 10, 12, 17, 19, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We refer the  interested reader to [5] for a comprehensive account of this literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Date: January 4, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' 35B65;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' 35J92;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' 35Q74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Transmission problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' p-Laplace operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' local bounded-  ness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' BMO gradient estimates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Log-Lipschitz regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  1  2  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' BIANCA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' PIMENTEL, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' URBANO  Developments concerning the regularity of the solutions to transmission  problems are much more recent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' In [14], the authors study a class of ellip-  tic equations in divergence form, with discontinuous coefficients, modelling  composite materials with closely spaced interfacial boundaries, such as fibre-  reinforced structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  The main result in that paper is the local H¨older  continuity for the gradient of the solutions, with estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' The findings  in [14] are relevant from the applied perspective since the gradient of a so-  lution accounts for the stress of the material, and estimating it shows the  stresses remain uniformly bounded, even when fibres are arbitrarily close to  each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' The vectorial counterpart of the results in [14] appeared in [13],  where regularity estimates for higher-order derivatives of the solutions are  obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' See also the developments reported in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  A further layer of analysis concerns the proximity of sub-regions in lim-  iting scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' In [1], the authors examine a domain containing two sub-  regions, which are ε-apart, for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Within each sub-region, the  diffusion process is given by a divergence-form equation with a diffusivity  coefficient A ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' In the remainder of the domain, the diffusivity is also  constant but equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  By setting A = +∞, the authors examine the  case of perfect conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' The remarkable fact about this model is that  estimates on the gradient of the solutions deteriorate as the two regions ap-  proach each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' In [1], the authors obtain blow-up rates for the gradient  norm in terms of ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We also notice the findings reported in [2] extend  those results to the context of multiple inclusions and also treat the case of  perfect insulation A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We also refer the reader to [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  More recently, the analysis of transmission problems focused on the ge-  ometry of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' The minimum requirements on the transmission  interface yielding regularity properties for the solutions are particularly in-  teresting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' In [7], the authors consider a domain split into two sub-regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Inside each sub-region, the solution of the problem is required to be a har-  monic function, and a flux condition is prescribed along the interface sepa-  rating the sub-regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' By resorting to a representation formula for harmonic  functions, the authors establish the existence of solutions to the problem and  prove that solutions are of class C0,Log−Lip across the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' In addition,  under the assumption that the interface is locally of class C1,α, they prove  the solutions are of class C1,α within each sub-region, up to the transmission  interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' This fact follows from a new stability result allowing the argument  A DEGENERATE TRANSMISSION PROBLEM  3  to import information from the case of flat interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' In [20], the authors  extend the analysis in [7] to the context of fully nonlinear elliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Under the assumption that the interface is of class C1,α, they prove that  solutions are of class C0,α across, and C1,α up to the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Further-  more, if the interface is of class C2,α, then solutions became C2,α-regular,  also up to the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' The findings in [20] rely on a new Aleksandrov-  Bakelman-Pucci estimate and variants of the maximum principle and the  Harnack inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Our gist in this paper is to extend the results of [7] to the nonlinear de-  generate case of p-harmonic functions in each sub-region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We first prove  that weak solutions, properly defined and whose existence follows from well-  known methods, are locally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' The proof combines delicate inequali-  ties with the careful choice of auxiliary test functions and a cut-off argument  to produce a variant of the weak Harnack inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Working under a C1  interface geometry, our main contribution is an integral estimate for the  gradient, leading to regularity in BMO–spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' As a corollary, we infer that  solutions are of class C0,Log−Lip across the fixed interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' This transmis-  sion problem driven by the degenerate p-Laplace operator presents genuine  difficulties compared to the Laplacian’s linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Firstly, the operator  lacks representation formulas, and the strategy developed in [7] is no longer  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Secondly, the degenerate nature of the problem rules out the ap-  proach put forward in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Consequently, one must develop new machinery  to examine the regularity of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Another fundamental question concerns the optimal regularity up to the  interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' As mentioned before, results of this type appear in the recent  works [7] and [20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' see also [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' In the context of the p-Laplacian opera-  tor, the problem remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We believe the analysis of the boundary  behaviour of p-harmonic functions may yield helpful information in this di-  rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  The remainder of this article is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Section 2 contains the  precise formulation of the problem, details the existence of a unique solution  and gathers basic material used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' In Section 3, we put forward  the proof of the local boundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' The proof of the BMO–regularity and  its consequences is the object of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  4  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' BIANCA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' PIMENTEL, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' URBANO  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Setting of the problem and auxiliary results  In this section, we precisely state our transmission problem, introduce the  notion of a weak solution and prove its existence and uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We then  collect several auxiliary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Let Ω ⊂ Rd be a bounded domain and fix Ω1 ⋐ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Define Ω2 := Ω \\ Ω1  and consider the interface Γ := ∂Ω1, which we assume is a (d − 1)-surface  of class C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' For a function u : Ω → R, we set  u1 := u  ��  Ω1  and  u2 := u  ��  Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Note that we necessarily have u1 = u2 on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Denoting with ν the unit  normal vector to Γ pointing inwards to Ω1, we write  (ui)ν = ∂ui  ∂ν = Dui · ν,  i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  For p > 2 and g ∈ L∞(Γ), we consider the problem of finding a function  u : Ω → R satisfying  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1)  \uf8f1  \uf8f2  \uf8f3  div(|Du1|p−2Du1) = 0  in  Ω1  div(|Du2|p−2Du2) = 0  in  Ω2,  and the additional conditions  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='2)  \uf8f1  \uf8f2  \uf8f3  u = 0  on  ∂Ω  |Du2|p−2(u2)ν − |Du1|p−2(u1)ν = g  on  Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  The precise definition of solution we have in mind is the object of the  following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' A function u ∈ W 1,p  0 (Ω) is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='2) if  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='3)  �  Ω  |Du|p−2Du · Dv dx =  �  Γ  gv dHd−1,  ∀ v ∈ W 1,p  0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  We use the Hausdorff measure Hd−1 in the surface integral to emphasise  that ∆pu is a measure supported along the interface, and we write  −∆pu = g dHd−1��  Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  To justify the former definition, we multiply both equations in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1) by a  test function ϕ ∈ C∞  c (Ω), and formally integrate by parts to get  �  Ω1  |Du1|p−2Du1 · Dϕ dx = −  �  Γ  �  |Du1|p−2Du1 · ν  �  ϕ dHd−1  A DEGENERATE TRANSMISSION PROBLEM  5  and  �  Ω2  |Du2|p−2Du2 · Dϕ dx =  �  Γ  �  |Du2|p−2Du2 · ν  �  ϕ dHd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Adding and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='2), we obtain, by density,  �  Ω  |Du|p−2Du · Dϕ dx =  �  Γ  gϕ dHd−1,  ∀ ϕ ∈ W 1,p  0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Existence and uniqueness of a weak solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' To prove the exis-  tence of a unique weak solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='2), we resort to standard methods  in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Indeed, the operator A : W 1,p  0 (Ω) → W −1,p′(Ω) defined by  ⟨Au, v⟩ :=  �  Ω  |Du|p−2Du · Dv dx  is bounded, hemicontinuous, strictly monotone and coercive, and hence it  is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Since, due to the trace theorem and Poincar´e’s inequality, the  right-hand side in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='3) defines an element in W −1,p′(Ω), we obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Additionally, we remark that the weak solution is the global minimiser of  the functional I : W 1,p  0 (Ω) → R defined by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='4)  I(u) = 1  p  �  Ω  |Du|p dx −  �  Γ  gu dHd−1,  whose Euler-Lagrange equation, in its weak formulation, is precisely (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Auxiliary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We now collect some auxiliary material which will  be instrumental in the proofs of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We start with a technical  inequality (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' [18, Lemma 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let p > 0, and N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let also a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' , aN, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' , qN be real  numbers such that 0 < ai < ∞ and 0 ≤ qi < p, for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Suppose that z, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' , zN are positive real numbers satisfying  zp ≤  N  �  i=1  aizqi  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Then there exists C > 0 such that  z ≤ C  N  �  i=1  aγi  i  where γi = (p − qi)−1, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Finally, C = C(N, p, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' , qN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Although standard in the field, the following result lacks detailed proof  in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We include it here for completeness and future reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  6  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' BIANCA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' PIMENTEL, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' URBANO  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Fix R0 > 0 and let φ : [0, R0] → [0, ∞) be a non-decreasing  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Suppose there exist constants C1, α, β > 0, and C2, µ ≥ 0, with  β < α, satisfying  φ(r) ≤ C1  �� r  R  �α  + µ  �  φ(R) + C2Rβ,  for every 0 < r ≤ R ≤ R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Then, for every σ ≤ β, there exists µ0 =  µ0(C1, α, β, σ) such that, if µ < µ0, for every 0 < r ≤ R ≤ R0, we have  φ(r) ≤ C3  � r  R  �σ�  φ(R) + C2Rσ�  ,  where C3 = C3(C1, α, β, σ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Moreover,  φ(r) ≤ C4rσ,  where C4 = C4(C2, C3, R0, φ(R0), σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' For clarity, we split the proof into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  First, an induction  argument leads to an inequality at discrete scales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' then, we pass to the  continuous case and conclude the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Step 1 - We want to verify that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='5)  φ(θn+1R) ≤ θ(n+1)δφ(R) + C2θnβRβ  n  �  j=0  θj(δ−β),  for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We notice it suffices to prove the estimate for σ = β and  work in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' For 0 < θ < 1 and 0 < R ≤ R0 the assumption of the  lemma yields  φ(θR) ≤ C1  ��θR  R  �α  + µ  �  φ(R) + C2Rβ = θαC1(1 + µθ−α)φ(R) + C2Rβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Choose θ ∈ (0, 1) such that 2C1θα = θδ with β < δ < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Notice that θ  depends only on C1, α, δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Take µ0 > 0 such that µ0θ−α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' For every  R ≤ R0 we then have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='6)  φ(θR) ≤ θδφ(R) + C2Rβ  and the base case follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Suppose the statement has already been verified  for some k ∈ N, k ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' then  φ(θkR) ≤ θkδφ(R) + C2θ(k−1)βRβ  k−1  �  j=0  θj(δ−β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  A DEGENERATE TRANSMISSION PROBLEM  7  Thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='6), we have  φ(θk+1R) = φ  �  θk(θR)  �  ≤ θkδφ(θR) + C2θ(k−1)β(θR)β  k−1  �  j=0  θj(δ−β)  ≤ θkδ�  θδφ(R) + C2Rβ�  + C2θkβRβ  k−1  �  j=0  θj(δ−β)  = θ(k+1)δφ(R) + C2θkδRβ + C2θkβRβ  k−1  �  j=0  θj(δ−β)  = θ(k+1)δφ(R) + C2θkβRβ  k  �  j=0  θj(δ−β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Hence, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='5) holds for every k ∈ N, and the induction argument is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Step 2 - Next, we pass from the discrete to the continuous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' In partic-  ular, we claim that  φ(r) ≤ C3  � r  R  �β�  φ(R) + C2Rβ�  ,  for every 0 < r ≤ R ≤ R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Indeed,  φ(θk+1R) ≤θ(k+1)δφ(R) + C2θkβRβ  1  1 − θδ−β  =θ(k+1)δφ(R) + C2Rβ θ(k+1)β  θβ − θδ  ≤C3θ(k+1)β�  φ(R) + C2Rβ�  ,  for every k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Taking k ∈ N such that θk+2R ≤ r < θk+1R, up to  relabeling the constant C3, we get  φ(r) ≤φ(θk+1R) ≤ C3θ(k+1)β�  φ(R) + C2Rβ�  =C3θ(k+2)βθ−β�  φ(R) + C2Rβ�  ≤C3  � r  R  �β�  φ(R) + C2Rβ�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Finally, one notices  φ(r) ≤ C3  1  Rβ  0  �  φ(R0) + C2Rβ  0  �  rβ =: C4rβ,  and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  □  8  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' BIANCA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' PIMENTEL, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' URBANO  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Local boundedness  In this section, we prove the local boundedness for the weak solutions to  the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Our argument is inspired by the one put forward in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Theorem 1 (Local Boundedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let u ∈ W 1,p  0 (Ω) be the weak solu-  tion to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Then for any BR := BR(x0) ⋐ Ω, there exists C =  C  �  d, p, R, ∥g∥L∞(Γ)  �  > 0 such that  ∥u∥L∞(BR/2) ≤ CR− d  p �  ∥u∥Lp(BR) + R  d  p +1∥g∥L∞(Γ)  �  and  ∥Du∥Lp(BR/2) ≤ CR−1�  ∥u∥Lp(BR) + R  d  p +1∥g∥L∞(Γ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Fix R > 0 such that BR ⋐ Ω and set k := R∥g∥L∞(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Define  u : Ω → R as  u(x) := |u(x)| + k  for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Fix q ≥ 1 and ℓ > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' For t ∈ R, denote t := |t| + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' To ease  the presentation, we split the remainder of the proof into four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Step 1 - Let F : [k, ∞) → R be defined as  F(s) :=  \uf8f1  \uf8f2  \uf8f3  sq  if  k ≤ s ≤ ℓ  qℓq−1s − (q − 1)ℓq  if  ℓ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Then F ∈ C1�  [k, ∞)  �  and F ∈ C∞�  [k, ∞) \\ {ℓ}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let G : R → R be defined  as  G(t) := sgn(t)  �  F(t)F ′(t)p−1 − qp−1kβ�  ,  ∀t ∈ R,  where β = p(q − 1) + 1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' A simple computation yields  G′(t) =  \uf8f1  \uf8f2  \uf8f3  q−1βF ′(t)p  if  |t| < ℓ − k  F ′(t)p  if  |t| > ℓ − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Notice that  |G(u)| ≤ F(u)F ′(u)p−1  and  uF ′(u) ≤ qF(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Step 2 - In this step, we introduce auxiliary test functions, which build upon  the former inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Fix 0 < r < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let η ∈ C∞  c (BR), 0 ≤ η ≤ 1, η = 1  in Br, |Dη| ≤ (R − r)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let v = ηpG(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Since G ∈ C1�  R \\ {±(ℓ − k)}  �  is  A DEGENERATE TRANSMISSION PROBLEM  9  continuous, with bounded derivative, it follows that G(u) ∈ W 1,p(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Hence  v is an admissible test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We have  Dv =  \uf8f1  \uf8f2  \uf8f3  pηp−1G(u)Dη + ηpG′(u)Du  if  u ̸= ±(ℓ − k)  pηp−1G(u)Dη  if  u = ±(ℓ − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Set w(x) = F  �  u(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Notice that q−1β ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' hence G′(u) ≤ q−1βF ′(u)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Notice also that |Du| = |Du|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Using the trace theorem and the Poincar´e inequality, we get  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1)  �  Ω  |Du|p−2Du · Dv dx ≤ ∥g∥L∞(Γ)  �  Γ  |v| dHd−1 ≤ C  �  Ω  |Dv| dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Now we estimate the left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1) from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We get  �  B1  |Du|p−2Du · Dv dx =  �  B1  |Du|p−2Du ·  �  pηp−1G(u)Dη + ηpG′(u)Du  �  dx  =p  �  B1  ηp−1G(u)|Du|p−2Du · Dη dx  +  �  B1  ηpG′(u)|Du|p dx  ≥ − p  �  B1  ηp−1F(u)F ′(u)p−1|Du|p−1|Dη| dx  +  �  B1  ηpF ′(u)p|Du|p dx  = − p  �  B1  ηp−1w|Dw|p−1|Dη| dx  +  �  B1  ηp|Dw|p dx  ≥ − p∥wDη∥Lp(B1)∥ηDw∥p−1  Lp(B1) + ∥ηDw∥p  Lp(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='2)  We also control the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1) by computing  C  �  B1  |Dv| dx =C  �  B1  up−1  up−1 |pηp−1G(u)Dη + ηpG′(u)Du| dx  ≤Ck1−pp  �  B1  up−1ηp−1|G(u)Dη| dx  + Ck1−p  �  B1  up−1ηpG′(u)|Du| dx  ≤C  �  B1  up−1ηp−1F(u)F ′(u)p−1|Dη| dx  10  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' BIANCA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' PIMENTEL, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' URBANO  + Cq−1β  �  B1  up−1ηpF ′(u)p|Du| dx  ≤C  �  B1  ηp−1qp−1F(u)p−1F(u)|Dη| dx  + Cq−1β  �  B1  qp−1F(u)p−1ηpF ′(u)|Du| dx  =Cqp−1  �  B1  (ηw)p−1w|Dη| dx  + Cqp−2β  �  B1  (ηw)p−1η|Dw| dx  ≤Cqp−1∥ηw∥p−1  Lp(B1)∥wDη∥Lp(B1)  + Cqp−2β∥ηw∥p−1  Lp(B1)∥ηDw∥Lp(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='3)  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1), combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='3) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='2), we get  ∥ηDw∥p  Lp(Ω) ≤p∥wDη∥Lp(Ω)∥ηDw∥p−1  Lp(Ω)  + Cqp−1∥ηw∥p−1  Lp(Ω)∥wDη∥Lp(Ω)  + Cqp−1∥ηw∥p−1  Lp(Ω)∥ηDw∥Lp(Ω),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='4)  where we have used  β = pq − p + 1 ≤ pq − p + q ≤ pq + q = (p + 1)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Step 3 - Set  z = ∥ηDw∥Lp(Ω)  ∥wDη∥Lp(Ω)  ,  ζ =  ∥ηw∥Lp(Ω)  ∥wDη∥Lp(Ω)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  By dividing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='4) for ∥wDη∥p  Lp(Ω), we have  zp ≤pzp−1 + Cqp−1 ∥ηw∥p−1  Lp(Ω)  ∥wDη∥p−1  Lp(Ω)  + Cqp−1 ∥ηw∥p−1  Lp(Ω)  ∥wDη∥p−1  Lp(Ω)  ∥ηDw∥Lp(Ω)  ∥wDη∥Lp(Ω)  =pzp−1 + Cqp−1ζp−1 + Cqp−1ζp−1z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  An application of Lemma 1, implies  z ≤ C  �  p + q  p−1  p ζ  p−1  p  + qζ  �  ≤ Cq(1 + ζ),  giving  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='5)  ∥ηDw∥Lp(Ω) ≤ Cq  �  ∥ηw∥Lp(Ω) + ∥wDη∥Lp(Ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  A DEGENERATE TRANSMISSION PROBLEM  11  Using the Sobolev inequality, we get  ∥ηw∥Lp∗ (Ω) ≤C∥D(ηw)∥Lp(Ω)  ≤C  �  ∥wDη∥Lp(Ω) + ∥ηDw∥Lp(Ω)  �  ≤C  �  ∥wDη∥Lp(Ω) + Cq  �  ∥ηw∥Lp(Ω) + ∥wDη∥Lp(Ω)  ��  and so  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='6)  ∥ηw∥Lp∗(Ω) ≤ Cq  �  ∥ηw∥Lp(Ω) + ∥wDη∥Lp(Ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Recall that η = 1 in Br and |Dη| ≤ (R − r)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='5) becomes  ∥Dw∥Lp(Br) ≤Cq  �� �  BR  wp dx  � 1  p  +  1  R − r  � �  BR  wp dx  � 1  p  �  =Cq∥w∥Lp(BR)  �  1 +  1  R − r  �  =CqR − r + 1  R − r  ∥w∥Lp(BR)  ≤Cqdiam(B1) + 1  R − r  ∥w∥Lp(BR)  ≤Cq  1  R − r∥w∥Lp(BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Similarly, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='6) becomes  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='7)  ∥w∥Lp∗(Br) ≤ Cq  1  R − r∥w∥Lp(BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  We claim that Fℓ ≤ Fℓ+1, for every ℓ ∈ N, ℓ > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' The only non-trivial case  is when ℓ < t ≤ ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' In this case, we have  Fℓ(t) = qℓq−1t − (q − 1)ℓq  and  Fℓ+1(t) = tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Let f : (ℓ, ℓ + 1] → R be defined by  f(t) = tq − qℓq−1t + (q − 1)ℓq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  We have f ′(t) = qtq−1 − qℓq−1 > 0, for every t ∈ (ℓ, ℓ + 1], and hence f is an  increasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Since limt→ℓ f(t) = 0, we have f ≥ 0 in (ℓ, ℓ + 1], and  so Fℓ ≤ Fℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Letting ℓ → ∞ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='7), since 0 ≤ Fℓ ≤ Fℓ+1 for every ℓ ∈ N,  12  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' BIANCA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' PIMENTEL, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' URBANO  ℓ > k, by the Monotone Convergence Theorem, we obtain  � �  Br  uqp∗ dx  � 1  p∗  ≤ Cq  1  R − r  � �  BR  uqp dx  � 1  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Set  s := qp  and  γ := p∗/p = d/(d − p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  then  � �  Br  usγ dx  � 1  pγ  ≤ Cq  1  R − r  � �  BR  us dx  � 1  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Raising both sides of the previous inequality to p/s, one gets  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='8)  � �  Br  usγ dx  � 1  sγ  ≤ C  p  s  �s  p  � p  s �  1  R − r  � p  s  � �  BR  us dx  � 1  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Set sj = sγj and rj = r + 2−j(R − r), for every j ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Iterating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='8),  which holds for every s ≥ p, we have  � �  Brj+1  usjγ dx  �  1  sjγ  ≤C  p  sj  �sj  p  � p  sj 2  p  sj (j+1)�  1  R − r  � p  sj  � �  Brj  usj dx  � 1  sj  =C  p  sj−1γ  �sj−1γ  p  �  p  sj−1γ  2  p  sj−1γ (j+1)�  1  R − r  �  p  sj−1γ  ×  � �  Brj  usj−1γ dx  �  1  sj−1γ  ≤C(j, p, s, d)  �  1  R − r  � p  s  �j  k=0 γ−k� �  BR  us dx  � 1  s  ,  where  C(j, p, s, d) := C  p  s  �j  k=0 γ−k�s  p  � p  s  �j  k=0 γ−k  γ  p  s  �j  k=0 kγ−k2  p  s  �j  k=0(k+1)γ−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Notice that r < rj, for every j ∈ N0, the series are convergent and in  particular �∞  k=0 γ−k = d/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' By letting j → ∞, we get  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='9)  sup  Br  u ≤ C  �  1  (R − r)d  �  BR  us dx  � 1  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Step 4 - Now, we can choose some parameters in the former inequalities to  complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' By choosing q = 1, setting r := R/2, and recalling that  u = |u| + k, we get  ∥u∥L∞(BR/2) ≤∥u∥L∞(BR/2) ≤ CR− d  p �  ∥u∥Lp(BR) + R  d  p k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  A DEGENERATE TRANSMISSION PROBLEM  13  The second inequality in the theorem follows by setting q = 1 and r := R/2  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='7), obtaining  ∥Du∥Lp(BR/2) =∥Du∥Lp(BR/2)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='10)  ≤CR−1∥u∥Lp(BR)  ≤CR−1�  ∥u∥Lp(BR) + ∥k∥Lp(BR)  �  ≤CR−1�  ∥u∥Lp(BR) + R  d  p k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Gradient regularity estimates in BMO–spaces  In this section, we prove the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We start with  a lemma, where we denote by W 1,p  f  (Ω) the affine space of functions w ∈  W 1,p(Ω) such that  w − f ∈ W 1,p  0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let w ∈ W 1,p(BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let also h ∈ W 1,p  w (BR) be such that ∆ph = 0  in BR, in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Then there exists C = C(d, p) > 0 such that  �  BR  |Dw|p − |Dh|p dx ≥ C  �  BR  |D(w − h)|p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Fix τ ∈ [0, 1] and define vτ := τw + (1 − τ)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We have  �  BR  |Dw|p − |Dh|p dx =  � 1  0  d  dτ  � �  BR  |Dvτ|p dx  �  dτ  =p  � 1  0  �  BR  |Dvτ|p−2Dvτ · D(w − h) dx dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1)  Since h ∈ W 1,p  w (BR) and ∆ph = 0 in BR, we also get  �  BR  |Dh|p−2Dh · D(w − h) dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Hence,  p  � 1  0  �  BR  |Dvτ|p−2Dvτ · D(w − h) dx dτ  =p  � 1  0  �  BR  (|Dvτ|p−2Dvτ − |Dh|p−2Dh) · D(w − h) dx dτ  =p  � 1  0  1  τ  �  BR  (|Dvτ|p−2Dvτ − |Dh|p−2Dh) · D(vτ − h) dx dτ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='2)  14  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' BIANCA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' PIMENTEL, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' URBANO  where the last equality relies on the fact that vτ −h = τ(w −h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Combining  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='2), and using a standard monotonicity property (recalling p >  2), we obtain  �  BR  |Dw|p − |Dh|p dx ≥C  � 1  0  1  τ  �  BR  |D(vτ − h)|p dx dτ  =C  � 1  0  τ p−1 dτ  �  BR  |D(w − h)|p dx  =C  �  BR  |D(w − h)|p dx,  and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  □  The following result concerns the decay of the excess of the gradient of  p-harmonic functions with respect to its average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Its proof can be found in  [15, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let h ∈ W 1,p(BR) be a weak solution of the p-Laplace  equation in BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Then there exist constants C = C(d, p) > 0 and α ∈ (0, 1)  such that, for every r ∈ (0, R], we have  �  Br  |Dh − (Dh)r|p dx ≤ C  � r  R  �d+pα �  BR  |Dh − (Dh)R|p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  The next proposition provides a control on the decay of the excess for  arbitrary Sobolev functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let w ∈ W 1,p(BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let also h ∈ W 1,p  w (BR) satisfy ∆ph = 0  in BR, in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Then there exists C = C(d, p) > 0 such that, for  every 0 < r ≤ R, we have  �  Br  |Dw − (Dw)r|p dx ≤C  � r  R  �d+pα �  BR  |Dw − (Dw)R|p dx  + C  �  BR  |Dw − Dh|p dx,  where α ∈ (0, 1) is the same as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let r ∈ (0, R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' We have  �  Br  |Dw − (Dw)r|p dx ≤2p−1  �  Br  |Dw − (Dh)r|p dx  + 2p−1  �  Br  |(Dw)r − (Dh)r|p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='3)  A DEGENERATE TRANSMISSION PROBLEM  15  Similarly,  �  Br  |Dw − (Dh)r|p dx ≤2p−1  �  Br  |Dw − Dh|p dx  + 2p−1  �  Br  |Dh − (Dh)r|p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='4)  Applying H¨older’s inequality, we get  �  Br  |(Dw)r − (Dh)r|p dx =|Br|  ����  1  |Br|  �  Br  Dw − Dh dx  ����  p  ≤|Br|1−p  �  |Br|  p−1  p  � �  Br  |Dw − Dh|p dx  � 1  p  �p  =  �  Br  |Dw − Dh|p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='5)  Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='5), we obtain  �  Br  |Dw − (Dw)r|p dx ≤C  �  Br  |Dh − (Dh)r|p dx  + C  �  Br  |Dw − Dh|p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Changing the roles of w and h, and integrating in the ball BR, we get  �  BR  |Dh − (Dh)R|p dx ≤C  �  BR  |Dw − (Dw)R|p dx  + C  �  BR  |Dw − Dh|p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='6)  Now, Proposition 1 implies  �  Br  |Dw − (Dw)r|p dx ≤C  � r  R  �d+pα �  BR  |Dh − (Dh)R|p dx  + C  �  BR  |Dw − Dh|p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='7)  Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='6) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='7), we finally get  �  Br  |Dw − (Dw)r|p dx ≤C  � r  R  �d+pα �  BR  |Dw − (Dw)R|p dx  + C  � r  R  �d+pα �  BR  |Dw − Dh|p dx  + C  �  BR  |Dw − Dh|p dx  16  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' BIANCA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' PIMENTEL, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' URBANO  ≤C  � r  R  �d+pα �  BR  |Dw − (Dw)R|p dx  + C  �  BR  |Dw − Dh|p dx  and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  □  We now state and prove our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Theorem 2 (Gradient regularity in BMO–spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let u ∈ W 1,p  0 (Ω) be the  weak solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Then Du ∈ BMOloc(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Moreover, for every  Ω′ ⋐ Ω,  ∥Du∥BMO(Ω′) ≤ C,  where C = C(p, d, ∥g∥L∞(Γ), diam(Ω), dist(Ω′, ∂Ω)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let x0 ∈ Γ, and let R > 0 be such that BR := BR(x0) ⋐ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let  h ∈ W 1,p  u (BR) be the weak solution of ∆ph = 0 in BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Since h = u on ∂BR  in the trace sense, we can extend h in Ω \\ BR such that h = u in Ω \\ BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  This implies that h ∈ W 1,p  0 (Ω) and hence, since u is the global minimizer of  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='4), we have  1  p  �  Ω  |Du|p dx −  �  Γ  gu dHd−1 ≤ 1  p  �  Ω  |Dh|p dx −  �  Γ  gh dHd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='8)  Set ΓR = BR ∩ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Since h = u in Ω \\ BR, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='8) becomes  1  p  �  BR  |Du|p dx −  �  ΓR  gu dHd−1 ≤ 1  p  �  BR  |Dh|p dx −  �  ΓR  gh dHd−1  from which, applying the Trace Theorem, H¨older’s inequality and Poincar´e’s  inequality, follows  1  p  �  BR  |Du|p dx − 1  p  �  BR  |Dh|p dx ≤  �  ΓR  gu dHd−1 −  �  ΓR  gh dHd−1  ≤∥g∥L∞(Γ)  �  ΓR  |u − h| dHd−1  ≤C  �  BR  |u − h| dx + C  �  BR  |D(u − h)| dx  ≤CRd p−1  p ∥u − h∥Lp(BR)  + CRd p−1  p ∥D(u − h)∥Lp(BR)  ≤CRd p−1  p ∥D(u − h)∥Lp(BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='9)  A DEGENERATE TRANSMISSION PROBLEM  17  Let us consider the left-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Using Lemma 3, we get  1  p  �  BR  |Du|p dx − 1  p  �  BR  |Dh|p dx ≥C(d, p)  �  BR  |Du − Dh|p dx  =C∥D(u − h)∥p  Lp(BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='10)  Combining now (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='9) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='10), we get  ∥D(u − h)∥p−1  Lp(BR) ≤ CRd p−1  p  and, raising both sides to p/(p − 1), we obtain  �  BR  |D(u − h)|p dx ≤ CRd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  From Proposition 2, we get  �  Br  |Du−(Du)r|p dx ≤ C  � r  R  �d+pα �  BR  |Du−(Du)R|p dx+CRd, ∀r ∈ (0, R]  and, applying Lemma 2, we reach  �  Br  |Du − (Du)r|p dx ≤ Crd,  ∀r ∈ (0, R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Hence, Du ∈ BMOloc(Ω) and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  □  As a corollary to Theorem 2, we obtain a modulus of continuity for the  solution u in C0,Log−Lip-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Corollary 1 (Log-Lipschitz continuity estimates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Let u ∈ W 1,p  0 (Ω) be the  weak solution to problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Then u ∈ C0,Log−Lip  loc  (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Moreover,  for every Ω′ ⋐ Ω,  ∥u∥C0,Log−Lip(Ω′) ≤ C  �  ∥u∥L∞(Ω) + ∥g∥L∞(Γ)  �  ,  where C = C(p, d, diam(Ω), dist(Ω′, ∂Ω)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Indeed, a function whose partial derivatives are in BMO belongs to the  Zygmund class (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Because functions in the latter have a C0,Log−Lip  modulus of continuity, the corollary follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' VB supported by the Centre for Mathematics of the Univer-  sity of Coimbra (UIDB/00324/2020, funded by the Portuguese Government through  FCT/MCTES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  18  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' BIANCA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' PIMENTEL, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' URBANO  EP partially supported by the Centre for Mathematics of the University of Coimbra  (UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES)  and by FAPERJ (grants E26/200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='002/2018 and E26/201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='390/2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  JMU partially supported by the King Abdullah University of Science and Tech-  nology (KAUST) and by the Centre for Mathematics of the University of Coimbra  (UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  References  [1] Ellen Shiting Bao, YanYan Li, and Biao Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
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+page_content=' A priori estimates and solvability of second order quasilinear  elliptic equations in a composite domain with nonlinear boundary condition and con-  jugacy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Trudy Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Steklov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=', 103:15–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' (loose errata), 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  [5] Mikhail V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Borsuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Transmission problems for elliptic second-order equations in non-  smooth domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Frontiers in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Birkh¨auser/Springer Basel AG, Basel,  2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  [6] Marc Briane, Yves Capdeboscq, and Luc Nguyen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Interior regularity estimates in high  conductivity homogenization and application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=', 207(1):75–  137, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  [7] Luis Caffareli, Mar´ıa Soria-Carro, and Pablo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Stinga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Regularity for C1,α interface  transmission problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=', 240(1):265–294, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  [8] Sergio Campanato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Sul problema di M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Picone relativo all’equilibrio di un corpo  elastico incastrato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Ricerche Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=', 6:125–149, 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  [9] Sergio Campanato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Sui problemi al contorno per sistemi di equazioni differenziali  lineari del tipo dell’elasticit`a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Scuola Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Pisa Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' (3), 13:223–  258, 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  [10] Sergio Campanato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Sui problemi al contorno per sistemi di equazioni differenziali  lineari del tipo dell’elasticit`a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Scuola Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Pisa Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
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+page_content=' The natural generalisation of the natural conditions of Ladyzhen-  skaya and Ural’tseva for elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
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+page_content=' Sur un probl`eme nouveau pour l’´equation lin´eaire aux d´eriv´ees  partielles de la th´eorie math´ematique classique de l’´elasticit´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
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+page_content=' Local behaviour of solutions of quasi-linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
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+page_content=' Estimates in Lp of solutions of elliptic equations with discontinuous  coefficients and satisfying general boundary conditions and conjugacy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
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+page_content='  [20] Mar´ıa Soria-Carro and Pablo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Stinga, Regularity of viscosity solutions to fully  nonlinear elliptic transmission problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' arXiv Preprint, arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='13772,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  [21] Guido Stampacchia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Su un problema relativo alle equazioni di tipo ellittico del secondo  ordine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Ricerche Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
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+page_content='  [22] Antoni Zygmund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Trigonometric series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
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+page_content=' Cambridge University Press, Cam-  bridge, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='  University of Coimbra, CMUC, Department of Mathematics, 3001-501 Coim-  bra, Portugal  Email address: vincenzo@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='pt  University of Coimbra, CMUC, Department of Mathematics, 3001-501 Coim-  bra, Portugal and Pontifical Catholic University of Rio de Janeiro – PUC-  Rio, 22451-900 G´avea, Rio de Janeiro-RJ, Brazil  Email address: edgard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='pimentel@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='pt  King Abdullah University of Science and Technology (KAUST), Computer,  Electrical and Mathematical Sciences and Engineering Division (CEMSE),  Thuwal 23955-6900, Saudi Arabia and University of Coimbra, CMUC, Depart-  ment of Mathematics, 3001-501 Coimbra, Portugal  Email address: miguel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
+page_content='urbano@kaust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AzT4oBgHgl3EQfOvtl/content/2301.01171v1.pdf'}
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+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Yunlong Hou 1 Vincent Y. F. Tan 1 2 3 Zixin Zhong 4
+Abstract
+Motivated by concerns about making online decisions that incur undue amount of risk at each time step, in this
+paper, we formulate the probably anytime-safe stochastic combinatorial semi-bandits problem. In this problem,
+the agent is given the option to select a subset of size at most K from a set of L ground items. Each item is
+associated to a certain mean reward as well as a variance that represents its risk. To mitigate the risk that the agent
+incurs, we require that with probability at least 1 − δ, over the entire horizon of time T, each of the choices that
+the agent makes should contain items whose sum of variances does not exceed a certain variance budget. We call
+this probably anytime-safe constraint. Under this constraint, we design and analyze an algorithm PASCOMBUCB
+that minimizes the regret over the horizon of time T. By developing accompanying information-theoretic lower
+bounds, we show under both the problem-dependent and problem-independent paradigms, PASCOMBUCB is
+almost asymptotically optimal. Our problem setup, the proposed PASCOMBUCB algorithm, and novel analyses
+are applicable to domains such as recommendation systems and transportation in which an agent is allowed to
+choose multiple items at a single time step and wishes to control the risk over the whole time horizon.
+1. Introduction
+Audrey, a burgeoning social media influencer, makes profits by posting advertisements (ads) under her account. The
+advertiser pays her only if an ad is clicked. Having taken a class in online optimization, Audrey aims to leverage the theory
+of bandit algorithms to design an exploration-exploitation strategy to ensure that the expected number of clicks of the ads
+she has posted is maximized. Since the platform is space-limited, Audrey can only post no more than K out of L available
+ads everyday. Some of these ads, however, include an innocuous-looking lottery or voucher that asks the viewer of the
+social media platform to provide personal information that may lead to fraud or information leakage. If a user clicks it and
+becomes a victim of fraud, this may damage Audrey’s reputation. Audrey thus has to be circumspect in which and how
+many ads she posts.
+On the one hand, Audrey wants to post as many ads with what she believes have high click-through rates as possible; the
+expected reward she obtains is then the sum of expected rewards of the individual ads. On the other hand, she should balance
+this with the total risk of the ads that are posted over a period of time; similarly, the risk of a set of ads posted is modeled as
+the sum of the risks of the individual ads. How should Audrey plan the posts of her ads over a period of time to learn their
+individual expected rewards and risks to ensure that her total expected reward is maximized and, at the same time, with
+high probability, the risk incurred at any point in time in her exploration-exploitation strategy is bounded by some fixed
+permissible threshold?
+In addition to influencers like Audrey, online platforms that make profits by advertising such as YouTube and TikTok also
+encounter similar problems. We are therefore motivated to formulate the probably anytime-safe stochastic combinatorial
+semi-bandits problem which is a regret minimization problem with an anytime safety constraint. More precisely, we aim to
+design and analyze the performance of an algorithm that, with high probability, ensures that the risk (as measured by the
+variance) at any time step is below a given threshold and whose regret is minimized.
+Literature review. There is a large body of works that take risk into account while conducting the exploration and/or
+1Department of Mathematics, National University of Singapore, Singapore 2Department of Electrical and Computer Engineering,
+National University of Singapore, Singapore 3Institute of Operations Research and Analytics, National University of Singapore, Singapore
+4Department of Computing Science, University of Alberta, Canada. Correspondence to: Zixin Zhong <zixin.zhong@u.nus.edu>.
+Copyright 2023 by the author(s).
+arXiv:2301.13393v1  [cs.LG]  31 Jan 2023
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+exploitation of the unknown reward distributions in the stochastic multi-armed bandits (MABs) literature.
+Under the risk-constrained pure exploration framework, Hou et al. (2023) and David et al. (2018) attempted to identify the
+optimal arm within those low-risk (based on their variances or α-quantiles) arms with probability at least 1 − δ.
+Under the risk-aware regret minimization setup, Sani et al. (2012), Vakili & Zhao (2016) and Zhu & Tan (2020) consider
+the mean-variance as the measure to be minimized over a fixed time horizon. Cassel et al. (2018) provided a general
+and systematic instruction to analyzing risk-aware MABs, i.e., the risk was incorporated in the Empirical Distribution
+Performance Measure and the U-UCB algorithm is adopted to perform “proxy regret minimization”. While these risk-aware
+algorithms reduce the overall risk during the exploration and exploitation process, the risk is not strictly enforced to be
+below a prescribed threshold; rather the risk measure is penalized within the objective function, similarly to a Lagrangian.
+Another setup similar to the risk-aware setup is the constrained bandits regret minimization. Mahdavi et al. (2012) required
+that the number of times the constraint can only be violated is at most sublinear in the horizon T. Kagrecha et al. (2023)
+proposed a CVaR constraint and performed exploration on the feasible arm, followed by exploration among the feasible arm
+set. Unlike our formulation, these algorithm are permitted to sample risky arms during exploration.
+A more stringent constraint can be found in the literature on conservative bandits (Wu et al., 2016), which requires the
+cumulative return at any time step to be above a constant fraction of the return resulting from repeatedly sampling the base
+arm. Kazerouni et al. (2017) extended this setup to conservative contextual linear bandits and this was further improved by
+Garcelon et al. (2020). A similar problem is bandits with knapsacks (Badanidiyuru et al., 2018), which imposes a budget on
+the cumulative consumed resources and the algorithm stops when the budget is depleted.
+The most stringent constraint can be found in the safe bandits problem. Khezeli & Bitar (2020) and Moradipari et al. (2020)
+presented the SEGE, SCLUCB, and SCLTS algorithms to tackle this problem. This problem demands that the expected
+reward of the pulled arm at each time step to be greater than a prescribed threshold with high probability, also known as the
+“stagewise safety constraint”. The authors utilized the convexity (and continuity) of the arm set and performed exploration
+around the explored arms, starting from a baseline arm. This correlation among the arms generally does not hold under the
+combinatorial semi-bandits setup.
+For the (unconstrained) combinatorial semi-bandits (CSB) setup, Chen et al. (2013) presented a UCB-type algorithm
+COMUCB1 to balance the trade-off between exploration and exploitation. Kveton et al. (2015b) improved the analysis of
+COMUCB1 and achieved a tight upper bound (within a specific set of instances). Kveton et al. (2014) introduced matroid
+structure to CSB and leveraged the matroid structure to design and analyze a greedy algorithm OMM. The risk-aware CSB
+problem is less studied by the community. Ayyagari & Dukkipati (2021) utilized CVaR as the risk-aware measure within the
+CSB problem, where the risk constraint was not explicitly specified.
+We observe that the existing literature mentioned above are not directly applicable to Audrey, while our setting (described
+formally below) dovetails neatly with her problem. Audrey can utilize our algorithm to sequentially and adaptively select
+different sets of ads everyday and almost always (i.e., with high probability) avoids sets of ads with unacceptably high risks.
+Beyond any specific applications, we believe that this problem is of fundamental theoretical importance in the broad context
+of regret minimization in combinatorial multi-armed bandits.
+Main Contributions. In probably anytime-safe stochastic combinatorial semi-bandits, there are L items with different
+reward distributions. At each time step, a random reward is generated from each item’s distribution. Based on the previous
+observations, the learning agent selects a solution at each time step. A solution consists of at most K items. The expected
+return (variance) of a solution is the summation of the reward (variance) of its constituents. Given T ∈ N, the agent aims
+to maximize the cumulative return over T time steps and ensure that with probability 1 − δ the variance of all selected
+solutions are below a given threshold.
+The key challenge of regret minimization under the probably anytime-safe stochastic combinatorial semi-bandits lies in
+handling two distinct tasks—we seek optimality in the mean and safeness in the variance of each chosen solution. Our
+first contribution is design and analysis of the Probably Anytime-Safe Combinatorial UCB (or PASCOMBUCB) algorithm.
+We also derive a problem-dependent upper bound on its regret, which involves a hardness parameter H(∆(Λ)). We see
+that H(∆(Λ)) characterizes the effectiveness of ascertaining the safety of potential solutions in the regret. To assess the
+optimality of PASCOMBUCB, we prove an accompanying problem-dependent lower bound on the regret of any variance-
+constrained consistent algorithm. The upper and lower problem-dependent bounds match in almost all the parameters
+(except in K). Additionally, we show that if δT decays exponentially fast in T, the problem-dependent regret cannot be
+logarithmic in T.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+We further present a problem-independent upper bound on the regret of PASCOMBUCB and a lower bound for any
+algorithm. Just as the problem-dependent bounds, these bounds also match in almost all the parameters.
+In summary, this paper is the first to explore the regret minimization problem in the combinatorial bandits with an anytime
+constraint on the variance. When δ → 1 and ¯σ2 is large (so that the optimal safe solution is the one with the highest mean
+regardless of safety considerations), our problem reduces to the standard combinatorial semi-bandits (Kveton et al., 2015a),
+and the regret incurred by the safety constraint vanishes, resulting in the same upper bound as the unconstrained case.
+Furthermore, the framework and analysis of PASCOMBUCB can be extended to other risk measures as long as there are
+appropriate concentration bounds, e.g., Bhat & Prashanth (2019) or Chang & Tan (2022) enables us to use CVaR or certain
+continuous functions as risk measures within the generic PASCOMBUCB framework.
+2. Problem Setup
+Given a positive integer m, we let [m] := {1, 2, . . . , m}. An instance of a variance-constrained stochastic combinatorial
+semi-bandit is a tuple Λ = (E, AK, ν, ¯σ2). We describe the four elements of Λ in the following. Firstly, the finite set E = [L]
+is known as the ground set in which each i ∈ E is known as an item. Secondly, the family AK ⊂ {S ∈ 2E : |S| ≤ K} is
+a collection of subsets of E with cardinality at most K. Each element S ∈ AK is known as a solution and AK satisfies
+the condition that all subsets of S ∈ AK remain solutions, i.e., AK is downward-closed. Thirdly, the vector of probability
+distributions ν = (ν1, ν2, . . . , νL) contains σ2-sub-Gaussian distributions {νi}i∈E with means {µi}i∈E and variances
+{σ2
+i }i∈E. The final element of an instance ¯σ2 > 0 denotes the permissible upper bound on the variance. To avoid trivialities,
+we assume that ¯σ2 > σ2 and K ≥ 2.
+The return of item i ∈ E is the random variable Wi with distribution νi. The (stochastic) return of a solution S ∈ AK is
+�
+i∈S Wi where W ∼ ν. The expected return and variance of S ∈ AK are
+µS :=
+�
+i∈S
+µi
+and
+σ2
+S :=
+�
+i∈S
+σ2
+i
+respectively. We further assume that every instance Λ satisfies σ2
+S ̸= ¯σ2 for all S ∈ AK and each distribution νi is supported
+in the interval [0, 1].
+We define S := {S ∈ AK : σ2
+S < ¯σ2} to be the safe set which contains all the safe solutions. Let the complement
+of S be the unsafe set Sc. Denote the optimal safe solution as S⋆ := arg max{µS : S ∈ S} with return µ⋆. For
+simplicity, we assume that S⋆ is unique. Denote the suboptimal set B := {S ∈ AK : µS < µ⋆} and the risky set
+R := {S ∈ AK : µS ≥ µ⋆, S ̸= S⋆}. For a solution S, let the mean gap ∆S := µ⋆ − µS and the variance gap
+∆v
+S := |σ2
+S − ¯σ2|.
+An instance Λ, time horizon T ∈ N and confidence parameter δ ∈ (0, 1) are specified. An agent, who knows E, AK and
+¯σ2 but not the vector of probability distributions ν, interacts adaptively with the instance over T time steps as follows. At
+time step t ∈ [T], the agent uses a stochastic function πt that selects a solution St ∈ AK based on the observation history
+Ht−1 := ((Ss, {Wi(s)}i∈Ss))s∈[t−1]. In other words, St = πt(Ht−1) is a stochastic function of the history Ht−1. The
+agent receives the random return �
+i∈St Wi(t), where {W(s) = {Wi(s)}i∈E}s∈[T ] are i.i.d. according to ν across time.
+The weights of the selected items {Wi(t) : i ∈ St} are observed by the agent at each time t ∈ [T]. The collection of
+stochastic functions π = {πt}t∈[T ] is known as the agent’s policy.
+The goal of the agent is to minimize the expected cumulative regret (or simply regret) Reg(T) over the horizon T, subject to
+a certain risk constraint. More precisely, the regret suffered by a policy π employed by the agent is defined as
+Regπ(T) := Eπ
+� T
+�
+t=1
+� �
+i∈S⋆
+Wi(t) −
+�
+i∈St
+Wi(t)
+��
+The policy π should satisfy the condition that all the solutions chosen {Sπ
+t }t∈[T ] ⊂ AK are safe with probability at least
+1 − δ, i.e.,
+Pπ
+�
+∀ t ∈ [T], Sπ
+t ∈ S
+�
+≥ 1 − δ.
+(1)
+This is referred to as the probably anytime-safe constraint.
+In the problem-dependent lower bounds, we will refer to a certain class of “good” policies that operate as the time horizon
+T → ∞ and the probability of being safe in the sense of (1) tends to 1. This is formalized in the following.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Definition 2.1. Fix an instance ν and a vanishing sequence {δT }∞
+T =1 ⊂ (0, 1). An policy π = {πt}∞
+t=1 is said to be a
+{δT }∞
+T =1-variance-constrained consistent algorithm if
+• Regπ(T) = o(T a) for all a > 0 and
+• Pπ
+�
+∀ t ∈ [T], Sπ
+t ∈ S
+�
+≥ 1 − δT .
+We often omit the superscripts π in Regπ, Sπ
+t (or Aπ
+t and Aπ
+t,r in PASCOMBUCB) and the subscripts π in the probabilities
+and expectations if there is no risk of confusion.
+3. Our Algorithm: PASCOMBUCB
+Our algorithm Probably Anytime-Safe Combinatorial UCB (or PASCOMBUCB) is presented in Algorithm 1. PAS-
+COMBUCBis delicately designed to satisfy the probably anytime-safe constraint. In particular, we apply (and analyze)
+the GREEDY-SPLIT subroutine in Line 11; this subroutine has not been involved in an algorithm designed for standard
+combinatorial semi-bandits such as COMBUCB1 (Chen et al., 2013).
+Algorithm 1 PASCOMBUCB
+1: Input: An instance Λ (with unknown ν), the horizon T and the confidence parameter δ ∈ (0, 1).
+2: Set phase counter p = 1 and time step counter t = 1.
+3: while ∃ i ∈ E such that Ti(p − 1) < 2 do
+4:
+Pull Ap =arg maxS:|S|≤q |{i∈S : Ti(p − 1)<2}|.
+5:
+p ← p + 1, t ← t + 1.
+6: end while
+7: Update the sample mean, sample variance and confidence bounds according to (4).
+8: Update the empirically safe set Sp and possibly safe set ¯Sp according to (5) and (6) respectively.
+9: while t < T do
+10:
+Find a solution Ap =arg maxA∈ ¯Sp−1 U µ
+A(p−1).
+11:
+Invoke GREEDY-SPLIT to split the solution Ap into np sub-solutions {Ap,1, . . . , Ap,np} ⊂ Sp−1.
+12:
+Set np ← min{np, T − count}.
+13:
+Choose solution {Ap,1, . . . , Ap,np}.
+14:
+Update the statistics of all solutions based on (4).
+15:
+Update the empirical sets based on (5) and (6).
+16:
+Set t = t + np and p = p + 1,
+17: end while
+Statistics. Since each item i ∈ E is σ2-sub-Gaussian, any solution that contains at most q := ⌊ ¯σ2
+σ2 ⌋ items is safe with
+probability (w.p.) 1. We call such a solution absolutely safe. Algorithm 1 (PASCOMBUCB) is conducted in phases, where
+each phase consists of multiple time steps and each item can be pulled at most once during each phase. Thus we adopt a
+different notation “A” to denote the solution in our algorithm. Define Ti(p) := �p
+s=1 1{i ∈ Ap} as the number of times
+item i is pulled up to and including phase p. Denote the sample mean and sample variance of item i at phase p as
+ˆµi(p) :=
+1
+Ti(p)
+p
+�
+s=1
+Wi(s) · 1{i ∈ As},
+and
+ˆσ2
+i (p) :=
+1
+Ti(p)
+p
+�
+s=1
+(Wi(s) − ˆµi(p))2 · 1{i ∈ As}.
+The bound based on the Law of Iterated Logarithms (LIL) is used to construct the confidence radii. For a fixed ϵ ∈ (0, 1),
+define lil(t, ρ) := (1 + √ϵ)
+�
+1+ϵ
+2t ln
+� ln((1+ϵ)t)
+ρ
+��1/2
+and denote the confidence radius for the mean as
+α(t) := lil(t, ωµ),
+(2)
+where ωµ is a parameter to be chosen. The confidence radii for the variance are asymmetric about the empirical variance
+and are parameterized by ωv and ω′
+v that may not necessarily be the same. They are defined as
+βu(t) := 3 · lil(t, ωv)
+and
+βl(t) := 3 · lil(t, ω′
+v).
+(3)
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+We denote the upper and lower confidence bounds (UCB and LCB) for the mean of item i as
+U µ
+i (p) := ˆµi(p) + α(Ti(p))
+and
+Lµ
+i (p) := ˆµi(p) − α(Ti(p))
+respectively. The UCB and LCB for the variance of item i are defined as
+U v
+i (p) := min{ˆσ2
+i (p) + βu(Ti(p)), σ2}
+and
+Lv
+i (p) := max{ˆσ2
+i (p) − βl(Ti(p)), 0}
+respectively. With the sample mean, sample variance, and confidence bounds for the items, we define the following statistics
+for all solution S ∈ AK:
+ˆµS(p) =
+�
+i∈S
+ˆµi(p),
+ˆσ2
+S(p) =
+�
+i∈S
+ˆσ2
+i (p),
+U µ
+S (p) =
+�
+i∈S
+U µ
+i (p),
+Lµ
+S(p) =
+�
+i∈S
+Lµ
+i (p),
+(4)
+U v
+S(p) =
+�
+i∈S
+U v
+i (p),
+Lv
+S(p) =
+�
+i∈s
+Lv
+i (p).
+Denote the empirically safe set as
+Sp := {S ∈ AK : U v
+S(p) < ¯σ2}
+(5)
+and the possibly safe set as
+¯Sp := {S ∈ AK : Lv
+S(p) < ¯σ2}.
+(6)
+The solutions in St and ¯St are called empirically safe and possibly safe solutions respectively.
+Dynamics. In the initialization stage (lines 3 to 6), PASCOMBUCB greedily pulls the absolutely safe solutions. When each
+item has been pulled at least twice, this stage is terminated. After initialization, during phase p, PASCOMBUCB firstly
+identifies a solution Ap = arg maxA∈ ¯Sp U µ
+A(p−1) via an optimization oracle (Line 10). It then calls a subroutine GREEDY-
+SPLIT to greedily partition the solution Ap into empirically safe sub-solutions (Line 11, see Figure 1 for illustration).
+Subsequently, these solutions are chosen and the stochastic rewards from the corresponding items are observed (Line 13).
+Lastly, the empirical estimates, the confidence bounds, and the empirical sets are updated (Lines 14 and 15).
+Algorithm 2 GREEDY-SPLIT
+1: Input: A solution Ap and the upper confidence bound on the variance U v(p − 1) at phase p − 1.
+2: Set np = 1, s = 1 and Ap,1 = ∅.
+3: Index the items in Ap by i1, . . . , i|Ap|.
+4: while s ≤ |Ap| do
+5:
+if U v
+Ap,np (p − 1) + U v
+is(p − 1) ≤ ¯σ2 then
+6:
+Set Ap,np ← Ap,np ∪ {is}.
+7:
+else
+8:
+np ← np + 1 and Ap,np = {is}.
+9:
+end if
+10:
+s ← s + 1.
+11: end while
+12: return {Ap,1, . . . , Ap,np}.
+Illustration. Figures 2 and 3 illustrate the regret accumulated during phase p and over the whole T horizon respectively. As
+shown in Figure 2, the regret accumulated during phase p can be decomposed into two parts
+np
+�
+r=1
+(µ⋆ − µAp,r) = ∆Ap + µ⋆(np − 1)
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Variance
+Solution
+Figure 1. A diagram of a split to a solution A containing 5 items.
+Mean
+Instantaneous regret  
+due to suboptimality
+Instantaneous regret  
+due to safeness-checking
+Figure 2. Solution Ap is split into np = 3 sub-solutions, the instantaneous regret at phase p can be divided into the instantaneous regret
+due to suboptimality and the instantaneous regret due to safeness checking.
+where ∆Ap is the phase-wise (instantaneous) regret due to suboptimality and µ⋆(np − 1) is the regret due to safeness-
+checking; the latter term results from the safeness constraint. At the beginning, since the upper confidence bounds of the
+variances of all solutions are large, each solution will be split into up to 2Q sub-solutions, where Q := ⌈ K
+q ⌉, and hence the
+regret due to safeness checking can be large. As the algorithm progresses, we obtain more observations of items and get
+more confident about their variances (U v
+i (p) decreases). Hence, during some later phase, it suffices to split some solutions
+into fewer sub-solutions and the regret due to safeness-checking reduces. Furthermore, when most items are sampled
+sufficiently many times, the unsafe solutions are excluded from the possibly safe set ¯Sp, and the only contribution to the
+regret is via the suboptimality of the solution Ap.
+Remark 3.1. • The confidence parameter ω′
+v is solely a parameter of PASCOMBUCB; its choice does not rely on the
+confidence parameter δ and only affects Lv
+S(p), the lower confidence bound of the variance, which determines when we
+ascertain a solution to be unsafe. The choice of ωv depends on δ and it influences U v
+S(p), the upper confidence bound of
+the variance, which guides PASCOMBUCB to split the solution to satisfy the probably anytime-safe constraint. The other
+parameters ωv and ω′
+v determine the confidence radii of variances and do not necessarily have to be the same.
+• Indexing the items in Line 3 of GREEDY-SPLIT can be done arbitrarily, i.e., it does not require any specific order of the
+items. As such, GREEDY-SPLIT is an efficient greedy algorithm. We note that finding the optimal order that leads to the
+minimum number of sub-solutions np is a combinatorial problem which is generally hard to solve.
+4. Problem-dependent Bounds
+For simplicity, when a time horizon T and a confidence parameter δ = δT are given, we set the confidence parameters
+ωµ = ω′
+v =
+1
+T 2 and ωv = δT
+T 2 .
+We introduce various suboptimality gaps that contribute to the regret due to the suboptimality.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Phase
+Instantaneous Regret
+2
+...
+...
+......
+Instantaneous regret due to suboptimality
+Instantaneous regret due to safeness-checking
+......
+Figure 3. An illustration of the instantaneous regret yielded by PASCOMBUCB. As the variances of the items are more determined, less
+regret due to safeness-checking is generated.
+• for i ∈ E \ S⋆, let the minimum safe-suboptimal gap be
+∆i,S∩B,min :=
+min
+S∋i,S∈S∩B ∆S;
+• for i ∈ E, let the minimum unsafe-suboptimal gap be
+∆i,Sc∩B,min :=
+min
+S∋i, S∈Sc∩B ∆S;
+and let the tension parameter between the mean gap ∆S and variance gap ∆v
+S be
+ci :=
+max
+S∋i, S∈Sc∩B
+�
+∆S
+max{∆S, ∆v
+S/3}
+�2
+.
+We also define following safeness gaps that induce the conservative sampling strategy to guarantee the probably anytime-safe
+constraint. For i ∈ E, and
+• for the risky set R, define the minimum unsafeness gap: ∆v
+i,R := minS∋i,S∈R ∆v
+S.
+• for the safe and suboptimal set S ∩ B, let
+Ψi,S∩B :=
+max
+S∋i, S∈S∩B min
+�ln T
+∆2
+S
+, 9 ln(T/δT )
+(∆v
+S)2
+�
+which characterizes the order of the number of times this item i needs to be sampled in order to identify the suboptimality
+of all safe and suboptimal solutions A ∋ i while satisfying the safeness constraint. We further define a variant of Ψi,S∩B
+as
+Ψ′
+i,S∩B :=
+max
+S∋i,S∈S∩B min
+�ln T
+∆2
+S
+, 9 ln(1/δT )
+(∆v
+S)2
+�
+which will be used to characterize the lower bound.
+• for the unsafe and suboptimal set Sc ∩ B, let
+Φi,Sc∩B :=
+max
+S∋i, S∈Sc∩B min
+�ln T
+∆2
+S
+, 9 ln T
+(∆v
+S)2
+�
+which characterizes the hardness of identifying the unsafeness of suboptimality of all unsafe and suboptimal solutions that
+contain item i.
+Define ξ(ω) := 2+ϵ
+ϵ
+�
+ω
+ln(1+ϵ)
+�1+ϵ, where ϵ ∈ (0, 1) is fixed.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+4.1. Problem-dependent Upper Bound
+Theorem 4.1 (Problem-dependent upper bound). Let Λ = (E, AK, ν, ¯σ2) be an instance and let {δT }∞
+T =1 ∈ o(1) be a
+sequence that satisfies ln(1/δT ) = o(T b) for all b > 0 (i.e., {δT } is not exponentially decaying). Then, PASCOMBUCB is
+a {δT }∞
+T =1-variance-constrained consistent algorithm. More precisely, given a time budget T, the probably anytime-safe
+constraint is satisfied and the regret of PASCOMBUCB Reg(T) is upper bounded by
+min {Tµ⋆, Reg1(T) + Reg2(T)} + Reg3(T),
+where
+Reg1(T) = O
+�
+�
+i∈E\S⋆
+K ln T
+∆i,S∩B,min
++
+�
+i∈E
+ciK ln T
+∆i,Sc∩B,min
+�
+Reg2(T) = 2µ⋆H (∆(Λ)) ,
+Reg3(T) = 2µ⋆(L + 1)
+where ∆(Λ) = {∆v
+S⋆} ∪ {∆v
+i,R, Ψi,S∩B, Φi,Sc∩B}i∈E and H (∆(Λ)) := H(1, Λ) is defined in (26) in App. B.4.
+Remark 4.2. If the gaps in ∆(Λ) are sufficiently small and δT = T −λ for a fixed λ > 0,
+H (∆(Λ)) = O
+�(λ + 1)K2 ln T
+(∆v
+S⋆)2
++ K
+�
+i∈E
+�
+ln T
+(∆v
+i,R)2 + max
+S∋i,
+S∈S∩B
+min
+�ln T
+∆2
+S
+, (λ + 1) ln T
+(∆v
+S)2
+�
++ Φi,Sc∩B
+��
+.
+See (26) for more details of this calculation.
+The first term Tµ⋆ in the regret bound provides a na¨ıve upper bound for the expected regret conditional on the variance
+constraint holds. The order of the regret (o(T a) for all a > 0) implies the regret will be asymptotically bounded by the
+second term when the time budget T is sufficiently large. The second term is comprised of two parts—the regret due to
+suboptimality Reg1(T) and the regret due to safeness-checking Reg2(T). The intuition for the regret due to suboptimality
+Reg1(T) is that
+• Each item in any safe and suboptimal solution will be sampled O(
+K ln T
+∆2
+i,S∩B,min ) times to ascertain the suboptimality of all
+safe and suboptimal solutions which this item belongs to.
+• Each item in an unsafe and suboptimal solution S will be sampled O
+�
+K ln T
+max{∆S,∆v
+S/3}2
+�
+times to ascertain either the
+suboptimality or the unsafeness of solution S. As this should be done for all the unsafe and suboptimal solutions, we
+need take the maximum of the above time complexity. More precisely, when ci = 1, suboptimality identification of the
+unsafe and suboptimal solutions to which item i belongs dominates the regret; and when ci < 1, the ascertaining of the
+unsafeness dominates the regret.
+The intuition for the regret due to safeness checking Reg2(T) is that the function provides an upper bound for the number
+of time steps needed for guaranteeing the safeness of all the solutions. PASCOMBUCB achieves this in a judicious manner
+since it does not check the safeness of all the solutions at the start, followed by exploration and exploitation of the possibly
+high-return safe solutions. Instead, it takes advantage of the fact that when a (safe or unsafe) suboptimal solution is
+ascertained to be suboptimal, its safeness can be disregarded, as reflected in the terms Ψi,S∩B and Ψi,Sc∩B. In addition, it
+will not sample an unsafe solution if it is identified as unsafe w.p. at least 1 − 2ξ(ω′
+v). The last term Reg3(T) corresponds
+to the regret due to failure of the “good” event and at the initialization stage.
+4.2. Problem-dependent Lower Bound
+Theorem 4.3 (Problem-dependent lower bound). Let {δT }∞
+T =1 ∈ o(1) be a sequence that satisfies ln(1/δT ) = o(T b) for
+all b > 0. There exists an instance Λ such that for any {δT }T ∈N-variance-constrained consistent algorithm π, the regret is
+lower bounded by
+Ω
+� �
+i∈E
+ln T
+∆i,S∩B,min
+�
++ µ⋆
+K · Ω
+�K ln(1/δT )
+(∆v
+S⋆)2
++
+�
+i∈E
+�
+Ψ′
+i,S∩B +
+ln T
+(∆v
+i,R)2 + Φi,Sc∩B
+��
+.
+With Theorem 4.3, the tightness of the problem-dependent upper bound can be ascertained for polynomially decay-
+ing {δT }T ∈N.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Corollary 4.4 (Tightness of problem-dependent bounds). Let δT = T −λ with a fixed λ > 0, the regret
+Reg(T) ∈ Ω
+� �
+i∈E
+ln T
+∆i,S∩B,min
++ µ⋆
+K2 H (∆(Λ))
+�
+∩ O
+� �
+i∈E
+K ln T
+∆i,S∩B,min
++ µ⋆H (∆(Λ))
+��
+where H (∆(Λ)) is defined in Remark 4.2. The upper bound above is achieved by PASCOMBUCB.
+Under different rates of decay of {δT }T ∈N (see App. D for the cases where ln(1/δT ) = ω(ln T) and o(ln T)), the upper
+bound of the regret due to suboptimality Reg1(T) (the first term in the total regret) and the upper bound of the regret
+due to safeness-checking Reg2(T) (the latter term) match their corresponding lower bounds up to factors of K and K2
+respectively; this gap is acceptable as K (e.g., number of ads displayed) is usually small relative to L (total number of
+ads). We consider general instances where all the items are independent and the gap in Reg1(T) can be closed when that
+the items are correlated, as in the lower bound for the unconstrained combinatorial bandits in Kveton et al. (2015a). This
+assumption also allows us to remove a factor of K from the gap of Reg2(T). One may naturally wonder whether we can
+tolerate a much more stringent probably anytime-safe constraint. The following theorem (with b = 1) indicates no algorithm
+is {δT }T ∈N-variance-constrained consistent if δT decays exponentially fast in T.
+Theorem 4.5 (Impossibility result). Let {δT }∞
+T =1 ∈ o(1) be a sequence that satisfies the following condition. There exists
+b > 0 such that ln(1/δT ) = Ω(T b). For instance Λ, the regret of any algorithm is lower bounded by Ω(T b).
+5. Problem-independent Bounds
+We can derive a problem-independent upper bound on the regret of PASCOMBUCB from the problem-dependent one in
+Theorem 4.1 with some delicate calculations.
+Theorem 5.1 (Problem-independent Upper Bound). Let {δT }∞
+T =1 ∈ o(1) be a sequence that satisfies ln(1/δT ) = o(T b)
+for all b > 0. If T > L, for any instance Λ with variance gaps lower bounded by ∆v ≤ minS∈AK ∆v
+S, the regret of
+PASCOMBUCB is upper bounded by
+O
+�√
+KLT ln T + LK2
+(∆v)2 ln
+� 1
+δT
+��
+.
+Theorem 5.2 (Problem-independent lower bound). Let the minimum variance gap be ∆v := minS∈AK ∆v
+S. When
+K3 ≥ L2, we have
+Reg(T) = Ω
+�√
+KLT + min
+�
+L
+(∆v)2 ln
+� 1
+δT
+�
+, T
+��
+.
+Remark 5.3. The assumption that the variance gaps of all solutions are lower bounded by ∆v is needed to achieve a
+non-vacuous problem-independent bound, hence, somewhat unconventionally, it appears in our “problem-independent”
+bounds. Given any algorithm and time budget T, the variance gap of S⋆ can be arbitrarily small if ∆v is not bounded away
+from zero, so the min in Theorem 5.2 will be dominated by the linear term T, and hence, no algorithm can attain sublinear
+regret.
+The above results allow us to investigate the tightness of problem-independent bounds.
+Corollary 5.4 (Tightness of problem-independent bounds). Let K3 ≤ L2, and {δT }∞
+T =1 ∈ o(1) satisfies ln(1/δT ) = o(T b)
+for all b > 0. We have
+Reg(T) ∈ Ω
+�√
+KLT +
+L
+(∆v)2 ln
+� 1
+δT
+��
+∩ O
+�√
+KLT ln T + LK2
+(∆v)2 ln
+� 1
+δT
+��
+.
+The upper bound is achieved by PASCOMBUCB.
+We observe that the gap between the upper and lower bounds is manifested on
+√
+ln T and K2. The presence of
+√
+ln T is not
+unexpected as it is also involved in the gap between the bounds on the regret for the (unconstrained) combinatorial bandits
+(Kveton et al., 2015a). Besides, the term K2 is induced by the design of PASCOMBUCB. During each phase, we select and
+sample solutions which are disjoint subsets of Ap, and hence one item is sample at most once during one phase. However,
+we believe that it is possible to sample some items more than once during one phase, which will help reduce the regret but
+requires more delicate analysis. We view this as a promising venue for future work.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+6. Proof Sketch of the Problem-Dependent Upper Bound (Theorem 4.1)
+Assume that PASCOMBUCBhas processed T ′ phases with T time steps, we have P[T ′ ≤ T] = 1 since each phase is
+composed by multiple time steps. Denote the expected regret of PASCOMBUCBwith p phases as E[R(p)]. The expected
+regret of PASCOMBUCBafter T time steps is
+E[R(T ′)] := E
+� T ′
+�
+p=1
+np
+�
+r=1
+(µ⋆ − µAp,r)
+�
+.
+In the proof of Theorem 4.1, we first show a regret decomposition lemma (Lemma 6.1) that separates the total regret into the
+regret due to suboptimality E[R1(T ′)], the regret due to safeness-checking E[R2(T ′)] and the regret due to the failure of the
+“good” event and the initialization. Then we upper bound R1(T ′) and R2(T ′) separately. To elucidate the dependence of the
+regret on the confidence parameters ωµ, ωv and ω′
+v, we retain these notations henceforth.
+For p ∈ [T], i ∈ E, define the “good” events that the sample mean and the sample variance are near their ground
+truths: Eµ
+i,Ti(p) := {ˆµi(p) − α(Ti(p)) ≤ µi ≤ ˆµi(p) + α(Ti(p))} and Ev
+i,Ti(p)(ρ) := {ˆσ2
+i (p) − 3 · lil(Ti(p), ρ) ≤ σ2
+i ≤
+ˆσ2
+i (p) + 3 · lil(Ti(p), ρ)} and
+Ei,Ti(p) := Eµ
+i,Ti(p) ∩ Ev
+i,Ti(p)(ωv) ∩ Ev
+i,Ti(p)(ω′
+v)
+E :=
+�
+i∈E
+�
+p∈[T ′]
+Ei,Ti(p−1)
+For r ∈ [Q − 1], define Up(r) := {U v
+Ap(p − 1) > r¯σ2}. When event Up(r) occurs at phase p, it indicates at least r + 1
+sub-solutions are needed in order to sample the items in Ap and guarantee the safeness constraint.
+Lemma 6.1. Assume that PASCOMBUCBhas processed T ′ phases with T time steps, the expected regret of PASCOM-
+BUCBcan be decomposed into three parts as follows
+E[R(T ′)] ≤ E[R1(T ′)|E] + E[R2(T ′)|E] + R3(T)
+where
+R1(T ′) :=
+T ′
+�
+p=1
+1{Ap ∈ B}∆Ap
+R2(T ′) := µ⋆
+T ′
+�
+p=1
+�
+2
+Q−1
+�
+r=1
+1{Up(r)}
+�
+R3(T) := 2µ⋆L
+�
+1 + T
+�
+ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′
+v
+��
+In Lemma 6.1, the first term R1(T ′) is the (high-probability) regret due to suboptimality, in the sense that only the mean
+gaps of the suboptimal solutions contribute to R1(T). The second term R2(T ′) is called the (high-probability) regret due
+to safeness-checking, since it depends on the variance gaps and goes to 0 if ¯σ2 is sufficiently large. The last term R3(T)
+contains the regret from the initialization stage and the regret results from the failure of the “good” event E.
+The regret due to suboptimality can be bounded in terms of the minimum safe/unsafe-suboptimal gaps as follows.
+Lemma 6.2. Conditioned on event E, the regret due to suboptimality R1(T) can be bounded by
+O
+�
+�
+i∈E\S⋆
+K
+∆i,S∩B,min
+ln 1
+ωµ
++
+�
+i∈E
+ciK
+∆i,Sc∩B,min
+ln 1
+ω′v
+�
+.
+The regret due to safeness-checking involves more critical parameters of the instance and we encode them in T ′
+r′ and
+H(r′, Λ) for r′ ∈ [Q] (see Figure 5), which are defined formally in (25) and (26) respectively.
+Lemma 6.3. On the event E, if T ′ ∈ [T ′
+r′, T ′
+r′−1) then
+R2(T ′) ≤ 2µ⋆[T ′(r′ − 1) + H(r′, Λ)] ≤ 2µ⋆H(1, Λ)
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Phase
+Instantaneous Regret
+2
+...
+...
+......
+Instantaneous regret due to safeness-checking
+......
+......
+Upper bound
+Figure 4. We assume the algorithm will sample those solutions with large U v
+A(p), i.e., those phases in which more sub-solutions are
+sampled are moved forward (the dark red ones). Based on this, an upper bound can be derived (the thick black lines).
+Phase
+Instantaneous Regret
+2
+......
+......
+Upper bound
+Figure 5. An illustration of the upper bound of R2(T ′) for phase T ′
+Q−2 ≤ T ′ < T ′
+Q−3. When r′ = 1, 2µ⋆H(1, Λ) is the area below the
+thick line, i.e., the upper bound for R2(T ′) regardless of T ′.
+To obtain the upper bound for R2(T ′), we assume the algorithm will sample those solutions with large U v
+A(p) in ¯Sp, which
+will be split into the most many sub-solutions (see Figure 4). Furthermore, for r′ = Q − 1, Q − 2, . . . , 1, we derive an upper
+bound for the number of phases in which event Up(r′) ∩ (Up(r′ + 1))c occurs (at most 2r′ + 1 sub-solutions are being pulled
+in these phases). To be more specific (see Figure 5), for r′ = Q − 1, we compute the maximum number of phases T ′
+Q−1
+in which at most 2Q − 1 sub-solutions are sampled. Then for r′ = Q − 2, we compute the maximum number of phases
+T ′
+Q−2 − T ′
+Q−1 in which at most 2Q − 3 sub-solutions are sampled. We continuously do this before the time budget runs out.
+As T ′ increases, r′ decreases and H(r′, Λ) increases. When r′ = 1, i.e. T ′ ≥ T ′
+1, H(1, Λ) is an upper bound for the total
+number of sub-solutions being pulled (up to a constant) for the safeness-checking. It can also be regarded as the price for the
+probably anytime-safe constraint and the upper bound for the regret due to safeness-checking remains an instance-dependent
+constant 2µ⋆H(1, Λ) when T ′ ≥ T ′
+1. More detailed discussions are postponed to Step 3 in the proof in App. B.4.
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+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Appendices
+In App. A, we list 3 useful lemmas concerning the LIL concentration bound. In App. B, we present detailed proofs of the
+upper bounds. In App. C, the proofs of the lower bounds are presented. In App. D, a corollary characterizing the tightness
+of the upper bound in Theorem 4.1 is presented.
+A. Auxiliary results
+Lemma A.1 (Lemma 3 in (Jamieson et al., 2014)). Let {Xi}∞
+i=1 be a sequence of i.i.d. centered sub-Gaussian random
+variables with scale parameter σ. Fix any ϵ ∈ (0, 1) and δ ∈ (0, ln(1 + ϵ)/e). Then one has
+P
+�
+∀ t∈N :
+t
+�
+s=1
+Xs ≤(1+√ϵ)
+�
+2σ2 (1+ϵ) t ln
+�ln ((1+ϵ)t)
+δ
+��
+≥ 1 − ξ(δ),
+where ξ(δ) := 2+ϵ
+ϵ
+�
+δ
+ln(1+ϵ)
+�1+ϵ.
+Lemma A.2. For t ≥ 1, ϵ ∈ (0, 1), ω ∈ (0, 1] and u > 0, let γ :=
+(1+ϵ)(1+√ϵ)2
+2
+, c :=
+(u·s)2
+γ
+=
+2(u·s)2
+(1+ϵ)(1+√ϵ)2 and
+m :=
+γ
+u2·s2
+�
+2 ln 1
+ω + ln ln+ 1
+s2 + ln 2γ(1+ϵ)
+u2
+�
+. If t > m, it holds that
+s > lil(t, ω) = (1 + √ϵ)
+�
+1 + ϵ
+2t
+ln
+�ln ((1+ϵ)t)
+ω
+�
+.
+Proof of Lemma A.2. Note that fact that
+u · s ≤ (1 + √ϵ)
+�
+1 + ϵ
+2t
+ln
+�ln ((1+ϵ)t)
+ω
+�
+⇐⇒ c =
+2(u · s)2
+(1 + ϵ)(1 + √ϵ)2 ≤ 1
+t ln
+�ln ((1+ϵ)t)
+ω
+�
+According to the computations in Jamieson et al. (2014) equation (1), i.e.,
+1
+t ln
+�ln((1 + ϵ)t)
+ω
+�
+≥ c′ ⇒ t ≤ 1
+c′ ln
+�2 ln((1 + ϵ)/(c′ω))
+ω
+�
+for t ≥ 1, ϵ ∈ (0, 1), c′ > 0, ω ∈ (0, 1]. We take c′ = c, thus
+t ≤ 1
+c ln
+�2 ln((1 + ϵ)/(cω))
+ω
+�
+= 1
+c
+�
+ln 2
+ω + ln
+�
+ln γ(1 + ϵ)
+u2 · ω
++ ln 1
+s2
+��
+(a)
+≤ 1
+c
+�
+ln 2
+ω + ln γ(1 + ϵ)
+u2 · ω
++ ln ln+
+1
+s2
+�
+=
+γ
+u2 · s2
+�
+2 ln 1
+ω + ln ln+
+1
+s2 + ln 2γ(1 + ϵ)
+u2
+�
+= m
+where we adopt ln(x + y) ≤ x + ln ln+ y, ∀x, y ∈ R+ in (a). Therefore, if t > m, we must have
+u · s > (1 + √ϵ)
+�
+1 + ϵ
+2t
+ln
+�ln ((1+ϵ)t)
+ω
+�
+.
+Lemma A.3. With the choice of the confidence radii in (2) and (3), for all i ∈ E, we have
+P [∀ p∈N : |ˆµi(p) − µi| ≤ α(Ti(p))] ≥ 1 − 2ξ(ωµ)
+(7)
+P
+�
+∀ p∈N : |ˆσ2
+i (p) − σ2
+i | ≤ βu(Ti(p))
+�
+≥ 1 − 4ξ(ωv)
+(8)
+P
+�
+∀ p∈N : |ˆσ2
+i (p) − σ2
+i | ≤ βl(Ti(p))
+�
+≥ 1 − 4ξ(ω′
+v)
+(9)
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Proof. Note the fact that any distribution supported on [0, 1] is 1/4-sub-Gaussian. By a direct application of Lemma A.1 to
+the sample mean ˆµi(p) and the sample second moment ˆ
+M2,i(p) :=
+1
+Ti(p)
+�p
+s=1 Wi(s)21{i ∈ As} of arm i ∈ [L], (7) can
+be derived and
+P [∀ p∈N : |µi − ˆµi(p)| ≤ lil(Ti(p), ω′
+v)] ≥ 1 − 2ξ(ω′
+v),
+and
+P
+�
+∀ p∈N : | ˆ
+M2,i(p) − (µ2
+i + σ2
+i )| ≤ lil(Ti(p), ω′
+v)
+�
+≥ 1 − 2ξ(ω′
+v)
+Since the rewards are in [0, 1], |µ2
+i − ˆµ2
+i (p)| = |µi + ˆµi(p)| · |µi − ˆµi(p)| ≤ 2 · lil(Ti(p), ω′
+v). Using this and the triangle
+inequality, we obtain for every p ≥ 1,
+|ˆσ2
+i (p) − σ2
+i | = |µ2
+i − ˆµ2
+i (p)| + |(µ2
+i + σ2
+i ) − ˆ
+M2,i(p)|
+≤ 2 · lil(Ti(p), ωv) + lil(Ti(p), ωv) = βu(Ti(p)).
+Therefore, (8) is proved. (9) can be similarly obtained.
+B. Proof of the Upper Bound
+B.1. Proof scheme of the problem-dependent upper bound
+In this subsection, we provide technical lemmas that can upper bound the components in R1(T ′) and R2(T ′).
+Note that at phase p, the identified solution Ap belongs to one of the 4 disjoint sets: (1) Ap = S⋆; (2) S ∩ B; (3) R and (4)
+Sc ∩ B, i.e.
+1 = 1 {Ap = S⋆} + 1 {Ap ∈ S ∩ B} + 1 {Ap ∈ R} + 1 {Ap ∈ Sc ∩ B}
+and 1 {Ap ∈ B} = 1 {Ap ∈ S ∩ B} + 1 {Ap ∈ Sc ∩ B}. Define two events (the F events) that connect the instance and
+the confidence radii
+Fµ
+p :=
+�
+∆Ap ≤ 2
+�
+i∈Ap\S⋆
+α(Ti(p − 1))
+�
+Fp(x, ρ) :=
+�
+x ≤ 2
+�
+i∈Ap
+lil(Ti(p − 1), ρ)
+�
+where x is a constant and ω is a confidence parameter. When Ap ∈ B, it indicates solution Ap has not been sampled
+sufficiently many times and its suboptimality has not been ascertained. When Ap ∈ Sc, it implies the unsafeness of Ap has
+not been recognized. We formalize this in the following lemma.
+Lemma B.1. Conditional on the event E, given any p ∈ [T], we have
+• S⋆ ∈ ¯Sp−1;
+• If Ap ∈ S ∩ B,
+1 {Ap ∈ S ∩ B} ≤ 1
+�
+Fµ
+p
+�
+;
+• If Ap ∈ R,
+1 {Ap ∈ R} ≤ 1
+�
+Fp
+�∆v
+Ap
+3
+, ω′
+v
+��
+;
+• If Ap ∈ Sc ∩ B,
+1 {Ap ∈ Sc ∩ B} ≤ 1
+�
+Fµ
+p , Fp
+�∆v
+Ap
+3
+, ω′
+v
+��
+.
+Proof of Lemma B.1. By the design of PASCOMBUCB, Ap ∈ ¯Sp−1.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+(1) We firstly prove that S ∈ ¯Sp−1, ∀S ∈ S. On the event E, we have
+Lv
+S(p − 1) =
+�
+i∈A
+max{ˆσ2
+i (p − 1) − βl(Ti(p − 1)), 0}
+≤
+�
+i∈A
+max{σ2
+i , 0}
+= σ2
+S < ¯σ2
+Thus, S ∈ ¯Sp−1, and in particular, S⋆ ∈ ¯Sp−1.
+(2) If Ap ∈ B, according to the sampling strategy in Line 10 of PASCOMBUCBand S⋆ ∈ ¯Sp−1, we have U µ
+S⋆(p − 1) ≤
+U µ
+Ap(p − 1) which indicates �
+i∈S⋆\Ap U µ
+i (p − 1) ≤ �
+i∈Ap\S⋆ U µ
+i (p − 1). Thus,
+�
+i∈S⋆\Ap
+µi ≤
+�
+i∈S⋆\Ap
+U µ
+i (p − 1)
+≤
+�
+i∈Ap\S⋆
+U µ
+i (p − 1)
+≤
+�
+i∈Ap\S⋆
+µi + 2αi(Ti(p − 1))
+=⇒
+∆Ap ≤ 2
+�
+i∈Ap\S⋆
+α(Ti(p − 1))
+(10)
+(3) If Ap ∈ Sc, according to the sampling strategy, we have Ap ∈ ¯Sp−1 which indicates Lv
+Ap(p − 1) = �
+i∈Ap Lv
+i (p − 1) <
+¯σ2. Thus,
+¯σ2 > ˆσ2
+Ap(p − 1) −
+�
+i∈Ap
+βl(Ti(p − 1))
+≥ σ2
+Ap − 2
+�
+i∈Ap
+βl(Ti(p − 1))
+=⇒
+∆v
+Ap ≤ 2
+�
+i∈Ap\S⋆
+βl(Ti(p − 1))
+(11)
+Note that if Ap ∈ S ∩ B, according to (10),
+1 {Ap ∈ S ∩ B} ≤ 1
+�
+Fµ
+p
+�
+.
+If Ap ∈ R ⊂ Sc, by (11)
+1 {Ap ∈ R} ≤ 1
+�
+Fp
+�∆v
+Ap
+3
+, ω′
+v
+��
+.
+If Ap ∈ Sc ∩ B, by (11) and (10)
+1 {Ap ∈ Sc ∩ B} ≤ 1
+�
+Fµ
+p , Fp
+�∆v
+Ap
+3
+, ω′
+v
+��
+At phase p, we define two sequences of mutually-exclusive events {Gµ
+j,p}j∈N and {Gj,p(x, ω}j∈N (the G events) which can
+further bound the number of times Fµ
+p and Fv
+p (x, ω) occur respectively. These events are indexed by two strictly-decreasing
+sequences of constants:
+a1 > a2 > . . . > ak > . . .
+1 > b1 > b2 > . . . > bk > . . .
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+where limj→∞ aj = limj→∞ bj = 0. For simplicity, we set aj =
+4
+9j−2 , bj =
+1
+4j , ∀j ∈ N and denote the constant
+C = �
+j∈N
+aj
+bj = 259.2. For x ∈ R+ and ω ∈ (0, ln(1 + ϵ)/e), define
+mj(x, ω) := aj · γK2
+x2
+�
+2 ln 1
+ω + ln ln+
+1
+x2 + D
+�
+and mj(x, ω) := ∞ otherwise, where (1) γ =
+(1+ϵ)(1+√ϵ)2
+2
+and ϵ is the constant in the confidence bounds (3), (2)
+ln ln+(x) = ln ln x if x ≥ e and it equals to 0 otherwise, (3) D = ln
+�
+324K2(1 + ϵ)2(1 + √ϵ)2�
+. Denote
+Gµ
+j,p :=
+�
+i ∈ Ap \ S⋆ : Ti(p − 1) ≤ mj(∆Ap, ωµ)
+�
+and
+Gj,p(x, ω) := {i ∈ Ap : Ti(p − 1) ≤ mj(x, ω)}
+as the sets of items that were not chosen sufficiently often. For j ∈ N, the events at phase p are sequentially defined as
+Gµ
+j,p :=
+�
+at least bjK items in Ap \ S⋆ were chosen
+at most mj(∆Ap, ωµ) times
+� �
+�
+�
+�
+k∈[j−1]
+Gµ
+k,p
+�
+�
+c
+=
+� ��Gµ
+j,p
+�� ≥ bjK
+� �
+�
+�
+�
+k∈[j−1]
+Gµ
+k,p
+�
+�
+c
+and
+Gj,p(x, ω) :=
+�
+at least bjK items in Ap were chosen
+at most mj(x, ω) times
+� �
+�
+�
+�
+k∈[j−1]
+Gk,p(x, ω)
+�
+�
+c
+=
+�
+|Gj,p(x, ω)| ≥ bjK
+� �
+�
+�
+�
+k∈[j−1]
+Gk,p(x, ω)
+�
+�
+c
+Lemma B.2. With our choice of {aj}j∈N and {bj}j∈N,
+• if Fµ
+p occurs, Gµ
+j,p occurs for some j, i.e.
+1
+�
+Fµ
+p
+�
+≤ 1
+�
+�
+�
+�
+j∈N
+Gµ
+j,p
+�
+�
+� .
+• if Fp(x, ω) occurs, Gj,p(x, ω) occurs for some j, i.e.
+1 {Fp(x, ω)} ≤ 1
+�
+�
+�
+�
+j∈N
+Gj,p(x, ω)
+�
+�
+� .
+(12)
+Proof of Lemma B.2. We prove (12) in the following and the other statement can be proved by the same procedures.
+To ease the notations, we omit the parameters x, ω and p in Fp(x, ω), Gj,p(x, ω) and mj(x, ω), since they are fixed when
+given Fp(x, ω). The event Gj can be rewritten as
+Gj =
+�
+|Gj| ≥ bjK
+� �
+�
+�
+�
+k∈[j−1]
+Gc
+k
+�
+�
+=
+�
+|Gj| ≥ bjK
+� �
+�
+�
+�
+k∈[j−1]
+�
+|Gk| < bkK
+�
+�
+�
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+The statement is proved by contradiction. We assume that when F (Fp(x, ω)) occurs, none of event Gj occurs. Hence,
+�
+� �
+j∈N
+Gj
+�
+�
+c
+=
+�
+j∈N
+Gc
+j
+=
+�
+j∈N
+�
+�
+�
+|Gj| < bjK
+� �
+�
+�
+�
+k∈[j−1]
+�
+|Gk| ≥ bkK
+�
+�
+�
+�
+�
+=
+�
+j∈N
+�
+|Gj| < bjK
+�
+Let ¯Gj := Ap \ Gj and define G0 = Ap. According to the definition of Gj, we have Gj ⊂ Gj−1 and ¯Gj−1 ⊂ ¯Gj, ∀j ∈ N.
+Because limj→∞ mj = 0, there exists j0 such that ¯Gj = Ap, ∀j ≥ j0. Therefore, we can write Ap by the “telescoping”
+sum, i.e., Ap = ∪j∈N
+� ¯Gj \ ¯Gj−1
+�
+. Fp(x, ω) indicates
+x ≤ 2
+�
+i∈Ap
+lil(Ti(p − 1), ω)
+(13)
+= 2
+�
+j∈N
+�
+i∈ ¯
+Gj\ ¯
+Gj−1
+lil(Ti(p − 1), ω)
+Note that for i ∈ ¯Gj \ ¯Gj−1 = Gj−1 \ Gj, we have Ti(p − 1) ∈ (mj, mj−1]. By Lemma A.2 with parameters
+t = Ti(p − 1), s = x, u =
+�
+1
+ajK2 and note a1 > aj, we have
+�
+1
+ajK2 · x > lil(Ti(p − 1), ω).
+Note our choice of aj and bj satisfy
+2
+�
+j∈N
+bj−1 − bj
+√aj
+≤ 1
+Thus, (13) can be further bounded by
+x ≤ 2
+�
+j∈N
+�
+i∈ ¯
+Gj\ ¯
+Gj−1
+lil(Ti(p − 1), ω)
+< 2
+�
+j∈N
+| ¯Gj \ ¯Gj−1|
+�
+1
+ajK2 · x
+≤ 2
+�
+j∈N
+(bj−1 − bj)K
+√ajK
+x
+≤ x
+which constitutes a contradiction. Thus when F occurs, there must exists j ∈ N such that Gj occurs.
+When none of the Gµ
+j,p (resp. Gj,p(x, ω)) occurs, Fµ
+p (resp. Fp(x, ω)) must not occur, which indicates all of the items in Ap
+have been sampled sufficiently many times such that the suboptimality (resp. unsafeness) of Ap is identified, thus Ap will
+not been sampled in future phases.
+As there will be multiple F events happening, we provide the following useful lemma that merges all F events.
+Lemma B.3. Given two confidence parameters ω1 ≥ ω2 ∈ (0, ln(1 + ϵ)/e)
+• If Fµ
+p occurs, then Fp(∆Ap, ωµ) occurs.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+• If both events Fp(x, ω1) and Fp(y, ω2) occur, then event Fp
+�
+max{x,
+�
+ln
+1
+ω1
+ln
+1
+ω2 y}, ω1
+�
+occurs.
+Proof of Lemma B.3. If Fµ
+p occurs, we have
+∆Ap ≤ 2
+�
+i∈Ap\S⋆
+α(Ti(p − 1)) ≤ 2
+�
+i∈Ap
+α(Ti(p − 1)) = 2
+�
+i∈Ap
+lil(Ti(p − 1), ωµ)
+Thus, Fp(∆Ap, ωµ) occurs.
+For the second statement, notice the fact that
+lil(Ti(p − 1), ω1)
+lil(Ti(p − 1), ω2) =
+(1 + √ϵ)
+�
+1+ϵ
+2Ti(p−1) ln
+� ln((1+ϵ)Ti(p−1))
+ω1
+��1/2
+(1 + √ϵ)
+�
+1+ϵ
+2t ln
+� ln((1+ϵ)Ti(p−1))
+ω2
+��1/2
+=
+�
+�
+�
+�ln 1
+ω1 + ln ln((1 + ϵ)Ti(p − 1))
+ln 1
+ω2 + ln ln((1 + ϵ)Ti(p − 1))
+(a)
+≥
+�
+�
+�
+�ln 1
+ω1
+ln 1
+ω2
+=: ρ
+where (a) utilizes the trick that a+c
+b+c ≥ a
+b , ∀a, b, c ∈ R+ and a ≤ b. When event Fp(y, ω2) occurs,
+y ≤ 2
+�
+i∈Ap
+lil(Ti(p − 1), ω2) ≤ 2
+�
+i∈Ap
+1
+ω lil(Ti(p − 1), ω1)
+=⇒
+ρ · y ≤ 2
+�
+i∈Ap
+lil(Ti(p − 1), ω1)
+=⇒
+Fp(ρy, ω1)
+Thus if both events Fp(x, ω1) and Fp(y, ω2) occur,
+we must have that event Fp (max{x, ρy}, ω1))
+=
+Fp
+�
+max{x,
+�
+ln
+1
+ω1
+ln
+1
+ω2 y}, ω1)
+�
+occurs.
+B.2. Upper bound decomposition
+Lemma 6.1. Assume that PASCOMBUCBhas processed T ′ phases with T time steps, the expected regret of PASCOM-
+BUCBcan be decomposed into three parts as follows
+E[R(T ′)] ≤ E[R1(T ′)|E] + E[R2(T ′)|E] + R3(T)
+where
+R1(T ′) :=
+T ′
+�
+p=1
+1{Ap ∈ B}∆Ap
+R2(T ′) := µ⋆
+T ′
+�
+p=1
+�
+2
+Q−1
+�
+r=1
+1{Up(r)}
+�
+R3(T) := 2µ⋆L
+�
+1 + T
+�
+ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′
+v
+��
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Proof of Lemma 6.1. The expected regret can be decomposed as:
+E [R(T ′)] = E
+�
+�
+T ′
+�
+p=1
+np
+�
+r=1
+(µ⋆ − µAp,r)
+�
+�
+= E
+�
+�
+T ′
+�
+p=1
+np
+�
+r=1
+(µ⋆ − µAp,r)1 {E}
+�
+� + E
+�
+�
+T ′
+�
+p=1
+np
+�
+r=1
+(µ⋆ − µAp,r)1 {Ec}
+�
+�
+= E
+�
+�
+T ′
+�
+p=1
+np
+�
+r=1
+(µ⋆ − µAp,r)
+����E
+�
+� P[E] + E
+�
+�
+T ′
+�
+p=1
+np
+�
+r=1
+(µ⋆ − µAp,r)
+����Ec
+�
+� P[Ec]
+In the initialization stage, it will take at most 2L time steps, since each pulled solution Ap contains at least one item i with
+Ti(p) < 2. Thus the regret is at most 2L · µ⋆.
+The expected regret when the good events fail can be upper bounded by
+E
+�
+�
+T ′
+�
+p=1
+np
+�
+r=1
+(µ⋆ − µAp,r)
+����Ec
+�
+� P [Ec] ≤ E
+�
+�
+T ′
+�
+p=1
+np
+�
+r=1
+µ⋆
+����Ec
+�
+� P[Ec]
+≤ E
+� T
+�
+t=1
+µ⋆
+����Ec
+�
+L · 2(ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′
+v))
+≤ 2µ⋆TL · (ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′
+v))
+where P[Ec] can be bounded using Lemma A.3.
+Conditional on the good event E, the high-probability regret can be upper bounded by
+ˆR(T ′) :=
+T ′
+�
+p=1
+np
+�
+r=1
+(µ⋆ − µAp,r)
+(14)
+=
+T ′
+�
+p=1
+�
+∆Ap + µ⋆(np − 1)
+�
+(a)
+≤
+T ′
+�
+p=1
+�
+∆Ap + µ⋆
+Q−1
+�
+r=0
+�
+2 · 1{U v
+Ap(p − 1) > r¯σ2} − 2
+��
+=
+T ′
+�
+p=1
+�
+∆Ap + µ⋆
+Q−1
+�
+r=1
+2 · 1{U v
+Ap(p − 1) > r¯σ2}
+�
+where (a) makes use of Lemma B.4 and the fact if U v
+Ap(p − 1) ∈ ((m − 1)¯σ2, m¯σ2], then m = �Q−1
+r=0 1{U v
+Ap(p − 1) >
+r¯σ2} = �Q−1
+r=0 1{Up(r)}. Note the fact that only the suboptimal solutions yields positive mean gap and that the negative
+mean gap of the risky solutions can be upper bounded by 0, thus the above equation can be further divided into two parts
+T ′
+�
+p=1
+�
+∆Ap + µ⋆
+Q−1
+�
+r=1
+2 · 1{U v
+Ap(p − 1) > r¯σ2}
+�
+≤
+T ′
+�
+p=1
+1{Ap ∈ B}∆Ap +
+T ′
+�
+p=1
+µ⋆
+Q−1
+�
+r=1
+2 · 1{U v
+Ap(p − 1) > r¯σ2}
+=: R1(T ′) + R2(T ′)
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+In conclusion, by summarizing the regret from the initialization stage, the regret due to failure of the good event and the
+high-probability regret, the expected regret can be bounded by
+E[R(T ′)] ≤ E[R1(T ′)|E]P[E] + E[R2(T ′)|E]P[E] + 2µ⋆ · TL (ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′
+v)) + 2µ⋆L
+≤ E[R1(T ′)|E] + E[R2(T ′)|E] + 2µ⋆ · TL (ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′
+v)) + 2µ⋆L
+= E[R1(T ′)|E] + E[R2(T ′)|E] + R3(T)
+B.3. Regret due to suboptimality
+Lemma 6.2. Conditioned on event E, the regret due to suboptimality R1(T) can be bounded by
+O
+�
+�
+i∈E\S⋆
+K
+∆i,S∩B,min
+ln 1
+ωµ
++
+�
+i∈E
+ciK
+∆i,Sc∩B,min
+ln 1
+ω′v
+�
+.
+Proof of Lemma 6.2. Notice the fact that the suboptimal solution Ap can be safe, i.e. Ap ∈ S ∩ B, or unsafe, i.e.,
+Ap ∈ Sc ∩ B, we upper bound the regret under these the two scenarios separately.
+Case 1: Ap ∈ S ∩ B
+By the definition of the event Gµ
+j,p, we have
+|Gµ
+j,p| =
+���
+i ∈ Ap \ S⋆ : Ti(p − 1) ≤ mj(∆Ap, ωµ)
+��� ≤ bjK
+which indicates
+1
+�
+Ap ∈ S ∩ B, Gµ
+j,p
+�
+≤
+1
+bjK
+�
+i∈Ap\S⋆
+1
+�
+Ap ∈ S ∩ B, Ti(p − 1) ≤ mj(∆Ap, ωµ)
+�
+.
+Given an item i ∈ E \ S⋆, assume it is included vi solutions in S ∩ B, we index them according to the decreasing order of
+their mean gaps, i.e. {Ai,k}k∈[vi] with ∆Ai,1 ≥ . . . ≥ ∆Ai,vi . Therefore,
+T
+�
+p=1
+1 {Ap ∈ S ∩ B} ∆Ap · 1 {E}
+(a)
+≤
+T
+�
+p=1
+1
+�
+Ap ∈ S ∩ B, Fµ
+p
+�
+∆Ap
+(b)
+≤
+T
+�
+p=1
+�
+j∈N
+1
+�
+Ap ∈ S ∩ B, Gµ
+j,p
+�
+∆Ap
+≤
+T
+�
+p=1
+�
+j∈N
+1
+bjK
+�
+i∈Ap\S⋆
+1
+�
+Ap ∈ S ∩ B, Ti(p − 1) ≤ mj(∆Ap, ωµ)
+�
+∆Ap
+≤
+T
+�
+p=1
+�
+j∈N
+1
+bjK
+�
+i∈E\S⋆
+�
+k∈[vi]
+1
+�
+Ai,k = Ap, i ∈ Ap, Ti(p − 1) ≤ mj(∆Ai,k, ωµ)
+�
+∆Ai,k
+≤
+�
+i∈E\S⋆
+�
+j∈N
+T
+�
+p=1
+�
+k∈[vi]
+∆Ai,k
+bjK 1
+�
+Ai,k = Ap, i ∈ Ai,k, Ti(p − 1) ≤ aj · γK2
+∆2
+Ai,k
+�
+2 ln 1
+ωµ
++ ln ln+
+1
+∆2
+Ai,k
++ D
+��
+(c)
+≤
+�
+i∈E\S⋆
+�
+j∈N
+T
+�
+p=1
+�
+k∈[vi]
+∆Ai,k
+bjK 1
+�
+Ai,k = Ap, i ∈ Ai,k, Ti(p − 1) ≤ aj · γK2
+∆2
+Ai,k
+�
+2 ln 1
+ωµ
++ ln ln+
+1
+∆2
+Ai,vi
++ D
+��
+(d)
+≤
+�
+i∈E\S⋆
+�
+j∈N
+aj · γK2
+bjK
+�
+2 ln 1
+ωµ
++ ln ln+
+1
+∆2
+Ai,vi
++ D
+� �
+1
+∆Ai,vi
++
+vi−1
+�
+k=2
+∆Ai,k+1
+�
+1
+∆2
+Ai,k+1
+−
+1
+∆2
+Ai,k
+��
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+≤
+�
+i∈E\S⋆
+2CγK
+∆Ai,vi
+�
+2 ln 1
+ωµ
++ ln ln+
+1
+∆2
+Ai,vi
++ D
+�
+where (a) and b make use of Lemma B.1 and Lemma B.2, (c) is obtained by relaxing ln ln+
+1
+∆Ai,k to ln ln+
+1
+∆Ai,vi , (d) is
+obtained by solving the optimization problem.
+Case 2: Ap ∈ Sc ∩ B
+For the case where ωµ ≤ ω′
+v, denote ¯ω :=
+�
+ln
+1
+ω′v
+ln
+1
+ωµ . Given an item i ∈ E, assume it is included vi solutions in
+Sc ∩ B, we index them according to the decreasing order of their gaps ¯∆A := max
+�
+¯ω∆A, ∆v
+A
+3
+�
+, i.e. {Ai,k}k∈[vi] with
+¯∆Ai,1 ≥ . . . ≥ ¯∆Ai,vi. Denote ci := maxk∈[vi]
+� ∆Ai,k
+¯∆Ai,k
+�2
+, ki = arg maxk∈[vi] ∆Ai,k and di = mink∈[vi] ∆Ai,k, i.e., the
+minimum mean gap.
+We have
+T
+�
+p=1
+1 {Ap ∈ Sc ∩ B} ∆Ap · 1 {E}
+(a)
+≤
+T
+�
+p=1
+1
+�
+Ap ∈ Sc ∩ B, Fµ
+p , Fp
+�∆v
+Ap
+3
+, ω′
+v
+��
+∆Ap
+(b)
+≤
+T
+�
+p=1
+1
+�
+Ap ∈ Sc ∩ B, Fp
+�
+max
+�
+¯ω∆Ap,
+∆v
+Ap
+3
+�
+, ω′
+v
+��
+∆Ap
+(c)
+≤
+T
+�
+p=1
+�
+j∈N
+1
+�
+Ap ∈ Sc ∩ B, Gj,p
+�
+max
+�
+¯ω∆µ,
+∆v
+Ap
+3
+�
+, ω′
+v
+��
+∆Ap
+≤
+T
+�
+p=1
+�
+j∈N
+1
+bjK
+�
+i∈Ap
+1
+�
+Ap ∈ Sc ∩ B, Ti(p − 1) ≤ mj
+� ¯∆Ap, ω′
+v
+��
+∆Ap
+=
+T
+�
+p=1
+�
+j∈N
+1
+bjK
+�
+i∈E\S⋆
+�
+k∈[vi]
+1
+�
+Ai,k = Ap, i ∈ Ap, Ti(p − 1) ≤ mj
+� ¯∆Ap, ω′
+v
+��
+∆Ai,k
+≤
+�
+i∈E
+�
+j∈N
+T
+�
+p=1
+�
+k∈[vi]
+∆Ai,k
+bjK 1
+�
+Ai,k = Ap, i ∈ Ai,k, Ti(p − 1) ≤ aj · γK2
+¯∆2
+Ai,k
+�
+2 ln 1
+ω′v
++ ln ln+
+1
+¯∆2
+Ai,k
++ D
+��
+(d)
+≤
+�
+i∈E
+�
+j∈N
+T
+�
+p=1
+�
+k∈[vi]
+∆Ai,k
+bjK 1
+�
+Ai,k = Ap, i ∈ Ai,k, Ti(p − 1) ≤ aj · γK2
+¯∆2
+Ai,k
+�
+2 ln 1
+ω′v
++ ln ln+
+1
+¯∆2
+Ai,vi
++ D
+��
+(15)
+(e)
+≤
+�
+i∈E
+�
+j∈N
+aj · γK2
+bjK
+�
+2 ln 1
+ω′v
++ ln ln+
+1
+¯∆2
+Ai,vi
++ D
+� � ci
+di
++ ci ·
+� 1
+di
+−
+1
+∆Ai,ki
+��
+≤
+�
+i∈E
+2ci · CγK
+di
+�
+2 ln 1
+ω′v
++ ln ln+
+1
+¯∆2
+Ai,vi
++ D
+�
+where (a) comes from Lemma B.1, (b) results from Lemma B.3, (c) is due to Lemma B.2, (d) is obtained by relaxing
+ln ln+
+1
+¯∆2
+Ai,k to ln ln+
+1
+¯∆2
+Ai,vi and (e) is achieved by solving the optimization problem in (15).
+In conclusion, for i ∈ E \ S⋆, denote
+∆i,S∩B,min :=
+min
+S∋i,S∈S∩B ∆S
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+For i ∈ E, denote
+∆i,Sc∩B,min :=
+min
+S∋i,S∈Sc∩B ∆S
+¯∆′
+i,Sc∩B :=
+min
+S∋i,S∈Sc∩B max{¯ω∆S, ∆v
+S/3}
+ci :=
+max
+S∋i,S∈Sc∩B
+�
+∆S
+max{¯ω∆S, ∆v
+S/3}
+�2
+The regret due to suboptimality can be upper bounded by
+R1(T) ≤
+�
+i∈E\S⋆
+2CγK
+∆i,S∩B,min
+�
+2 ln 1
+ωµ
++ ln ln+
+1
+∆2
+i,S∩B,min
++ D
+�
++
+�
+i∈E
+2ci · CγK
+∆i,Sc∩B,min
+�
+2 ln 1
+ω′v
++ ln ln+
+1
+( ¯∆′
+i,Sc∩B)2 + D
+�
+B.4. Regret due to safeness-checking
+We firstly introduce two more technical lemmas in order to upper bound the regret. We characterize the number of
+sub-solutions np in Algorithm 2 by the following lemma.
+Lemma B.4. At any phase p, we have P[np ≤ Q] = 1. Furthermore, if U v
+Ap(p − 1) ∈ ((m − 1)¯σ2, m¯σ2] for some m ∈ N,
+then m ≤ np ≤ 2m − 1.
+Proof of Lemma B.4. Recall that an absolutely safe solution S is safe w.p. 1 and S ∈ Sp−1, thus |Ap,r| ≥ q, ∀r ∈ [np − 1].
+If np > Q, i.e. np ≥ Q + 1, then
+K ≥ |Ap| =
+np
+�
+r=1
+|Ap,r| >
+Q
+�
+r=1
+|Ap,r| ≥ Q · q ≥ K
+which constitutes a contradiction. Therefore, we have P[np ≤ Q] = 1.
+If U v
+Ap(p − 1) ∈ ((m − 1)¯σ2, m¯σ2] for some m ∈ N+, we sequentially define {j1, j2, . . .} ⊂ [|Ap|] as follows: denote
+j0 := 0 and let j1 be the integer such that
+j1−1
+�
+s=1
+U v
+is(p − 1) ≤ ¯σ2
+and
+j1
+�
+s=1
+U v
+is(p − 1) > ¯σ2.
+Then let j2 > j1 be the integer such that
+j2−1
+�
+s=j1+1
+U v
+is(p − 1) ≤ ¯σ2
+and
+j2
+�
+s=j1+1
+U v
+is(p − 1) > ¯σ2.
+...
+The last integer jk = |Ap| satisfies
+0 <
+jk
+�
+s=jk−1+1
+U v
+is(p − 1) ≤ ¯σ2.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+If k ≥ m + 1, we must have
+U v
+At(p − 1) =
+k
+�
+l=1
+jl
+�
+s=jl−1+1
+U v
+is(p − 1)
+≥
+m
+�
+l=1
+jl
+�
+s=jl−1+1
+U v
+is(p − 1) > m¯σ2
+which contradicts with U v
+Ap(p − 1) ∈ ((m − 1)¯σ2, m¯σ2]. Hence, k ≤ m. Then we construct the sub-solutions by
+Ap,l = {ijl−1+1, . . . , ijl−1},
+∀l ∈ [k − 1]
+and
+Ap,k = {ijk−1+1, . . . , ijk}.
+There are k − 1 items {ijl : l ∈ [k − 1]} left which will compose at most k − 1 additional sub-solutions. In conclusion, we
+need at most 2m − 1 sub-solutions, i.e., np ≤ 2m − 1. Obviously, we need at least m sub-solutions since ¯σ2 > σ2. So
+m ≤ np ≤ 2m − 1.
+Due to the fact that the upper confidence of any solution S satisfies U v
+S(p) ≤ Q¯σ2, thus np can at most be 2Q − 1.
+Lemma 6.1 implies the key to upper bound the regret due to safeness-checking is to upper bound �Q−1
+r=1 1{Up(r)} over the
+horizon T. From the definition of Up(r) := {U v
+Ap(p − 1) > r¯σ2}, for r1, r2 ∈ [Q − 1] with r1 > r2, event Up(r1) indicates
+event Up(r2). Thus in order to upper bound �Q−1
+r=1 1{Up(r)}, it suffices to upper bound 1{Up(r)} for r ∈ [Q − 1]. To be
+more specific, given l ∈ [Q − 1], in order to compute the maximum number of times �Q−1
+r=1 1{Up(r)} ≥ l, we only need to
+compute the maximum number of times event Up(l) occurs.
+In the following lemma, we show a necessary condition (in terms of event Fp(x, ω)) for event Up(r).
+Lemma B.5. On the event E,
+• for Ap ∈ S:
+1 {Up(r)} ≤ 1
+�
+Fp
+�
+(r − 1)¯σ2 + ∆v
+Ap
+3
+, ωv
+��
+• for Ap ∈ Sc
+1 {Up(r)} ≤ 1
+�
+Fp
+�
+(r − 1)¯σ2 − ∆v
+Ap
+3
+, ωv
+��
+Proof of Lemma B.5. The proof is straightforward
+U v
+Ap(p − 1) > r¯σ2
+(a)
+⇒
+ˆσ2
+Ap(p − 1) +
+�
+i∈Ap
+βu(Ti(p − 1)) > r¯σ2
+(b)
+⇒
+σ2
+Ap + 2
+�
+i∈Ap
+βu(Ti(p − 1)) > r¯σ2
+⇒
+2
+�
+i∈Ap
+3 · lil(Ti(p − 1), ωv) > (r − 1)¯σ2 + (¯σ2 − σ2
+Ap)
+where (a) is due to the definition of the confidence bounds for a solution (4), and (b) utilizes the event �
+i∈Ap Ei,Ti(p−1).
+For Ap
+∈ S, the above event is equivalent to Fp
+�
+(r−1)¯σ2+∆v
+Ap
+3
+, ωv
+�
+; and for Ap
+∈ Sc, it is equivalent to
+Fp
+�
+(r−1)¯σ2−∆v
+Ap
+3
+, ωv
+�
+.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+The above lemma and Lemma B.1 upper bound the components in R2(T ′) by the F events.
+We are now ready to bound the regret due to safeness checking.
+Lemma 6.3. On the event E, if T ′ ∈ [T ′
+r′, T ′
+r′−1) then
+R2(T ′) ≤ 2µ⋆[T ′(r′ − 1) + H(r′, Λ)] ≤ 2µ⋆H(1, Λ)
+Proof of Lemma 6.3. From Lemma 6.1, the high-probability regret due to safeness-checking is
+R2(T ′) = µ⋆
+T ′
+�
+p=1
+�
+2
+Q−1
+�
+r=1
+1 {Up(r)}
+�
+= 2µ⋆
+Q−1
+�
+r=1
+T ′
+�
+p=1
+1 {Up(r)}
+In the following, given r ∈ [Q − 1], we are going to upper bound �T ′
+p=1 1{Up(r)} conditional on event E. When
+U v
+Ap(p − 1) > r¯σ2 holds, there are at most 2r + 1 solutions being chosen at phase p, i.e. np ≤ 2r + 1, according to
+Lemma B.4. Therefore, we are also deriving an upper bound for number of phases in which there are at most 2r + 1
+sub-solutions being sampled.
+The proof scheme is planned as follows: in Step 1, we decompose the event Up(r) into 4 events according to where Ap lies,
+i.e. (1) Ap = S⋆; (2) S ∩ B; (3) R and (4) Sc ∩ B. We will upper bound the regret under each of these cases.
+In Step 2, we apply Lemma B.1 to upper bound the number of times a solution A can be selected via the events Fµ
+p and
+Fp(x, ω). Because there will be multiple Fµ
+p and Fp(x, ω) events in the indicator function, Lemma B.3 will be adopted to
+merge them into one event. After that, Lemma B.2 is utilized to bridge the number of times a solution is identified to the
+number of times of an item is sampled. At the end of this step, we conclude the number of times 1{Up(r)} occurs under the
+four cases.
+In Step 3, we upper bound �Q−1
+r=1
+�T ′
+p=1 1 {Up(r)} based on the results from Step 2.
+Step 1: We decompose �T ′
+p=1 1{Up(r)} conditional on event E into four parts:
+T ′
+�
+p=1
+1{Up(r)}
+≤
+T ′
+�
+p=1
+1
+�
+U v
+Ap(p − 1) > r¯σ2�
+=
+T ′
+�
+p=1
+�
+1 {Ap = S⋆} + 1 {Ap ∈ S ∩ B} + 1 {Ap ∈ R} + 1 {Ap ∈ Sc ∩ B}
+�
+· 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+=
+T ′
+�
+p=1
+1 {Ap = S⋆} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
++
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
++
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
++
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B}
+�
+1
+�
+U v
+Ap(p − 1) > r¯σ2�
+Step 2: For each of the scenarios, we firstly upper bound the regret by the “F events” and they can be further bounded in
+terms of “G events”.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Case 1: Ap = S⋆
+T ′
+�
+p=1
+1 {Ap = S⋆} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+(a)
+≤
+T ′
+�
+p=1
+1 {Ap = S⋆} 1
+�
+Fp
+�
+(r − 1)¯σ2 + ∆v
+Ap
+3
+, ωv
+��
+(b)
+≤
+T ′
+�
+p=1
+�
+j∈N
+1 {Ap = S⋆} 1
+�
+Gj,p
+�
+(r − 1)¯σ2 + ∆v
+Ap
+3
+, ωv
+��
+(c)
+≤
+T ′
+�
+p=1
+�
+j∈N
+1 {Ap = S⋆}
+1
+bjK
+�
+i∈Ap
+1
+�
+Ti(p − 1) ≤ mj
+�
+(r − 1)¯σ2 + ∆v
+Ap
+3
+, ωv
+��
+=
+�
+i∈S⋆
+�
+j∈N
+T ′
+�
+p=1
+1
+bjK 1
+�
+Ti(p − 1) ≤ mj
+�(r − 1)¯σ2 + ∆v
+S⋆
+3
+, ωv
+��
+≤
+�
+i∈S⋆
+�
+j∈N
+1
+bjK mj
+�(r − 1)¯σ2 + ∆v
+S⋆
+3
+, ωv
+�
+=
+�
+i∈S⋆
+�
+j∈N
+aj · γK2
+bjK
+9
+((r − 1)¯σ2 + ∆v
+S⋆)2
+�
+2 ln 1
+ωv
++ ln ln+
+1
+((r − 1)¯σ2 + ∆v
+S⋆)2 + D
+�
+≤
+�
+i∈S⋆
+C ·
+9 · γK
+((r − 1)¯σ2 + ∆v
+S⋆)2
+�
+2 ln 1
+ωv
++ ln ln+
+1
+∆v
+S⋆2 + D
+�
+where (a) utilizes Lemma B.5, (b) makes use of Lemma B.2 and (c) follows the definition of Gj,p. For simplicity, we denote
+gS⋆(r, ∆v
+S⋆) = C ·
+9 · γK
+((r − 1)¯σ2 + ∆v
+S⋆)2
+�
+2 ln 1
+ωv
++ ln ln+
+1
+∆v
+S⋆2 + D
+�
+.
+Thus
+T ′
+�
+p=1
+1 {Ap = S⋆} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+≤
+�
+i∈S⋆
+gS⋆(r, ∆v
+S⋆)
+Case 2: Ap ∈ S ∩ B
+Under this case, there will be a comparison between ωµ and ωv thus we denote ˜ω =
+�
+ln
+1
+ωµ
+ln
+1
+ωv . For i ∈ E, denote
+¯∆i,S∩B := minS∋i,S∈S∩B max
+�
+∆S
+√
+ln(1/ωµ),
+∆v
+S
+3√
+ln(1/ωv)
+�
+which is achieved by solution Si,S∩B, and assume ¯∆i,S∩B ∈
+(
+ri¯σ2
+3√
+ln(1/ωv),
+(ri+1)¯σ2
+3√
+ln(1/ωv)] for some ri ∈ N.
+Scenario 1: ωµ ≥ ωv
+We firstly deal with the case where ωµ ≥ ωv, i.e., ˜ω ≤ 1. We have
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+(a)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+Fµ
+p , Fp
+�
+(r − 1)¯σ2 + ∆v
+Ap
+3
+, ωv
+��
+(b)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+Fp
+�
+∆Ap, ωµ
+�
+, Fp
+�
+(r − 1)¯σ2 + ∆v
+Ap
+3
+, ωv
+��
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+(c)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+Fp
+�
+max
+�
+∆Ap, ˜ω
+(r − 1)¯σ2 + ∆v
+Ap
+3
+�
+, ωµ
+��
+(d)
+≤
+T ′
+�
+p=1
+�
+j∈N
+1 {Ap ∈ S ∩ B} 1
+�
+Gj,p
+�
+max
+�
+∆Ap, ˜ω
+(r − 1)¯σ2 + ∆v
+Ap
+3
+�
+, ωµ
+��
+(e)
+≤
+T ′
+�
+p=1
+�
+j∈N
+1 {Ap ∈ S ∩ B}
+1
+bjK
+�
+i∈Ap
+1
+�
+Ti(p − 1) ≤ mj
+�
+max
+�
+∆Ap, ˜ω
+(r − 1)¯σ2 + ∆v
+Ap
+3
+�
+, ωµ
+��
+=
+�
+i∈E
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+∆Ap, ˜ω
+(r − 1)¯σ2 + ∆v
+Ap
+3
+�
+, ωµ
+��
+(16)
+where (a) utilizes Lemma B.5, (b) and (c) make use of Lemma B.3, (d) is due to Lemma B.2 and (e) follows the definition
+of Gj,p.
+Given i ∈ E,
+(1) if r ≥ ri + 2, for any S ∈ S ∩ B that contains item i,
+max
+�
+∆S, ˜ω (r − 1)¯σ2 + ∆v
+S
+3
+�
+≥ ˜ω (r − 1)¯σ2
+3
+≥ ˜ω (ri + 1)¯σ2
+3
+≥
+�
+ln 1
+ωµ
+· ¯∆i,S∩B.
+Thus,
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+∆Ap, ˜ω
+(r − 1)¯σ2 + ∆v
+Ap
+3
+�
+, ωµ
+��
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+˜ω (r − 1)¯σ2
+3
+, ωµ
+��
+=
+�
+j∈N
+1
+bjK mj
+�
+˜ω (r − 1)¯σ2
+3
+, ωµ
+�
+=
+�
+j∈N
+aj · γK2
+bjK
+9
+((r − 1)¯σ2)2
+�
+2 ln 1
+ωv
++ 1
+˜ω2 ln ln+
+9
+(˜ω(r − 1)¯σ2)2 + 1
+˜ω2 D
+�
+≤ C ·
+9 · γK
+((r − 1)¯σ2)2
+�
+2 ln 1
+ωv
++ 1
+˜ω2 ln ln+
+1
+ln(1/ωµ) ¯∆2
+i,S∩B
++ 1
+˜ω2 D
+�
+= C ·
+γK
+(
+(r−1)¯σ2
+3√
+ln(1/ωv))2
+�
+2 +
+1
+ln(1/ωµ) ln ln+
+1
+ln(1/ωµ) ¯∆2
+i,S∩B
++
+1
+ln(1/ωµ)D
+�
+(2) if r ≤ ri + 1, for any S ∈ S ∩ B that contains item i
+max
+�
+∆S, ˜ω (r − 1)¯σ2 + ∆v
+S
+3
+�
+≥ max
+�
+∆S, ˜ω ∆v
+S
+3
+�
+≥
+�
+ln 1
+ωµ
+· ¯∆i,S∩B.
+Thus,
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+∆Ap, ˜ω
+(r − 1)¯σ2 + ∆v
+Ap
+3
+�
+, ωµ
+��
+≤
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+��
+ln 1
+ωµ
+· ¯∆i,S∩B, ωµ
+��
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+(a)
+≤
+�
+j∈N
+1
+bjK mj
+��
+ln 1
+ωµ
+· ¯∆i,S∩B, ωµ
+�
+=
+�
+j∈N
+aj · γK2
+bjK
+1
+(
+�
+ln
+1
+ωµ · ¯∆i,S∩B)2
+�
+�2 ln 1
+ωµ
++ ln ln+
+1
+(
+�
+ln
+1
+ωµ · ¯∆i,S∩B)2 + D
+�
+�
+≤ C ·
+γK
+¯∆2
+i,S∩B
+�
+2 +
+1
+ln(1/ωµ) ln ln+
+1
+ln(1/ωµ) ¯∆2
+i,S∩B
++
+1
+ln(1/ωµ)D
+�
+where (a) is achieved by sampling Ai,S∩B. Therefore, if we denote
+gS∩B,1(r, ¯∆i,S∩B) :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+CγK
+(
+(r−1)¯σ2
+3√
+ln(1/ωv))2
+�
+2 +
+1
+ln(1/ωµ) ln ln+
+1
+ln(1/ωµ) ¯∆2
+i,S∩B
++
+1
+ln(1/ωµ)D
+�
+, r ≥
+�
+3
+�
+ln(1/ωv) · ¯∆i,S∩B
+¯σ2
+�
++ 2
+CγK
+¯∆2
+i,S∩B
+�
+2 +
+1
+ln(1/ωµ) ln ln+
+1
+ln(1/ωµ) ¯∆2
+i,S∩B
++
+1
+ln(1/ωµ)D
+�
+, r ≤
+�
+3
+�
+ln(1/ωv) · ¯∆i,S∩B
+¯σ2
+�
++ 1
+then (16) can be upper bounded by
+�
+i∈E
+gS∩B,1(r, ¯∆i,S∩B)
+Scenario 2: ωµ ≤ ωv
+For the case where ωµ ≤ ωv, i.e., ˜ω ≥ 1. We have
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+(a)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+Fµ
+p , Fp
+�
+(r − 1)¯σ2 + ∆v
+Ap
+3
+, ωv
+��
+(b)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+Fp
+�
+∆Ap, ωµ
+�
+, Fp
+�
+(r − 1)¯σ2 + ∆v
+Ap
+3
+, ωv
+��
+(c)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+Fp
+�
+max
+�
+∆Ap
+˜ω ,
+(r − 1)¯σ2 + ∆v
+Ap
+3
+�
+, ωv
+��
+(d)
+≤
+T ′
+�
+p=1
+�
+j∈N
+1 {Ap ∈ S ∩ B} 1
+�
+Gj,p
+�
+max
+�
+∆Ap
+˜ω ,
+(r − 1)¯σ2 + ∆v
+Ap
+3
+�
+, ωv
+��
+(e)
+≤
+T ′
+�
+p=1
+�
+j∈N
+1 {Ap ∈ S ∩ B}
+1
+bjK
+�
+i∈Ap
+1
+�
+Ti(p − 1) ≤ mj
+�
+max
+�
+∆Ap
+˜ω ,
+(r − 1)¯σ2 + ∆v
+Ap
+3
+�
+, ωv
+��
+=
+�
+i∈E
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+∆Ap
+˜ω ,
+(r − 1)¯σ2 + ∆v
+Ap
+3
+�
+, ωv
+��
+(17)
+Given i ∈ E,
+(1) if r ≥ ri + 2, for any S ∈ S ∩ B that contains item i,
+max
+�∆S
+˜ω , (r − 1)¯σ2 + ∆v
+S
+3
+�
+≥ (r − 1)¯σ2
+3
+≥ (ri + 1)¯σ2
+3
+≥
+�
+ln 1
+ωv
+· ¯∆i,S∩B.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Thus,
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+∆Ap
+˜ω ,
+(r − 1)¯σ2 + ∆v
+Ap
+3
+�
+, ωv
+��
+≤
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�(r − 1)¯σ2
+3
+, ωv
+��
+=
+�
+j∈N
+1
+bjK mj
+�(r − 1)¯σ2
+3
+, ωv
+�
+=
+�
+j∈N
+aj · γK2
+bjK
+9
+((r − 1)¯σ2)2
+�
+2 ln 1
+ωv
++ ln ln+
+9
+((r − 1)¯σ2)2 + D
+�
+≤ C ·
+9 · γK
+((r − 1)¯σ2)2
+�
+2 ln 1
+ωv
++ ln ln+
+1
+ln(1/ωv) ¯∆2
+i,S∩B
++ D
+�
+= C ·
+γK
+(
+(r−1)¯σ2
+3√
+ln(1/ωv))2
+�
+2 +
+1
+ln(1/ωv) ln ln+
+1
+ln(1/ωv) ¯∆2
+i,S∩B
++
+1
+ln(1/ωv)D
+�
+(2) if r ≤ ri + 1, for any S ∈ S ∩ B that contains item i,
+max
+�∆S
+˜ω , (r − 1)¯σ2 + ∆v
+S
+3
+�
+≥ max
+�∆S
+˜ω , ∆v
+S
+3
+�
+≥
+�
+ln 1
+ωv
+· ¯∆i,S∩B.
+Thus,
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+∆Ap
+˜ω ,
+(r − 1)¯σ2 + ∆v
+Ap
+3
+�
+, ωv
+��
+≤
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+��
+ln 1
+ωv
+· ¯∆i,S∩B, ωv
+��
+=
+�
+j∈N
+1
+bjK mj
+��
+ln 1
+ωv
+· ¯∆i,S∩B, ωv
+�
+=
+�
+j∈N
+aj · γK2
+bjK
+1
+(
+�
+ln 1
+ωv · ¯∆i,S∩B)2
+�
+�2 ln 1
+ωv
++ ln ln+
+1
+(
+�
+ln 1
+ωv · ¯∆i,S∩B)2 + D
+�
+�
+≤ C ·
+γK
+¯∆2
+i,S∩B
+�
+2 +
+1
+ln(1/ωv) ln ln+
+1
+ln(1/ωv) ¯∆2
+i,S∩B
++
+1
+ln(1/ωv)D
+�
+Therefore, if we denote
+gS∩B,2(r, ¯∆i,S∩B) :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+CγK
+(
+(r−1)¯σ2
+3√
+ln(1/ωv))2
+�
+2 +
+1
+ln(1/ωv) ln ln+
+1
+ln(1/ωv) ¯∆2
+i,S∩B
++
+1
+ln(1/ωv)D
+�
+, r ≥
+�
+3
+�
+ln(1/ωv) · ¯∆i,S∩B
+¯σ2
+�
++ 2
+CγK
+¯∆2
+i,S∩B
+�
+2 +
+1
+ln(1/ωv) ln ln+
+1
+ln(1/ωv) ¯∆2
+i,S∩B
++
+1
+ln(1/ωv)D
+�
+, r ≤
+�
+3
+�
+ln(1/ωv) · ¯∆i,S∩B
+¯σ2
+�
++ 1
+then (17) can be upper bounded by
+�
+i∈E
+gS∩B,2(r, ¯∆i,S∩B).
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+In conclusion, for i ∈ E, we denote ¯∆i,S∩B := minS∋i,S∈S∩B max
+�
+∆S
+√
+ln(1/ωµ),
+∆v
+A
+3√
+ln(1/ωv)
+�
+, ωµv := max{ωµ, ωv} and
+gS∩B(r, ¯∆i,S∩B) :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+CγK
+(
+(r−1)¯σ2
+3√
+ln(1/ωv))2
+�
+2 +
+1
+ln(1/ωµv)
+�
+ln ln+
+1
+ln(1/ωµv) ¯∆2
+i,S∩B
++ D
+��
+, r ≥
+�
+3
+�
+ln(1/ωv) · ¯∆i,S∩B
+¯σ2
+�
++ 2
+CγK
+¯∆2
+i,S∩B
+�
+2 +
+1
+ln(1/ωµv)
+�
+ln ln+
+1
+ln(1/ωµv) ¯∆2
+i,S∩B
++ D
+��
+, r ≤
+�
+3
+�
+ln(1/ωv) · ¯∆i,S∩B
+¯σ2
+�
++ 1
+then
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+≤
+�
+i∈E
+gS∩B(r, ¯∆i,S∩B)
+Case 3: Ap ∈ R
+Under this case, there will be a comparison between ωv and ω′
+v thus we denote ¯ω =
+�
+ln
+1
+ω′v
+ln
+1
+ωv and ωsum :=
+�
+ln 1
+ω′v +
+�
+ln 1
+ωv .
+For i ∈ E, denote ∆v
+i,R := minS∋i,S∈R ∆v
+S and assume ∆v
+i,R ∈ (ri
+¯ω
+¯ω+1 ¯σ2, (ri + 1)
+¯ω
+¯ω+1 ¯σ2].
+Scenario 1: ωv ≤ ω′
+v For the case ωv ≤ ω′
+v, we have ¯ω ≤ 1.
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+(a)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+Fp
+�∆v
+Ap
+3
+, ω′
+v
+�
+, Fp
+�
+(r − 1)¯σ2 − ∆v
+Ap
+3
+, ωv
+��
+(b)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+Fp
+�
+max
+�
+∆v
+Ap
+3
+, ¯ω ·
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ω′
+v
+��
+(c)
+≤
+T ′
+�
+p=1
+�
+j∈N
+1 {Ap ∈ R} 1
+�
+Gj,p
+�
+max
+�
+∆v
+Ap
+3
+, ¯ω ·
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ω′
+v
+��
+(d)
+≤
+T ′
+�
+p=1
+�
+j∈N
+1 {Ap ∈ R}
+1
+bjK
+�
+i∈Ap
+1
+�
+Ti(p − 1) ≤ mj
+�
+max
+�
+∆v
+Ap
+3
+, ¯ω ·
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ω′
+v
+��
+=
+�
+i∈E
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+∆v
+Ap
+3
+, ¯ω ·
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ω′
+v
+��
+(18)
+where (a) utilizes Lemma B.5, (b) makes use of Lemma B.3, (c) is due to Lemma B.2 and (d) follows the definition of Gj,p.
+Given i ∈ E,
+(1) if r ≥ ri + 2, for any S ∈ R that contains item i,
+max
+�∆v
+S
+3 , ¯ω · (r − 1)¯σ2 − ∆v
+S
+3
+�
+≥ max
+�∆v
+S
+3 ,
+¯ω
+1 + ¯ω · (r − 1)¯σ2
+3
+�
+≥
+¯ω
+1 + ¯ω · (r − 1)¯σ2
+3
+≥
+∆v
+i,R
+3
+where the first inequality uses the fact that for x, y ∈ R+ and z ∈ [0, 1], , max{x, y} ≥ max{x, xz + y(1 − z)}. We take
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+x =
+∆v
+Ap
+3 , y = ¯ω ·
+(r−1)¯σ2−∆v
+Ap
+3
+and z =
+¯ω
+1+¯ω. Thus,
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+∆v
+Ap
+3
+, ¯ω ·
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ω′
+v
+��
+≤
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+¯ω
+1 + ¯ω · (r − 1)¯σ2
+3
+, ω′
+v
+��
+=
+�
+j∈N
+1
+bjK mj
+�
+¯ω
+1 + ¯ω · (r − 1)¯σ2
+3
+, ω′
+v
+�
+=
+�
+j∈N
+aj · γK2
+bjK
+1
+(
+¯ω
+1+¯ω · (r−1)¯σ2
+3
+)2
+�
+2 ln 1
+ω′v
++ ln ln+
+1
+(
+¯ω
+1+¯ω · (r−1)¯σ2
+3
+)2 + D
+�
+≤
+CγK
+(
+¯ω
+1+¯ω · (r−1)¯σ2
+3
+)2
+�
+2 ln 1
+ω′v
++ ln ln+
+1
+(
+¯ω
+1+¯ω · (r−1)¯σ2
+3
+)2 + D
+�
+≤
+CγK
+( (r−1)¯σ2
+3ωsum )2
+�
+�
+�2 +
+1
+ln(1/ω′v)
+�
+�
+�ln ln+
+1
+ln(1/ω′v)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+�
+(2) if r ≤ ri + 1, for any S ∈ R that contains item i,
+max
+�∆v
+S
+3 , ¯ω · (r − 1)¯σ2 − ∆v
+S
+3
+�
+≥ max
+�∆v
+S
+3 ,
+¯ω
+1 + ¯ω · (r − 1)¯σ2
+3
+�
+≥
+∆v
+i,R
+3
+Thus,
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+∆v
+Ap
+3
+, ¯ω ·
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ω′
+v
+��
+≤
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�∆v
+i,R
+3
+, ω′
+v
+��
+=
+�
+j∈N
+1
+bjK mj
+�∆v
+i,R
+3
+, ω′
+v
+�
+=
+�
+j∈N
+aj · γK2
+bjK
+1
+(
+∆v
+i,R
+3
+)2
+�
+2 ln 1
+ω′v
++ ln ln+
+1
+(
+∆v
+i,R
+3
+)2 + D
+�
+≤
+CγK
+(
+∆v
+i,R
+3
+)2
+�
+2 ln 1
+ω′v
++ ln ln+
+1
+(
+∆v
+i,R
+3
+)2 + D
+�
+=
+CγK
+(
+∆v
+i,R
+3√
+ln(1/ω′v))2
+�
+�
+�2 +
+1
+ln(1/ω′v)
+�
+�
+�ln ln+
+1
+ln(1/ω′v)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+�
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Therefore, if we denote
+gR,1(r, ∆v
+i,R) :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+CγK
+( (r−1)¯σ2
+3ωsum )2
+�
+�
+�2 +
+1
+ln(1/ω′v)
+�
+�
+�ln ln+
+1
+ln(1/ω′v)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+� , r ≥
+�
+ωsum · ∆v
+i,R
+�
+ln(1/ω′v)¯σ2
+�
++ 2
+CγK
+(
+∆v
+i,R
+3√
+ln(1/ω′
+v))2
+�
+�
+�2 +
+1
+ln(1/ω′v)
+�
+�
+�ln ln+
+1
+ln(1/ω′v)(
+∆v
+i,R
+3√
+ln(1/ω′
+v))2 + D
+�
+�
+�
+�
+�
+� , r ≤
+�
+ωsum · ∆v
+i,R
+�
+ln(1/ω′v)¯σ2
+�
++ 1
+then (18) can be upper bounded by
+�
+i∈E
+gR,1(r, ∆v
+i,R).
+Scenario 2: ωv ≥ ω′
+v For the case ωv ≥ ω′
+v, we have ¯ω ≥ 1.
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+(a)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+Fp
+�∆v
+Ap
+3
+, ω′
+v
+�
+, Fp
+�
+(r − 1)¯σ2 − ∆v
+Ap
+3
+, ωv
+��
+(b)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+Fp
+�
+max
+�
+1
+¯ω
+∆v
+Ap
+3
+,
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ωv
+��
+(c)
+≤
+T ′
+�
+p=1
+�
+j∈N
+1 {Ap ∈ R} 1
+�
+Gj,p
+�
+max
+�
+1
+¯ω
+∆v
+Ap
+3
+,
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ωv
+��
+(d)
+≤
+T ′
+�
+p=1
+�
+j∈N
+1 {Ap ∈ R}
+1
+bjK
+�
+i∈Ap
+1
+�
+Ti(p − 1) ≤ mj
+�
+max
+�
+1
+¯ω
+∆v
+Ap
+3
+,
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ωv
+��
+=
+�
+i∈E
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+1
+¯ω
+∆v
+Ap
+3
+,
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ωv
+��
+(19)
+where (a) utilizes Lemma B.5, (b) makes use of Lemma B.3, (c) is due to Lemma B.2 and (d) follows the definition of Gj,p.
+Given i ∈ E,
+(1) if r ≥ ri + 2, for any S ∈ R that contains item i,
+max
+� 1
+¯ω
+∆v
+S
+3 , (r − 1)¯σ2 − ∆v
+S
+3
+�
+≥ max
+� 1
+¯ω
+∆v
+S
+3 ,
+¯ω
+1 + ¯ω · (r − 1)¯σ2
+3
+�
+≥
+1
+1 + ¯ω · (r − 1)¯σ2
+3
+≥ 1
+¯ω
+∆v
+i,R
+3
+where the first inequality uses the fact that for x, y ∈ R+ and z ∈ [0, 1], , max{x, y} ≥ max{x, xz + y(1 − z)}. We take
+x = 1
+¯ω
+∆v
+Ap
+3 , y =
+(r−1)¯σ2−∆v
+Ap
+3
+and z =
+¯ω
+1+¯ω. Thus,
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+1
+¯ω
+∆v
+Ap
+3
+,
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ωv
+��
+≤
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+1
+1 + ¯ω · (r − 1)¯σ2
+3
+, ωv
+��
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+=
+�
+j∈N
+1
+bjK mj
+�
+1
+1 + ¯ω · (r − 1)¯σ2
+3
+, ωv
+�
+=
+�
+j∈N
+aj · γK2
+bjK
+1
+(
+1
+1+¯ω · (r−1)¯σ2
+3
+)2
+�
+2 ln 1
+ωv
++ ln ln+
+1
+(
+1
+1+¯ω · (r−1)¯σ2
+3
+)2 + D
+�
+≤
+CγK
+(
+1
+1+¯ω · (r−1)¯σ2
+3
+)2
+�
+2 ln 1
+ωv
++ ln ln+
+1
+(
+1
+1+¯ω · (r−1)¯σ2
+3
+)2 + D
+�
+≤
+CγK
+( (r−1)¯σ2
+3ωsum )2
+�
+�
+�2 +
+1
+ln(1/ωv)
+�
+�
+�ln ln+
+1
+ln(1/ωv)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+�
+(2) if r ≤ ri + 1, for any S ∈ R that contains item i,
+max
+� 1
+¯ω
+∆v
+S
+3 , (r − 1)¯σ2 − ∆v
+S
+3
+�
+≥ max
+� 1
+¯ω
+∆v
+A
+3 ,
+¯ω
+1 + ¯ω · (r − 1)¯σ2
+3
+�
+≥ 1
+¯ω
+∆v
+i,R
+3
+Thus,
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+1
+¯ω
+∆v
+Ap
+3
+,
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ωv
+��
+≤
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+� 1
+¯ω
+∆v
+i,R
+3
+, ωv
+��
+=
+�
+j∈N
+1
+bjK mj
+� 1
+¯ω
+∆v
+i,R
+3
+, ωv
+�
+=
+�
+j∈N
+aj · γK2
+bjK
+1
+( 1
+¯ω
+∆v
+i,R
+3
+)2
+�
+2 ln 1
+ωv
++ ln ln+
+1
+( 1
+¯ω
+∆v
+i,R
+3
+)2 + D
+�
+≤
+CγK
+( 1
+¯ω
+∆v
+i,R
+3
+)2
+�
+2 ln 1
+ωv
++ ln ln+
+1
+( 1
+¯ω
+∆v
+i,R
+3
+)2 + D
+�
+=
+CγK
+(
+∆v
+i,R
+3√
+ln(1/ω′v))2
+�
+�
+�2 +
+1
+ln(1/ωv)
+�
+�
+�ln ln+
+1
+ln(1/ωv)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+�
+Therefore, if we denote
+gR,2(r, ∆v
+i,R) :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+CγK
+( (r−1)¯σ2
+3ωsum )2
+�
+�
+�2 +
+1
+ln(1/ωv)
+�
+�
+�ln ln+
+1
+ln(1/ωv)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+� , r ≥
+�
+ωsum · ∆v
+i,R
+�
+ln(1/ω′v)¯σ2
+�
++ 2
+CγK
+(
+∆v
+i,R
+3√
+ln(1/ω′v))2
+�
+�
+�2 +
+1
+ln(1/ωv)
+�
+�
+�ln ln+
+1
+ln(1/ωv)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+� , r ≤
+�
+ωsum · ∆v
+i,R
+�
+ln(1/ω′v)¯σ2
+�
++ 1
+then (19) can be upper bounded by
+�
+i∈E
+gR,2(r, ∆v
+i,R).
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+In conclusion, for i ∈ E, we denote ∆v
+i,R := minS∋i,S∈R ∆v
+S, ωvv′ := max{ωv, ω′
+v}, ωsum :=
+�
+ln 1
+ω′
+v +
+�
+ln 1
+ωv and
+gR(r, ∆v
+i,R) :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+CγK
+( (r−1)¯σ2
+3ωsum )2
+�
+�
+�2 +
+1
+ln(1/ωvv′)
+�
+�
+�ln ln+
+1
+ln(1/ωvv′)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+� , r ≥
+�
+ωsum · ∆v
+i,R
+�
+ln(1/ω′v)¯σ2
+�
++ 2
+CγK
+(
+∆v
+i,R
+3√
+ln(1/ω′v))2
+�
+�
+�2 +
+1
+ln(1/ωvv′)
+�
+�
+�ln ln+
+1
+ln(1/ωvv′)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+� , r ≤
+�
+ωsum · ∆v
+i,R
+�
+ln(1/ω′v)¯σ2
+�
++ 1
+then
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+≤
+�
+i∈E
+gR(r, ∆v
+i,R)
+Case 4: Ap ∈ Sc ∩ B
+Under this case, there will be a comparison among ωµ, ωv and ω′
+v thus we denote ωmax = max{ωµ, ωv, ω′
+v} and ω1 =
+�
+ln
+1
+ωmax
+ln
+1
+ωµ , ω2 =
+�
+ln
+1
+ωmax
+ln
+1
+ωv , ω3 =
+�
+ln
+1
+ωmax
+ln
+1
+ω′v
+. For S ∈ Sc ∩ B, we denote ¯∆S := max
+�
+ω1∆S, ω3
+∆v
+S
+3
+�
+.
+For i ∈ E, denote
+¯∆i,Sc∩B :=
+min
+S∋i,S∈Sc∩B max
+�
+∆S
+�
+ln(1/ωµ)
+,
+∆v
+S
+3
+�
+ln(1/ω′v)
+�
+=
+�
+1
+ln(1/ωmax)
+min
+S∋i,S∈Sc∩B
+¯∆S.
+and assume ¯∆i,Sc∩B ∈ (ri
+¯σ2/3
+√
+ln(1/ωv)+√
+ln(1/ω′v), (ri + 1)
+¯σ2/3
+√
+ln(1/ωv)+√
+ln(1/ω′v)]
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B}
+�
+1
+�
+U v
+Ap(p − 1) > r¯σ2�
+· 1 {E}
+(a)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B} 1
+�
+Fµ
+p , Fp
+�∆v
+Ap
+3
+, ω′
+v
+�
+, Fp
+�
+(r − 1)¯σ2 − ∆v
+Ap
+3
+, ωv
+��
+(b)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B} 1
+�
+Fp
+�
+∆Ap, ωµ
+�
+, Fp
+�∆v
+Ap
+3
+, ω′
+v
+�
+, Fp
+�
+(r − 1)¯σ2 − ∆v
+Ap
+3
+, ωv
+��
+(c)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B} 1
+�
+Fp
+�
+max
+�
+ω1∆Ap, ω3
+∆v
+Ap
+3
+, ω2
+(r − 1)¯σ2 − ∆v
+Ap
+3
+�
+, ωmax
+��
+(d)
+≤
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B} 1
+�
+Fp
+�
+max
+�
+max
+�
+ω1∆Ap, ω3
+∆v
+Ap
+3
+�
+, ω2
+(r − 1)¯σ2
+3
+− ω2
+ω3
+max
+�
+ω1∆Ap, ω3
+∆v
+Ap
+3
+��
+, ωmax
+��
+(e)
+≤
+T ′
+�
+p=1
+�
+j∈N
+1 {Ap ∈ Sc ∩ B} 1
+�
+Gj,p
+�
+max
+�
+¯∆Ap, ω2
+(r − 1)¯σ2
+3
+− ω2
+ω3
+¯∆Ap
+�
+, ωmax
+��
+(f)
+≤
+T ′
+�
+p=1
+�
+j∈N
+1 {Ap ∈ Sc ∩ B}
+1
+bjK
+�
+i∈Ap
+1
+�
+Ti(p − 1) ≤ mj
+�
+max
+�
+¯∆Ap, ω2
+(r − 1)¯σ2
+3
+− ω2
+ω3
+¯∆Ap
+�
+, ωmax
+��
+=
+�
+i∈E
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+¯∆Ap, ω2
+(r − 1)¯σ2
+3
+− ω2
+ω3
+¯∆Ap
+�
+, ωmax
+��
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Given i ∈ E
+(1) if r ≥ ri + 2, for any S ∈ Sc ∩ B that contains item i,
+max
+�
+¯∆S, ω2
+(r − 1)¯σ2
+3
+− ω2
+ω3
+¯∆S
+�
+≥ max
+�
+¯∆S,
+ω2ω3
+ω2 + ω3
+(r − 1)¯σ2
+3
+�
+≥
+ω2ω3
+ω2 + ω3
+(r − 1)¯σ2
+3
+≥
+�
+ln(1/ωmax) ¯∆i,Sc∩R
+where the first inequality uses the fact that for x, y ∈ R+ and z ∈ [0, 1], , max{x, y} ≥ max{x, xz + y(1 − z)}. We take
+x = ¯∆Ap, y = ω2
+(r−1)¯σ2
+3
+− ω2
+ω3 ¯∆Ap and z =
+ω2
+ω2+ω3 .
+Similar to the computations for the case Ap ∈ R, we have
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+¯∆Ap, ω2
+(r − 1)¯σ2
+3
+− ω2
+ω3
+¯∆Ap
+�
+, ωmax
+��
+≤
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+� ω2ω3
+ω2 + ω3
+(r − 1)¯σ2
+3
+, ωmax
+��
+≤
+CγK
+( ω2ω3
+ω2+ω3
+(r−1)¯σ2
+3
+)2
+�
+2 ln
+1
+ωmax
++ ln ln+
+1
+( ω2ω3
+ω2+ω3
+(r−1)¯σ2
+3
+)2 + D
+�
+≤
+CγK
+( ω2ω3
+ω2+ω3
+(r−1)¯σ2
+3
+)2
+�
+2 ln
+1
+ωmax
++ ln ln+
+1
+ln(1/ωmax)( ¯∆i,Sc∩R)2 + D
+�
+=
+CγK
+(
+(r−1)¯σ2/3
+√
+ln(1/ωv)+√
+ln(1/ω′v))2
+�
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆i,Sc∩R)2 + D
+��
+(2) if r ≤ ri + 1, for any S ∈ Sc ∩ B that contains item i,
+max
+�
+¯∆S, ω2
+(r − 1)¯σ2
+3
+− ω2
+ω3
+¯∆S
+�
+≥ max
+�
+¯∆S,
+ω2ω3
+ω2 + ω3
+(r − 1)¯σ2
+3
+�
+≥ ¯∆S ≥
+�
+ln(1/ωmax) ¯∆i,Sc∩R
+Similar to the computations for the case Ap ∈ R, we have
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+�
+max
+�
+¯∆Ap, ω2
+(r − 1)¯σ2
+3
+− ω2
+ω3
+¯∆Ap
+�
+, ωmax
+��
+≤
+�
+j∈N
+1
+bjK
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B} 1
+�
+i ∈ Ap, Ti(p − 1) ≤ mj
+��
+ln(1/ωmax) ¯∆i,Sc∩R, ωmax
+��
+≤
+CγK
+(
+�
+ln(1/ωmax) ¯∆i,Sc∩B)2
+�
+2 ln
+1
+ωmax
++ ln ln+
+1
+(
+�
+ln(1/ωmax) ¯∆i,Sc∩B)2 + D
+�
+=
+CγK
+( ¯∆i,Sc∩B)2
+�
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆i,Sc∩R)2 + D
+��
+In conclusion, for i ∈ E, we denote
+¯∆i,Sc∩B :=
+min
+S∋i,S∈Sc∩B max
+�
+∆S
+�
+ln(1/ωµ)
+,
+∆v
+S
+3
+�
+ln(1/ω′v)
+�
+.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+and ωmax := max{ωµ, ωv, ω′
+v} and ωsum :=
+�
+ln 1
+ω′
+v +
+�
+ln 1
+ωv and
+gSc∩B(r, ¯∆i,Sc∩B) :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+CγK
+( (r−1)¯σ2
+3ωsum )2
+�
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆i,Sc∩R)2 + D
+��
+, r ≥
+�ωsum · ¯∆i,Sc∩B
+¯σ2/3
+�
++ 2
+CγK
+( ¯∆i,Sc∩R)2
+�
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆i,Sc∩R)2 + D
+��
+, r ≤
+�ωsum · ¯∆i,Sc∩B
+¯σ2/3
+�
++ 1
+then
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+≤
+�
+i∈E
+gSc∩B(r, ¯∆i,Sc∩B)
+Conclusion of Step 2:
+For S⋆: denote
+gS⋆(r, ∆v
+S⋆) :=
+9 · CγK
+((r − 1)¯σ2 + ∆v
+S⋆)2
+�
+2 ln 1
+ωv
++ ln ln+
+1
+∆v
+S⋆2 + D
+�
+(20)
+we have
+T ′
+�
+p=1
+1 {Ap = S⋆} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+≤
+�
+i∈S⋆
+gS⋆(r, ∆v
+S⋆)
+For S ∩ B: for i ∈ E, denote ¯∆i,S∩B := minS∋i,S∈S∩B max
+�
+∆S
+√
+ln(1/ωµ),
+∆v
+S
+3√
+ln(1/ωv)
+�
+, ωµv := max{ωµ, ωv} and
+gS∩B(r, ¯∆i,S∩B)
+(21)
+:=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+CγK
+(
+(r−1)¯σ2
+3√
+ln(1/ωv))2
+�
+2 +
+1
+ln(1/ωµv)
+�
+ln ln+
+1
+ln(1/ωµv) ¯∆2
+i,S∩B
++ D
+��
+, r ≥
+����
+¯∆i,S∩B
+¯σ2
+3√
+ln(1/ωv)
+���� + 2
+CγK
+¯∆2
+i,S∩B
+�
+2 +
+1
+ln(1/ωµv)
+�
+ln ln+
+1
+ln(1/ωµv) ¯∆2
+i,S∩B
++ D
+��
+, r ≤
+����
+¯∆i,S∩B
+¯σ2
+3√
+ln(1/ωv)
+���� + 1
+then
+T ′
+�
+p=1
+1 {Ap ∈ S ∩ B} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+≤
+�
+i∈E
+gS∩B(r, ¯∆i,S∩B)
+For R: for i ∈ E, denote ∆v
+i,R := minS∋i,S∈R ∆v
+S, ωvv′ := max{ωv, ω′
+v}, ωsum :=
+�
+ln 1
+ω′v +
+�
+ln 1
+ωv and
+gR(r, ∆v
+i,R)
+(22)
+:=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+CγK
+( (r−1)¯σ2
+3ωsum )2
+�
+�
+�2 +
+1
+ln(1/ωvv′)
+�
+�
+�ln ln+
+1
+ln(1/ωvv′)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+� , r ≥
+�
+ωsum · ∆v
+i,R
+�
+ln(1/ω′v)¯σ2
+�
++ 2
+CγK
+(
+∆v
+i,R
+3√
+ln(1/ω′v))2
+�
+�
+�2 +
+1
+ln(1/ωvv′)
+�
+�
+�ln ln+
+1
+ln(1/ωvv′)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+� , r ≤
+�
+ωsum · ∆v
+i,R
+�
+ln(1/ω′v)¯σ2
+�
++ 1
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+then
+T ′
+�
+p=1
+1 {Ap ∈ R} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+≤
+�
+i∈E
+gR(r, ∆v
+i,R)
+For Sc ∩ B: for i ∈ E, denote
+¯∆i,Sc∩B :=
+min
+S∋i,S∈Sc∩B max
+�
+∆S
+�
+ln(1/ωµ)
+,
+∆v
+S
+3
+�
+ln(1/ω′v)
+�
+.
+and ωmax := max{ωµ, ωv, ω′
+v} and ωsum :=
+�
+ln 1
+ω′v +
+�
+ln 1
+ωv and
+gSc∩B(r, ¯∆i,Sc∩B)
+(23)
+:=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+CγK
+( (r−1)¯σ2
+3ωsum )2
+�
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆i,Sc∩B)2 + D
+��
+, r ≥
+�ωsum · ¯∆i,Sc∩B
+¯σ2/3
+�
++ 2
+CγK
+( ¯∆i,Sc∩B)2
+�
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆i,Sc∩B)2 + D
+��
+, r ≤
+�ωsum · ¯∆i,Sc∩B
+¯σ2/3
+�
++ 1
+then
+T ′
+�
+p=1
+1 {Ap ∈ Sc ∩ B} 1
+�
+U v
+Ap(p − 1) > r¯σ2�
+≤
+�
+i∈E
+gSc∩B(r, ¯∆i,Sc∩B)
+Step 3:
+According to the results in Step 2, given r ∈ [Q − 1], the event Up(r) can happen at most
+T ′
+�
+p=1
+1 {Up(r)}
+(24)
+≤ min
+�
+T ′,
+�
+i∈S⋆
+gS⋆(r, ∆v
+S⋆) +
+�
+i∈E
+gS∩B(r, ¯∆i,S∩B) +
+�
+i∈E
+gR(r, ∆v
+i,R) +
+�
+i∈E
+gSc∩B(r, ¯∆i,Sc∩B)
+�
+phases, where the g functions are defined in (20), (21), (22) and (23). By Lemma B.4, (1) event Up(r) indicates r + 1 ≤ np,
+thus (24) also indicates at most in this number of phases there are at least r + 1 being pulled at each phase. (2) event
+Up(r) ∩ Up(r + 1)c indicates r + 1 ≤ np ≤ 2r + 1, i.e., there are at least r + 1 and at most 2r + 1 solutions being pulled at
+each phase. For r ∈ [Q], we denote
+T ′
+r :=
+�
+i∈S⋆
+gS⋆(r, ∆v
+S⋆) +
+�
+i∈E
+gS∩B(r, ¯∆i,S∩B) +
+�
+i∈E
+gR(r, ∆v
+i,R) +
+�
+i∈E
+gSc∩B(r, ¯∆i,Sc∩B),
+r ∈ [Q − 1]
+(25)
+and T ′
+Q := 0, T ′
+0 = ∞. Note that the g functions are increasing as r decreases, so if there exists an r′ ∈ [Q], such that
+T ′ ∈ [T ′
+r′, T ′
+r′−1) (or T ′ ≤ T ′
+r′−1) , then
+Q−1
+�
+r=1
+T ′
+�
+p=1
+1 {Up(r)}
+=
+r′−1
+�
+r=1
+T ′
+�
+p=1
+1 {Up(r)} +
+Q−1
+�
+r=r′
+T ′
+�
+p=1
+1 {Up(r)}
+≤ T ′ · (r′ − 1) +
+Q−1
+�
+r=r′
+�
+i∈S⋆
+gS⋆(r, ∆v
+S⋆) +
+�
+i∈E
+gS∩B(r, ¯∆i,S∩B) +
+�
+i∈E
+gR(r, ∆v
+i,R) +
+�
+i∈E
+gSc∩B(r, ¯∆i,Sc∩B)
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+= T ′ · (r′ − 1) +
+Q−1
+�
+r=r′
+T ′
+r
+≤ T ′ · (r′ − 1) +
+�
+i∈S⋆
+hS⋆(r′, ∆v
+S⋆) +
+�
+i∈E
+hS∩B(r′, ¯∆i,S∩B) +
+�
+i∈E
+hR(r′, ∆v
+i,R) +
+�
+i∈E
+hSc∩B(r′, ¯∆i,Sc∩B)
+= T ′ · (r′ − 1) + H(r′, Λ)
+where
+H(r′, Λ) :=
+�
+i∈S⋆
+hS⋆(r′, ∆v
+S⋆) +
+�
+i∈E
+hS∩B(r′, ¯∆i,S∩B) +
+�
+i∈E
+hR(r′, ∆v
+i,R) +
+�
+i∈E
+hSc∩B(r′, ¯∆i,Sc∩B)
+(26)
+and the h functions are defined at the end of the proof. This indicates, when T ′ ≤ T ′
+r′−1, the upper bound of the high-
+probability regret due to safeness-checking R2(T ′) (hence the the total regret) is lower bounded by a linear function with
+slope r′ − 1 . In particular, the upper bound of R2(T ′) is lower bounded by a linear function when T ′ ≤ T ′
+1 and it remains a
+constant when T ′ > T ′
+1. We can compute an upper bound for the number of solutions being pulled during these T ′ phases
+when T ′ ∈ [T ′
+r′, T ′
+r′−1):
+T ′
+Q−1 · (2Q − 1) + (T ′
+Q−2 − T ′
+Q−1) · (2Q − 3) + · · · + (T ′
+r′ − T ′
+r′+1) · (2r′ + 1) + (T ′ − T ′
+r′)(2r′ − 1)
+= T ′ + 2
+�
+T ′
+Q−1 · (Q − 1) + (T ′
+Q−2 − T ′
+Q−1) · (Q − 2) + · · · + (T ′
+r′ − T ′
+r′+1) · r′ + (T ′ − T ′
+r′)(r′ − 1)
+�
+= (2r′ − 1)T ′ + 2
+Q−1
+�
+r=r′
+T ′
+r
+≤ (2r′ − 1)T ′ + 2H(r′, Λ)
+Thus, the number of pulled solutions is at most 2H(1, Λ).1
+In conclusion, the regret due to safeness-checking can be upper bounded by
+R2(T ′) = 2µ⋆
+Q−1
+�
+r=1
+T ′
+�
+p=1
+1 {Up(r)}
+≤ 2µ⋆T ′ · (r′ − 1)
++ 2µ⋆
+� �
+i∈S⋆
+hS⋆(r′, ∆v
+S⋆) +
+�
+i∈E
+hS∩B(r′, ¯∆i,S∩B) +
+�
+i∈E
+hR(r′, ∆v
+i,R) +
+�
+i∈E
+hSc∩B(r′, ¯∆i,Sc∩B)
+�
+= 2µ⋆T ′ · (r′ − 1) + 2µ⋆ · H(r′, Λ)
+where T ′ ∈ [T ′
+r′, T ′
+r′−1) with T ′
+r′ defined in (25), for i ∈ E, ¯∆i,S∩B := minS∋i,S∈S∩B max
+�
+∆S
+√
+ln(1/ωµ),
+∆v
+S
+3√
+ln(1/ωv)
+�
+,
+∆v
+i,R := minS∋i,S∈R ∆v
+S and ¯∆i,Sc∩B := minS∋i,S∈Sc∩B max
+�
+∆S
+√
+ln(1/ωµ),
+∆v
+S
+3√
+ln(1/ω′v)
+�
+.
+The h functions:
+(For convenience, we restate the notations: ωµv := max{ωµ, ωv}, ωvv′ := max{ωv, ω′
+v}, ωmax := max{ωµ, ωv, ω′
+v} and
+ωsum :=
+�
+ln 1
+ω′v +
+�
+ln 1
+ωv .)
+For S⋆: for each i ∈ S⋆
+hS⋆(r′, ∆v
+S⋆) :=
+Q−1
+�
+r=r′
+gS⋆(r, ∆v
+S⋆)
+(27)
+1Note that the upper bound for the regret is 2µ⋆T ′ · (r′ − 1) + 2µ⋆ · H(r′, Λ) and the upper bound for the number of solutions is
+(2r′ − 1)T ′ + 2H(r′, Λ), which indicates we can roughly use min{Tµ⋆, 2µ⋆H(r′, Λ)} to bound the regret due to safeness-checking
+with T time steps.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+=
+Q−1
+�
+r=r′
+9 · CγK
+((r − 1)¯σ2 + ∆v
+S⋆)2
+�
+2 ln 1
+ωv
++ ln ln+
+1
+∆v
+S⋆2 + D
+�
+≤
+�
+�
+�
+�
+�
+�
+�
+18 · CγK
+(r′ − 1)¯σ4
+�
+2 ln 1
+ωv
++ ln ln+
+1
+(∆v
+S⋆)2 + D
+�
+,
+r′ ≥ 2
+18 · CγK
+(∆v
+S⋆)2
+�
+2 ln 1
+ωv
++ ln ln+
+1
+(∆v
+S⋆)2 + D
+�
+,
+r′ = 1
+For S ∩ B: for each i ∈ E, there is a changing point
+�
+¯∆i,S∩B
+¯σ2
+3√
+ln(1/ωv)
+�
++ 2 in gS∩B(r, ¯∆i,S∩B), thus
+hS∩B(r′, ¯∆i,S∩B) :=
+Q−1
+�
+r=r′
+gS∩B(r, ¯∆i,S∩B)
+(28)
+≤
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0, r′ = Q
+(Q − r′) CγK
+¯∆2
+i,S∩B
+�
+2 +
+1
+ln(1/ωµv)
+�
+ln ln+
+1
+ln(1/ωµv) ¯∆2
+i,S∩B
++ D
+��
+,
+����
+¯∆i,S∩B
+¯σ2
+3√
+ln(1/ωv)
+���� ≥ Q − 3
+2 · CγK
+(r′ − 1)(
+¯σ2
+3√
+ln(1/ωv))2
+�
+2 +
+1
+ln(1/ωµv)
+�
+ln ln+
+1
+ln(1/ωµv) ¯∆2
+i,S∩B
++ D
+��
+,
+����
+¯∆i,S∩B
+¯σ2
+3√
+ln(1/ωv)
+���� + 2 ≤ r′ < Q − 1
+3 · CγK
+¯∆2
+i,S∩B
+�
+2 +
+1
+ln(1/ωµv)
+�
+ln ln+
+1
+ln(1/ωµv) ¯∆2
+i,S∩B
++ D
+��
+,
+����
+¯∆i,S∩B
+¯σ2
+3√
+ln(1/ωv)
+���� = 0, r′ = 1
+CγK
+�
+�
+4
+¯σ2
+3√
+ln(1/ωv) · ¯∆i,S∩B
+− r′ − 1
+¯∆2
+i,S∩B
+�
+�
+�
+2 +
+1
+ln(1/ωµv)
+�
+ln ln+
+1
+ln(1/ωµv) ¯∆2
+i,S∩B
++ D
+��
+, otherwise
+For R: for each i ∈ E, there is a changing point
+�
+ωsum·∆v
+i,R
+√
+ln(1/ω′v)¯σ2
+�
++ 2, thus
+hR(r′, ∆v
+i,R) :=
+Q−1
+�
+r=r′
+gR(r, ∆v
+i,R)
+(29)
+≤
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0, r′ = Q
+(Q − r′)
+CγK
+(
+∆v
+i,R
+3√
+ln(1/ω′v))2
+�
+�
+�2 +
+1
+ln(1/ωvv′)
+�
+�
+�ln ln+
+1
+ln(1/ωvv′)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+� ,
+�
+ωsum · ∆v
+i,R
+�
+ln(1/ω′v)¯σ2
+�
+≥ Q − 3
+2CγK
+(r′ − 1)(
+¯σ2
+3ωsum )2
+�
+�
+�2 +
+1
+ln(1/ωvv′)
+�
+�
+�ln ln+
+1
+ln(1/ωvv′)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+� ,
+�
+ωsum · ∆v
+i,R
+�
+ln(1/ω′v)¯σ2
+�
++ 2 ≤ r′ < Q − 1
+3CγK
+(
+∆v
+i,R
+3√
+ln(1/ω′
+v))2
+�
+�
+�2 +
+1
+ln(1/ωvv′)
+�
+�
+�ln ln+
+1
+ln(1/ωvv′)(
+∆v
+i,R
+3√
+ln(1/ω′
+v))2 + D
+�
+�
+�
+�
+�
+� ,
+�
+ωsum · ∆v
+i,R
+�
+ln(1/ω′v)¯σ2
+�
+= 0, r′ = 1
+CγK
+�
+�
+�
+3
+¯σ2
+3ωsum ·
+∆v
+i,R
+3√
+ln(1/ω′v)
+−
+r′ − 1
+(
+∆v
+i,R
+3√
+ln(1/ω′v))2
+�
+�
+�
+�
+�
+�2 +
+1
+ln(1/ωvv′)
+�
+�
+�ln ln+
+1
+ln(1/ωvv′)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+� , otherwise
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+For Sc ∩ B: for each i ∈ E, there is a changing point
+�
+ωsum· ¯∆i,Sc∩B
+¯σ2/3
+�
++ 2, thus
+hSc∩B(r′, ¯∆i,Sc∩B) :=
+Q−1
+�
+r=r′
+gSc∩B(r, ¯∆i,Sc∩B)
+(30)
+≤
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0, r′ = Q
+(Q − r′)
+CγK
+( ¯∆i,Sc∩B)2
+�
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆i,Sc∩B)2 + D
+��
+,
+�ωsum · ¯∆i,Sc∩B
+¯σ2/3
+�
+≥ Q − 3
+2CγK
+(r′ − 1)(
+¯σ2
+3ωsum )2
+�
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆i,Sc∩B)2 + D
+��
+,
+�ωsum · ¯∆i,Sc∩B
+¯σ2/3
+�
++ 2 ≤ r′ < Q − 1
+3CγK
+( ¯∆i,Sc∩B)2
+�
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆i,Sc∩B)2 + D
+��
+,
+�ωsum · ¯∆i,Sc∩B
+¯σ2/3
+�
+= 0, r′ = 1
+CγK
+�
+3
+¯σ2
+3ωsum ¯∆i,Sc∩B
+−
+r′ − 1
+( ¯∆i,Sc∩B)2
+� �
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆i,Sc∩B)2 + D
+��
+, otherwise
+B.5. Proofs of Theorem 4.1 and Theorem 5.1
+Theorem 4.1 (Problem-dependent upper bound). Let Λ = (E, AK, ν, ¯σ2) be an instance and let {δT }∞
+T =1 ∈ o(1) be a
+sequence that satisfies ln(1/δT ) = o(T b) for all b > 0 (i.e., {δT } is not exponentially decaying). Then, PASCOMBUCB is
+a {δT }∞
+T =1-variance-constrained consistent algorithm. More precisely, given a time budget T, the probably anytime-safe
+constraint is satisfied and the regret of PASCOMBUCB Reg(T) is upper bounded by
+min {Tµ⋆, Reg1(T) + Reg2(T)} + Reg3(T),
+where
+Reg1(T) = O
+�
+�
+i∈E\S⋆
+K ln T
+∆i,S∩B,min
++
+�
+i∈E
+ciK ln T
+∆i,Sc∩B,min
+�
+Reg2(T) = 2µ⋆H (∆(Λ)) ,
+Reg3(T) = 2µ⋆(L + 1)
+where ∆(Λ) = {∆v
+S⋆} ∪ {∆v
+i,R, Ψi,S∩B, Φi,Sc∩B}i∈E and H (∆(Λ)) := H(1, Λ) is defined in (26) in App. B.4.
+Proof of Theorem 4.1. According to Lemma 6.1, Lemma 6.2 and Lemma 6.3, and take ωµ = ω′
+v =
+1
+T 2 and ωv = δT
+T 2 , the
+expected regret of T ′ phases E[R(T ′)] can be upper bounded as
+E[R(T ′)] ≤ E[R1(T ′)|E] + E[R2(T ′)|E] + R3(T)
+(31)
+≤ O
+�
+� �
+i∈E\S⋆
+K
+∆i,S∩B,min
+ln 1
+ωµ
++
+�
+i∈E
+ciK
+∆i,Sc∩B,min
+ln 1
+ω′v
+�
+� + 2µ⋆ [T ′ · (r′ − 1) + H(r′, Λ)]
++ 2µ⋆L + 2µ⋆TL (ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′
+v))
+≤ O
+�
+� �
+i∈E\S⋆
+K
+∆i,S∩B,min
+ln 1
+T +
+�
+i∈E
+ciK
+∆i,Sc∩B,min
+ln 1
+T
+�
+� + 2µ⋆H(1, Λ)
++ 2µ⋆L + 2µ⋆TL
+�
+3ξ(1/T 2) + 2ξ(δT /T 2)
+�
+= Reg1(T) + Reg2(T) + Reg3(T)
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+On the other hand, the high-probability regret (14) can be na¨ıve bounded as
+ˆR(T ′) =
+T ′
+�
+p=1
+np
+�
+r=1
+(µ⋆ − µAp,r)
+≤
+T ′
+�
+p=1
+np
+�
+r=1
+µ⋆
+= Tµ⋆.
+Therefore, we have
+E[R(T ′)] ≤ E[ˆR(T ′)|E] + Reg3(T)
+(32)
+≤ Tµ⋆ + Reg3(T)
+(31) and (32) give the final upper bound with T time steps.
+Theorem 5.1 (Problem-independent Upper Bound). Let {δT }∞
+T =1 ∈ o(1) be a sequence that satisfies ln(1/δT ) = o(T b)
+for all b > 0. If T > L, for any instance Λ with variance gaps lower bounded by ∆v ≤ minS∈AK ∆v
+S, the regret of
+PASCOMBUCB is upper bounded by
+O
+�√
+KLT ln T + LK2
+(∆v)2 ln
+� 1
+δT
+��
+.
+Proof of Theorem 5.1. We firstly deal with the regret due to suboptimality R1(T ′). Let ∆µ be a constant that is to be chosen.
+R1(T ′) =
+T ′
+�
+p=1
+1{Ap ∈ B}∆Ap
+=
+T ′
+�
+p=1
+1{∆Ap ≥ ∆µ}1{Ap ∈ B}∆Ap +
+T ′
+�
+p=1
+1{∆Ap < ∆µ}1{Ap ∈ B}∆Ap
+≤
+T ′
+�
+p=1
+1{∆Ap ≥ ∆µ}1{Ap ∈ B}∆Ap + T · ∆µ
+where the second term makes use of the fact that T ′ ≤ T w.p. 1.
+The first term can be upper bounded by adopting the proof of Lemma 6.3 with the constraint that ∆Ap ≥ ∆µ. Thus, for
+i ∈ E \ S⋆, ∆i,S∩B,min ≥ ∆µ. For i ∈ E,
+∆i,Sc∩B,min ≥ ∆µ
+¯∆′
+i,Sc∩B ≥ max{¯ω∆µ, ∆v/3} = max{∆µ, ∆v/3} =: ∆µv
+ci ≤ 1
+¯ω2 = 1
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+where ¯ω :=
+�
+ln
+1
+ω′v
+ln
+1
+ωµ = 1. The regret due to suboptimality can be upper bounded by
+T ′
+�
+p=1
+1{∆Ap ≥ ∆µ}1{Ap ∈ B}∆Ap ≤
+�
+i∈E\S⋆
+2CγK
+∆i,S∩B,min
+�
+2 ln 1
+ωµ
++ ln ln+
+1
+∆2
+i,S∩B,min
++ D
+�
++
+�
+i∈E
+2ci · CγK
+∆i,Sc∩B,min
+�
+2 ln 1
+ω′v
++ ln ln+
+1
+( ¯∆′
+i,Sc∩B)2 + D
+�
+≤
+�
+i∈E\S⋆
+2CγK
+∆µ
+�
+2 ln 1
+ωµ
++ ln ln+
+1
+(∆µ)2 + D
+�
++
+�
+i∈E
+2CγK
+∆µ
+�
+2 ln 1
+ω′v
++ ln ln+
+1
+(∆µv)2 + D
+�
+≤L · 8CγK
+∆µ
+�
+2 ln T + ln ln+
+1
+(∆µ)2 + D
+�
+By taking ∆µ =
+�
+KL ln(T L)
+T
+, the regret due to suboptimality is bounded by
+R1(T ′) ≤
+T ′
+�
+p=1
+1{∆Ap ≥ ∆µ}1{Ap ∈ B}∆Ap + T · ∆µ
+≤ 16Cγ
+�
+KLT ln(TL) + 8Cγ
+�
+KLT
+ln T ln ln+
+T
+KL ln T + 8Cγ
+�
+KLT
+ln T D +
+�
+KLT ln(TL)
+= O(
+�
+KLT ln(TL))
+= O(
+√
+KLT ln T).
+where we utilize T ≥ L.
+We then cope with the regret due to safeness-checking R2(T ′). According to Lemma 6.3, we only need to upper bound
+H(1, Λ), i.e., the h functions defined in (27), (28), (29) and (30).
+For hS⋆(1, ∆v
+S⋆):
+hS⋆(1, ∆v
+S⋆) ≤ hS⋆(1, ∆v)
+= 18 · CγK
+(∆v)2
+�
+2 ln 1
+ωv
++ ln ln+
+1
+(∆v)2 + D
+�
+= O
+�
+K
+(∆v)2 ln 1
+ωv
+�
+= O
+�
+K
+(∆v)2 ln T
+δ
+�
+For hS∩B(1, ¯∆i,S∩B), define ∆µv := max
+�
+∆µ
+√
+ln(1/ωµ),
+∆v
+3√
+ln(1/ωv)
+�
+= max
+��
+KL
+T ,
+∆v
+3√
+ln(T L/δ)
+�
+. We have ¯∆i,S∩B ≥
+∆µv. The threshold
+�
+∆µv
+¯σ2
+3√
+ln(1/ωv)
+�
+=
+����
+max
+�
+3
+�
+KL ln(T L/δ)
+T
+,∆v
+�
+¯σ2
+���� .
+hS∩B(1, ¯∆i,S∩B) ≤ hS∩B(1, ∆µv)
+(33)
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(Q − 1) CγK
+(∆µv)2
+�
+2 +
+1
+ln(1/ωµv)
+�
+ln ln+
+1
+ln(1/ωµv)(∆µv)2 + D
+��
+,
+����
+∆µv
+¯σ2
+3√
+ln(1/ωv)
+���� ≥ Q − 3
+CγK
+4
+¯σ2
+3√
+ln(1/ωv) · ∆µv
+�
+2 +
+1
+ln(1/ωµv)
+�
+ln ln+
+1
+ln(1/ωµv)(∆µv)2 + D
+��
+, 0 <
+����
+∆µv
+¯σ2
+3√
+ln(1/ωv)
+���� < Q − 3
+3 · CγK
+(∆µv)2
+�
+2 +
+1
+ln(1/ωµv)
+�
+ln ln+
+1
+ln(1/ωµv)(∆µv)2 + D
+��
+,
+����
+∆µv
+¯σ2
+3√
+ln(1/ωv)
+���� = 0
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+O
+�
+(Q − 1)
+K
+(∆µv)2
+�
+,
+����
+∆µv
+¯σ2
+3√
+ln(1/ωv)
+���� ≥ Q − 3
+O
+�
+�
+K
+¯σ2
+3√
+ln(1/ωv) · ∆µv
+�
+� ,
+0 <
+����
+∆µv
+¯σ2
+3√
+ln(1/ωv)
+���� < Q − 3
+O
+�
+K
+(∆µv)2
+�
+,
+����
+∆µv
+¯σ2
+3√
+ln(1/ωv)
+���� = 0
+In particular, in the asymptotic case where T → ∞ and ln 1
+δ = ln 1
+δT = o(T b), ∀b > 0 (this includes the scenario where δ is
+fixed with respect to T), we have
+hS∩B(1, ∆µv) = O
+�
+K
+(∆µv)2
+�
+= O
+�
+K
+(∆v)2 ln T
+δ
+�
+For hR(1, ∆v
+i,R), we have ∆v
+i,R ≥ ∆v.
+Furthermore, ωsum =
+�
+ln(TL) +
+�
+ln(TL/δ) and the changing point
+�
+ωsum·∆v
+√
+ln(1/ω′v)¯σ2
+�
+=
+�
+(√
+ln(T L)+√
+ln(T L/δ))·∆v
+√
+ln(T L)¯σ2
+�
+.
+Thus
+hR(1, ∆v
+i,R) ≤ hR(r′, ∆v)
+(34)
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(Q − 1)
+CγK
+(
+∆v
+3√
+ln(1/ω′v))2
+�
+�2 +
+1
+ln(1/ωvv′)
+�
+�ln ln+
+1
+ln(1/ωvv′)(
+∆v
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+� ,
+�
+ωsum · ∆v
+�
+ln(1/ω′v)¯σ2
+�
+≥ Q − 3
+CγK
+3
+¯σ2
+3ωsum ·
+∆v
+i,R
+3√
+ln(1/ω′v)
+�
+�
+�2 +
+1
+ln(1/ωvv′)
+�
+�
+�ln ln+
+1
+ln(1/ωvv′)(
+∆v
+i,R
+3√
+ln(1/ω′v))2 + D
+�
+�
+�
+�
+�
+� , 0 <
+�
+ωsum · ∆v
+�
+ln(1/ω′v)¯σ2
+�
+≤ Q − 3
+3CγK
+(
+∆v
+3√
+ln(1/ω′
+v))2
+�
+�2 +
+1
+ln(1/ωvv′)
+�
+�ln ln+
+1
+ln(1/ωvv′)(
+∆v
+3√
+ln(1/ω′
+v))2 + D
+�
+�
+�
+� ,
+�
+ωsum · ∆v
+�
+ln(1/ω′v)¯σ2
+�
+= 0
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+O
+�
+�(Q − 1)
+K
+(
+∆v
+3√
+ln(1/ω′v))2
+�
+� ,
+�
+ωsum · ∆v
+�
+ln(1/ω′v)¯σ2
+�
+≥ Q − 3
+O
+�
+�
+�
+K
+¯σ2
+3ωsum ·
+∆v
+i,R
+3√
+ln(1/ω′v)
+�
+�
+� ,
+0 <
+�
+ωsum · ∆v
+�
+ln(1/ω′v)¯σ2
+�
+≤ Q − 3
+= O
+�
+�
+K
+(
+∆v
+3√
+ln(1/ω′v))2
+�
+� ,
+�
+ωsum · ∆v
+�
+ln(1/ω′v)¯σ2
+�
+= 0
+In particular, in the asymptotic case where T → ∞ and ln 1
+δ = ln 1
+δT = o(T b), ∀b > 0, we have
+hR(1, ∆v) = O
+� QK
+(∆v)2 ln 1
+ω′v
+�
+= O
+� QK
+(∆v)2 ln T
+�
+For Sc ∩ B: define ¯∆µv := max
+�
+∆µ
+√
+ln(1/ωµ),
+∆v
+3√
+ln(1/ω′v)
+�
+= max
+��
+KL
+T ,
+∆v
+3√
+ln(T L)
+�
+. We have ¯∆i,Sc∩B ≥ ¯∆µv and
+the changing point
+�
+ωsum· ¯∆µv
+¯σ2/3
+�
+=
+�
+(√
+ln(T L)+√
+ln(T L/δ))·∆µv
+¯σ2/3
+�
+thus
+hSc∩B(1, ¯∆i,Sc∩B) ≤ hSc∩B(1, ¯∆µv)
+(35)
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(Q − 1) CγK
+( ¯∆µv)2
+�
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆µv)2 + D
+��
+,
+�ωsum · ¯∆µv
+¯σ2/3
+�
+≥ Q − 3
+CγK
+�
+3
+¯σ2
+3ωsum ¯∆µv
+� �
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆µv)2 + D
+��
+, 0 <
+�ωsum · ¯∆µv
+¯σ2/3
+�
+< Q − 3
+3CγK
+( ¯∆µv)2
+�
+2 +
+1
+ln(1/ωmax)
+�
+ln ln+
+1
+ln(1/ωmax)( ¯∆µv)2 + D
+��
+,
+�ωsum · ¯∆µv
+¯σ2/3
+�
+= 0, r′ = 1
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+O
+�
+(Q − 1)
+K
+( ¯∆µv)2
+�
+,
+�ωsum · ¯∆µv
+¯σ2/3
+�
+≥ Q − 3
+O
+�
+K
+¯σ2
+3ωsum · ¯∆µv
+�
+,
+0 <
+�ωsum · ¯∆µv
+¯σ2/3
+�
+< Q − 3
+O
+�
+K
+( ¯∆µv)2
+�
+,
+�ωsum · ¯∆µv
+¯σ2/3
+�
+= 0, r′ = 1
+In particular, in the asymptotic case where T → ∞ and ln 1
+δ = ln 1
+δT = o(T b), ∀b > 0, we have
+hSc∩B(1, ¯∆µv) = O
+� QK
+(∆v)2 ln T
+�
+Lastly,
+R3(T) = 2µ⋆L + 2µ⋆TL (ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′
+v))
+≤ 2KL + Kδ + 2KTL · 4 · 2 + ϵ
+ϵ
+�
+1
+T 2
+ln(1 + ϵ)
+�1+ϵ
+≤ 2KL + Kδ + 4K · 2 + ϵ
+ϵ
+�
+1
+ln(1 + ϵ)
+�1+ϵ
+= O(1)
+where we utilize T > L and the O notation refers to the fact that the preceding term is bounded as a function of T.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Note that µ⋆ ≤ K, so Tµ⋆ + Kδ ≤ TK + Kδ.
+In conclusion, according to Theorem 4.1, for any T > L, the problem-independent upper bound is the minimum of
+TK + Kδ and
+O(
+√
+KLT ln T) + K
+� �
+i∈S⋆
+hS⋆(1, ∆v) +
+�
+i∈E
+hS∩B(1, ∆µv) +
+�
+i∈E
+hR(1, ∆v) +
+�
+i∈E
+hSc∩B(1, ¯∆µv)
+�
+≤ O(
+√
+KLT ln T) + K
+�
+KhS⋆(1, ∆v) + LhS∩B(1, ∆µv) + LhR(1, ∆v) + LhSc∩B(1, ¯∆µv)
+�
+= O(
+√
+KLT ln T) + O
+� K3
+(∆v)2 ln T
+δ
+�
++ KL
+�
+hS∩B(1, ∆µv) + hR(1, ∆v) + hSc∩B(1, ¯∆µv)
+�
+where the h functions are defined in (33), (34) and (35). In the asymptotic case where T → ∞ and ln 1
+δ = ln 1
+δT =
+o(T b), ∀b > 0 (this includes the scenario where δ is fixed with respect to T), the asymptotic problem-independent upper
+bound is
+O(
+√
+KLT ln T) + K
+��
+i∈E
+O
+�
+K
+(∆v)2 ln T
+δ
+�
++ O
+�
+i∈E
+� QK
+(∆v)2 ln T
+��
+= O(
+√
+KLT ln T) + O
+� LK2
+(∆v)2 ln 1
+δ
+�
+where we utilize
+√
+T ln T ≥
+QK2
+(∆v)2 ln T when T is sufficiently large.
+C. Proofs of the Lower Bounds
+C.1. Preliminaries
+Let KL(ν, ν′) denote the KL divergence between distributions ν and ν′, and
+d(x, y) := x ln
+�x
+y
+�
++ (1 − x) ln
+�1 − x
+1 − y
+�
+denote the Kullback–Leibler (KL) divergence between the Bernoulli distributions Bern(x) and Bern(y).
+Lemma C.1 (Pinsker’s and reverse Pinsker’s Inequality). Consider two probability mass functions PX, PY defined on the
+same discrete probability space A ⊂ [0, 1]. The following inequalities hold:
+|EX∼PX[X] − EY ∼PY [Y ]| ≤ δ(PX, PY ) ≤
+�
+1
+2KL(PX, PY ) ≤
+1
+√αY
+· δ(PX, PY ), .
+where δ(PX, PY ) := supA⊆A
+��
+a∈A PX(a) − �
+a∈A PY (a)
+�
+=
+1
+2
+�
+a∈A |PX(a) − PY (a)| is the total variational
+distance, and αY := mina∈A:PY (a)>0 Q(a).
+Lemma C.2 (Lemma 1 in Kaufmann et al. (2016)). Assume the distributions under instance Λ1 = (E, AK, ν(1), ¯σ2) and
+instance Λ2 = (E, AK, ν(2), ¯σ2) are mutually absolutely continuous. Given time budget T,
+L
+�
+i=1
+EΛ1[Ni(T)] · KL(ν(1)
+i
+, ν(2)
+i
+) ≥
+sup
+E∈HΛ1
+T
+d
+�
+PΛ1(E), PΛ2(E)
+�
+.
+where Ni(t) denotes the number of time steps item i is selected up to and including time step t and HΛ1
+T is all the possible
+events generated by instance Λ1 and algorithm π with T time steps.
+Lemma C.3. Let solution S containing |S| = m(q < m ≤ K) items be a safe solution under instance Λ1 =
+(E, AK, ν(1), ¯σ2).
+Each item in S is i.i.d.
+with reward distribution ν1 , mean µ1 and variance σ2
+1 < ¯σ2.
+De-
+fine event E(t,1) = {S is identified as safe after time step t}, E(t,2) = {S is chosen at least once after time step t}, and
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+E(t) = E(t,1) ∩ E(t,2). Assume there exists τ ≤ T such that PΛ1[E(τ)] ≥ 1 − δ and PΛ1[E(τ−1,1)] < 1 − δ. If τ exists, we
+have
+�
+i∈S
+EΛ1[Ni(τ)] ≥
+sup
+ν2∈E(ν1)
+d(δ, 1 − δ)
+KL(ν1, ν2).
+Furthermore,
+EΛ1[M(τ)] ≥
+sup
+ν2∈E(ν1)
+1
+|S| − 1 · d(δ, 1 − δ)
+KL(ν1, ν2) := T(ν(1)),
+where M(t) is the number of times that a solution S′ ⊂ S is sampled up to and include time step t and E(ν1) = {ν2 :
+the variance associated to ν2 is larger than ¯σ2/|S|}.
+Proof. With σ2
+2 > ¯σ2/|S|, we construct an alternative instance Λ2 = (E, AK, ν2, ¯σ2), under which each item in S is with
+reward distribution ν2 , mean µ2 and variance σ2
+2, while the distributions of other items remain unchanged.
+Define event E(t,1) = {S is identified as safe after time step t}, E(t,2) = {S is chosen at least once after time step t}, and
+E(t) = E(t,1) ∩ E(t,2). Assume there exists τ ≤ T such that PΛ1[E(τ)] ≥ 1 − δ and PΛ1[E(τ−1,1)] < 1 − δ. Since S is
+unsafe under instance Λ2 and all the solutions chosen {St}T
+t=1 ⊂ AK are safe with probability at least 1 − δ, we have
+PΛ2[E(t)] < δ for all t ≤ T.
+We now apply Lemma C.2 to obtain that
+�
+i∈S
+EΛ1[Ni(τ)] · KL(ν1, ν2) ≥ d(P1(Ec
+(τ)), P2(Ec
+(τ))) ≥ d(δ, 1 − δ) ⇒
+�
+i∈S
+EΛ1[Ni(τ)] ≥ d(δ, 1 − δ)
+KL(ν1, ν2).
+Since PΛ1[E(τ−1,1)] < 1 − δ, we can select at most m − 1 items at one time step among the first τ time steps. Therefore,
+EΛ1[M(τ)] ≥
+1
+m − 1
+�
+i∈P
+EΛ1[Ni(τ)] ≥
+1
+m − 1 · d(δ, 1 − δ)
+KL(ν1, ν2).
+Theorem 4.5 (Impossibility result). Let {δT }∞
+T =1 ∈ o(1) be a sequence that satisfies the following condition. There exists
+b > 0 such that ln(1/δT ) = Ω(T b). For instance Λ, the regret of any algorithm is lower bounded by Ω(T b).
+Proof. The proof is similar to the proof of Lemma C.3. We consider an alternative instance Λ2 = (E, AK, ν(2), ¯σ2) with
+the distributions of the items in the optimal safe solution S⋆ changed such that (assume the variances of all the items in S⋆
+are changed (increased))
+�
+i∈S⋆
+(σ(2)
+i
+)2 ≥ ¯σ2
+i.e., under instance Λ2, this solution S⋆ is unsafe (thus not optimal safe). The other items remain unchanged.
+By a similar argument as the proof of Lemma C.3, we have
+�
+i∈S⋆
+EΛ1[Ni(τ)] · KL(ν(1)
+i
+, ν(2)
+i
+) ≥ d(δ, 1 − δ)
+⇒
+�
+i∈S⋆
+EΛ1[Ni(τ)] ≥
+d(δ, 1 − δ)
+mini∈S⋆ KL(ν(1)
+i
+, ν(2)
+i
+)
+So the safeness checking of S⋆ will take Ω(ln
+1
+2.4δT ) = Ω(T b)
+Recall the probably anytime-safe constraint (1):
+P
+�
+∀ t ∈ [T], St ∈ S
+�
+≥ 1 − δT .
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+This indicates at time step t, for any solution S, if PH(0)
+t [S ∈ S] < 1−δT , S will not be selected at this time step. Otherwise,
+(1) is violated. Therefore, before the safeness of the optimal safe solution S is ascertained, it is not going to be sampled and
+the instantaneous regret will be lower bounded by minS∈S∩B ∆S.
+In conclusion, the regret is at least ln
+1
+2.4δT · minS∈S∩B ∆S = Ω(T b).
+We derive both the problem-dependent and problem independent lower bounds on the K-path semi-bandit problem. The
+items in the ground set are divided into L0 paths: P1, . . . , PL0, each of which contains K unique items. Path Pj contains
+items (j − 1)K + 1, . . . , jK. Without loss of generality, we assume that L/K is an integer. A set S is a solution if and
+only if S ⊂ Pj for some j. In other words, the solution set AK = {S : S ⊂ Pj, ∃j = 1, . . . , L0}. We let Ni(t) denote the
+number of time steps item i is selected up to and including time step t, Mj(t) denote the number of time steps a safe subset
+in path j is selected up to and including time step t, and Sj(t) denote the time steps when a safe subset in path j is selected,
+i.e.,
+Ni(t) =
+t
+�
+s=1
+1{i ∈ Ss},
+Mj(t) =
+t
+�
+s=1
+1{Ss ⊂ Pj, Ss is safe },
+Sj(t) = {1 ≤ s ≤ t : Ss ⊂ Pj, Ss is safe }.
+We let Reg[j](t) denote the regret accumulated in Sj(t). Since all the chosen solutions are safe with probability at least
+1 − δ, we have Reg(T) ≥ (1 − δ) · �L0
+j=1 Reg[j](T). In the following, we lower bound the regret accumulated in time steps
+where safe solutions are chosen.
+We will construct several instances to prove each of the lower bounds (which will be specified in the proof) such that
+under instance k (Λk = (E, AK, ν(k), ¯σ2)), the stochastic reward of items in path j (1 ≤ j ≤ L0) are i.i.d., i.e. ,
+ν(k)
+i
+= ν(k)
+Pj , ∀i ∈ Pj, which will be specified in each case. Under instance k, we define several other notations as follows:
+• Let Wi(t)(k) be the random reward of arm i at time step t.
+• Let S(k)
+t
+be the pulled solution at time step t, and H(j)
+t
+= {(S(j)
+s , {Wi(s)(k)}s∈S(k)
+s )}t
+s=1 be the sequence of selected
+solutions and observed rewards up to and including time step t.
+For simplicity, we abbreviate EH(k)
+T , PH(k)
+T , as Ek, Pk respectively.
+C.2. Problem-dependent Lower bound
+In order to provide a better understanding of the analysis, we first derive a lower bound with Gaussian distributions
+(unbounded) in Theorem C.4. With the same technique, we derive a lower bound with bounded Bernoulli distributions in
+Theorem 4.3, which corroborates with our problem setup.
+Theorem C.4 (Problem-dependent lower bound for sub-Gaussian instances). Let {δT }∞
+T =1 ∈ o(1) be a sequence that
+satisfies ln(1/δT ) = o(T b) for all b > 0. There exists an instance Λ, for any {δT }T ∈N-variance-constrained consistent
+algorithm π, the regret is lower bounded by
+Ω
+��
+i∈E
+ln T
+∆i,S∩B,min
+�
++ µ⋆
+K · Ω
+�K · ln(1/δT )
+(∆S⋆)2
++
+�
+i∈E
+�
+Ψ′
+i,S∩B +
+ln T
+(∆v
+i,R)2 + Φi,Sc∩B
+��
+.
+Proof. Given a vanishing sequence {δT }T ∈N, we consider a fixed {δT }T ∈N-variance-constrained consistent algorithm π on
+the K-path semi-bandit problem. For simplicity, the distributions of the items are assumed to be Gaussian in the proof, but
+the techniques can be applied to the Bernoulli case and we provide the instance design and the corresponding bound at the
+end of the proof.
+Under instance Λ0 (the base instance), σ2 = 2¯σ2
+K (so the absolutely safe solutions contain at most K/2 items) and the
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+distributions of the items are
+ν(0)
+i
+= N(µ(0)
+j , (σ2
+i )(0)) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ν(0)
+P1 = N(∆, ¯σ2 − ϵv
+K
+),
+i ∈ P1
+ν(0)
+Pj = N(∆ − ϵµ
+K , ¯σ2 − ϵv
+K
+),
+i ∈ Pj, 2 ≤ j ≤ L1 + 1
+ν(0)
+Pj = N(∆ + ϵµ
+K , ¯σ2 + ϵv
+K
+),
+i ∈ Pj, L1 + 2 ≤ j ≤ L2 + 1
+ν(0)
+Pj = N(∆ − ϵµ
+K , ¯σ2 + ϵv
+K
+),
+i ∈ Pj, L2 + 2 ≤ j ≤ L0
+where ϵµ < ∆
+K and ϵv are small positive constants (e.g. ϵµ =
+∆
+2K , ϵv ≤ ¯σ2
+K2 ), and L1 = L2 − L1 = L0 − L2 − 1 = L0−1
+3
+(assume it is an integer). In this case,
+• path 1 is an optimal safe path,
+• path 2 to path L1 + 1 are the safe and suboptimal paths,
+• path L1 + 2 to path L2 + 1 are the risky paths
+• path L2 + 2 to path L3 + 1 = L0 are the unsafe and suboptimal paths.
+In the following, we will compute the minimum regret yielded from each of the paths.
+Case 1: the optimal safe path P1
+In order to achieve o(T a), ∀a > 0 regret, any algorithm has to identify the safeness of the optimal safe solution P1 and
+sample P1 Ω(T) times, otherwise, the regret is linear. According to Lemma C.3, the expected number of time steps needed
+for the safeness identification of P1 is lower bounded by
+E0[M1(τ)] ≥
+sup
+ν(1)
+P1 ∈E(ν(0)
+P1 )
+1
+K − 1 · d(δT , 1 − δT )
+KL(ν(0)
+P1 , ν(1)
+P1 )
+:= T(ν(0)
+P1 )
+where E(ν(0)
+P1 ) = {ν(1)
+P1
+:
+the variance associated to ν(1)
+P1 is larger than ¯σ2/K}. In particular, we let instance Λ1 =
+(E, AK, ν(1), ¯σ2) with
+ν(1)
+i
+=
+�
+�
+�
+�
+�
+ν(1)
+P1 = N
+�
+∆, ¯σ2 + ϵv
+1
+K
+�
+,
+i ∈ P1
+ν(0)
+i
+,
+i /∈ P1
+where ϵv
+1 is an arbitrarily small constant. Thus, we can take supremum over ϵv
+1 > 0 and have
+T(ν(0)
+P1 ) ≥
+1
+K − 1 ·
+ln
+1
+2.4δT
+KL(N(∆, ¯σ2−ϵv
+K
+), N(∆, ¯σ2
+K ))
+≥
+1
+K − 1 · 4K2(¯σ2/K)2
+(ϵv)2
+ln
+1
+2.4δT
+= K(σ2)2
+(ϵv)2
+ln
+1
+2.4δT
+When a solution S ⊂ P1 is chosen before this time step, the instantaneous regret is lower bounded by ∆. The accumulative
+regret from P1 is lower bounded by
+Reg[1](T) ≥ ∆ · T(ν(0)
+P1 ) = Ω( K∆
+(∆v
+P1)2 ln 1
+δT
+)
+Case 2: the safe and suboptimal paths
+For any safe and suboptimal path Pj(2 ≤ j ≤ L1 + 1), we define instance Λj = (E, AK, ν(j), ¯σ2) with
+ν(j)
+i
+=
+�
+�
+�
+�
+�
+ν(j)
+Pj = N(∆ +
+ϵµ
+j
+K , ¯σ2 − ϵv
+K
+),
+i ∈ Pj
+ν(0)
+i
+,
+i /∈ Pj
+where ϵµ
+j < ∆ is an arbitrarily small constant. So Pj is the optimal safe solution under instance j.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Fix any item i ∈ Pj, consider the event Ej = {Ni(T) ≥ T
+2 }, under instance 1, Ej indicates the optimal safe solution P1
+is sampled less than T
+2 times; under instance j, Ec
+j indicates the optimal safe solution Pj is sampled less than T
+2 times.
+Therefore,
+RegΛ0(T) + RegΛj(T) ≥ T
+2 ϵµP0[Ej] + T
+2 ϵµ
+j Pj[Ec
+j ]
+≥ T
+2 min{ϵµ, ϵµ
+j }
+�
+P0[Ej] + Pj[Ec
+j ]
+�
+According to Lemma C.1 and Lemma C.2, we have
+P0[Ej] + Pj[Ec
+j ] ≥ 1
+2 exp{−KL(P1, Pj)}
+KL(P1, Pj) =
+L
+�
+i=1
+E0[Ni(T)] · KL
+�
+ν(0)
+i
+, ν(j)
+i
+�
+=
+�
+i∈Pj
+E0[Ni(T)] · KL
+�
+ν(0)
+i
+, ν(j)
+i
+�
+Thus
+RegΛ0(T) + RegΛj(T) ≥ T
+2 min{ϵµ, ϵµ
+j } · 1
+2 exp{−KL(P1, Pj)}
+⇐⇒
+ln
+�
+RegΛ0(T) + RegΛj(T)
+�
+ln T
+≥ 1 +
+min{ϵµ, ϵµ
+j }/4
+ln T
+− KL(P1, Pj)
+ln T
+(∗)
+=⇒
+KL(P1, Pj)
+ln T
+≥ 1
+⇐⇒
+�
+i∈Pj E0[Ni(T)]
+ln T
+≥
+1
+KL
+�
+ν(0)
+Pj , ν(j)
+Pj
+� = 2 ¯σ2−ϵv
+K
+(
+ϵµ+ϵµ
+j
+K
+)2
+(∗∗)
+=⇒
+�
+i∈Pj E0[Ni(T)]
+ln T
+≥ 2 ¯σ2−ϵv
+K
+( ϵµ
+K )2
+=: T(ν(0)
+Pj )
+where in (∗) we let T → ∞ and note that both RegΛ0(T) and RegΛj(T) are of order o(T a), ∀a > 0; in (∗∗) we take
+supremum over ϵµ
+j > 0.
+We also have to take the safeness constraint into consideration. According to Lemma C.3, in order to check the safeness of
+Pj, the items in Pj have to be sampled
+�
+i∈Pj
+EΛ0[Ni(τ)] ≥
+sup
+ν′
+Pj ∈E(ν(0)
+Pj )
+d(δT , 1 − δT )
+KL(ν(0)
+Pj , ν′
+Pj)
+.
+⇐⇒
+�
+i∈Pj EΛ0[Ni(τ)]
+ln T
+≥
+sup
+ν′
+Pj ∈E(ν(0)
+Pj )
+1
+KL(ν(0)
+Pj , ν′
+Pj)
+· d(δT , 1 − δT )
+ln T
+≥ 4K2(¯σ2/K)2
+(ϵv)2
+ln
+1
+2.4δT
+ln T
+:= Tsafe(ν(0)
+Pj ).
+If T(ν(0)
+Pj ) ≤ Tsafe(ν(0)
+Pj ), it indicates the suboptimality of Pj is identified before the safeness. Furthermore, whenever a
+solution S ⊂ Pj is sampled, S can have most K − 1 items. So the regret is lower bounded by
+Reg[j](T)
+ln T
+≥
+T(ν(0)
+Pj )
+K − 1 · (ϵµ + ∆ − ϵµ
+K ) ≥ 2K2 ¯σ2−ϵv
+K
+(ϵµ)2
+· (
+∆
+K − 1 + ϵµ
+K )
+=⇒
+Reg[j](T)
+ln T
+≥ 2K ¯σ2−ϵv
+K
+(ϵµ)2
+· (∆ + ϵµ) ≥ Kσ2/2
+(ϵµ)2 · (∆ + ϵµ)
+=⇒
+Reg[j](T) = Ω
+�
+K
+∆2
+Pj
+(∆Pj + ∆) ln T
+�
+(36)
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+If T(ν(0)
+Pj ) ≥ Tsafe(ν(0)
+Pj ), it indicates the suboptimality of Pj is identified after the safeness. Thus, whenever a solution
+S ⊂ Pj is sampled, S can have most K − 1 items before the safeness checking is finished. So the regret is lower bounded by
+Reg[j](T)
+ln T
+≥
+Tsafe(ν(0)
+Pj )
+K − 1
+· (ϵµ + ∆ − ϵµ
+K ) +
+T(ν(0)
+Pj ) − Tsafe(ν(0)
+Pj )
+K
+· ϵµ
+≥
+T(ν(0)
+Pj )
+K
+· ϵµ + (∆ − ϵµ
+K ) ·
+Tsafe(ν(0)
+Pj )
+K − 1
+≥ 2K ¯σ2−ϵv
+K
+(ϵµ)2
+· ϵµ + (∆ − ϵµ
+K ) ·
+1
+K − 1 · 4K2(¯σ2/K)2
+(ϵv)2
+ln
+1
+2.4δT
+ln T
+≥ Kσ2/2
+ϵµ
++ (∆ − ϵµ
+K ) ·
+1
+K − 1 · K2(σ2)2
+(ϵv)2
+ln
+1
+2.4δT
+ln T
+=⇒
+Reg[j](T) = Ω
+�
+K
+∆Pj
+ln T + (∆ − ∆Pj
+K )
+K
+(∆v
+Pj)2 ln 1
+δT
+�
+(37)
+Based on (36) and (37), the regret is
+Reg[j](T) = Ω
+�
+K
+∆Pj
+ln T + ∆ · min
+�
+K
+∆2
+Pj
+ln T,
+K
+(∆v
+Pj)2 ln 1
+δT
+��
+Case 3: the risky paths
+For any risky path Pj (L1 + 2 ≤ j ≤ L2 + 1, we define instance Λj = (E, AK, ν(j), ¯σ2) with
+ν(j)
+i
+=
+�
+�
+�
+�
+�
+ν(j)
+Pj = N(∆ + ϵµ
+K , ¯σ2 − ϵv
+j
+K
+),
+i ∈ Pj
+ν(0)
+i
+,
+i /∈ Pj
+where ϵv
+j is an arbitrarily small constant. So Pj is the optimal safe solution under instance j.
+Fix any item i ∈ Pj, consider the event Ej = {Ni(T) ≥ T
+2 }, under instance 1, Ej indicates the optimal safe solution P1
+is sampled less than T
+2 times; under instance j, Ec
+j indicates the optimal safe solution Pj is sampled less than T
+2 times.
+Therefore,
+RegΛ0(T) + RegΛj(T) ≥ T
+2
+�
+∆ − K − 1
+K
+ϵµ
+�
+P0[Ej] + T
+2 ϵµPj[Ec
+j ]
+≥ T
+2 ϵµ ·
+�
+P0[Ej] + Pj[Ec
+j ]
+�
+According to Lemma C.1 and Lemma C.2, we have
+P0[Ej] + Pj[Ec
+j ] ≥ 1
+2 exp{−KL(P1, Pj)}
+KL(P1, Pj) =
+L
+�
+i=1
+E0[Ni(T)] · KL
+�
+ν(0)
+i
+, ν(j)
+i
+�
+=
+�
+i∈Pj
+E0[Ni(T)] · KL
+�
+ν(0)
+i
+, ν(j)
+i
+�
+Thus
+RegΛ0(T) + RegΛj(T) ≥ T
+2 ϵµ · 1
+2 exp{−KL(P1, Pj)}
+⇐⇒
+ln
+�
+RegΛ0(T) + RegΛj(T)
+�
+ln T
+≥ 1 +
+min{ϵµ, ϵµ
+j }/4
+ln T
+− KL(P1, Pj)
+ln T
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+(∗)
+=⇒
+KL(P1, Pj)
+ln T
+≥ 1
+⇐⇒
+�
+i∈Pj E0[Ni(T)]
+ln T
+≥
+1
+KL
+�
+ν(0)
+Pj , ν(j)
+Pj
+� = K2(σ2)2
+(ϵv)2
+=: T(ν(0)
+Pj )
+where in (∗) we let T → ∞ and note that both RegΛ0(T) and RegΛj(T) are of order o(T a), ∀a > 0.
+Note that Pj is a risky path and if any solution S ⊂ Pj is sampled, |S| ≤ K − 1. Thus, E[Mj(T)] ≥
+T (ν(0)
+Pj )
+K−1 ln T. The
+regret is lower bounded by
+Reg[j](T) = Ω
+��
+∆ − K − 1
+K
+ϵµ
+�
+1
+K − 1
+K2(σ2)2
+(ϵv)2
+ln T
+�
+= Ω
+�
+K∆
+(∆v
+Pj)2 ln T
+�
+Case 4: the unsafe and suboptimal paths
+For any unsafe and suboptimal path Pj(L2 + 2 ≤ j ≤ L0), we define instance Λj = (E, AK, ν(j), ¯σ2) with
+ν(j)
+i
+=
+�
+�
+�
+�
+�
+ν(j)
+Pj = N(∆ +
+ϵµ
+j
+K , ¯σ2 − ϵv
+j
+K
+),
+i ∈ Pj
+ν(0)
+i
+,
+i /∈ Pj
+where ϵµ
+j < ∆, ϵv
+j < ¯σ2
+K are arbitrarily small constants. So Pj is the optimal safe solution under instance j.
+Fix any item i ∈ Pj, consider the event Ej = {Ni(T) ≥ T
+2 }, under instance 1, Ej indicates the optimal safe solution P1
+is sampled less than T
+2 times; under instance j, Ec
+j indicates the optimal safe solution Pj is sampled less than T
+2 times.
+Therefore,
+RegΛ0(T) + RegΛj(T) ≥ T
+2 ϵµP0[Ej] + T
+2 ϵµ
+j Pj[Ec
+j ]
+≥ T
+2 min{ϵµ, ϵµ
+j }
+�
+P0[Ej] + Pj[Ec
+j ]
+�
+According to Lemma C.1 and Lemma C.2, we have
+P0[Ej] + Pj[Ec
+j ] ≥ 1
+2 exp{−KL(P1, Pj)}
+KL(P1, Pj) =
+L
+�
+i=1
+E0[Ni(T)] · KL
+�
+ν(0)
+i
+, ν(j)
+i
+�
+=
+�
+i∈Pj
+E0[Ni(T)] · KL
+�
+ν(0)
+i
+, ν(j)
+i
+�
+Thus
+RegΛ0(T) + RegΛj(T) ≥ T
+2 min{ϵµ, ϵµ
+j } · 1
+2 exp{−KL(P1, Pj)}
+⇐⇒
+ln
+�
+RegΛ0(T) + RegΛj(T)
+�
+ln T
+≥ 1 +
+min{ϵµ, ϵµ
+j }/4
+ln T
+− KL(P1, Pj)
+ln T
+(∗)
+=⇒
+KL(P1, Pj)
+ln T
+≥ 1
+⇐⇒
+�
+i∈Pj E0[Ni(T)]
+ln T
+≥
+1
+KL
+�
+ν(0)
+Pj , ν(j)
+Pj
+� = 4K2
+�
+�
+�
+ϵv
+j + ϵv
+(¯σ2 − ϵv
+j )/K
+�2
++
+2(ϵµ
+j + ϵµ)2
+(¯σ2 − ϵv
+j )/K
+�
+�
+−1
+(∗∗)
+=⇒
+�
+i∈Pj E0[Ni(T)]
+ln T
+≥ 4K2
+�� ϵv
+σ2/2
+�2
++ 2(ϵµ)2
+σ2/2
+�−1
+=: T(ν(0)
+Pj )
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+where in (∗) we let T → ∞ and note that both RegΛ0(T) and RegΛj(T) are of order o(T a), ∀a > 0; in (∗∗) we take the
+supremum over ϵµ
+j > 0, ϵv
+j > 0.
+Note that Pj is an unsafe path and if any solution S ⊂ Pj is sampled, |S| ≤ K − 1. Thus, E[Mj(T)] ≥
+T (ν(0)
+Pj )
+K−1 ln T. The
+regret is lower bounded by
+Reg[j](T) ≥
+�
+∆ + K − 1
+K
+ϵµ
+� 4K2
+K − 1
+�� ϵv
+σ2/2
+�2
++ 2(ϵµ)2
+σ2/2
+�−1
+ln T
+= Ω
+� K∆ + Kϵµ
+max{ϵµ, ϵv}2 ln T
+�
+= Ω
+�
+min
+�
+K∆
+∆2
+Pj
+ln T,
+K∆
+(∆v
+Pj)2 ln T
+��
+In conclusion, the regret yielded from these paths is lower bounded by
+Reg(T)
+1 − δT
+≥ Reg[1](T) +
+L1+1
+�
+j=2
+Reg[j](T) +
+L2+1
+�
+j=L1+2
+Reg[j](T) +
+L0
+�
+j=L2+2
+Reg[j](T)
+≥ Ω
+� K∆
+(ϵv)2 ln 1
+δT
+�
++
+L1+1
+�
+j=2
+Ω
+�K
+ϵµ ln T + ∆ · min
+� K
+(ϵµ)2 ln T,
+K
+(ϵv)2 ln 1
+δT
+��
++
+L2+1
+�
+j=L1+2
+Ω
+� K∆
+(ϵv)2 ln T
+�
++
+L0
+�
+j=L2+2
+Ω
+�
+min
+� K∆
+(ϵµ)2 ln T, K∆
+(ϵv)2 ln T
+��
+.
+Note
+• L1 = L2 − L1 = L0 − L2 − 1 = L0−1
+3
+, L0 = L
+K and µ⋆ = K∆.
+• for S⋆, ϵv = ∆v
+S⋆.
+• for S ∈ S ∩ B and i ∈ S, we check that if S ⊂ Pj, 2 ≤ j ≤ L1 + 1 , ∆i,S∩B,min = ϵµ and
+Ψ′
+i,S∩B ≥ min
+�ln T
+∆2
+S
+, 9 ln(1/δT )
+(∆v
+S)2
+�
+.
+where the equality holds when S = Pj, 2 ≤ j ≤ L1 + 1.
+• for S ∈ R and i ∈ S, ϵv = ∆v
+i,R.
+• S ∈ Sc ∩ B and i ∈ S, we can easily check S = Pj for some L2 + 2 ≤ j ≤ L0, thus ∆i,Sc∩B,min = ϵµ and
+Φi,Sc∩B = min
+�ln T
+∆2
+S
+, 9 ln T
+(∆v
+S)2
+�
+.
+Therefore,
+Reg(T) ≥ Ω
+�
+L ln T
+∆i,S∩B,min
+�
++ µ⋆
+K · Ω
+�
+K · ln(1/δT )
+(∆S⋆)2
++
+�
+i∈E
+Ψ′
+i,S∩B +
+�
+i∈E
+ln T
+(∆v
+i,R)2 +
+�
+i∈E
+Φi,Sc∩B
+�
+.
+Theorem 4.3 (Problem-dependent lower bound). Let {δT }∞
+T =1 ∈ o(1) be a sequence that satisfies ln(1/δT ) = o(T b) for
+all b > 0. There exists an instance Λ such that for any {δT }T ∈N-variance-constrained consistent algorithm π, the regret is
+lower bounded by
+Ω
+� �
+i∈E
+ln T
+∆i,S∩B,min
+�
++ µ⋆
+K · Ω
+�K ln(1/δT )
+(∆v
+S⋆)2
++
+�
+i∈E
+�
+Ψ′
+i,S∩B +
+ln T
+(∆v
+i,R)2 + Φi,Sc∩B
+��
+.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Proof. We divide the whole ground set into several paths. Meanwhile, we define four sets E1, . . . , E4 such that Ei ∩Ej = ∅
+for any i ̸= j. P1 = E1, Pj ⊂ Ej for j = 2, 3, 4. For any j > 4, there exists k ̸= 1 such that Pj ⊂ Ek. For any path Pj, if
+Pj ⊂ E4, |Pj| = K2; otherwise, |Pj| = K1. Pj consists of arms
+j−1
+�
+i=1
+|Pi| + 1, . . . ,
+j−1
+�
+i=1
+|Pi| + K1 · 1{Pj ̸⊂ E4} + K2 · 1{Pj ⊂ E4}.
+The feasible solution set AK consists of all subsets of each path.
+We let Bern(a) denote the Bernoulli distribution with parameter a(a ∈ (0, 1)). Note that the variance of Bern(a) is a(1−a).
+We construct an instance with νi = Bern(µi). With ε1, ε2 > 0, we set
+µi = ∆
+∀i ∈ E1,
+µi = ∆ − ε1
+∀i ∈ E2,
+µi = ∆ + ε2
+∀i ∈ E3,
+µi = ∆ − ε3
+∀i ∈ E4.
+We let 2 ≤ K1 < K2 ≤ K, ∆ < 1/2 and
+K1∆(1 − ∆) < ¯σ2 (paths in E1 and E2 are safe),
+K1(∆ + ε2)(1 − ∆ − ε2) > ¯σ2 (paths in E3 are unsafe),
+K2(∆ − ε3)(1 − ∆ + ε3) > ¯σ (paths in E4 are unsafe),
+K2(∆ − ε3) < K1∆ (paths in E4 are suboptimal),
+(K1 − 1) · (∆ + ε2) < K1 · ∆ ⇔ (K1 − 1) · ε2 < ∆ ⇔ ε2 <
+∆
+K1 − 1 (paths in E3 are suboptimal or unsafe).
+The conditions above indicate that
+• P1 is the unique optimal safe set;
+• if Pj ⊂ E2, Pj and its subsets are safe but suboptimal;
+• if Pj ⊂ E3, Pj is risky, and its proper subsets are suboptimal;
+• if Pj ⊂ E4, Pj and its subsets are suboptimal, and Pj is unsafe.
+Let
+p1 := 1 −
+�
+1 − ¯σ2/K1
+2
+and p2 := 1 −
+�
+1 − ¯σ2/K2
+2
+,
+The relations between µi’s are as in the following figure:
+With a similar proof to that of Theorem C.4, we have, the accumulative regret from the optimal P1 is lower bounded by
+Reg[1](T) ≥ Ω( K1∆
+(∆v
+P1)2 ln 1
+δT
+);
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+the accumulative regret from a safe and suboptimal path in E2 is lower bounded by
+Reg[j](T) ≥ Ω
+�
+K1
+∆Pj
+ln T + ∆ · min
+�
+K1
+∆2
+Pj
+ln T,
+K1
+(∆v
+Pj)2 ln 1
+δT
+��
+;
+the accumulative regret from a risky path in E3 is lower bounded by
+Reg[j](T) ≥ Ω
+�
+K1∆
+(∆v
+Pj)2 ln T
+�
+;
+the accumulative regret from a unsafe and suboptimal path in E4 is lower bounded by
+Reg[j](T) ≥ Ω
+�
+min
+�
+K2∆
+∆2
+Pj
+ln T, K2∆
+(∆v
+Pj)2 ln T
+��
+.
+We let
+ϵµ = min
+j {∆Pj},
+ϵv = min
+j {∆v
+Pj};
+and set K1, K2, ε1, ε2, ε3 such that
+K1 > 3
+4 · K2,
+K2 = K,
+min
+j {∆Pj} < 2ϵµ,
+min
+j {∆v
+Pj} < 2ϵv.
+In conclusion, the regret yielded from these paths is lower bounded by
+Reg(T)
+1 − δT
+≥ Reg[1](T) +
+�
+Pj∈E2
+Reg[j](T) +
+�
+Pj∈E3
+Reg[j](T) +
+�
+Pj∈E4
+Reg[j](T)
+≥ Ω
+� K∆
+(ϵv)2 ln 1
+δT
+�
++ |E2| · Ω
+�K
+ϵµ ln T + ∆ · min
+� K
+(ϵµ)2 ln T,
+K
+(ϵv)2 ln 1
+δT
+��
++ |E3| · Ω
+� K∆
+(ϵv)2 ln T
+�
++ |E4| · Ω
+�
+min
+� K∆
+(ϵµ)2 ln T, K∆
+(ϵv)2 ln T
+��
+.
+Note that
+• µ⋆ = K1∆.
+• for S⋆, ϵv = ∆v
+S⋆.
+• for S ∈ S ∩ B and i ∈ S, we check that if S ⊂ Pj, j ∈ E2 , ∆i,S∩B,min = ϵµ and
+Ψ′
+i,S∩B ≥ min
+�ln T
+∆2
+S
+, 9 ln(1/δT )
+(∆v
+S)2
+�
+.
+where the equality holds when S = Pj, Pj ∈ E2.
+• for S ∈ R and i ∈ S, ϵv = ∆v
+i,R.
+• S ∈ Sc ∩ B and i ∈ S, we can easily check S = Pj for some Pj ∈ E4, thus ∆i,Sc∩B,min = ϵµ and
+Φi,Sc∩B = min
+�ln T
+∆2
+S
+, 9 ln T
+(∆v
+S)2
+�
+.
+Therefore,
+Reg(T) ≥ Ω
+�
+L ln T
+∆i,S∩B,min
+�
++ µ⋆
+K · Ω
+�
+K · ln(1/δT )
+(∆S⋆)2
++
+�
+i∈E
+Ψ′
+i,S∩B +
+�
+i∈E
+ln T
+(∆v
+i,R)2 +
+�
+i∈E
+Φi,Sc∩B
+�
+.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+C.3. Problem-independent Lower bound
+Theorem 5.2 (Problem-independent lower bound). Let the minimum variance gap be ∆v := minS∈AK ∆v
+S. When
+K3 ≥ L2, we have
+Reg(T) = Ω
+�√
+KLT + min
+�
+L
+(∆v)2 ln
+� 1
+δT
+�
+, T
+��
+.
+Proof. Since the rewards of items are bounded in [0, 1], the variance of each arm is at most 1/4. Therefore, when
+¯σ2 ∈ [K/4, ∞), any solution in AK is safe and there exists a generic lower bound (Kveton et al., 2015a). When
+¯σ2 ∈ (0, K/4), let
+¯a := 1 +
+�
+1 − ¯σ2/K
+2
+and a := 1 −
+�
+1 − ¯σ2/K
+2
+.
+We consider the instances containing items with Bernoulli reward distributions. We let Bern(a) denote the Bernoulli
+distribution with mean a. An item i ∈ [L] with reward distribution Bern(µi) is with variance µi(1 − µi), which is smaller
+than ¯σ2/K if and only if µi ∈ (0, a) ∪ (¯a, 1).
+We will construct 3 instances such that under instance k (0 ≤ k ≤ 2), the stochastic reward of an item in path j (1 ≤ j ≤ L0)
+is drawn from distribution ν(k)
+j
+= Bern(µ(k)
+j ), where µ(k)
+j
+will be specified in each case.
+Under instance 0, with µ0 < a, we let µ(0)
+j
+= µ0 for all j, i.e., the reward distribution of each item is Bern(µ0). Since
+µPj = Kµ0 and σ2
+Pj = Kµ0(1 − µ0) < ¯σ2 for all j, each path is an identical safe and optimal solution. Since all paths
+are equivalent under instance 0, we have E0[Mj(t)] = T/L0 for all j ∈ 1, . . . , L0, where E0 denote the expectation under
+instance 0.
+We next construct instance 1 such that
+µ(1)
+1
+= µ1,
+µ(1)
+j
+= µ0
+j ̸= 1,
+where µ0 < µ1 < a.2 Hence, P1 is the unique optimal safe solution under instance 1 while all other solutions are safe but
+suboptimal.
+With an analysis similar to that of Lemma 6.4 in Zhong et al. (2021), we can show that
+Lemma C.5. Let the reward distribution of item i be ν(j)
+i
+under instance j(j = 1, 2), then
+KL
+�
+H(1)
+T , H(2)
+T
+�
+=
+L
+�
+i=1
+E0[Ni(T)] · KL
+�
+ν(1)
+i
+, ν(2)
+i
+�
+.
+Hence, we have
+KL
+�
+H(0)
+T , H(1)
+T
+�
+=
+�
+i∈P1
+E0[Ni(T)] ·
+�
+ν(0)
+i
+, ν(1)
+i
+� (a)
+≤ K · E0[M1(t)] · d(µ0, µ1) = K · T
+L0
+· d(µ0, µ1),
+where (a) follows from the fact that at most K items are selected at one time step and the definition of instances.
+Next, we apply Pinsker’s Inequality to bound E1[M1(T)]. Lemma C.1 indicates that
+|E0[M1(T)] − E1[M1(T)]| ≤
+�
+1
+2KL
+�
+H(0)
+T , H(1)
+T
+�
+,
+⇒
+����E1[M1(T)] − T
+L0
+���� ≤
+�
+KT
+2L0
+· d(µ0, µ1).
+2In this proof, µ1 are µ2 are redefined to minimize clutter; the previous definitions of them not used.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+Moreover, since the paths Pj for j ̸= 1 are identical under instance 1, we have
+E1[Mj(T)] =
+1
+L0 − 1
+�
+T − E1[M1(T)]
+�
+≥
+1
+L0 − 1
+�
+T − T
+L0
+−
+�
+KT
+2L0
+· d(µ0, µ1)
+�
+:= M.
+To learn the regret incurred by P2 under instance 1, we need to take the effects of the safety constraint into consideration.
+Lemma C.3 indicates that
+• if M < T(ν0), at each of the first M time steps in S2(T), at most K − 1 items are pulled and regret of at least
+[K(µ1 − µ2) + µ2] · M is incurred, i.e.,
+Reg[2] ≥ [K(µ1 − µ2) + µ2] · M.
+• if M ≥ T(ν0), at each of the first T(ν0) time steps in S2(T), at most K − 1 items are pulled and regret of at least
+[K(µ1 − µ2) + µ2] · T(ν0) is incurred; besides, at the subsequent time steps in S2(T), regret of at least K(µ1 − µ2) ·
+[M − T(ν0)] is incurred, i.e.,
+Reg[2] ≥ [K(µ1 − µ2) + µ2] · T(ν0) + K(µ1 − µ2) · [M − T(ν0)] = K(µ1 − µ2) · M + µ2 · T(ν0).
+In short, we have
+Reg[2] ≥ K(µ1 − µ2) · M + µ2 · min{T(ν0), M}.
+We can lower bound Reg(j) for all j = 3, . . . , L0 with the same method.
+Besides, since
+E1[M1(T)] ≥ T −
+�
+KT
+2L0
+· d(µ0, µ1)
+and at most K − 1 items are selected at each of the first T(ν1) time steps in S1(T), we have
+Reg[1] ≥ µ1 · min
+�
+T(ν1), T −
+�
+KT
+2L0
+· d(µ0, µ1)
+�
+.
+Therefore, under instance 1, we have
+Reg(T)
+1 − δ
+≥
+L0
+�
+j=1
+Reg[j]
+≥ µ1 · min
+�
+T(ν1), T −
+�
+KT
+2L0
+· d(µ0, µ1)
+�
++ (L0 − 1) · [K(µ1 − µ2) · M + µ2 · min{T(ν0), M}]
+= µ1 · min
+�
+sup
+ν′
+1∈E(ν1)
+1
+K − 1 · d(δ, 1 − δ)
+KL(ν1, ν′
+1), T −
+�
+KT
+2L0
+· d(µ0, µ1)
+�
++ (L0 − 1) ·
+�
+K(µ1 − µ2) · M + µ2 · min
+�
+sup
+ν′
+0∈E(ν0)
+1
+K − 1 · d(δ, 1 − δ)
+KL(ν0, ν′
+0), M
+��
+where
+E(ν0) = {ν(0′) : the variance related to ν(0′) is larger than ¯σ2/K},
+E(ν1) = {ν(1′) : the variance related to ν(1′) is larger than ¯σ2/K},
+M =
+1
+L0 − 1
+�
+T − T
+L0
+−
+�
+KT
+2L0
+· d(µ0, µ1)
+�
+.
+
+Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits
+By Pinsker’s inequality (see Lemma C.1), for µi ≥ 1/2,
+d(µi, ¯a) ≤ (µi − ¯a)2 · (1 − µi − ¯a)2
+a(1 − µi − ¯a)2
+= [µi(1 − µi) − ¯σ2/K]2
+a(1 − µi − ¯a)2
+≤ [µi(1 − µi) − ¯σ2/K]2
+a(¯a − 1/2)2
+= [µi(1 − µi) − ¯σ2/K]2
+a(1/2 − a)2
+;
+for µi < 1/2,
+d(µi, a) ≤ (µi − a)2 · (1 − µi − a)2
+a(1 − µi − a)2
+= [µi(1 − µi) − ¯σ2/K]2
+a(1 − µi − a)2
+≤ [µi(1 − µi) − ¯σ2/K]2
+a(1/2 − a)2
+.
+Since ν0 = Bern(µ0), ν1 = Bern(µ1), and 0 < µ0 < µ1 < a < 1/2, we have
+sup
+ν′
+0∈E(ν0)
+1
+K − 1 · d(δ, 1 − δ)
+KL(ν0, ν′
+0) = d(δ, 1 − δ)
+K − 1
+·
+(1/2 − a)2
+[µ0(0 − µ0) − ¯σ2/K]2 ,
+sup
+ν′
+1∈E(ν1)
+1
+K − 1 · d(δ, 1 − δ)
+KL(ν1, ν′
+1) = d(δ, 1 − δ)
+K − 1
+·
+(1/2 − a)2
+[µ1(1 − µ1) − ¯σ2/K]2 .
+We define the minimum variance gap ∆v := minS∈A ∆v
+S. When K3 ≤ L2 and LK/T ≤ a2, we can let µ1 − µ2 =
+�
+L/KT. Then we have
+Reg(T) = Ω
+�
+min
+�
+1
+K − 1 · d(δ, 1 − δ)
+(∆v/K)2 , T
+�
++
+√
+KLT + L0 · min
+�
+1
+K − 1 · d(δ, 1 − δ)
+(∆v/K)2 , TK
+L
+��
+= Ω
+�
+min
+�
+K · d(δ, 1 − δ)
+(∆v/)2
+, T
+�
++
+√
+KLT + L · min
+�d(δ, 1 − δ)
+(∆v)2
+, T
+L
+��
+= Ω
+�√
+KLT + min
+�
+L · d(δ, 1 − δ)
+(∆v)2
+, T
+��
+.
+Moreover, we complete the proof with d(δ, 1 − δ) ≥ − ln(2.4δ) and δT = δ.
+D. Tightness of the Upper bound
+Corollary D.1 (Tightness of problem-dependent bounds). Let {δT }∞
+T =1 ∈ o(1) be a sequence that satisfies ln(1/δT ) =
+o(T b) for all b > 0,
+• if ln(1/δT ) ∈ o(ln T), in particular, if δT = δ0 > 0 for all T, the regret Reg(T) is
+Ω
+��
+i∈E
+ln T
+∆i,S∩B,min
++ µ⋆ ln T/K
+(∆v
+i,R)2
++ µ⋆
+K Φi,Sc∩B
+�
+∩ O
+��
+i∈E
+K ln T
+∆i,S∩B,min
++ µ⋆K ln T
+(∆v
+i,R)2 + µ⋆KΦi,Sc∩B
+�
+• if ln(1/δT ) = λ ln T, i.e., δT = T −λ with a fixed λ > 0, the regret Reg(T) is
+Ω
+��
+i∈E
+ln T
+∆i,S∩B,min
++ λµ⋆ ln T
+(∆v
+S⋆)2 + µ⋆
+K
+�
+i∈E
+�
+Ψ′
+i,S∩B +
+ln T
+(∆v
+i,R)2 + Φi,Sc∩B
+��
+∩ O
+��
+i∈E
+K ln T
+∆i,S∩B,min
++ λµ⋆K2 ln T
+(∆v
+S⋆)2
++ Kµ⋆ �
+i∈E
+�
+Ψi,S∩B +
+ln T
+(∆v
+i,R)2 + Φi,Sc∩B
+��
+where ln(1/δT ) in the Ψ and Ψ′ functions should be replaced by λ ln T.
+• if ln(1/δT ) ∈ ω(ln T), the regret Reg(T) is
+Reg(T) ∈ Ω
+�µ⋆ ln(1/δT )
+(∆v
+S⋆)2
+�
+∩ O
+�µ⋆K2 ln(1/δT )
+(∆v
+S⋆)2
+�
+The upper bounds above are achieved by PASCOMBUCB.
+
diff --git a/rNFQT4oBgHgl3EQftzab/content/tmp_files/load_file.txt b/rNFQT4oBgHgl3EQftzab/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3a7eb3efe23cf72a7f3327cf5f0bea479135ad9d
--- /dev/null
+++ b/rNFQT4oBgHgl3EQftzab/content/tmp_files/load_file.txt
@@ -0,0 +1,2058 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf,len=2057
+page_content='Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Yunlong Hou 1 Vincent Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Tan 1 2 3 Zixin Zhong 4  Abstract  Motivated by concerns about making online decisions that incur undue amount of risk at each time step, in this  paper, we formulate the probably anytime-safe stochastic combinatorial semi-bandits problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In this problem,  the agent is given the option to select a subset of size at most K from a set of L ground items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Each item is  associated to a certain mean reward as well as a variance that represents its risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' To mitigate the risk that the agent  incurs, we require that with probability at least 1 − δ, over the entire horizon of time T, each of the choices that  the agent makes should contain items whose sum of variances does not exceed a certain variance budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We call  this probably anytime-safe constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Under this constraint, we design and analyze an algorithm PASCOMBUCB  that minimizes the regret over the horizon of time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' By developing accompanying information-theoretic lower  bounds, we show under both the problem-dependent and problem-independent paradigms, PASCOMBUCB is  almost asymptotically optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Our problem setup, the proposed PASCOMBUCB algorithm, and novel analyses  are applicable to domains such as recommendation systems and transportation in which an agent is allowed to  choose multiple items at a single time step and wishes to control the risk over the whole time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Introduction  Audrey, a burgeoning social media influencer, makes profits by posting advertisements (ads) under her account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The  advertiser pays her only if an ad is clicked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Having taken a class in online optimization, Audrey aims to leverage the theory  of bandit algorithms to design an exploration-exploitation strategy to ensure that the expected number of clicks of the ads  she has posted is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Since the platform is space-limited, Audrey can only post no more than K out of L available  ads everyday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Some of these ads, however, include an innocuous-looking lottery or voucher that asks the viewer of the  social media platform to provide personal information that may lead to fraud or information leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' If a user clicks it and  becomes a victim of fraud, this may damage Audrey’s reputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Audrey thus has to be circumspect in which and how  many ads she posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  On the one hand, Audrey wants to post as many ads with what she believes have high click-through rates as possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' the  expected reward she obtains is then the sum of expected rewards of the individual ads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' On the other hand, she should balance  this with the total risk of the ads that are posted over a period of time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' similarly, the risk of a set of ads posted is modeled as  the sum of the risks of the individual ads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' How should Audrey plan the posts of her ads over a period of time to learn their  individual expected rewards and risks to ensure that her total expected reward is maximized and, at the same time, with  high probability, the risk incurred at any point in time in her exploration-exploitation strategy is bounded by some fixed  permissible threshold?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In addition to influencers like Audrey, online platforms that make profits by advertising such as YouTube and TikTok also  encounter similar problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We are therefore motivated to formulate the probably anytime-safe stochastic combinatorial  semi-bandits problem which is a regret minimization problem with an anytime safety constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' More precisely, we aim to  design and analyze the performance of an algorithm that, with high probability, ensures that the risk (as measured by the  variance) at any time step is below a given threshold and whose regret is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' There is a large body of works that take risk into account while conducting the exploration and/or  1Department of Mathematics, National University of Singapore, Singapore 2Department of Electrical and Computer Engineering,  National University of Singapore, Singapore 3Institute of Operations Research and Analytics, National University of Singapore, Singapore  4Department of Computing Science, University of Alberta, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Correspondence to: Zixin Zhong <zixin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='zhong@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='edu>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Copyright 2023 by the author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='13393v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='LG]  31 Jan 2023  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  exploitation of the unknown reward distributions in the stochastic multi-armed bandits (MABs) literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Under the risk-constrained pure exploration framework, Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2023) and David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2018) attempted to identify the  optimal arm within those low-risk (based on their variances or α-quantiles) arms with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Under the risk-aware regret minimization setup, Sani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2012), Vakili & Zhao (2016) and Zhu & Tan (2020) consider  the mean-variance as the measure to be minimized over a fixed time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Cassel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2018) provided a general  and systematic instruction to analyzing risk-aware MABs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', the risk was incorporated in the Empirical Distribution  Performance Measure and the U-UCB algorithm is adopted to perform “proxy regret minimization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' While these risk-aware  algorithms reduce the overall risk during the exploration and exploitation process, the risk is not strictly enforced to be  below a prescribed threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' rather the risk measure is penalized within the objective function, similarly to a Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Another setup similar to the risk-aware setup is the constrained bandits regret minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Mahdavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2012) required  that the number of times the constraint can only be violated is at most sublinear in the horizon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Kagrecha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2023)  proposed a CVaR constraint and performed exploration on the feasible arm, followed by exploration among the feasible arm  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Unlike our formulation, these algorithm are permitted to sample risky arms during exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  A more stringent constraint can be found in the literature on conservative bandits (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', 2016), which requires the  cumulative return at any time step to be above a constant fraction of the return resulting from repeatedly sampling the base  arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Kazerouni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2017) extended this setup to conservative contextual linear bandits and this was further improved by  Garcelon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' A similar problem is bandits with knapsacks (Badanidiyuru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', 2018), which imposes a budget on  the cumulative consumed resources and the algorithm stops when the budget is depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The most stringent constraint can be found in the safe bandits problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Khezeli & Bitar (2020) and Moradipari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2020)  presented the SEGE, SCLUCB, and SCLTS algorithms to tackle this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' This problem demands that the expected  reward of the pulled arm at each time step to be greater than a prescribed threshold with high probability, also known as the  “stagewise safety constraint”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The authors utilized the convexity (and continuity) of the arm set and performed exploration  around the explored arms, starting from a baseline arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' This correlation among the arms generally does not hold under the  combinatorial semi-bandits setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  For the (unconstrained) combinatorial semi-bandits (CSB) setup, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2013) presented a UCB-type algorithm  COMUCB1 to balance the trade-off between exploration and exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Kveton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2015b) improved the analysis of  COMUCB1 and achieved a tight upper bound (within a specific set of instances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Kveton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2014) introduced matroid  structure to CSB and leveraged the matroid structure to design and analyze a greedy algorithm OMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The risk-aware CSB  problem is less studied by the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ayyagari & Dukkipati (2021) utilized CVaR as the risk-aware measure within the  CSB problem, where the risk constraint was not explicitly specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We observe that the existing literature mentioned above are not directly applicable to Audrey, while our setting (described  formally below) dovetails neatly with her problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Audrey can utilize our algorithm to sequentially and adaptively select  different sets of ads everyday and almost always (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', with high probability) avoids sets of ads with unacceptably high risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Beyond any specific applications, we believe that this problem is of fundamental theoretical importance in the broad context  of regret minimization in combinatorial multi-armed bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Main Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In probably anytime-safe stochastic combinatorial semi-bandits, there are L items with different  reward distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' At each time step, a random reward is generated from each item’s distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Based on the previous  observations, the learning agent selects a solution at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' A solution consists of at most K items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The expected  return (variance) of a solution is the summation of the reward (variance) of its constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Given T ∈ N, the agent aims  to maximize the cumulative return over T time steps and ensure that with probability 1 − δ the variance of all selected  solutions are below a given threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The key challenge of regret minimization under the probably anytime-safe stochastic combinatorial semi-bandits lies in  handling two distinct tasks—we seek optimality in the mean and safeness in the variance of each chosen solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Our  first contribution is design and analysis of the Probably Anytime-Safe Combinatorial UCB (or PASCOMBUCB) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We also derive a problem-dependent upper bound on its regret, which involves a hardness parameter H(∆(Λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We see  that H(∆(Λ)) characterizes the effectiveness of ascertaining the safety of potential solutions in the regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' To assess the  optimality of PASCOMBUCB, we prove an accompanying problem-dependent lower bound on the regret of any variance-  constrained consistent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The upper and lower problem-dependent bounds match in almost all the parameters  (except in K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Additionally, we show that if δT decays exponentially fast in T, the problem-dependent regret cannot be  logarithmic in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  We further present a problem-independent upper bound on the regret of PASCOMBUCB and a lower bound for any  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Just as the problem-dependent bounds, these bounds also match in almost all the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In summary, this paper is the first to explore the regret minimization problem in the combinatorial bandits with an anytime  constraint on the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When δ → 1 and ¯σ2 is large (so that the optimal safe solution is the one with the highest mean  regardless of safety considerations), our problem reduces to the standard combinatorial semi-bandits (Kveton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', 2015a),  and the regret incurred by the safety constraint vanishes, resulting in the same upper bound as the unconstrained case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Furthermore, the framework and analysis of PASCOMBUCB can be extended to other risk measures as long as there are  appropriate concentration bounds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', Bhat & Prashanth (2019) or Chang & Tan (2022) enables us to use CVaR or certain  continuous functions as risk measures within the generic PASCOMBUCB framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Problem Setup  Given a positive integer m, we let [m] := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' An instance of a variance-constrained stochastic combinatorial  semi-bandit is a tuple Λ = (E, AK, ν, ¯σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We describe the four elements of Λ in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Firstly, the finite set E = [L]  is known as the ground set in which each i ∈ E is known as an item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Secondly, the family AK ⊂ {S ∈ 2E : |S| ≤ K} is  a collection of subsets of E with cardinality at most K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Each element S ∈ AK is known as a solution and AK satisfies  the condition that all subsets of S ∈ AK remain solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', AK is downward-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thirdly, the vector of probability  distributions ν = (ν1, ν2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , νL) contains σ2-sub-Gaussian distributions {νi}i∈E with means {µi}i∈E and variances  {σ2  i }i∈E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The final element of an instance ¯σ2 > 0 denotes the permissible upper bound on the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' To avoid trivialities,  we assume that ¯σ2 > σ2 and K ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The return of item i ∈ E is the random variable Wi with distribution νi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The (stochastic) return of a solution S ∈ AK is  �  i∈S Wi where W ∼ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The expected return and variance of S ∈ AK are  µS :=  �  i∈S  µi  and  σ2  S :=  �  i∈S  σ2  i  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We further assume that every instance Λ satisfies σ2  S ̸= ¯σ2 for all S ∈ AK and each distribution νi is supported  in the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We define S := {S ∈ AK : σ2  S < ¯σ2} to be the safe set which contains all the safe solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let the complement  of S be the unsafe set Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Denote the optimal safe solution as S⋆ := arg max{µS : S ∈ S} with return µ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For  simplicity, we assume that S⋆ is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Denote the suboptimal set B := {S ∈ AK : µS < µ⋆} and the risky set  R := {S ∈ AK : µS ≥ µ⋆, S ̸= S⋆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For a solution S, let the mean gap ∆S := µ⋆ − µS and the variance gap  ∆v  S := |σ2  S − ¯σ2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  An instance Λ, time horizon T ∈ N and confidence parameter δ ∈ (0, 1) are specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' An agent, who knows E, AK and  ¯σ2 but not the vector of probability distributions ν, interacts adaptively with the instance over T time steps as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' At  time step t ∈ [T], the agent uses a stochastic function πt that selects a solution St ∈ AK based on the observation history  Ht−1 := ((Ss, {Wi(s)}i∈Ss))s∈[t−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In other words, St = πt(Ht−1) is a stochastic function of the history Ht−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The  agent receives the random return �  i∈St Wi(t), where {W(s) = {Wi(s)}i∈E}s∈[T ] are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' according to ν across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The weights of the selected items {Wi(t) : i ∈ St} are observed by the agent at each time t ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The collection of  stochastic functions π = {πt}t∈[T ] is known as the agent’s policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The goal of the agent is to minimize the expected cumulative regret (or simply regret) Reg(T) over the horizon T, subject to  a certain risk constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' More precisely, the regret suffered by a policy π employed by the agent is defined as  Regπ(T) := Eπ  � T  �  t=1  � �  i∈S⋆  Wi(t) −  �  i∈St  Wi(t)  ��  The policy π should satisfy the condition that all the solutions chosen {Sπ  t }t∈[T ] ⊂ AK are safe with probability at least  1 − δ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=',  Pπ  �  ∀ t ∈ [T], Sπ  t ∈ S  �  ≥ 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (1)  This is referred to as the probably anytime-safe constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In the problem-dependent lower bounds, we will refer to a certain class of “good” policies that operate as the time horizon  T → ∞ and the probability of being safe in the sense of (1) tends to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' This is formalized in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fix an instance ν and a vanishing sequence {δT }∞  T =1 ⊂ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' An policy π = {πt}∞  t=1 is said to be a  {δT }∞  T =1-variance-constrained consistent algorithm if  Regπ(T) = o(T a) for all a > 0 and  Pπ  �  ∀ t ∈ [T], Sπ  t ∈ S  �  ≥ 1 − δT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We often omit the superscripts π in Regπ, Sπ  t (or Aπ  t and Aπ  t,r in PASCOMBUCB) and the subscripts π in the probabilities  and expectations if there is no risk of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Our Algorithm: PASCOMBUCB  Our algorithm Probably Anytime-Safe Combinatorial UCB (or PASCOMBUCB) is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' PAS-  COMBUCBis delicately designed to satisfy the probably anytime-safe constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In particular, we apply (and analyze)  the GREEDY-SPLIT subroutine in Line 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' this subroutine has not been involved in an algorithm designed for standard  combinatorial semi-bandits such as COMBUCB1 (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Algorithm 1 PASCOMBUCB  1: Input: An instance Λ (with unknown ν), the horizon T and the confidence parameter δ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  2: Set phase counter p = 1 and time step counter t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  3: while ∃ i ∈ E such that Ti(p − 1) < 2 do  4:  Pull Ap =arg maxS:|S|≤q |{i∈S : Ti(p − 1)<2}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  5:  p ← p + 1, t ← t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  6: end while  7: Update the sample mean, sample variance and confidence bounds according to (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  8: Update the empirically safe set Sp and possibly safe set ¯Sp according to (5) and (6) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  9: while t < T do  10:  Find a solution Ap =arg maxA∈ ¯Sp−1 U µ  A(p−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  11:  Invoke GREEDY-SPLIT to split the solution Ap into np sub-solutions {Ap,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , Ap,np} ⊂ Sp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  12:  Set np ← min{np, T − count}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  13:  Choose solution {Ap,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , Ap,np}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  14:  Update the statistics of all solutions based on (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  15:  Update the empirical sets based on (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  16:  Set t = t + np and p = p + 1,  17: end while  Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Since each item i ∈ E is σ2-sub-Gaussian, any solution that contains at most q := ⌊ ¯σ2  σ2 ⌋ items is safe with  probability (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We call such a solution absolutely safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Algorithm 1 (PASCOMBUCB) is conducted in phases, where  each phase consists of multiple time steps and each item can be pulled at most once during each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus we adopt a  different notation “A” to denote the solution in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Define Ti(p) := �p  s=1 1{i ∈ Ap} as the number of times  item i is pulled up to and including phase p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Denote the sample mean and sample variance of item i at phase p as  ˆµi(p) :=  1  Ti(p)  p  �  s=1  Wi(s) · 1{i ∈ As},  and  ˆσ2  i (p) :=  1  Ti(p)  p  �  s=1  (Wi(s) − ˆµi(p))2 · 1{i ∈ As}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The bound based on the Law of Iterated Logarithms (LIL) is used to construct the confidence radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For a fixed ϵ ∈ (0, 1),  define lil(t, ρ) := (1 + √ϵ)  �  1+ϵ  2t ln  � ln((1+ϵ)t)  ρ  ��1/2  and denote the confidence radius for the mean as  α(t) := lil(t, ωµ),  (2)  where ωµ is a parameter to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The confidence radii for the variance are asymmetric about the empirical variance  and are parameterized by ωv and ω′  v that may not necessarily be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' They are defined as  βu(t) := 3 · lil(t, ωv)  and  βl(t) := 3 · lil(t, ω′  v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (3)  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  We denote the upper and lower confidence bounds (UCB and LCB) for the mean of item i as  U µ  i (p) := ˆµi(p) + α(Ti(p))  and  Lµ  i (p) := ˆµi(p) − α(Ti(p))  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The UCB and LCB for the variance of item i are defined as  U v  i (p) := min{ˆσ2  i (p) + βu(Ti(p)), σ2}  and  Lv  i (p) := max{ˆσ2  i (p) − βl(Ti(p)), 0}  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' With the sample mean, sample variance, and confidence bounds for the items, we define the following statistics  for all solution S ∈ AK:  ˆµS(p) =  �  i∈S  ˆµi(p),  ˆσ2  S(p) =  �  i∈S  ˆσ2  i (p),  U µ  S (p) =  �  i∈S  U µ  i (p),  Lµ  S(p) =  �  i∈S  Lµ  i (p),  (4)  U v  S(p) =  �  i∈S  U v  i (p),  Lv  S(p) =  �  i∈s  Lv  i (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Denote the empirically safe set as  Sp := {S ∈ AK : U v  S(p) < ¯σ2}  (5)  and the possibly safe set as  ¯Sp := {S ∈ AK : Lv  S(p) < ¯σ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (6)  The solutions in St and ¯St are called empirically safe and possibly safe solutions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In the initialization stage (lines 3 to 6), PASCOMBUCB greedily pulls the absolutely safe solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When each  item has been pulled at least twice, this stage is terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' After initialization, during phase p, PASCOMBUCB firstly  identifies a solution Ap = arg maxA∈ ¯Sp U µ  A(p−1) via an optimization oracle (Line 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' It then calls a subroutine GREEDY-  SPLIT to greedily partition the solution Ap into empirically safe sub-solutions (Line 11, see Figure 1 for illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Subsequently, these solutions are chosen and the stochastic rewards from the corresponding items are observed (Line 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lastly, the empirical estimates, the confidence bounds, and the empirical sets are updated (Lines 14 and 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Algorithm 2 GREEDY-SPLIT  1: Input: A solution Ap and the upper confidence bound on the variance U v(p − 1) at phase p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  2: Set np = 1, s = 1 and Ap,1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  3: Index the items in Ap by i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , i|Ap|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  4: while s ≤ |Ap| do  5:  if U v  Ap,np (p − 1) + U v  is(p − 1) ≤ ¯σ2 then  6:  Set Ap,np ← Ap,np ∪ {is}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  7:  else  8:  np ← np + 1 and Ap,np = {is}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  9:  end if  10:  s ← s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  11: end while  12: return {Ap,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , Ap,np}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Figures 2 and 3 illustrate the regret accumulated during phase p and over the whole T horizon respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' As  shown in Figure 2, the regret accumulated during phase p can be decomposed into two parts  np  �  r=1  (µ⋆ − µAp,r) = ∆Ap + µ⋆(np − 1)  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Variance  Solution  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' A diagram of a split to a solution A containing 5 items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Mean  Instantaneous regret  due to suboptimality  Instantaneous regret  due to safeness-checking  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Solution Ap is split into np = 3 sub-solutions, the instantaneous regret at phase p can be divided into the instantaneous regret  due to suboptimality and the instantaneous regret due to safeness checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  where ∆Ap is the phase-wise (instantaneous) regret due to suboptimality and µ⋆(np − 1) is the regret due to safeness-  checking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' the latter term results from the safeness constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' At the beginning, since the upper confidence bounds of the  variances of all solutions are large, each solution will be split into up to 2Q sub-solutions, where Q := ⌈ K  q ⌉, and hence the  regret due to safeness checking can be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' As the algorithm progresses, we obtain more observations of items and get  more confident about their variances (U v  i (p) decreases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Hence, during some later phase, it suffices to split some solutions  into fewer sub-solutions and the regret due to safeness-checking reduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Furthermore, when most items are sampled  sufficiently many times, the unsafe solutions are excluded from the possibly safe set ¯Sp, and the only contribution to the  regret is via the suboptimality of the solution Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' • The confidence parameter ω′  v is solely a parameter of PASCOMBUCB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' its choice does not rely on the  confidence parameter δ and only affects Lv  S(p), the lower confidence bound of the variance, which determines when we  ascertain a solution to be unsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The choice of ωv depends on δ and it influences U v  S(p), the upper confidence bound of  the variance, which guides PASCOMBUCB to split the solution to satisfy the probably anytime-safe constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The other  parameters ωv and ω′  v determine the confidence radii of variances and do not necessarily have to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Indexing the items in Line 3 of GREEDY-SPLIT can be done arbitrarily, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', it does not require any specific order of the  items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' As such, GREEDY-SPLIT is an efficient greedy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We note that finding the optimal order that leads to the  minimum number of sub-solutions np is a combinatorial problem which is generally hard to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Problem-dependent Bounds  For simplicity, when a time horizon T and a confidence parameter δ = δT are given, we set the confidence parameters  ωµ = ω′  v =  1  T 2 and ωv = δT  T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We introduce various suboptimality gaps that contribute to the regret due to the suboptimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Phase  Instantaneous Regret  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='.  Instantaneous regret due to suboptimality  Instantaneous regret due to safeness-checking  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='.  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' An illustration of the instantaneous regret yielded by PASCOMBUCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' As the variances of the items are more determined, less  regret due to safeness-checking is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  for i ∈ E \\ S⋆, let the minimum safe-suboptimal gap be  ∆i,S∩B,min :=  min  S∋i,S∈S∩B ∆S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  for i ∈ E, let the minimum unsafe-suboptimal gap be  ∆i,Sc∩B,min :=  min  S∋i, S∈Sc∩B ∆S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  and let the tension parameter between the mean gap ∆S and variance gap ∆v  S be  ci :=  max  S∋i, S∈Sc∩B  �  ∆S  max{∆S, ∆v  S/3}  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We also define following safeness gaps that induce the conservative sampling strategy to guarantee the probably anytime-safe  constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For i ∈ E, and  for the risky set R, define the minimum unsafeness gap: ∆v  i,R := minS∋i,S∈R ∆v  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  for the safe and suboptimal set S ∩ B, let  Ψi,S∩B :=  max  S∋i, S∈S∩B min  �ln T  ∆2  S  , 9 ln(T/δT )  (∆v  S)2  �  which characterizes the order of the number of times this item i needs to be sampled in order to identify the suboptimality  of all safe and suboptimal solutions A ∋ i while satisfying the safeness constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We further define a variant of Ψi,S∩B  as  Ψ′  i,S∩B :=  max  S∋i,S∈S∩B min  �ln T  ∆2  S  , 9 ln(1/δT )  (∆v  S)2  �  which will be used to characterize the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  for the unsafe and suboptimal set Sc ∩ B, let  Φi,Sc∩B :=  max  S∋i, S∈Sc∩B min  �ln T  ∆2  S  , 9 ln T  (∆v  S)2  �  which characterizes the hardness of identifying the unsafeness of suboptimality of all unsafe and suboptimal solutions that  contain item i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Define ξ(ω) := 2+ϵ  ϵ  �  ω  ln(1+ϵ)  �1+ϵ, where ϵ ∈ (0, 1) is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Problem-dependent Upper Bound  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 (Problem-dependent upper bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let Λ = (E, AK, ν, ¯σ2) be an instance and let {δT }∞  T =1 ∈ o(1) be a  sequence that satisfies ln(1/δT ) = o(T b) for all b > 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', {δT } is not exponentially decaying).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Then, PASCOMBUCB is  a {δT }∞  T =1-variance-constrained consistent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' More precisely, given a time budget T, the probably anytime-safe  constraint is satisfied and the regret of PASCOMBUCB Reg(T) is upper bounded by  min {Tµ⋆, Reg1(T) + Reg2(T)} + Reg3(T),  where  Reg1(T) = O  �  �  i∈E\\S⋆  K ln T  ∆i,S∩B,min  +  �  i∈E  ciK ln T  ∆i,Sc∩B,min  �  Reg2(T) = 2µ⋆H (∆(Λ)) ,  Reg3(T) = 2µ⋆(L + 1)  where ∆(Λ) = {∆v  S⋆} ∪ {∆v  i,R, Ψi,S∩B, Φi,Sc∩B}i∈E and H (∆(Λ)) := H(1, Λ) is defined in (26) in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' If the gaps in ∆(Λ) are sufficiently small and δT = T −λ for a fixed λ > 0,  H (∆(Λ)) = O  �(λ + 1)K2 ln T  (∆v  S⋆)2  + K  �  i∈E  �  ln T  (∆v  i,R)2 + max  S∋i,  S∈S∩B  min  �ln T  ∆2  S  , (λ + 1) ln T  (∆v  S)2  �  + Φi,Sc∩B  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  See (26) for more details of this calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The first term Tµ⋆ in the regret bound provides a na¨ıve upper bound for the expected regret conditional on the variance  constraint holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The order of the regret (o(T a) for all a > 0) implies the regret will be asymptotically bounded by the  second term when the time budget T is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The second term is comprised of two parts—the regret due to  suboptimality Reg1(T) and the regret due to safeness-checking Reg2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The intuition for the regret due to suboptimality  Reg1(T) is that  Each item in any safe and suboptimal solution will be sampled O(  K ln T  ∆2  i,S∩B,min ) times to ascertain the suboptimality of all  safe and suboptimal solutions which this item belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Each item in an unsafe and suboptimal solution S will be sampled O  �  K ln T  max{∆S,∆v  S/3}2  �  times to ascertain either the  suboptimality or the unsafeness of solution S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' As this should be done for all the unsafe and suboptimal solutions, we  need take the maximum of the above time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' More precisely, when ci = 1, suboptimality identification of the  unsafe and suboptimal solutions to which item i belongs dominates the regret;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' and when ci < 1, the ascertaining of the  unsafeness dominates the regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The intuition for the regret due to safeness checking Reg2(T) is that the function provides an upper bound for the number  of time steps needed for guaranteeing the safeness of all the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' PASCOMBUCB achieves this in a judicious manner  since it does not check the safeness of all the solutions at the start, followed by exploration and exploitation of the possibly  high-return safe solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Instead, it takes advantage of the fact that when a (safe or unsafe) suboptimal solution is  ascertained to be suboptimal, its safeness can be disregarded, as reflected in the terms Ψi,S∩B and Ψi,Sc∩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In addition, it  will not sample an unsafe solution if it is identified as unsafe w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' at least 1 − 2ξ(ω′  v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The last term Reg3(T) corresponds  to the regret due to failure of the “good” event and at the initialization stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Problem-dependent Lower Bound  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3 (Problem-dependent lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let {δT }∞  T =1 ∈ o(1) be a sequence that satisfies ln(1/δT ) = o(T b) for  all b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' There exists an instance Λ such that for any {δT }T ∈N-variance-constrained consistent algorithm π, the regret is  lower bounded by  Ω  � �  i∈E  ln T  ∆i,S∩B,min  �  + µ⋆  K · Ω  �K ln(1/δT )  (∆v  S⋆)2  +  �  i∈E  �  Ψ′  i,S∩B +  ln T  (∆v  i,R)2 + Φi,Sc∩B  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  With Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3, the tightness of the problem-dependent upper bound can be ascertained for polynomially decay-  ing {δT }T ∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4 (Tightness of problem-dependent bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let δT = T −λ with a fixed λ > 0, the regret  Reg(T) ∈ Ω  � �  i∈E  ln T  ∆i,S∩B,min  + µ⋆  K2 H (∆(Λ))  �  ∩ O  � �  i∈E  K ln T  ∆i,S∩B,min  + µ⋆H (∆(Λ))  ��  where H (∆(Λ)) is defined in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The upper bound above is achieved by PASCOMBUCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Under different rates of decay of {δT }T ∈N (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' D for the cases where ln(1/δT ) = ω(ln T) and o(ln T)), the upper  bound of the regret due to suboptimality Reg1(T) (the first term in the total regret) and the upper bound of the regret  due to safeness-checking Reg2(T) (the latter term) match their corresponding lower bounds up to factors of K and K2  respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' this gap is acceptable as K (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', number of ads displayed) is usually small relative to L (total number of  ads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We consider general instances where all the items are independent and the gap in Reg1(T) can be closed when that  the items are correlated, as in the lower bound for the unconstrained combinatorial bandits in Kveton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' This  assumption also allows us to remove a factor of K from the gap of Reg2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' One may naturally wonder whether we can  tolerate a much more stringent probably anytime-safe constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The following theorem (with b = 1) indicates no algorithm  is {δT }T ∈N-variance-constrained consistent if δT decays exponentially fast in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='5 (Impossibility result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let {δT }∞  T =1 ∈ o(1) be a sequence that satisfies the following condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' There exists  b > 0 such that ln(1/δT ) = Ω(T b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For instance Λ, the regret of any algorithm is lower bounded by Ω(T b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Problem-independent Bounds  We can derive a problem-independent upper bound on the regret of PASCOMBUCB from the problem-dependent one in  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 with some delicate calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 (Problem-independent Upper Bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let {δT }∞  T =1 ∈ o(1) be a sequence that satisfies ln(1/δT ) = o(T b)  for all b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' If T > L, for any instance Λ with variance gaps lower bounded by ∆v ≤ minS∈AK ∆v  S, the regret of  PASCOMBUCB is upper bounded by  O  �√  KLT ln T + LK2  (∆v)2 ln  � 1  δT  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 (Problem-independent lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let the minimum variance gap be ∆v := minS∈AK ∆v  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When  K3 ≥ L2, we have  Reg(T) = Ω  �√  KLT + min  �  L  (∆v)2 ln  � 1  δT  �  , T  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The assumption that the variance gaps of all solutions are lower bounded by ∆v is needed to achieve a  non-vacuous problem-independent bound, hence, somewhat unconventionally, it appears in our “problem-independent”  bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Given any algorithm and time budget T, the variance gap of S⋆ can be arbitrarily small if ∆v is not bounded away  from zero, so the min in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 will be dominated by the linear term T, and hence, no algorithm can attain sublinear  regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The above results allow us to investigate the tightness of problem-independent bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4 (Tightness of problem-independent bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let K3 ≤ L2, and {δT }∞  T =1 ∈ o(1) satisfies ln(1/δT ) = o(T b)  for all b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We have  Reg(T) ∈ Ω  �√  KLT +  L  (∆v)2 ln  � 1  δT  ��  ∩ O  �√  KLT ln T + LK2  (∆v)2 ln  � 1  δT  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The upper bound is achieved by PASCOMBUCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We observe that the gap between the upper and lower bounds is manifested on  √  ln T and K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The presence of  √  ln T is not  unexpected as it is also involved in the gap between the bounds on the regret for the (unconstrained) combinatorial bandits  (Kveton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Besides, the term K2 is induced by the design of PASCOMBUCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' During each phase, we select and  sample solutions which are disjoint subsets of Ap, and hence one item is sample at most once during one phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' However,  we believe that it is possible to sample some items more than once during one phase, which will help reduce the regret but  requires more delicate analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We view this as a promising venue for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Proof Sketch of the Problem-Dependent Upper Bound (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1)  Assume that PASCOMBUCBhas processed T ′ phases with T time steps, we have P[T ′ ≤ T] = 1 since each phase is  composed by multiple time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Denote the expected regret of PASCOMBUCBwith p phases as E[R(p)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The expected  regret of PASCOMBUCBafter T time steps is  E[R(T ′)] := E  � T ′  �  p=1  np  �  r=1  (µ⋆ − µAp,r)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1, we first show a regret decomposition lemma (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1) that separates the total regret into the  regret due to suboptimality E[R1(T ′)], the regret due to safeness-checking E[R2(T ′)] and the regret due to the failure of the  “good” event and the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Then we upper bound R1(T ′) and R2(T ′) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' To elucidate the dependence of the  regret on the confidence parameters ωµ, ωv and ω′  v, we retain these notations henceforth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  For p ∈ [T], i ∈ E, define the “good” events that the sample mean and the sample variance are near their ground  truths: Eµ  i,Ti(p) := {ˆµi(p) − α(Ti(p)) ≤ µi ≤ ˆµi(p) + α(Ti(p))} and Ev  i,Ti(p)(ρ) := {ˆσ2  i (p) − 3 · lil(Ti(p), ρ) ≤ σ2  i ≤  ˆσ2  i (p) + 3 · lil(Ti(p), ρ)} and  Ei,Ti(p) := Eµ  i,Ti(p) ∩ Ev  i,Ti(p)(ωv) ∩ Ev  i,Ti(p)(ω′  v)  E :=  �  i∈E  �  p∈[T ′]  Ei,Ti(p−1)  For r ∈ [Q − 1], define Up(r) := {U v  Ap(p − 1) > r¯σ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When event Up(r) occurs at phase p, it indicates at least r + 1  sub-solutions are needed in order to sample the items in Ap and guarantee the safeness constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Assume that PASCOMBUCBhas processed T ′ phases with T time steps, the expected regret of PASCOM-  BUCBcan be decomposed into three parts as follows  E[R(T ′)] ≤ E[R1(T ′)|E] + E[R2(T ′)|E] + R3(T)  where  R1(T ′) :=  T ′  �  p=1  1{Ap ∈ B}∆Ap  R2(T ′) := µ⋆  T ′  �  p=1  �  2  Q−1  �  r=1  1{Up(r)}  �  R3(T) := 2µ⋆L  �  1 + T  �  ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′  v  ��  In Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1, the first term R1(T ′) is the (high-probability) regret due to suboptimality, in the sense that only the mean  gaps of the suboptimal solutions contribute to R1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The second term R2(T ′) is called the (high-probability) regret due  to safeness-checking, since it depends on the variance gaps and goes to 0 if ¯σ2 is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The last term R3(T)  contains the regret from the initialization stage and the regret results from the failure of the “good” event E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The regret due to suboptimality can be bounded in terms of the minimum safe/unsafe-suboptimal gaps as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Conditioned on event E, the regret due to suboptimality R1(T) can be bounded by  O  �  �  i∈E\\S⋆  K  ∆i,S∩B,min  ln 1  ωµ  +  �  i∈E  ciK  ∆i,Sc∩B,min  ln 1  ω′v  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The regret due to safeness-checking involves more critical parameters of the instance and we encode them in T ′  r′ and  H(r′, Λ) for r′ ∈ [Q] (see Figure 5), which are defined formally in (25) and (26) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' On the event E, if T ′ ∈ [T ′  r′, T ′  r′−1) then  R2(T ′) ≤ 2µ⋆[T ′(r′ − 1) + H(r′, Λ)] ≤ 2µ⋆H(1, Λ)  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Phase  Instantaneous Regret  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='.  Instantaneous regret due to safeness-checking  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='.  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='.  Upper bound  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We assume the algorithm will sample those solutions with large U v  A(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', those phases in which more sub-solutions are  sampled are moved forward (the dark red ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Based on this, an upper bound can be derived (the thick black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Phase  Instantaneous Regret  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='.  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='.  Upper bound  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' An illustration of the upper bound of R2(T ′) for phase T ′  Q−2 ≤ T ′ < T ′  Q−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When r′ = 1, 2µ⋆H(1, Λ) is the area below the  thick line, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', the upper bound for R2(T ′) regardless of T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  To obtain the upper bound for R2(T ′), we assume the algorithm will sample those solutions with large U v  A(p) in ¯Sp, which  will be split into the most many sub-solutions (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Furthermore, for r′ = Q − 1, Q − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , 1, we derive an upper  bound for the number of phases in which event Up(r′) ∩ (Up(r′ + 1))c occurs (at most 2r′ + 1 sub-solutions are being pulled  in these phases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' To be more specific (see Figure 5), for r′ = Q − 1, we compute the maximum number of phases T ′  Q−1  in which at most 2Q − 1 sub-solutions are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Then for r′ = Q − 2, we compute the maximum number of phases  T ′  Q−2 − T ′  Q−1 in which at most 2Q − 3 sub-solutions are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We continuously do this before the time budget runs out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  As T ′ increases, r′ decreases and H(r′, Λ) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When r′ = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' T ′ ≥ T ′  1, H(1, Λ) is an upper bound for the total  number of sub-solutions being pulled (up to a constant) for the safeness-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' It can also be regarded as the price for the  probably anytime-safe constraint and the upper bound for the regret due to safeness-checking remains an instance-dependent  constant 2µ⋆H(1, Λ) when T ′ ≥ T ′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' More detailed discussions are postponed to Step 3 in the proof in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
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+page_content='  Mahdavi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', Jin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', and Yang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Trading regret for efficiency: Online convex optimization with long term constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Journal of Machine Learning Research, 13(1):2503–2528, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Moradipari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', Thrampoulidis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', and Alizadeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Stage-wise conservative linear bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In Proceedings of the 34th  International Conference on Neural Information Processing Systems, volume 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 11191–11201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Curran Associates,  Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Sani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', Lazaric, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', and Munos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Risk-aversion in multi-armed bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In Proceedings of the 25th International  Conference on Neural Information Processing Systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 3275–3283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Curran Associates Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Vakili, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' and Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Risk-averse multi-armed bandit problems under mean-variance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' IEEE Journal of Selected  Topics in Signal Processing, 10(6):1093–1111, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', Shariff, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', Lattimore, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', and Szepesvari, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Conservative bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In Proceedings of the 33rd International  Conference on Machine Learning, volume 48, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 1254–1262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' PMLR, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Zhong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', Cheung, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', and Tan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thompson sampling algorithms for cascading bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Journal of Machine  Learning Research, 22(218):1–66, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Zhu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' and Tan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thompson sampling algorithms for mean-variance bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In Proceedings of the 37th International  Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 11599–11608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Appendices  In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' A, we list 3 useful lemmas concerning the LIL concentration bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' B, we present detailed proofs of the  upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' C, the proofs of the lower bounds are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' D, a corollary characterizing the tightness  of the upper bound in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Auxiliary results  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 (Lemma 3 in (Jamieson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', 2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let {Xi}∞  i=1 be a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' centered sub-Gaussian random  variables with scale parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fix any ϵ ∈ (0, 1) and δ ∈ (0, ln(1 + ϵ)/e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Then one has  P  �  ∀ t∈N :  t  �  s=1  Xs ≤(1+√ϵ)  �  2σ2 (1+ϵ) t ln  �ln ((1+ϵ)t)  δ  ��  ≥ 1 − ξ(δ),  where ξ(δ) := 2+ϵ  ϵ  �  δ  ln(1+ϵ)  �1+ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For t ≥ 1, ϵ ∈ (0, 1), ω ∈ (0, 1] and u > 0, let γ :=  (1+ϵ)(1+√ϵ)2  2  , c :=  (u·s)2  γ  =  2(u·s)2  (1+ϵ)(1+√ϵ)2 and  m :=  γ  u2·s2  �  2 ln 1  ω + ln ln+ 1  s2 + ln 2γ(1+ϵ)  u2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' If t > m, it holds that  s > lil(t, ω) = (1 + √ϵ)  �  1 + ϵ  2t  ln  �ln ((1+ϵ)t)  ω  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Note that fact that  u · s ≤ (1 + √ϵ)  �  1 + ϵ  2t  ln  �ln ((1+ϵ)t)  ω  �  ⇐⇒ c =  2(u · s)2  (1 + ϵ)(1 + √ϵ)2 ≤ 1  t ln  �ln ((1+ϵ)t)  ω  �  According to the computations in Jamieson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2014) equation (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=',  1  t ln  �ln((1 + ϵ)t)  ω  �  ≥ c′ ⇒ t ≤ 1  c′ ln  �2 ln((1 + ϵ)/(c′ω))  ω  �  for t ≥ 1, ϵ ∈ (0, 1), c′ > 0, ω ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We take c′ = c, thus  t ≤ 1  c ln  �2 ln((1 + ϵ)/(cω))  ω  �  = 1  c  �  ln 2  ω + ln  �  ln γ(1 + ϵ)  u2 · ω  + ln 1  s2  ��  (a)  ≤ 1  c  �  ln 2  ω + ln γ(1 + ϵ)  u2 · ω  + ln ln+  1  s2  �  =  γ  u2 · s2  �  2 ln 1  ω + ln ln+  1  s2 + ln 2γ(1 + ϵ)  u2  �  = m  where we adopt ln(x + y) ≤ x + ln ln+ y, ∀x, y ∈ R+ in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Therefore, if t > m, we must have  u · s > (1 + √ϵ)  �  1 + ϵ  2t  ln  �ln ((1+ϵ)t)  ω  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' With the choice of the confidence radii in (2) and (3), for all i ∈ E, we have  P [∀ p∈N : |ˆµi(p) − µi| ≤ α(Ti(p))] ≥ 1 − 2ξ(ωµ)  (7)  P  �  ∀ p∈N : |ˆσ2  i (p) − σ2  i | ≤ βu(Ti(p))  �  ≥ 1 − 4ξ(ωv)  (8)  P  �  ∀ p∈N : |ˆσ2  i (p) − σ2  i | ≤ βl(Ti(p))  �  ≥ 1 − 4ξ(ω′  v)  (9)  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Note the fact that any distribution supported on [0, 1] is 1/4-sub-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' By a direct application of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 to  the sample mean ˆµi(p) and the sample second moment ˆ  M2,i(p) :=  1  Ti(p)  �p  s=1 Wi(s)21{i ∈ As} of arm i ∈ [L], (7) can  be derived and  P [∀ p∈N : |µi − ˆµi(p)| ≤ lil(Ti(p), ω′  v)] ≥ 1 − 2ξ(ω′  v),  and  P  �  ∀ p∈N : | ˆ  M2,i(p) − (µ2  i + σ2  i )| ≤ lil(Ti(p), ω′  v)  �  ≥ 1 − 2ξ(ω′  v)  Since the rewards are in [0, 1], |µ2  i − ˆµ2  i (p)| = |µi + ˆµi(p)| · |µi − ˆµi(p)| ≤ 2 · lil(Ti(p), ω′  v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Using this and the triangle  inequality, we obtain for every p ≥ 1,  |ˆσ2  i (p) − σ2  i | = |µ2  i − ˆµ2  i (p)| + |(µ2  i + σ2  i ) − ˆ  M2,i(p)|  ≤ 2 · lil(Ti(p), ωv) + lil(Ti(p), ωv) = βu(Ti(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Therefore, (8) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (9) can be similarly obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Proof of the Upper Bound  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Proof scheme of the problem-dependent upper bound  In this subsection, we provide technical lemmas that can upper bound the components in R1(T ′) and R2(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Note that at phase p, the identified solution Ap belongs to one of the 4 disjoint sets: (1) Ap = S⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2) S ∩ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (3) R and (4)  Sc ∩ B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  1 = 1 {Ap = S⋆} + 1 {Ap ∈ S ∩ B} + 1 {Ap ∈ R} + 1 {Ap ∈ Sc ∩ B}  and 1 {Ap ∈ B} = 1 {Ap ∈ S ∩ B} + 1 {Ap ∈ Sc ∩ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Define two events (the F events) that connect the instance and  the confidence radii  Fµ  p :=  �  ∆Ap ≤ 2  �  i∈Ap\\S⋆  α(Ti(p − 1))  �  Fp(x, ρ) :=  �  x ≤ 2  �  i∈Ap  lil(Ti(p − 1), ρ)  �  where x is a constant and ω is a confidence parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When Ap ∈ B, it indicates solution Ap has not been sampled  sufficiently many times and its suboptimality has not been ascertained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When Ap ∈ Sc, it implies the unsafeness of Ap has  not been recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We formalize this in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Conditional on the event E, given any p ∈ [T], we have  S⋆ ∈ ¯Sp−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  If Ap ∈ S ∩ B,  1 {Ap ∈ S ∩ B} ≤ 1  �  Fµ  p  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  If Ap ∈ R,  1 {Ap ∈ R} ≤ 1  �  Fp  �∆v  Ap  3  , ω′  v  ��  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  If Ap ∈ Sc ∩ B,  1 {Ap ∈ Sc ∩ B} ≤ 1  �  Fµ  p , Fp  �∆v  Ap  3  , ω′  v  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' By the design of PASCOMBUCB, Ap ∈ ¯Sp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  (1) We firstly prove that S ∈ ¯Sp−1, ∀S ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' On the event E, we have  Lv  S(p − 1) =  �  i∈A  max{ˆσ2  i (p − 1) − βl(Ti(p − 1)), 0}  ≤  �  i∈A  max{σ2  i , 0}  = σ2  S < ¯σ2  Thus, S ∈ ¯Sp−1, and in particular, S⋆ ∈ ¯Sp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (2) If Ap ∈ B, according to the sampling strategy in Line 10 of PASCOMBUCBand S⋆ ∈ ¯Sp−1, we have U µ  S⋆(p − 1) ≤  U µ  Ap(p − 1) which indicates �  i∈S⋆\\Ap U µ  i (p − 1) ≤ �  i∈Ap\\S⋆ U µ  i (p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus,  �  i∈S⋆\\Ap  µi ≤  �  i∈S⋆\\Ap  U µ  i (p − 1)  ≤  �  i∈Ap\\S⋆  U µ  i (p − 1)  ≤  �  i∈Ap\\S⋆  µi + 2αi(Ti(p − 1))  =⇒  ∆Ap ≤ 2  �  i∈Ap\\S⋆  α(Ti(p − 1))  (10)  (3) If Ap ∈ Sc, according to the sampling strategy, we have Ap ∈ ¯Sp−1 which indicates Lv  Ap(p − 1) = �  i∈Ap Lv  i (p − 1) <  ¯σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus,  ¯σ2 > ˆσ2  Ap(p − 1) −  �  i∈Ap  βl(Ti(p − 1))  ≥ σ2  Ap − 2  �  i∈Ap  βl(Ti(p − 1))  =⇒  ∆v  Ap ≤ 2  �  i∈Ap\\S⋆  βl(Ti(p − 1))  (11)  Note that if Ap ∈ S ∩ B, according to (10),  1 {Ap ∈ S ∩ B} ≤ 1  �  Fµ  p  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  If Ap ∈ R ⊂ Sc, by (11)  1 {Ap ∈ R} ≤ 1  �  Fp  �∆v  Ap  3  , ω′  v  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  If Ap ∈ Sc ∩ B, by (11) and (10)  1 {Ap ∈ Sc ∩ B} ≤ 1  �  Fµ  p , Fp  �∆v  Ap  3  , ω′  v  ��  At phase p, we define two sequences of mutually-exclusive events {Gµ  j,p}j∈N and {Gj,p(x, ω}j∈N (the G events) which can  further bound the number of times Fµ  p and Fv  p (x, ω) occur respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' These events are indexed by two strictly-decreasing  sequences of constants:  a1 > a2 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' > ak > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  1 > b1 > b2 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' > bk > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  where limj→∞ aj = limj→∞ bj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For simplicity, we set aj =  4  9j−2 , bj =  1  4j , ∀j ∈ N and denote the constant  C = �  j∈N  aj  bj = 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For x ∈ R+ and ω ∈ (0, ln(1 + ϵ)/e), define  mj(x, ω) := aj · γK2  x2  �  2 ln 1  ω + ln ln+  1  x2 + D  �  and mj(x, ω) := ∞ otherwise, where (1) γ =  (1+ϵ)(1+√ϵ)2  2  and ϵ is the constant in the confidence bounds (3), (2)  ln ln+(x) = ln ln x if x ≥ e and it equals to 0 otherwise, (3) D = ln  �  324K2(1 + ϵ)2(1 + √ϵ)2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Denote  Gµ  j,p :=  �  i ∈ Ap \\ S⋆ : Ti(p − 1) ≤ mj(∆Ap, ωµ)  �  and  Gj,p(x, ω) := {i ∈ Ap : Ti(p − 1) ≤ mj(x, ω)}  as the sets of items that were not chosen sufficiently often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For j ∈ N, the events at phase p are sequentially defined as  Gµ  j,p :=  �  at least bjK items in Ap \\ S⋆ were chosen  at most mj(∆Ap, ωµ) times  � �  �  �  �  k∈[j−1]  Gµ  k,p  �  �  c  =  � ��Gµ  j,p  �� ≥ bjK  � �  �  �  �  k∈[j−1]  Gµ  k,p  �  �  c  and  Gj,p(x, ω) :=  �  at least bjK items in Ap were chosen  at most mj(x, ω) times  � �  �  �  �  k∈[j−1]  Gk,p(x, ω)  �  �  c  =  �  |Gj,p(x, ω)| ≥ bjK  � �  �  �  �  k∈[j−1]  Gk,p(x, ω)  �  �  c  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' With our choice of {aj}j∈N and {bj}j∈N,  if Fµ  p occurs, Gµ  j,p occurs for some j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  1  �  Fµ  p  �  ≤ 1  �  �  �  �  j∈N  Gµ  j,p  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  if Fp(x, ω) occurs, Gj,p(x, ω) occurs for some j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  1 {Fp(x, ω)} ≤ 1  �  �  �  �  j∈N  Gj,p(x, ω)  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (12)  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We prove (12) in the following and the other statement can be proved by the same procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  To ease the notations, we omit the parameters x, ω and p in Fp(x, ω), Gj,p(x, ω) and mj(x, ω), since they are fixed when  given Fp(x, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The event Gj can be rewritten as  Gj =  �  |Gj| ≥ bjK  � �  �  �  �  k∈[j−1]  Gc  k  �  �  =  �  |Gj| ≥ bjK  � �  �  �  �  k∈[j−1]  �  |Gk| < bkK  �  �  �  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  The statement is proved by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We assume that when F (Fp(x, ω)) occurs, none of event Gj occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Hence,  �  � �  j∈N  Gj  �  �  c  =  �  j∈N  Gc  j  =  �  j∈N  �  �  �  |Gj| < bjK  � �  �  �  �  k∈[j−1]  �  |Gk| ≥ bkK  �  �  �  �  �  =  �  j∈N  �  |Gj| < bjK  �  Let ¯Gj := Ap \\ Gj and define G0 = Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' According to the definition of Gj, we have Gj ⊂ Gj−1 and ¯Gj−1 ⊂ ¯Gj, ∀j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Because limj→∞ mj = 0, there exists j0 such that ¯Gj = Ap, ∀j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Therefore, we can write Ap by the “telescoping”  sum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', Ap = ∪j∈N  � ¯Gj \\ ¯Gj−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp(x, ω) indicates  x ≤ 2  �  i∈Ap  lil(Ti(p − 1), ω)  (13)  = 2  �  j∈N  �  i∈ ¯  Gj\\ ¯  Gj−1  lil(Ti(p − 1), ω)  Note that for i ∈ ¯Gj \\ ¯Gj−1 = Gj−1 \\ Gj, we have Ti(p − 1) ∈ (mj, mj−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 with parameters  t = Ti(p − 1), s = x, u =  �  1  ajK2 and note a1 > aj, we have  �  1  ajK2 · x > lil(Ti(p − 1), ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Note our choice of aj and bj satisfy  2  �  j∈N  bj−1 − bj  √aj  ≤ 1  Thus, (13) can be further bounded by  x ≤ 2  �  j∈N  �  i∈ ¯  Gj\\ ¯  Gj−1  lil(Ti(p − 1), ω)  < 2  �  j∈N  | ¯Gj \\ ¯Gj−1|  �  1  ajK2 · x  ≤ 2  �  j∈N  (bj−1 − bj)K  √ajK  x  ≤ x  which constitutes a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus when F occurs, there must exists j ∈ N such that Gj occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  When none of the Gµ  j,p (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Gj,p(x, ω)) occurs, Fµ  p (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp(x, ω)) must not occur, which indicates all of the items in Ap  have been sampled sufficiently many times such that the suboptimality (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' unsafeness) of Ap is identified, thus Ap will  not been sampled in future phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  As there will be multiple F events happening, we provide the following useful lemma that merges all F events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Given two confidence parameters ω1 ≥ ω2 ∈ (0, ln(1 + ϵ)/e)  If Fµ  p occurs, then Fp(∆Ap, ωµ) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  If both events Fp(x, ω1) and Fp(y, ω2) occur, then event Fp  �  max{x,  �  ln  1  ω1  ln  1  ω2 y}, ω1  �  occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' If Fµ  p occurs, we have  ∆Ap ≤ 2  �  i∈Ap\\S⋆  α(Ti(p − 1)) ≤ 2  �  i∈Ap  α(Ti(p − 1)) = 2  �  i∈Ap  lil(Ti(p − 1), ωµ)  Thus, Fp(∆Ap, ωµ) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  For the second statement, notice the fact that  lil(Ti(p − 1), ω1)  lil(Ti(p − 1), ω2) =  (1 + √ϵ)  �  1+ϵ  2Ti(p−1) ln  � ln((1+ϵ)Ti(p−1))  ω1  ��1/2  (1 + √ϵ)  �  1+ϵ  2t ln  � ln((1+ϵ)Ti(p−1))  ω2  ��1/2  =  �  �  �  �ln 1  ω1 + ln ln((1 + ϵ)Ti(p − 1))  ln 1  ω2 + ln ln((1 + ϵ)Ti(p − 1))  (a)  ≥  �  �  �  �ln 1  ω1  ln 1  ω2  =: ρ  where (a) utilizes the trick that a+c  b+c ≥ a  b , ∀a, b, c ∈ R+ and a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When event Fp(y, ω2) occurs,  y ≤ 2  �  i∈Ap  lil(Ti(p − 1), ω2) ≤ 2  �  i∈Ap  1  ω lil(Ti(p − 1), ω1)  =⇒  ρ · y ≤ 2  �  i∈Ap  lil(Ti(p − 1), ω1)  =⇒  Fp(ρy, ω1)  Thus if both events Fp(x, ω1) and Fp(y, ω2) occur,  we must have that event Fp (max{x, ρy}, ω1))  =  Fp  �  max{x,  �  ln  1  ω1  ln  1  ω2 y}, ω1)  �  occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Upper bound decomposition  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Assume that PASCOMBUCBhas processed T ′ phases with T time steps, the expected regret of PASCOM-  BUCBcan be decomposed into three parts as follows  E[R(T ′)] ≤ E[R1(T ′)|E] + E[R2(T ′)|E] + R3(T)  where  R1(T ′) :=  T ′  �  p=1  1{Ap ∈ B}∆Ap  R2(T ′) := µ⋆  T ′  �  p=1  �  2  Q−1  �  r=1  1{Up(r)}  �  R3(T) := 2µ⋆L  �  1 + T  �  ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′  v  ��  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The expected regret can be decomposed as:  E [R(T ′)] = E  �  �  T ′  �  p=1  np  �  r=1  (µ⋆ − µAp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='r)  �  �  = E  �  �  T ′  �  p=1  np  �  r=1  (µ⋆ − µAp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='r)1 {E}  �  � + E  �  �  T ′  �  p=1  np  �  r=1  (µ⋆ − µAp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='r)1 {Ec}  �  �  = E  �  �  T ′  �  p=1  np  �  r=1  (µ⋆ − µAp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='r)  ����E  �  � P[E] + E  �  �  T ′  �  p=1  np  �  r=1  (µ⋆ − µAp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='r)  ����Ec  �  � P[Ec]  In the initialization stage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' it will take at most 2L time steps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' since each pulled solution Ap contains at least one item i with  Ti(p) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus the regret is at most 2L · µ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The expected regret when the good events fail can be upper bounded by  E  �  �  T ′  �  p=1  np  �  r=1  (µ⋆ − µAp,r)  ����Ec  �  � P [Ec] ≤ E  �  �  T ′  �  p=1  np  �  r=1  µ⋆  ����Ec  �  � P[Ec]  ≤ E  � T  �  t=1  µ⋆  ����Ec  �  L · 2(ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′  v))  ≤ 2µ⋆TL · (ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′  v))  where P[Ec] can be bounded using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Conditional on the good event E, the high-probability regret can be upper bounded by  ˆR(T ′) :=  T ′  �  p=1  np  �  r=1  (µ⋆ − µAp,r)  (14)  =  T ′  �  p=1  �  ∆Ap + µ⋆(np − 1)  �  (a)  ≤  T ′  �  p=1  �  ∆Ap + µ⋆  Q−1  �  r=0  �  2 · 1{U v  Ap(p − 1) > r¯σ2} − 2  ��  =  T ′  �  p=1  �  ∆Ap + µ⋆  Q−1  �  r=1  2 · 1{U v  Ap(p − 1) > r¯σ2}  �  where (a) makes use of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4 and the fact if U v  Ap(p − 1) ∈ ((m − 1)¯σ2, m¯σ2], then m = �Q−1  r=0 1{U v  Ap(p − 1) >  r¯σ2} = �Q−1  r=0 1{Up(r)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Note the fact that only the suboptimal solutions yields positive mean gap and that the negative  mean gap of the risky solutions can be upper bounded by 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' thus the above equation can be further divided into two parts  T ′  �  p=1  �  ∆Ap + µ⋆  Q−1  �  r=1  2 · 1{U v  Ap(p − 1) > r¯σ2}  �  ≤  T ′  �  p=1  1{Ap ∈ B}∆Ap +  T ′  �  p=1  µ⋆  Q−1  �  r=1  2 · 1{U v  Ap(p − 1) > r¯σ2}  =: R1(T ′) + R2(T ′)  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  In conclusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' by summarizing the regret from the initialization stage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' the regret due to failure of the good event and the  high-probability regret,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' the expected regret can be bounded by  E[R(T ′)] ≤ E[R1(T ′)|E]P[E] + E[R2(T ′)|E]P[E] + 2µ⋆ · TL (ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′  v)) + 2µ⋆L  ≤ E[R1(T ′)|E] + E[R2(T ′)|E] + 2µ⋆ · TL (ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′  v)) + 2µ⋆L  = E[R1(T ′)|E] + E[R2(T ′)|E] + R3(T)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Regret due to suboptimality  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Conditioned on event E, the regret due to suboptimality R1(T) can be bounded by  O  �  �  i∈E\\S⋆  K  ∆i,S∩B,min  ln 1  ωµ  +  �  i∈E  ciK  ∆i,Sc∩B,min  ln 1  ω′v  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Notice the fact that the suboptimal solution Ap can be safe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ap ∈ S ∩ B, or unsafe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=',  Ap ∈ Sc ∩ B, we upper bound the regret under these the two scenarios separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Case 1: Ap ∈ S ∩ B  By the definition of the event Gµ  j,p, we have  |Gµ  j,p| =  ���  i ∈ Ap \\ S⋆ : Ti(p − 1) ≤ mj(∆Ap, ωµ)  ��� ≤ bjK  which indicates  1  �  Ap ∈ S ∩ B, Gµ  j,p  �  ≤  1  bjK  �  i∈Ap\\S⋆  1  �  Ap ∈ S ∩ B, Ti(p − 1) ≤ mj(∆Ap, ωµ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Given an item i ∈ E \\ S⋆, assume it is included vi solutions in S ∩ B, we index them according to the decreasing order of  their mean gaps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' {Ai,k}k∈[vi] with ∆Ai,1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ≥ ∆Ai,vi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  T  �  p=1  1 {Ap ∈ S ∩ B} ∆Ap · 1 {E}  (a)  ≤  T  �  p=1  1  �  Ap ∈ S ∩ B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fµ  p  �  ∆Ap  (b)  ≤  T  �  p=1  �  j∈N  1  �  Ap ∈ S ∩ B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Gµ  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p  �  ∆Ap  ≤  T  �  p=1  �  j∈N  1  bjK  �  i∈Ap\\S⋆  1  �  Ap ∈ S ∩ B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj(∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ)  �  ∆Ap  ≤  T  �  p=1  �  j∈N  1  bjK  �  i∈E\\S⋆  �  k∈[vi]  1  �  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k = Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj(∆Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ)  �  ∆Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  ≤  �  i∈E\\S⋆  �  j∈N  T  �  p=1  �  k∈[vi]  ∆Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  bjK 1  �  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k = Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' i ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ aj · γK2  ∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  �  2 ln 1  ωµ  + ln ln+  1  ∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  + D  ��  (c)  ≤  �  i∈E\\S⋆  �  j∈N  T  �  p=1  �  k∈[vi]  ∆Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  bjK 1  �  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k = Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' i ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ aj · γK2  ∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  �  2 ln 1  ωµ  + ln ln+  1  ∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='vi  + D  ��  (d)  ≤  �  i∈E\\S⋆  �  j∈N  aj · γK2  bjK  �  2 ln 1  ωµ  + ln ln+  1  ∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='vi  + D  � �  1  ∆Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='vi  +  vi−1  �  k=2  ∆Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k+1  �  1  ∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k+1  −  1  ∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  ��  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  ≤  �  i∈E\\S⋆  2CγK  ∆Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='vi  �  2 ln 1  ωµ  + ln ln+  1  ∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='vi  + D  �  where (a) and b make use of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2, (c) is obtained by relaxing ln ln+  1  ∆Ai,k to ln ln+  1  ∆Ai,vi , (d) is  obtained by solving the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Case 2: Ap ∈ Sc ∩ B  For the case where ωµ ≤ ω′  v, denote ¯ω :=  �  ln  1  ω′v  ln  1  ωµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Given an item i ∈ E, assume it is included vi solutions in  Sc ∩ B, we index them according to the decreasing order of their gaps ¯∆A := max  �  ¯ω∆A, ∆v  A  3  �  , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' {Ai,k}k∈[vi] with  ¯∆Ai,1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ≥ ¯∆Ai,vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Denote ci := maxk∈[vi]  � ∆Ai,k  ¯∆Ai,k  �2  , ki = arg maxk∈[vi] ∆Ai,k and di = mink∈[vi] ∆Ai,k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', the  minimum mean gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We have  T  �  p=1  1 {Ap ∈ Sc ∩ B} ∆Ap · 1 {E}  (a)  ≤  T  �  p=1  1  �  Ap ∈ Sc ∩ B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fµ  p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp  �∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  ∆Ap  (b)  ≤  T  �  p=1  1  �  Ap ∈ Sc ∩ B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp  �  max  �  ¯ω∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  ∆Ap  (c)  ≤  T  �  p=1  �  j∈N  1  �  Ap ∈ Sc ∩ B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p  �  max  �  ¯ω∆µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  ∆Ap  ≤  T  �  p=1  �  j∈N  1  bjK  �  i∈Ap  1  �  Ap ∈ Sc ∩ B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  � ¯∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  ∆Ap  =  T  �  p=1  �  j∈N  1  bjK  �  i∈E\\S⋆  �  k∈[vi]  1  �  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k = Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  � ¯∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  ∆Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  ≤  �  i∈E  �  j∈N  T  �  p=1  �  k∈[vi]  ∆Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  bjK 1  �  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k = Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' i ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ aj · γK2  ¯∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  �  2 ln 1  ω′v  + ln ln+  1  ¯∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  + D  ��  (d)  ≤  �  i∈E  �  j∈N  T  �  p=1  �  k∈[vi]  ∆Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  bjK 1  �  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k = Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' i ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ aj · γK2  ¯∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='k  �  2 ln 1  ω′v  + ln ln+  1  ¯∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='vi  + D  ��  (15)  (e)  ≤  �  i∈E  �  j∈N  aj · γK2  bjK  �  2 ln 1  ω′v  + ln ln+  1  ¯∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='vi  + D  � � ci  di  + ci ·  � 1  di  −  1  ∆Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='ki  ��  ≤  �  i∈E  2ci · CγK  di  �  2 ln 1  ω′v  + ln ln+  1  ¯∆2  Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='vi  + D  �  where (a) comes from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1, (b) results from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3, (c) is due to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2, (d) is obtained by relaxing  ln ln+  1  ¯∆2  Ai,k to ln ln+  1  ¯∆2  Ai,vi and (e) is achieved by solving the optimization problem in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In conclusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' for i ∈ E \\ S⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' denote  ∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='min :=  min  S∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈S∩B ∆S  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  For i ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' denote  ∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='min :=  min  S∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈Sc∩B ∆S  ¯∆′  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B :=  min  S∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈Sc∩B max{¯ω∆S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  S/3}  ci :=  max  S∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈Sc∩B  �  ∆S  max{¯ω∆S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  S/3}  �2  The regret due to suboptimality can be upper bounded by  R1(T) ≤  �  i∈E\\S⋆  2CγK  ∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='min  �  2 ln 1  ωµ  + ln ln+  1  ∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='min  + D  �  +  �  i∈E  2ci · CγK  ∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='min  �  2 ln 1  ω′v  + ln ln+  1  ( ¯∆′  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2 + D  �  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Regret due to safeness-checking  We firstly introduce two more technical lemmas in order to upper bound the regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We characterize the number of  sub-solutions np in Algorithm 2 by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' At any phase p, we have P[np ≤ Q] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Furthermore, if U v  Ap(p − 1) ∈ ((m − 1)¯σ2, m¯σ2] for some m ∈ N,  then m ≤ np ≤ 2m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Recall that an absolutely safe solution S is safe w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 1 and S ∈ Sp−1, thus |Ap,r| ≥ q, ∀r ∈ [np − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  If np > Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' np ≥ Q + 1, then  K ≥ |Ap| =  np  �  r=1  |Ap,r| >  Q  �  r=1  |Ap,r| ≥ Q · q ≥ K  which constitutes a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Therefore, we have P[np ≤ Q] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  If U v  Ap(p − 1) ∈ ((m − 1)¯σ2, m¯σ2] for some m ∈ N+, we sequentially define {j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='} ⊂ [|Ap|] as follows: denote  j0 := 0 and let j1 be the integer such that  j1−1  �  s=1  U v  is(p − 1) ≤ ¯σ2  and  j1  �  s=1  U v  is(p − 1) > ¯σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Then let j2 > j1 be the integer such that  j2−1  �  s=j1+1  U v  is(p − 1) ≤ ¯σ2  and  j2  �  s=j1+1  U v  is(p − 1) > ¯σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The last integer jk = |Ap| satisfies  0 <  jk  �  s=jk−1+1  U v  is(p − 1) ≤ ¯σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  If k ≥ m + 1, we must have  U v  At(p − 1) =  k  �  l=1  jl  �  s=jl−1+1  U v  is(p − 1)  ≥  m  �  l=1  jl  �  s=jl−1+1  U v  is(p − 1) > m¯σ2  which contradicts with U v  Ap(p − 1) ∈ ((m − 1)¯σ2, m¯σ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Hence, k ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Then we construct the sub-solutions by  Ap,l = {ijl−1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , ijl−1},  ∀l ∈ [k − 1]  and  Ap,k = {ijk−1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , ijk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  There are k − 1 items {ijl : l ∈ [k − 1]} left which will compose at most k − 1 additional sub-solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In conclusion, we  need at most 2m − 1 sub-solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', np ≤ 2m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Obviously, we need at least m sub-solutions since ¯σ2 > σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' So  m ≤ np ≤ 2m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Due to the fact that the upper confidence of any solution S satisfies U v  S(p) ≤ Q¯σ2, thus np can at most be 2Q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 implies the key to upper bound the regret due to safeness-checking is to upper bound �Q−1  r=1 1{Up(r)} over the  horizon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' From the definition of Up(r) := {U v  Ap(p − 1) > r¯σ2}, for r1, r2 ∈ [Q − 1] with r1 > r2, event Up(r1) indicates  event Up(r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus in order to upper bound �Q−1  r=1 1{Up(r)}, it suffices to upper bound 1{Up(r)} for r ∈ [Q − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' To be  more specific, given l ∈ [Q − 1], in order to compute the maximum number of times �Q−1  r=1 1{Up(r)} ≥ l, we only need to  compute the maximum number of times event Up(l) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In the following lemma, we show a necessary condition (in terms of event Fp(x, ω)) for event Up(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' On the event E,  for Ap ∈ S:  1 {Up(r)} ≤ 1  �  Fp  �  (r − 1)¯σ2 + ∆v  Ap  3  , ωv  ��  for Ap ∈ Sc  1 {Up(r)} ≤ 1  �  Fp  �  (r − 1)¯σ2 − ∆v  Ap  3  , ωv  ��  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The proof is straightforward  U v  Ap(p − 1) > r¯σ2  (a)  ⇒  ˆσ2  Ap(p − 1) +  �  i∈Ap  βu(Ti(p − 1)) > r¯σ2  (b)  ⇒  σ2  Ap + 2  �  i∈Ap  βu(Ti(p − 1)) > r¯σ2  ⇒  2  �  i∈Ap  3 · lil(Ti(p − 1), ωv) > (r − 1)¯σ2 + (¯σ2 − σ2  Ap)  where (a) is due to the definition of the confidence bounds for a solution (4), and (b) utilizes the event �  i∈Ap Ei,Ti(p−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  For Ap  ∈ S, the above event is equivalent to Fp  �  (r−1)¯σ2+∆v  Ap  3  , ωv  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' and for Ap  ∈ Sc, it is equivalent to  Fp  �  (r−1)¯σ2−∆v  Ap  3  , ωv  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  The above lemma and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 upper bound the components in R2(T ′) by the F events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We are now ready to bound the regret due to safeness checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' On the event E, if T ′ ∈ [T ′  r′, T ′  r′−1) then  R2(T ′) ≤ 2µ⋆[T ′(r′ − 1) + H(r′, Λ)] ≤ 2µ⋆H(1, Λ)  Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1, the high-probability regret due to safeness-checking is  R2(T ′) = µ⋆  T ′  �  p=1  �  2  Q−1  �  r=1  1 {Up(r)}  �  = 2µ⋆  Q−1  �  r=1  T ′  �  p=1  1 {Up(r)}  In the following, given r ∈ [Q − 1], we are going to upper bound �T ′  p=1 1{Up(r)} conditional on event E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When  U v  Ap(p − 1) > r¯σ2 holds, there are at most 2r + 1 solutions being chosen at phase p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' np ≤ 2r + 1, according to  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Therefore, we are also deriving an upper bound for number of phases in which there are at most 2r + 1  sub-solutions being sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The proof scheme is planned as follows: in Step 1, we decompose the event Up(r) into 4 events according to where Ap lies,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (1) Ap = S⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2) S ∩ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (3) R and (4) Sc ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We will upper bound the regret under each of these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In Step 2, we apply Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 to upper bound the number of times a solution A can be selected via the events Fµ  p and  Fp(x, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Because there will be multiple Fµ  p and Fp(x, ω) events in the indicator function, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3 will be adopted to  merge them into one event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' After that, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 is utilized to bridge the number of times a solution is identified to the  number of times of an item is sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' At the end of this step, we conclude the number of times 1{Up(r)} occurs under the  four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In Step 3, we upper bound �Q−1  r=1  �T ′  p=1 1 {Up(r)} based on the results from Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Step 1: We decompose �T ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p=1 1{Up(r)} conditional on event E into four parts:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='T ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1{Up(r)}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='T ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='U v  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Ap(p − 1) > r¯σ2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='T ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 {Ap = S⋆} + 1 {Ap ∈ S ∩ B} + 1 {Ap ∈ R} + 1 {Ap ∈ Sc ∩ B}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='U v  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Ap(p − 1) > r¯σ2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='T ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 {Ap = S⋆} 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='U v  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Ap(p − 1) > r¯σ2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='T ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 {Ap ∈ S ∩ B} 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='U v  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Ap(p − 1) > r¯σ2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='T ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 {Ap ∈ R} 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='U v  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Ap(p − 1) > r¯σ2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='T ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 {Ap ∈ Sc ∩ B}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='U v  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Ap(p − 1) > r¯σ2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Step 2: For each of the scenarios,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we firstly upper bound the regret by the “F events” and they can be further bounded in  terms of “G events”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Case 1: Ap = S⋆  T ′  �  p=1  1 {Ap = S⋆} 1  �  U v  Ap(p − 1) > r¯σ2�  (a)  ≤  T ′  �  p=1  1 {Ap = S⋆} 1  �  Fp  �  (r − 1)¯σ2 + ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (b)  ≤  T ′  �  p=1  �  j∈N  1 {Ap = S⋆} 1  �  Gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p  �  (r − 1)¯σ2 + ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (c)  ≤  T ′  �  p=1  �  j∈N  1 {Ap = S⋆}  1  bjK  �  i∈Ap  1  �  Ti(p − 1) ≤ mj  �  (r − 1)¯σ2 + ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  =  �  i∈S⋆  �  j∈N  T ′  �  p=1  1  bjK 1  �  Ti(p − 1) ≤ mj  �(r − 1)¯σ2 + ∆v  S⋆  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  ≤  �  i∈S⋆  �  j∈N  1  bjK mj  �(r − 1)¯σ2 + ∆v  S⋆  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  �  =  �  i∈S⋆  �  j∈N  aj · γK2  bjK  9  ((r − 1)¯σ2 + ∆v  S⋆)2  �  2 ln 1  ωv  + ln ln+  1  ((r − 1)¯σ2 + ∆v  S⋆)2 + D  �  ≤  �  i∈S⋆  C ·  9 · γK  ((r − 1)¯σ2 + ∆v  S⋆)2  �  2 ln 1  ωv  + ln ln+  1  ∆v  S⋆2 + D  �  where (a) utilizes Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='5, (b) makes use of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 and (c) follows the definition of Gj,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For simplicity, we denote  gS⋆(r, ∆v  S⋆) = C ·  9 · γK  ((r − 1)¯σ2 + ∆v  S⋆)2  �  2 ln 1  ωv  + ln ln+  1  ∆v  S⋆2 + D  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Thus  T ′  �  p=1  1 {Ap = S⋆} 1  �  U v  Ap(p − 1) > r¯σ2�  ≤  �  i∈S⋆  gS⋆(r, ∆v  S⋆)  Case 2: Ap ∈ S ∩ B  Under this case, there will be a comparison between ωµ and ωv thus we denote ˜ω =  �  ln  1  ωµ  ln  1  ωv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For i ∈ E, denote  ¯∆i,S∩B := minS∋i,S∈S∩B max  �  ∆S  √  ln(1/ωµ),  ∆v  S  3√  ln(1/ωv)  �  which is achieved by solution Si,S∩B, and assume ¯∆i,S∩B ∈  (  ri¯σ2  3√  ln(1/ωv),  (ri+1)¯σ2  3√  ln(1/ωv)] for some ri ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Scenario 1: ωµ ≥ ωv  We firstly deal with the case where ωµ ≥ ωv, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', ˜ω ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We have  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  U v  Ap(p − 1) > r¯σ2�  (a)  ≤  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  Fµ  p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp  �  (r − 1)¯σ2 + ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (b)  ≤  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  Fp  �  ∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp  �  (r − 1)¯σ2 + ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  (c)  ≤  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  Fp  �  max  �  ∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ˜ω  (r − 1)¯σ2 + ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  ��  (d)  ≤  T ′  �  p=1  �  j∈N  1 {Ap ∈ S ∩ B} 1  �  Gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p  �  max  �  ∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ˜ω  (r − 1)¯σ2 + ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  ��  (e)  ≤  T ′  �  p=1  �  j∈N  1 {Ap ∈ S ∩ B}  1  bjK  �  i∈Ap  1  �  Ti(p − 1) ≤ mj  �  max  �  ∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ˜ω  (r − 1)¯σ2 + ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  ��  =  �  i∈E  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  ∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ˜ω  (r − 1)¯σ2 + ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  ��  (16)  where (a) utilizes Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='5, (b) and (c) make use of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3, (d) is due to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 and (e) follows the definition  of Gj,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Given i ∈ E,  (1) if r ≥ ri + 2, for any S ∈ S ∩ B that contains item i,  max  �  ∆S, ˜ω (r − 1)¯σ2 + ∆v  S  3  �  ≥ ˜ω (r − 1)¯σ2  3  ≥ ˜ω (ri + 1)¯σ2  3  ≥  �  ln 1  ωµ  ¯∆i,S∩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  ∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ˜ω  (r − 1)¯σ2 + ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  ��  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  ˜ω (r − 1)¯σ2  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  ��  =  �  j∈N  1  bjK mj  �  ˜ω (r − 1)¯σ2  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  �  =  �  j∈N  aj · γK2  bjK  9  ((r − 1)¯σ2)2  �  2 ln 1  ωv  + 1  ˜ω2 ln ln+  9  (˜ω(r − 1)¯σ2)2 + 1  ˜ω2 D  �  ≤ C ·  9 · γK  ((r − 1)¯σ2)2  �  2 ln 1  ωv  + 1  ˜ω2 ln ln+  1  ln(1/ωµ) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  + 1  ˜ω2 D  �  = C ·  γK  (  (r−1)¯σ2  3√  ln(1/ωv))2  �  2 +  1  ln(1/ωµ) ln ln+  1  ln(1/ωµ) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  +  1  ln(1/ωµ)D  �  (2) if r ≤ ri + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' for any S ∈ S ∩ B that contains item i  max  �  ∆S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ˜ω (r − 1)¯σ2 + ∆v  S  3  �  ≥ max  �  ∆S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ˜ω ∆v  S  3  �  ≥  �  ln 1  ωµ  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  ∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ˜ω  (r − 1)¯σ2 + ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  ��  ≤  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  ��  ln 1  ωµ  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  ��  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  (a)  ≤  �  j∈N  1  bjK mj  ��  ln 1  ωµ  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  �  =  �  j∈N  aj · γK2  bjK  1  (  �  ln  1  ωµ · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B)2  �  �2 ln 1  ωµ  + ln ln+  1  (  �  ln  1  ωµ · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B)2 + D  �  �  ≤ C ·  γK  ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  �  2 +  1  ln(1/ωµ) ln ln+  1  ln(1/ωµ) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  +  1  ln(1/ωµ)D  �  where (a) is achieved by sampling Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Therefore, if we denote  gS∩B,1(r, ¯∆i,S∩B) :=  �  �  �  �  �  �  �  �  �  �  �  �  �  CγK  (  (r−1)¯σ2  3√  ln(1/ωv))2  �  2 +  1  ln(1/ωµ) ln ln+  1  ln(1/ωµ) ¯∆2  i,S∩B  +  1  ln(1/ωµ)D  �  , r ≥  �  3  �  ln(1/ωv) · ¯∆i,S∩B  ¯σ2  �  + 2  CγK  ¯∆2  i,S∩B  �  2 +  1  ln(1/ωµ) ln ln+  1  ln(1/ωµ) ¯∆2  i,S∩B  +  1  ln(1/ωµ)D  �  , r ≤  �  3  �  ln(1/ωv) · ¯∆i,S∩B  ¯σ2  �  + 1  then (16) can be upper bounded by  �  i∈E  gS∩B,1(r, ¯∆i,S∩B)  Scenario 2: ωµ ≤ ωv  For the case where ωµ ≤ ωv, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', ˜ω ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We have  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  U v  Ap(p − 1) > r¯σ2�  (a)  ≤  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  Fµ  p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp  �  (r − 1)¯σ2 + ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (b)  ≤  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  Fp  �  ∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp  �  (r − 1)¯σ2 + ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (c)  ≤  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  Fp  �  max  �  ∆Ap  ˜ω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (r − 1)¯σ2 + ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (d)  ≤  T ′  �  p=1  �  j∈N  1 {Ap ∈ S ∩ B} 1  �  Gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p  �  max  �  ∆Ap  ˜ω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (r − 1)¯σ2 + ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (e)  ≤  T ′  �  p=1  �  j∈N  1 {Ap ∈ S ∩ B}  1  bjK  �  i∈Ap  1  �  Ti(p − 1) ≤ mj  �  max  �  ∆Ap  ˜ω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (r − 1)¯σ2 + ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  =  �  i∈E  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  ∆Ap  ˜ω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (r − 1)¯σ2 + ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (17)  Given i ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (1) if r ≥ ri + 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' for any S ∈ S ∩ B that contains item i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  max  �∆S  ˜ω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (r − 1)¯σ2 + ∆v  S  3  �  ≥ (r − 1)¯σ2  3  ≥ (ri + 1)¯σ2  3  ≥  �  ln 1  ωv  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  ∆Ap  ˜ω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (r − 1)¯σ2 + ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  ≤  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �(r − 1)¯σ2  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  =  �  j∈N  1  bjK mj  �(r − 1)¯σ2  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  �  =  �  j∈N  aj · γK2  bjK  9  ((r − 1)¯σ2)2  �  2 ln 1  ωv  + ln ln+  9  ((r − 1)¯σ2)2 + D  �  ≤ C ·  9 · γK  ((r − 1)¯σ2)2  �  2 ln 1  ωv  + ln ln+  1  ln(1/ωv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  + D  �  = C ·  γK  (  (r−1)¯σ2  3√  ln(1/ωv))2  �  2 +  1  ln(1/ωv) ln ln+  1  ln(1/ωv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  +  1  ln(1/ωv)D  �  (2) if r ≤ ri + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' for any S ∈ S ∩ B that contains item i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  max  �∆S  ˜ω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (r − 1)¯σ2 + ∆v  S  3  �  ≥ max  �∆S  ˜ω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  S  3  �  ≥  �  ln 1  ωv  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  ∆Ap  ˜ω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (r − 1)¯σ2 + ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  ≤  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  ��  ln 1  ωv  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  =  �  j∈N  1  bjK mj  ��  ln 1  ωv  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  �  =  �  j∈N  aj · γK2  bjK  1  (  �  ln 1  ωv · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B)2  �  �2 ln 1  ωv  + ln ln+  1  (  �  ln 1  ωv · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B)2 + D  �  �  ≤ C ·  γK  ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  �  2 +  1  ln(1/ωv) ln ln+  1  ln(1/ωv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  +  1  ln(1/ωv)D  �  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' if we denote  gS∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B) :=  �  �  �  �  �  �  �  �  �  �  �  �  �  CγK  (  (r−1)¯σ2  3√  ln(1/ωv))2  �  2 +  1  ln(1/ωv) ln ln+  1  ln(1/ωv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  +  1  ln(1/ωv)D  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≥  �  3  �  ln(1/ωv) · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  ¯σ2  �  + 2  CγK  ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  �  2 +  1  ln(1/ωv) ln ln+  1  ln(1/ωv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  +  1  ln(1/ωv)D  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≤  �  3  �  ln(1/ωv) · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  ¯σ2  �  + 1  then (17) can be upper bounded by  �  i∈E  gS∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  In conclusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' for i ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we denote ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B := minS∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈S∩B max  �  ∆S  √  ln(1/ωµ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ∆v  A  3√  ln(1/ωv)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµv := max{ωµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv} and  gS∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B) :=  �  �  �  �  �  �  �  �  �  �  �  �  �  CγK  (  (r−1)¯σ2  3√  ln(1/ωv))2  �  2 +  1  ln(1/ωµv)  �  ln ln+  1  ln(1/ωµv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≥  �  3  �  ln(1/ωv) · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  ¯σ2  �  + 2  CγK  ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  �  2 +  1  ln(1/ωµv)  �  ln ln+  1  ln(1/ωµv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≤  �  3  �  ln(1/ωv) · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  ¯σ2  �  + 1  then  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  U v  Ap(p − 1) > r¯σ2�  ≤  �  i∈E  gS∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B)  Case 3: Ap ∈ R  Under this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' there will be a comparison between ωv and ω′  v thus we denote ¯ω =  �  ln  1  ω′v  ln  1  ωv and ωsum :=  �  ln 1  ω′v +  �  ln 1  ωv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  For i ∈ E, denote ∆v  i,R := minS∋i,S∈R ∆v  S and assume ∆v  i,R ∈ (ri  ¯ω  ¯ω+1 ¯σ2, (ri + 1)  ¯ω  ¯ω+1 ¯σ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Scenario 1: ωv ≤ ω′  v For the case ωv ≤ ω′  v, we have ¯ω ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  T ′  �  p=1  1 {Ap ∈ R} 1  �  U v  Ap(p − 1) > r¯σ2�  (a)  ≤  T ′  �  p=1  1 {Ap ∈ R} 1  �  Fp  �∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp  �  (r − 1)¯σ2 − ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (b)  ≤  T ′  �  p=1  1 {Ap ∈ R} 1  �  Fp  �  max  �  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯ω ·  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  (c)  ≤  T ′  �  p=1  �  j∈N  1 {Ap ∈ R} 1  �  Gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p  �  max  �  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯ω ·  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  (d)  ≤  T ′  �  p=1  �  j∈N  1 {Ap ∈ R}  1  bjK  �  i∈Ap  1  �  Ti(p − 1) ≤ mj  �  max  �  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯ω ·  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  =  �  i∈E  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ R} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯ω ·  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  (18)  where (a) utilizes Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='5, (b) makes use of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3, (c) is due to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 and (d) follows the definition of Gj,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Given i ∈ E,  (1) if r ≥ ri + 2, for any S ∈ R that contains item i,  max  �∆v  S  3 , ¯ω · (r − 1)¯σ2 − ∆v  S  3  �  ≥ max  �∆v  S  3 ,  ¯ω  1 + ¯ω · (r − 1)¯σ2  3  �  ≥  ¯ω  1 + ¯ω · (r − 1)¯σ2  3  ≥  ∆v  i,R  3  where the first inequality uses the fact that for x, y ∈ R+ and z ∈ [0, 1], , max{x, y} ≥ max{x, xz + y(1 − z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We take  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  x =  ∆v  Ap  3 , y = ¯ω ·  (r−1)¯σ2−∆v  Ap  3  and z =  ¯ω  1+¯ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ R} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯ω ·  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  ≤  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ R} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  ¯ω  1 + ¯ω · (r − 1)¯σ2  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  =  �  j∈N  1  bjK mj  �  ¯ω  1 + ¯ω · (r − 1)¯σ2  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  �  =  �  j∈N  aj · γK2  bjK  1  (  ¯ω  1+¯ω · (r−1)¯σ2  3  )2  �  2 ln 1  ω′v  + ln ln+  1  (  ¯ω  1+¯ω · (r−1)¯σ2  3  )2 + D  �  ≤  CγK  (  ¯ω  1+¯ω · (r−1)¯σ2  3  )2  �  2 ln 1  ω′v  + ln ln+  1  (  ¯ω  1+¯ω · (r−1)¯σ2  3  )2 + D  �  ≤  CγK  ( (r−1)¯σ2  3ωsum )2  �  �  �2 +  1  ln(1/ω′v)  �  �  �ln ln+  1  ln(1/ω′v)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  �  (2) if r ≤ ri + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' for any S ∈ R that contains item i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  max  �∆v  S  3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯ω · (r − 1)¯σ2 − ∆v  S  3  �  ≥ max  �∆v  S  3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ¯ω  1 + ¯ω · (r − 1)¯σ2  3  �  ≥  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ R} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯ω ·  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  ≤  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ R} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  ��  =  �  j∈N  1  bjK mj  �∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  �  =  �  j∈N  aj · γK2  bjK  1  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  )2  �  2 ln 1  ω′v  + ln ln+  1  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  )2 + D  �  ≤  CγK  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  )2  �  2 ln 1  ω′v  + ln ln+  1  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  )2 + D  �  =  CγK  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2  �  �  �2 +  1  ln(1/ω′v)  �  �  �ln ln+  1  ln(1/ω′v)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  �  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' if we denote  gR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R) :=  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  CγK  ( (r−1)¯σ2  3ωsum )2  �  �  �2 +  1  ln(1/ω′v)  �  �  �ln ln+  1  ln(1/ω′v)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≥  �  ωsum · ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  �  ln(1/ω′v)¯σ2  �  + 2  CγK  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′  v))2  �  �  �2 +  1  ln(1/ω′v)  �  �  �ln ln+  1  ln(1/ω′v)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′  v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≤  �  ωsum · ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  �  ln(1/ω′v)¯σ2  �  + 1  then (18) can be upper bounded by  �  i∈E  gR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Scenario 2: ωv ≥ ω′  v For the case ωv ≥ ω′  v, we have ¯ω ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  T ′  �  p=1  1 {Ap ∈ R} 1  �  U v  Ap(p − 1) > r¯σ2�  (a)  ≤  T ′  �  p=1  1 {Ap ∈ R} 1  �  Fp  �∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp  �  (r − 1)¯σ2 − ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (b)  ≤  T ′  �  p=1  1 {Ap ∈ R} 1  �  Fp  �  max  �  1  ¯ω  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (c)  ≤  T ′  �  p=1  �  j∈N  1 {Ap ∈ R} 1  �  Gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p  �  max  �  1  ¯ω  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (d)  ≤  T ′  �  p=1  �  j∈N  1 {Ap ∈ R}  1  bjK  �  i∈Ap  1  �  Ti(p − 1) ≤ mj  �  max  �  1  ¯ω  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  =  �  i∈E  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ R} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  1  ¯ω  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (19)  where (a) utilizes Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='5, (b) makes use of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3, (c) is due to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 and (d) follows the definition of Gj,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Given i ∈ E,  (1) if r ≥ ri + 2, for any S ∈ R that contains item i,  max  � 1  ¯ω  ∆v  S  3 , (r − 1)¯σ2 − ∆v  S  3  �  ≥ max  � 1  ¯ω  ∆v  S  3 ,  ¯ω  1 + ¯ω · (r − 1)¯σ2  3  �  ≥  1  1 + ¯ω · (r − 1)¯σ2  3  ≥ 1  ¯ω  ∆v  i,R  3  where the first inequality uses the fact that for x, y ∈ R+ and z ∈ [0, 1], , max{x, y} ≥ max{x, xz + y(1 − z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We take  x = 1  ¯ω  ∆v  Ap  3 , y =  (r−1)¯σ2−∆v  Ap  3  and z =  ¯ω  1+¯ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ R} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  1  ¯ω  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  ≤  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ R} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  1  1 + ¯ω · (r − 1)¯σ2  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  =  �  j∈N  1  bjK mj  �  1  1 + ¯ω · (r − 1)¯σ2  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  �  =  �  j∈N  aj · γK2  bjK  1  (  1  1+¯ω · (r−1)¯σ2  3  )2  �  2 ln 1  ωv  + ln ln+  1  (  1  1+¯ω · (r−1)¯σ2  3  )2 + D  �  ≤  CγK  (  1  1+¯ω · (r−1)¯σ2  3  )2  �  2 ln 1  ωv  + ln ln+  1  (  1  1+¯ω · (r−1)¯σ2  3  )2 + D  �  ≤  CγK  ( (r−1)¯σ2  3ωsum )2  �  �  �2 +  1  ln(1/ωv)  �  �  �ln ln+  1  ln(1/ωv)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  �  (2) if r ≤ ri + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' for any S ∈ R that contains item i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  max  � 1  ¯ω  ∆v  S  3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (r − 1)¯σ2 − ∆v  S  3  �  ≥ max  � 1  ¯ω  ∆v  A  3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ¯ω  1 + ¯ω · (r − 1)¯σ2  3  �  ≥ 1  ¯ω  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ R} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  1  ¯ω  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  ≤  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ R} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  � 1  ¯ω  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  =  �  j∈N  1  bjK mj  � 1  ¯ω  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  �  =  �  j∈N  aj · γK2  bjK  1  ( 1  ¯ω  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  )2  �  2 ln 1  ωv  + ln ln+  1  ( 1  ¯ω  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  )2 + D  �  ≤  CγK  ( 1  ¯ω  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  )2  �  2 ln 1  ωv  + ln ln+  1  ( 1  ¯ω  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3  )2 + D  �  =  CγK  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2  �  �  �2 +  1  ln(1/ωv)  �  �  �ln ln+  1  ln(1/ωv)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  �  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' if we denote  gR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R) :=  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  CγK  ( (r−1)¯σ2  3ωsum )2  �  �  �2 +  1  ln(1/ωv)  �  �  �ln ln+  1  ln(1/ωv)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≥  �  ωsum · ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  �  ln(1/ω′v)¯σ2  �  + 2  CγK  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2  �  �  �2 +  1  ln(1/ωv)  �  �  �ln ln+  1  ln(1/ωv)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≤  �  ωsum · ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  �  ln(1/ω′v)¯σ2  �  + 1  then (19) can be upper bounded by  �  i∈E  gR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  In conclusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' for i ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we denote ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R := minS∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈R ∆v  S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωvv′ := max{ωv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωsum :=  �  ln 1  ω′  v +  �  ln 1  ωv and  gR(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R) :=  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  CγK  ( (r−1)¯σ2  3ωsum )2  �  �  �2 +  1  ln(1/ωvv′)  �  �  �ln ln+  1  ln(1/ωvv′)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≥  �  ωsum · ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  �  ln(1/ω′v)¯σ2  �  + 2  CγK  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2  �  �  �2 +  1  ln(1/ωvv′)  �  �  �ln ln+  1  ln(1/ωvv′)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≤  �  ωsum · ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  �  ln(1/ω′v)¯σ2  �  + 1  then  T ′  �  p=1  1 {Ap ∈ R} 1  �  U v  Ap(p − 1) > r¯σ2�  ≤  �  i∈E  gR(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R)  Case 4: Ap ∈ Sc ∩ B  Under this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' there will be a comparison among ωµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv and ω′  v thus we denote ωmax = max{ωµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v} and ω1 =  �  ln  1  ωmax  ln  1  ωµ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω2 =  �  ln  1  ωmax  ln  1  ωv ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω3 =  �  ln  1  ωmax  ln  1  ω′v  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For S ∈ Sc ∩ B, we denote ¯∆S := max  �  ω1∆S, ω3  ∆v  S  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  For i ∈ E, denote  ¯∆i,Sc∩B :=  min  S∋i,S∈Sc∩B max  �  ∆S  �  ln(1/ωµ)  ,  ∆v  S  3  �  ln(1/ω′v)  �  =  �  1  ln(1/ωmax)  min  S∋i,S∈Sc∩B  ¯∆S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  and assume ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B ∈ (ri  ¯σ2/3  √  ln(1/ωv)+√  ln(1/ω′v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (ri + 1)  ¯σ2/3  √  ln(1/ωv)+√  ln(1/ω′v)]  T ′  �  p=1  1 {Ap ∈ Sc ∩ B}  �  1  �  U v  Ap(p − 1) > r¯σ2�  1 {E}  (a)  ≤  T ′  �  p=1  1 {Ap ∈ Sc ∩ B} 1  �  Fµ  p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp  �∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp  �  (r − 1)¯σ2 − ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (b)  ≤  T ′  �  p=1  1 {Ap ∈ Sc ∩ B} 1  �  Fp  �  ∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp  �∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Fp  �  (r − 1)¯σ2 − ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv  ��  (c)  ≤  T ′  �  p=1  1 {Ap ∈ Sc ∩ B} 1  �  Fp  �  max  �  ω1∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω3  ∆v  Ap  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω2  (r − 1)¯σ2 − ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωmax  ��  (d)  ≤  T ′  �  p=1  1 {Ap ∈ Sc ∩ B} 1  �  Fp  �  max  �  max  �  ω1∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω3  ∆v  Ap  3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω2  (r − 1)¯σ2  3  − ω2  ω3  max  �  ω1∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω3  ∆v  Ap  3  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωmax  ��  (e)  ≤  T ′  �  p=1  �  j∈N  1 {Ap ∈ Sc ∩ B} 1  �  Gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p  �  max  �  ¯∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω2  (r − 1)¯σ2  3  − ω2  ω3  ¯∆Ap  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωmax  ��  (f)  ≤  T ′  �  p=1  �  j∈N  1 {Ap ∈ Sc ∩ B}  1  bjK  �  i∈Ap  1  �  Ti(p − 1) ≤ mj  �  max  �  ¯∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω2  (r − 1)¯σ2  3  − ω2  ω3  ¯∆Ap  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωmax  ��  =  �  i∈E  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ Sc ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  ¯∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω2  (r − 1)¯σ2  3  − ω2  ω3  ¯∆Ap  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωmax  ��  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Given i ∈ E  (1) if r ≥ ri + 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' for any S ∈ Sc ∩ B that contains item i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  max  �  ¯∆S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω2  (r − 1)¯σ2  3  − ω2  ω3  ¯∆S  �  ≥ max  �  ¯∆S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ω2ω3  ω2 + ω3  (r − 1)¯σ2  3  �  ≥  ω2ω3  ω2 + ω3  (r − 1)¯σ2  3  ≥  �  ln(1/ωmax) ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩R  where the first inequality uses the fact that for x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' y ∈ R+ and z ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' max{x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' y} ≥ max{x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' xz + y(1 − z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We take  x = ¯∆Ap, y = ω2  (r−1)¯σ2  3  − ω2  ω3 ¯∆Ap and z =  ω2  ω2+ω3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Similar to the computations for the case Ap ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we have  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ Sc ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  ¯∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω2  (r − 1)¯σ2  3  − ω2  ω3  ¯∆Ap  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωmax  ��  ≤  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ Sc ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  � ω2ω3  ω2 + ω3  (r − 1)¯σ2  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωmax  ��  ≤  CγK  ( ω2ω3  ω2+ω3  (r−1)¯σ2  3  )2  �  2 ln  1  ωmax  + ln ln+  1  ( ω2ω3  ω2+ω3  (r−1)¯σ2  3  )2 + D  �  ≤  CγK  ( ω2ω3  ω2+ω3  (r−1)¯σ2  3  )2  �  2 ln  1  ωmax  + ln ln+  1  ln(1/ωmax)( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩R)2 + D  �  =  CγK  (  (r−1)¯σ2/3  √  ln(1/ωv)+√  ln(1/ω′v))2  �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩R)2 + D  ��  (2) if r ≤ ri + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' for any S ∈ Sc ∩ B that contains item i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  max  �  ¯∆S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω2  (r − 1)¯σ2  3  − ω2  ω3  ¯∆S  �  ≥ max  �  ¯∆S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ω2ω3  ω2 + ω3  (r − 1)¯σ2  3  �  ≥ ¯∆S ≥  �  ln(1/ωmax) ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩R  Similar to the computations for the case Ap ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we have  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ Sc ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  �  max  �  ¯∆Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω2  (r − 1)¯σ2  3  − ω2  ω3  ¯∆Ap  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωmax  ��  ≤  �  j∈N  1  bjK  T ′  �  p=1  1 {Ap ∈ Sc ∩ B} 1  �  i ∈ Ap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Ti(p − 1) ≤ mj  ��  ln(1/ωmax) ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωmax  ��  ≤  CγK  (  �  ln(1/ωmax) ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2  �  2 ln  1  ωmax  + ln ln+  1  (  �  ln(1/ωmax) ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2 + D  �  =  CγK  ( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2  �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩R)2 + D  ��  In conclusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' for i ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we denote  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B :=  min  S∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈Sc∩B max  �  ∆S  �  ln(1/ωµ)  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ∆v  S  3  �  ln(1/ω′v)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  and ωmax := max{ωµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v} and ωsum :=  �  ln 1  ω′  v +  �  ln 1  ωv and  gSc∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B) :=  �  �  �  �  �  �  �  �  �  CγK  ( (r−1)¯σ2  3ωsum )2  �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩R)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≥  �ωsum · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B  ¯σ2/3  �  + 2  CγK  ( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩R)2  �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩R)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≤  �ωsum · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B  ¯σ2/3  �  + 1  then  T ′  �  p=1  1 {Ap ∈ Sc ∩ B} 1  �  U v  Ap(p − 1) > r¯σ2�  ≤  �  i∈E  gSc∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)  Conclusion of Step 2:  For S⋆: denote  gS⋆(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  S⋆) :=  9 · CγK  ((r − 1)¯σ2 + ∆v  S⋆)2  �  2 ln 1  ωv  + ln ln+  1  ∆v  S⋆2 + D  �  (20)  we have  T ′  �  p=1  1 {Ap = S⋆} 1  �  U v  Ap(p − 1) > r¯σ2�  ≤  �  i∈S⋆  gS⋆(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  S⋆)  For S ∩ B: for i ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' denote ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B := minS∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈S∩B max  �  ∆S  √  ln(1/ωµ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ∆v  S  3√  ln(1/ωv)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωµv := max{ωµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv} and  gS∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B)  (21)  :=  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  CγK  (  (r−1)¯σ2  3√  ln(1/ωv))2  �  2 +  1  ln(1/ωµv)  �  ln ln+  1  ln(1/ωµv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≥  ����  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  ¯σ2  3√  ln(1/ωv)  ���� + 2  CγK  ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  �  2 +  1  ln(1/ωµv)  �  ln ln+  1  ln(1/ωµv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≤  ����  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  ¯σ2  3√  ln(1/ωv)  ���� + 1  then  T ′  �  p=1  1 {Ap ∈ S ∩ B} 1  �  U v  Ap(p − 1) > r¯σ2�  ≤  �  i∈E  gS∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B)  For R: for i ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' denote ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R := minS∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈R ∆v  S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωvv′ := max{ωv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωsum :=  �  ln 1  ω′v +  �  ln 1  ωv and  gR(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R)  (22)  :=  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  CγK  ( (r−1)¯σ2  3ωsum )2  �  �  �2 +  1  ln(1/ωvv′)  �  �  �ln ln+  1  ln(1/ωvv′)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≥  �  ωsum · ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  �  ln(1/ω′v)¯σ2  �  + 2  CγK  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2  �  �  �2 +  1  ln(1/ωvv′)  �  �  �ln ln+  1  ln(1/ωvv′)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≤  �  ωsum · ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  �  ln(1/ω′v)¯σ2  �  + 1  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  then  T ′  �  p=1  1 {Ap ∈ R} 1  �  U v  Ap(p − 1) > r¯σ2�  ≤  �  i∈E  gR(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R)  For Sc ∩ B: for i ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' denote  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B :=  min  S∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈Sc∩B max  �  ∆S  �  ln(1/ωµ)  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ∆v  S  3  �  ln(1/ω′v)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  and ωmax := max{ωµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ωv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ω′  v} and ωsum :=  �  ln 1  ω′v +  �  ln 1  ωv and  gSc∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)  (23)  :=  �  �  �  �  �  �  �  �  �  CγK  ( (r−1)¯σ2  3ωsum )2  �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≥  �ωsum · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B  ¯σ2/3  �  + 2  CγK  ( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2  �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r ≤  �ωsum · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B  ¯σ2/3  �  + 1  then  T ′  �  p=1  1 {Ap ∈ Sc ∩ B} 1  �  U v  Ap(p − 1) > r¯σ2�  ≤  �  i∈E  gSc∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)  Step 3:  According to the results in Step 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' given r ∈ [Q − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' the event Up(r) can happen at most  T ′  �  p=1  1 {Up(r)}  (24)  ≤ min  �  T ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  i∈S⋆  gS⋆(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  S⋆) +  �  i∈E  gS∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B) +  �  i∈E  gR(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R) +  �  i∈E  gSc∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)  �  phases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' where the g functions are defined in (20),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (21),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (22) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' By Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4, (1) event Up(r) indicates r + 1 ≤ np,  thus (24) also indicates at most in this number of phases there are at least r + 1 being pulled at each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2) event  Up(r) ∩ Up(r + 1)c indicates r + 1 ≤ np ≤ 2r + 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', there are at least r + 1 and at most 2r + 1 solutions being pulled at  each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For r ∈ [Q], we denote  T ′  r :=  �  i∈S⋆  gS⋆(r, ∆v  S⋆) +  �  i∈E  gS∩B(r, ¯∆i,S∩B) +  �  i∈E  gR(r, ∆v  i,R) +  �  i∈E  gSc∩B(r, ¯∆i,Sc∩B),  r ∈ [Q − 1]  (25)  and T ′  Q := 0, T ′  0 = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Note that the g functions are increasing as r decreases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' so if there exists an r′ ∈ [Q],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' such that  T ′ ∈ [T ′  r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' T ′  r′−1) (or T ′ ≤ T ′  r′−1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' then  Q−1  �  r=1  T ′  �  p=1  1 {Up(r)}  =  r′−1  �  r=1  T ′  �  p=1  1 {Up(r)} +  Q−1  �  r=r′  T ′  �  p=1  1 {Up(r)}  ≤ T ′ · (r′ − 1) +  Q−1  �  r=r′  �  i∈S⋆  gS⋆(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  S⋆) +  �  i∈E  gS∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B) +  �  i∈E  gR(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R) +  �  i∈E  gSc∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  = T ′ · (r′ − 1) +  Q−1  �  r=r′  T ′  r  ≤ T ′ · (r′ − 1) +  �  i∈S⋆  hS⋆(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  S⋆) +  �  i∈E  hS∩B(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B) +  �  i∈E  hR(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R) +  �  i∈E  hSc∩B(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)  = T ′ · (r′ − 1) + H(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Λ)  where  H(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Λ) :=  �  i∈S⋆  hS⋆(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  S⋆) +  �  i∈E  hS∩B(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B) +  �  i∈E  hR(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R) +  �  i∈E  hSc∩B(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)  (26)  and the h functions are defined at the end of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' This indicates, when T ′ ≤ T ′  r′−1, the upper bound of the high-  probability regret due to safeness-checking R2(T ′) (hence the the total regret) is lower bounded by a linear function with  slope r′ − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In particular, the upper bound of R2(T ′) is lower bounded by a linear function when T ′ ≤ T ′  1 and it remains a  constant when T ′ > T ′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We can compute an upper bound for the number of solutions being pulled during these T ′ phases  when T ′ ∈ [T ′  r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' T ′  r′−1):  T ′  Q−1 · (2Q − 1) + (T ′  Q−2 − T ′  Q−1) · (2Q − 3) + · · · + (T ′  r′ − T ′  r′+1) · (2r′ + 1) + (T ′ − T ′  r′)(2r′ − 1)  = T ′ + 2  �  T ′  Q−1 · (Q − 1) + (T ′  Q−2 − T ′  Q−1) · (Q − 2) + · · · + (T ′  r′ − T ′  r′+1) · r′ + (T ′ − T ′  r′)(r′ − 1)  �  = (2r′ − 1)T ′ + 2  Q−1  �  r=r′  T ′  r  ≤ (2r′ − 1)T ′ + 2H(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Λ)  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' the number of pulled solutions is at most 2H(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1  In conclusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' the regret due to safeness-checking can be upper bounded by  R2(T ′) = 2µ⋆  Q−1  �  r=1  T ′  �  p=1  1 {Up(r)}  ≤ 2µ⋆T ′ · (r′ − 1)  + 2µ⋆  � �  i∈S⋆  hS⋆(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  S⋆) +  �  i∈E  hS∩B(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B) +  �  i∈E  hR(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R) +  �  i∈E  hSc∩B(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)  �  = 2µ⋆T ′ · (r′ − 1) + 2µ⋆ · H(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Λ)  where T ′ ∈ [T ′  r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' T ′  r′−1) with T ′  r′ defined in (25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' for i ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B := minS∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈S∩B max  �  ∆S  √  ln(1/ωµ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ∆v  S  3√  ln(1/ωv)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R := minS∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈R ∆v  S and ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B := minS∋i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∈Sc∩B max  �  ∆S  √  ln(1/ωµ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ∆v  S  3√  ln(1/ω′v)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The h functions:  (For convenience, we restate the notations: ωµv := max{ωµ, ωv}, ωvv′ := max{ωv, ω′  v}, ωmax := max{ωµ, ωv, ω′  v} and  ωsum :=  �  ln 1  ω′v +  �  ln 1  ωv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=')  For S⋆: for each i ∈ S⋆  hS⋆(r′, ∆v  S⋆) :=  Q−1  �  r=r′  gS⋆(r, ∆v  S⋆)  (27)  1Note that the upper bound for the regret is 2µ⋆T ′ · (r′ − 1) + 2µ⋆ · H(r′, Λ) and the upper bound for the number of solutions is  (2r′ − 1)T ′ + 2H(r′, Λ), which indicates we can roughly use min{Tµ⋆, 2µ⋆H(r′, Λ)} to bound the regret due to safeness-checking  with T time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  =  Q−1  �  r=r′  9 · CγK  ((r − 1)¯σ2 + ∆v  S⋆)2  �  2 ln 1  ωv  + ln ln+  1  ∆v  S⋆2 + D  �  ≤  �  �  �  �  �  �  �  18 · CγK  (r′ − 1)¯σ4  �  2 ln 1  ωv  + ln ln+  1  (∆v  S⋆)2 + D  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  r′ ≥ 2  18 · CγK  (∆v  S⋆)2  �  2 ln 1  ωv  + ln ln+  1  (∆v  S⋆)2 + D  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  r′ = 1  For S ∩ B: for each i ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' there is a changing point  �  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  ¯σ2  3√  ln(1/ωv)  �  + 2 in gS∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' thus  hS∩B(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B) :=  Q−1  �  r=r′  gS∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B)  (28)  ≤  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r′ = Q  (Q − r′) CγK  ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  �  2 +  1  ln(1/ωµv)  �  ln ln+  1  ln(1/ωµv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ����  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  ¯σ2  3√  ln(1/ωv)  ���� ≥ Q − 3  2 · CγK  (r′ − 1)(  ¯σ2  3√  ln(1/ωv))2  �  2 +  1  ln(1/ωµv)  �  ln ln+  1  ln(1/ωµv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ����  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  ¯σ2  3√  ln(1/ωv)  ���� + 2 ≤ r′ < Q − 1  3 · CγK  ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  �  2 +  1  ln(1/ωµv)  �  ln ln+  1  ln(1/ωµv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ����  ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  ¯σ2  3√  ln(1/ωv)  ���� = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r′ = 1  CγK  �  �  4  ¯σ2  3√  ln(1/ωv) · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  − r′ − 1  ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  �  �  �  2 +  1  ln(1/ωµv)  �  ln ln+  1  ln(1/ωµv) ¯∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B  + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' otherwise  For R: for each i ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' there is a changing point  �  ωsum·∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  √  ln(1/ω′v)¯σ2  �  + 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' thus  hR(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R) :=  Q−1  �  r=r′  gR(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R)  (29)  ≤  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r′ = Q  (Q − r′)  CγK  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2  �  �  �2 +  1  ln(1/ωvv′)  �  �  �ln ln+  1  ln(1/ωvv′)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  ωsum · ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  �  ln(1/ω′v)¯σ2  �  ≥ Q − 3  2CγK  (r′ − 1)(  ¯σ2  3ωsum )2  �  �  �2 +  1  ln(1/ωvv′)  �  �  �ln ln+  1  ln(1/ωvv′)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  ωsum · ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  �  ln(1/ω′v)¯σ2  �  + 2 ≤ r′ < Q − 1  3CγK  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′  v))2  �  �  �2 +  1  ln(1/ωvv′)  �  �  �ln ln+  1  ln(1/ωvv′)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′  v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  ωsum · ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  �  ln(1/ω′v)¯σ2  �  = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r′ = 1  CγK  �  �  �  3  ¯σ2  3ωsum ·  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v)  −  r′ − 1  (  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2  �  �  �  �  �  �2 +  1  ln(1/ωvv′)  �  �  �ln ln+  1  ln(1/ωvv′)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' otherwise  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  For Sc ∩ B: for each i ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' there is a changing point  �  ωsum· ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B  ¯σ2/3  �  + 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' thus  hSc∩B(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B) :=  Q−1  �  r=r′  gSc∩B(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)  (30)  ≤  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r′ = Q  (Q − r′)  CγK  ( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2  �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �ωsum · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B  ¯σ2/3  �  ≥ Q − 3  2CγK  (r′ − 1)(  ¯σ2  3ωsum )2  �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �ωsum · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B  ¯σ2/3  �  + 2 ≤ r′ < Q − 1  3CγK  ( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2  �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �ωsum · ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B  ¯σ2/3  �  = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r′ = 1  CγK  �  3  ¯σ2  3ωsum ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B  −  r′ − 1  ( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2  � �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' otherwise  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Proofs of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 (Problem-dependent upper bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let Λ = (E, AK, ν, ¯σ2) be an instance and let {δT }∞  T =1 ∈ o(1) be a  sequence that satisfies ln(1/δT ) = o(T b) for all b > 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', {δT } is not exponentially decaying).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Then, PASCOMBUCB is  a {δT }∞  T =1-variance-constrained consistent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' More precisely, given a time budget T, the probably anytime-safe  constraint is satisfied and the regret of PASCOMBUCB Reg(T) is upper bounded by  min {Tµ⋆, Reg1(T) + Reg2(T)} + Reg3(T),  where  Reg1(T) = O  �  �  i∈E\\S⋆  K ln T  ∆i,S∩B,min  +  �  i∈E  ciK ln T  ∆i,Sc∩B,min  �  Reg2(T) = 2µ⋆H (∆(Λ)) ,  Reg3(T) = 2µ⋆(L + 1)  where ∆(Λ) = {∆v  S⋆} ∪ {∆v  i,R, Ψi,S∩B, Φi,Sc∩B}i∈E and H (∆(Λ)) := H(1, Λ) is defined in (26) in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' According to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' and take ωµ = ω′  v =  1  T 2 and ωv = δT  T 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' the  expected regret of T ′ phases E[R(T ′)] can be upper bounded as  E[R(T ′)] ≤ E[R1(T ′)|E] + E[R2(T ′)|E] + R3(T)  (31)  ≤ O  �  � �  i∈E\\S⋆  K  ∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='min  ln 1  ωµ  +  �  i∈E  ciK  ∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='min  ln 1  ω′v  �  � + 2µ⋆ [T ′ · (r′ − 1) + H(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Λ)]  + 2µ⋆L + 2µ⋆TL (ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′  v))  ≤ O  �  � �  i∈E\\S⋆  K  ∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='min  ln 1  T +  �  i∈E  ciK  ∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='min  ln 1  T  �  � + 2µ⋆H(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Λ)  + 2µ⋆L + 2µ⋆TL  �  3ξ(1/T 2) + 2ξ(δT /T 2)  �  = Reg1(T) + Reg2(T) + Reg3(T)  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' the high-probability regret (14) can be na¨ıve bounded as  ˆR(T ′) =  T ′  �  p=1  np  �  r=1  (µ⋆ − µAp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='r)  ≤  T ′  �  p=1  np  �  r=1  µ⋆  = Tµ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Therefore, we have  E[R(T ′)] ≤ E[ˆR(T ′)|E] + Reg3(T)  (32)  ≤ Tµ⋆ + Reg3(T)  (31) and (32) give the final upper bound with T time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 (Problem-independent Upper Bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let {δT }∞  T =1 ∈ o(1) be a sequence that satisfies ln(1/δT ) = o(T b)  for all b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' If T > L, for any instance Λ with variance gaps lower bounded by ∆v ≤ minS∈AK ∆v  S, the regret of  PASCOMBUCB is upper bounded by  O  �√  KLT ln T + LK2  (∆v)2 ln  � 1  δT  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We firstly deal with the regret due to suboptimality R1(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let ∆µ be a constant that is to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  R1(T ′) =  T ′  �  p=1  1{Ap ∈ B}∆Ap  =  T ′  �  p=1  1{∆Ap ≥ ∆µ}1{Ap ∈ B}∆Ap +  T ′  �  p=1  1{∆Ap < ∆µ}1{Ap ∈ B}∆Ap  ≤  T ′  �  p=1  1{∆Ap ≥ ∆µ}1{Ap ∈ B}∆Ap + T · ∆µ  where the second term makes use of the fact that T ′ ≤ T w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The first term can be upper bounded by adopting the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3 with the constraint that ∆Ap ≥ ∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus, for  i ∈ E \\ S⋆, ∆i,S∩B,min ≥ ∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For i ∈ E,  ∆i,Sc∩B,min ≥ ∆µ  ¯∆′  i,Sc∩B ≥ max{¯ω∆µ, ∆v/3} = max{∆µ, ∆v/3} =: ∆µv  ci ≤ 1  ¯ω2 = 1  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  where ¯ω :=  �  ln  1  ω′v  ln  1  ωµ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The regret due to suboptimality can be upper bounded by  T ′  �  p=1  1{∆Ap ≥ ∆µ}1{Ap ∈ B}∆Ap ≤  �  i∈E\\S⋆  2CγK  ∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='min  �  2 ln 1  ωµ  + ln ln+  1  ∆2  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='min  + D  �  +  �  i∈E  2ci · CγK  ∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='min  �  2 ln 1  ω′v  + ln ln+  1  ( ¯∆′  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B)2 + D  �  ≤  �  i∈E\\S⋆  2CγK  ∆µ  �  2 ln 1  ωµ  + ln ln+  1  (∆µ)2 + D  �  +  �  i∈E  2CγK  ∆µ  �  2 ln 1  ω′v  + ln ln+  1  (∆µv)2 + D  �  ≤L · 8CγK  ∆µ  �  2 ln T + ln ln+  1  (∆µ)2 + D  �  By taking ∆µ =  �  KL ln(T L)  T  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' the regret due to suboptimality is bounded by  R1(T ′) ≤  T ′  �  p=1  1{∆Ap ≥ ∆µ}1{Ap ∈ B}∆Ap + T · ∆µ  ≤ 16Cγ  �  KLT ln(TL) + 8Cγ  �  KLT  ln T ln ln+  T  KL ln T + 8Cγ  �  KLT  ln T D +  �  KLT ln(TL)  = O(  �  KLT ln(TL))  = O(  √  KLT ln T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  where we utilize T ≥ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We then cope with the regret due to safeness-checking R2(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' According to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3, we only need to upper bound  H(1, Λ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', the h functions defined in (27), (28), (29) and (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  For hS⋆(1, ∆v  S⋆):  hS⋆(1, ∆v  S⋆) ≤ hS⋆(1, ∆v)  = 18 · CγK  (∆v)2  �  2 ln 1  ωv  + ln ln+  1  (∆v)2 + D  �  = O  �  K  (∆v)2 ln 1  ωv  �  = O  �  K  (∆v)2 ln T  δ  �  For hS∩B(1, ¯∆i,S∩B), define ∆µv := max  �  ∆µ  √  ln(1/ωµ),  ∆v  3√  ln(1/ωv)  �  = max  ��  KL  T ,  ∆v  3√  ln(T L/δ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We have ¯∆i,S∩B ≥  ∆µv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The threshold  �  ∆µv  ¯σ2  3√  ln(1/ωv)  �  =  ����  max  �  3  �  KL ln(T L/δ)  T  ,∆v  �  ¯σ2  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  hS∩B(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='S∩B) ≤ hS∩B(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆µv)  (33)  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  (Q − 1) CγK  (∆µv)2  �  2 +  1  ln(1/ωµv)  �  ln ln+  1  ln(1/ωµv)(∆µv)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ����  ∆µv  ¯σ2  3√  ln(1/ωv)  ���� ≥ Q − 3  CγK  4  ¯σ2  3√  ln(1/ωv) · ∆µv  �  2 +  1  ln(1/ωµv)  �  ln ln+  1  ln(1/ωµv)(∆µv)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 0 <  ����  ∆µv  ¯σ2  3√  ln(1/ωv)  ���� < Q − 3  3 · CγK  (∆µv)2  �  2 +  1  ln(1/ωµv)  �  ln ln+  1  ln(1/ωµv)(∆µv)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ����  ∆µv  ¯σ2  3√  ln(1/ωv)  ���� = 0  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  O  �  (Q − 1)  K  (∆µv)2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ����  ∆µv  ¯σ2  3√  ln(1/ωv)  ���� ≥ Q − 3  O  �  �  K  ¯σ2  3√  ln(1/ωv) · ∆µv  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  0 <  ����  ∆µv  ¯σ2  3√  ln(1/ωv)  ���� < Q − 3  O  �  K  (∆µv)2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ����  ∆µv  ¯σ2  3√  ln(1/ωv)  ���� = 0  In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' in the asymptotic case where T → ∞ and ln 1  δ = ln 1  δT = o(T b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∀b > 0 (this includes the scenario where δ is  fixed with respect to T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we have  hS∩B(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆µv) = O  �  K  (∆µv)2  �  = O  �  K  (∆v)2 ln T  δ  �  For hR(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we have ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R ≥ ∆v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Furthermore, ωsum =  �  ln(TL) +  �  ln(TL/δ) and the changing point  �  ωsum·∆v  √  ln(1/ω′v)¯σ2  �  =  �  (√  ln(T L)+√  ln(T L/δ))·∆v  √  ln(T L)¯σ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Thus  hR(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R) ≤ hR(r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v)  (34)  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  (Q − 1)  CγK  (  ∆v  3√  ln(1/ω′v))2  �  �2 +  1  ln(1/ωvv′)  �  �ln ln+  1  ln(1/ωvv′)(  ∆v  3√  ln(1/ω′v))2 + D  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  ωsum · ∆v  �  ln(1/ω′v)¯σ2  �  ≥ Q − 3  CγK  3  ¯σ2  3ωsum ·  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v)  �  �  �2 +  1  ln(1/ωvv′)  �  �  �ln ln+  1  ln(1/ωvv′)(  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v))2 + D  �  �  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 0 <  �  ωsum · ∆v  �  ln(1/ω′v)¯σ2  �  ≤ Q − 3  3CγK  (  ∆v  3√  ln(1/ω′  v))2  �  �2 +  1  ln(1/ωvv′)  �  �ln ln+  1  ln(1/ωvv′)(  ∆v  3√  ln(1/ω′  v))2 + D  �  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  ωsum · ∆v  �  ln(1/ω′v)¯σ2  �  = 0  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  O  �  �(Q − 1)  K  (  ∆v  3√  ln(1/ω′v))2  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  ωsum · ∆v  �  ln(1/ω′v)¯σ2  �  ≥ Q − 3  O  �  �  �  K  ¯σ2  3ωsum ·  ∆v  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='R  3√  ln(1/ω′v)  �  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  0 <  �  ωsum · ∆v  �  ln(1/ω′v)¯σ2  �  ≤ Q − 3  = O  �  �  K  (  ∆v  3√  ln(1/ω′v))2  �  � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �  ωsum · ∆v  �  ln(1/ω′v)¯σ2  �  = 0  In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' in the asymptotic case where T → ∞ and ln 1  δ = ln 1  δT = o(T b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∀b > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we have  hR(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∆v) = O  � QK  (∆v)2 ln 1  ω′v  �  = O  � QK  (∆v)2 ln T  �  For Sc ∩ B: define ¯∆µv := max  �  ∆µ  √  ln(1/ωµ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ∆v  3√  ln(1/ω′v)  �  = max  ��  KL  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ∆v  3√  ln(T L)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We have ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B ≥ ¯∆µv and  the changing point  �  ωsum· ¯∆µv  ¯σ2/3  �  =  �  (√  ln(T L)+√  ln(T L/δ))·∆µv  ¯σ2/3  �  thus  hSc∩B(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='Sc∩B) ≤ hSc∩B(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆µv)  (35)  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  (Q − 1) CγK  ( ¯∆µv)2  �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆µv)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �ωsum · ¯∆µv  ¯σ2/3  �  ≥ Q − 3  CγK  �  3  ¯σ2  3ωsum ¯∆µv  � �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆µv)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 0 <  �ωsum · ¯∆µv  ¯σ2/3  �  < Q − 3  3CγK  ( ¯∆µv)2  �  2 +  1  ln(1/ωmax)  �  ln ln+  1  ln(1/ωmax)( ¯∆µv)2 + D  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �ωsum · ¯∆µv  ¯σ2/3  �  = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r′ = 1  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  O  �  (Q − 1)  K  ( ¯∆µv)2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �ωsum · ¯∆µv  ¯σ2/3  �  ≥ Q − 3  O  �  K  ¯σ2  3ωsum · ¯∆µv  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  0 <  �ωsum · ¯∆µv  ¯σ2/3  �  < Q − 3  O  �  K  ( ¯∆µv)2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  �ωsum · ¯∆µv  ¯σ2/3  �  = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' r′ = 1  In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' in the asymptotic case where T → ∞ and ln 1  δ = ln 1  δT = o(T b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∀b > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we have  hSc∩B(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯∆µv) = O  � QK  (∆v)2 ln T  �  Lastly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  R3(T) = 2µ⋆L + 2µ⋆TL (ξ(ωµ) + 2ξ(ωv) + 2ξ(ω′  v))  ≤ 2KL + Kδ + 2KTL · 4 · 2 + ϵ  ϵ  �  1  T 2  ln(1 + ϵ)  �1+ϵ  ≤ 2KL + Kδ + 4K · 2 + ϵ  ϵ  �  1  ln(1 + ϵ)  �1+ϵ  = O(1)  where we utilize T > L and the O notation refers to the fact that the preceding term is bounded as a function of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Note that µ⋆ ≤ K, so Tµ⋆ + Kδ ≤ TK + Kδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In conclusion, according to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1, for any T > L, the problem-independent upper bound is the minimum of  TK + Kδ and  O(  √  KLT ln T) + K  � �  i∈S⋆  hS⋆(1, ∆v) +  �  i∈E  hS∩B(1, ∆µv) +  �  i∈E  hR(1, ∆v) +  �  i∈E  hSc∩B(1, ¯∆µv)  �  ≤ O(  √  KLT ln T) + K  �  KhS⋆(1, ∆v) + LhS∩B(1, ∆µv) + LhR(1, ∆v) + LhSc∩B(1, ¯∆µv)  �  = O(  √  KLT ln T) + O  � K3  (∆v)2 ln T  δ  �  + KL  �  hS∩B(1, ∆µv) + hR(1, ∆v) + hSc∩B(1, ¯∆µv)  �  where the h functions are defined in (33), (34) and (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In the asymptotic case where T → ∞ and ln 1  δ = ln 1  δT =  o(T b), ∀b > 0 (this includes the scenario where δ is fixed with respect to T), the asymptotic problem-independent upper  bound is  O(  √  KLT ln T) + K  ��  i∈E  O  �  K  (∆v)2 ln T  δ  �  + O  �  i∈E  � QK  (∆v)2 ln T  ��  = O(  √  KLT ln T) + O  � LK2  (∆v)2 ln 1  δ  �  where we utilize  √  T ln T ≥  QK2  (∆v)2 ln T when T is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Proofs of the Lower Bounds  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Preliminaries  Let KL(ν, ν′) denote the KL divergence between distributions ν and ν′, and  d(x, y) := x ln  �x  y  �  + (1 − x) ln  �1 − x  1 − y  �  denote the Kullback–Leibler (KL) divergence between the Bernoulli distributions Bern(x) and Bern(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 (Pinsker’s and reverse Pinsker’s Inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Consider two probability mass functions PX, PY defined on the  same discrete probability space A ⊂ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The following inequalities hold:  |EX∼PX[X] − EY ∼PY [Y ]| ≤ δ(PX, PY ) ≤  �  1  2KL(PX, PY ) ≤  1  √αY  δ(PX, PY ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  where δ(PX, PY ) := supA⊆A  ��  a∈A PX(a) − �  a∈A PY (a)  �  =  1  2  �  a∈A |PX(a) − PY (a)| is the total variational  distance, and αY := mina∈A:PY (a)>0 Q(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 (Lemma 1 in Kaufmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Assume the distributions under instance Λ1 = (E, AK, ν(1), ¯σ2) and  instance Λ2 = (E, AK, ν(2), ¯σ2) are mutually absolutely continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Given time budget T,  L  �  i=1  EΛ1[Ni(T)] · KL(ν(1)  i  , ν(2)  i  ) ≥  sup  E∈HΛ1  T  d  �  PΛ1(E), PΛ2(E)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  where Ni(t) denotes the number of time steps item i is selected up to and including time step t and HΛ1  T is all the possible  events generated by instance Λ1 and algorithm π with T time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let solution S containing |S| = m(q < m ≤ K) items be a safe solution under instance Λ1 =  (E, AK, ν(1), ¯σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Each item in S is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  with reward distribution ν1 , mean µ1 and variance σ2  1 < ¯σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  De-  fine event E(t,1) = {S is identified as safe after time step t}, E(t,2) = {S is chosen at least once after time step t}, and  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  E(t) = E(t,1) ∩ E(t,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Assume there exists τ ≤ T such that PΛ1[E(τ)] ≥ 1 − δ and PΛ1[E(τ−1,1)] < 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' If τ exists, we  have  �  i∈S  EΛ1[Ni(τ)] ≥  sup  ν2∈E(ν1)  d(δ, 1 − δ)  KL(ν1, ν2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Furthermore,  EΛ1[M(τ)] ≥  sup  ν2∈E(ν1)  1  |S| − 1 · d(δ, 1 − δ)  KL(ν1, ν2) := T(ν(1)),  where M(t) is the number of times that a solution S′ ⊂ S is sampled up to and include time step t and E(ν1) = {ν2 :  the variance associated to ν2 is larger than ¯σ2/|S|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' With σ2  2 > ¯σ2/|S|, we construct an alternative instance Λ2 = (E, AK, ν2, ¯σ2), under which each item in S is with  reward distribution ν2 , mean µ2 and variance σ2  2, while the distributions of other items remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Define event E(t,1) = {S is identified as safe after time step t}, E(t,2) = {S is chosen at least once after time step t}, and  E(t) = E(t,1) ∩ E(t,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Assume there exists τ ≤ T such that PΛ1[E(τ)] ≥ 1 − δ and PΛ1[E(τ−1,1)] < 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Since S is  unsafe under instance Λ2 and all the solutions chosen {St}T  t=1 ⊂ AK are safe with probability at least 1 − δ, we have  PΛ2[E(t)] < δ for all t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We now apply Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 to obtain that  �  i∈S  EΛ1[Ni(τ)] · KL(ν1, ν2) ≥ d(P1(Ec  (τ)), P2(Ec  (τ))) ≥ d(δ, 1 − δ) ⇒  �  i∈S  EΛ1[Ni(τ)] ≥ d(δ, 1 − δ)  KL(ν1, ν2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Since PΛ1[E(τ−1,1)] < 1 − δ, we can select at most m − 1 items at one time step among the first τ time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Therefore,  EΛ1[M(τ)] ≥  1  m − 1  �  i∈P  EΛ1[Ni(τ)] ≥  1  m − 1 · d(δ, 1 − δ)  KL(ν1, ν2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='5 (Impossibility result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let {δT }∞  T =1 ∈ o(1) be a sequence that satisfies the following condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' There exists  b > 0 such that ln(1/δT ) = Ω(T b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For instance Λ, the regret of any algorithm is lower bounded by Ω(T b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The proof is similar to the proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We consider an alternative instance Λ2 = (E, AK, ν(2), ¯σ2) with  the distributions of the items in the optimal safe solution S⋆ changed such that (assume the variances of all the items in S⋆  are changed (increased))  �  i∈S⋆  (σ(2)  i  )2 ≥ ¯σ2  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', under instance Λ2, this solution S⋆ is unsafe (thus not optimal safe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The other items remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  By a similar argument as the proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3, we have  �  i∈S⋆  EΛ1[Ni(τ)] · KL(ν(1)  i  , ν(2)  i  ) ≥ d(δ, 1 − δ)  ⇒  �  i∈S⋆  EΛ1[Ni(τ)] ≥  d(δ, 1 − δ)  mini∈S⋆ KL(ν(1)  i  , ν(2)  i  )  So the safeness checking of S⋆ will take Ω(ln  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4δT ) = Ω(T b)  Recall the probably anytime-safe constraint (1):  P  �  ∀ t ∈ [T], St ∈ S  �  ≥ 1 − δT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  This indicates at time step t, for any solution S, if PH(0)  t [S ∈ S] < 1−δT , S will not be selected at this time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Otherwise,  (1) is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Therefore, before the safeness of the optimal safe solution S is ascertained, it is not going to be sampled and  the instantaneous regret will be lower bounded by minS∈S∩B ∆S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In conclusion, the regret is at least ln  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4δT · minS∈S∩B ∆S = Ω(T b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We derive both the problem-dependent and problem independent lower bounds on the K-path semi-bandit problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The  items in the ground set are divided into L0 paths: P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , PL0, each of which contains K unique items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Path Pj contains  items (j − 1)K + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , jK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Without loss of generality, we assume that L/K is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' A set S is a solution if and  only if S ⊂ Pj for some j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In other words, the solution set AK = {S : S ⊂ Pj, ∃j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , L0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We let Ni(t) denote the  number of time steps item i is selected up to and including time step t, Mj(t) denote the number of time steps a safe subset  in path j is selected up to and including time step t, and Sj(t) denote the time steps when a safe subset in path j is selected,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=',  Ni(t) =  t  �  s=1  1{i ∈ Ss},  Mj(t) =  t  �  s=1  1{Ss ⊂ Pj, Ss is safe },  Sj(t) = {1 ≤ s ≤ t : Ss ⊂ Pj, Ss is safe }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We let Reg[j](t) denote the regret accumulated in Sj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Since all the chosen solutions are safe with probability at least  1 − δ, we have Reg(T) ≥ (1 − δ) · �L0  j=1 Reg[j](T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In the following, we lower bound the regret accumulated in time steps  where safe solutions are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We will construct several instances to prove each of the lower bounds (which will be specified in the proof) such that  under instance k (Λk = (E, AK, ν(k), ¯σ2)), the stochastic reward of items in path j (1 ≤ j ≤ L0) are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ,  ν(k)  i  = ν(k)  Pj , ∀i ∈ Pj, which will be specified in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Under instance k, we define several other notations as follows:  Let Wi(t)(k) be the random reward of arm i at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Let S(k)  t  be the pulled solution at time step t, and H(j)  t  = {(S(j)  s , {Wi(s)(k)}s∈S(k)  s )}t  s=1 be the sequence of selected  solutions and observed rewards up to and including time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  For simplicity, we abbreviate EH(k)  T , PH(k)  T , as Ek, Pk respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Problem-dependent Lower bound  In order to provide a better understanding of the analysis, we first derive a lower bound with Gaussian distributions  (unbounded) in Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' With the same technique, we derive a lower bound with bounded Bernoulli distributions in  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3, which corroborates with our problem setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4 (Problem-dependent lower bound for sub-Gaussian instances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let {δT }∞  T =1 ∈ o(1) be a sequence that  satisfies ln(1/δT ) = o(T b) for all b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' There exists an instance Λ, for any {δT }T ∈N-variance-constrained consistent  algorithm π, the regret is lower bounded by  Ω  ��  i∈E  ln T  ∆i,S∩B,min  �  + µ⋆  K · Ω  �K · ln(1/δT )  (∆S⋆)2  +  �  i∈E  �  Ψ′  i,S∩B +  ln T  (∆v  i,R)2 + Φi,Sc∩B  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Given a vanishing sequence {δT }T ∈N, we consider a fixed {δT }T ∈N-variance-constrained consistent algorithm π on  the K-path semi-bandit problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For simplicity, the distributions of the items are assumed to be Gaussian in the proof, but  the techniques can be applied to the Bernoulli case and we provide the instance design and the corresponding bound at the  end of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Under instance Λ0 (the base instance),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' σ2 = 2¯σ2  K (so the absolutely safe solutions contain at most K/2 items) and the  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  distributions of the items are  ν(0)  i  = N(µ(0)  j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (σ2  i )(0)) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  ν(0)  P1 = N(∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯σ2 − ϵv  K  ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  i ∈ P1  ν(0)  Pj = N(∆ − ϵµ  K ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯σ2 − ϵv  K  ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  i ∈ Pj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 2 ≤ j ≤ L1 + 1  ν(0)  Pj = N(∆ + ϵµ  K ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯σ2 + ϵv  K  ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  i ∈ Pj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' L1 + 2 ≤ j ≤ L2 + 1  ν(0)  Pj = N(∆ − ϵµ  K ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ¯σ2 + ϵv  K  ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  i ∈ Pj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' L2 + 2 ≤ j ≤ L0  where ϵµ < ∆  K and ϵv are small positive constants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ϵµ =  ∆  2K , ϵv ≤ ¯σ2  K2 ), and L1 = L2 − L1 = L0 − L2 − 1 = L0−1  3  (assume it is an integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In this case,  path 1 is an optimal safe path,  path 2 to path L1 + 1 are the safe and suboptimal paths,  path L1 + 2 to path L2 + 1 are the risky paths  path L2 + 2 to path L3 + 1 = L0 are the unsafe and suboptimal paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In the following, we will compute the minimum regret yielded from each of the paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Case 1: the optimal safe path P1  In order to achieve o(T a), ∀a > 0 regret, any algorithm has to identify the safeness of the optimal safe solution P1 and  sample P1 Ω(T) times, otherwise, the regret is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' According to Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3, the expected number of time steps needed  for the safeness identification of P1 is lower bounded by  E0[M1(τ)] ≥  sup  ν(1)  P1 ∈E(ν(0)  P1 )  1  K − 1 · d(δT , 1 − δT )  KL(ν(0)  P1 , ν(1)  P1 )  := T(ν(0)  P1 )  where E(ν(0)  P1 ) = {ν(1)  P1  :  the variance associated to ν(1)  P1 is larger than ¯σ2/K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' In particular, we let instance Λ1 =  (E, AK, ν(1), ¯σ2) with  ν(1)  i  =  �  �  �  �  �  ν(1)  P1 = N  �  ∆, ¯σ2 + ϵv  1  K  �  ,  i ∈ P1  ν(0)  i  ,  i /∈ P1  where ϵv  1 is an arbitrarily small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus, we can take supremum over ϵv  1 > 0 and have  T(ν(0)  P1 ) ≥  1  K − 1 ·  ln  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4δT  KL(N(∆, ¯σ2−ϵv  K  ), N(∆, ¯σ2  K ))  ≥  1  K − 1 · 4K2(¯σ2/K)2  (ϵv)2  ln  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4δT  = K(σ2)2  (ϵv)2  ln  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4δT  When a solution S ⊂ P1 is chosen before this time step, the instantaneous regret is lower bounded by ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The accumulative  regret from P1 is lower bounded by  Reg[1](T) ≥ ∆ · T(ν(0)  P1 ) = Ω( K∆  (∆v  P1)2 ln 1  δT  )  Case 2: the safe and suboptimal paths  For any safe and suboptimal path Pj(2 ≤ j ≤ L1 + 1), we define instance Λj = (E, AK, ν(j), ¯σ2) with  ν(j)  i  =  �  �  �  �  �  ν(j)  Pj = N(∆ +  ϵµ  j  K , ¯σ2 − ϵv  K  ),  i ∈ Pj  ν(0)  i  ,  i /∈ Pj  where ϵµ  j < ∆ is an arbitrarily small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' So Pj is the optimal safe solution under instance j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Fix any item i ∈ Pj, consider the event Ej = {Ni(T) ≥ T  2 }, under instance 1, Ej indicates the optimal safe solution P1  is sampled less than T  2 times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' under instance j, Ec  j indicates the optimal safe solution Pj is sampled less than T  2 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Therefore,  RegΛ0(T) + RegΛj(T) ≥ T  2 ϵµP0[Ej] + T  2 ϵµ  j Pj[Ec  j ]  ≥ T  2 min{ϵµ, ϵµ  j }  �  P0[Ej] + Pj[Ec  j ]  �  According to Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 and Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we have  P0[Ej] + Pj[Ec  j ] ≥ 1  2 exp{−KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj)}  KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj) =  L  �  i=1  E0[Ni(T)] · KL  �  ν(0)  i  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ν(j)  i  �  =  �  i∈Pj  E0[Ni(T)] · KL  �  ν(0)  i  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ν(j)  i  �  Thus  RegΛ0(T) + RegΛj(T) ≥ T  2 min{ϵµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ϵµ  j } · 1  2 exp{−KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj)}  ⇐⇒  ln  �  RegΛ0(T) + RegΛj(T)  �  ln T  ≥ 1 +  min{ϵµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ϵµ  j }/4  ln T  − KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj)  ln T  (∗)  =⇒  KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj)  ln T  ≥ 1  ⇐⇒  �  i∈Pj E0[Ni(T)]  ln T  ≥  1  KL  �  ν(0)  Pj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ν(j)  Pj  � = 2 ¯σ2−ϵv  K  (  ϵµ+ϵµ  j  K  )2  (∗∗)  =⇒  �  i∈Pj E0[Ni(T)]  ln T  ≥ 2 ¯σ2−ϵv  K  ( ϵµ  K )2  =: T(ν(0)  Pj )  where in (∗) we let T → ∞ and note that both RegΛ0(T) and RegΛj(T) are of order o(T a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∀a > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' in (∗∗) we take  supremum over ϵµ  j > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We also have to take the safeness constraint into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' According to Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3, in order to check the safeness of  Pj, the items in Pj have to be sampled  �  i∈Pj  EΛ0[Ni(τ)] ≥  sup  ν′  Pj ∈E(ν(0)  Pj )  d(δT , 1 − δT )  KL(ν(0)  Pj , ν′  Pj)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  ⇐⇒  �  i∈Pj EΛ0[Ni(τ)]  ln T  ≥  sup  ν′  Pj ∈E(ν(0)  Pj )  1  KL(ν(0)  Pj , ν′  Pj)  d(δT , 1 − δT )  ln T  ≥ 4K2(¯σ2/K)2  (ϵv)2  ln  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4δT  ln T  := Tsafe(ν(0)  Pj ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  If T(ν(0)  Pj ) ≤ Tsafe(ν(0)  Pj ), it indicates the suboptimality of Pj is identified before the safeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Furthermore, whenever a  solution S ⊂ Pj is sampled, S can have most K − 1 items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' So the regret is lower bounded by  Reg[j](T)  ln T  ≥  T(ν(0)  Pj )  K − 1 · (ϵµ + ∆ − ϵµ  K ) ≥ 2K2 ¯σ2−ϵv  K  (ϵµ)2  (  ∆  K − 1 + ϵµ  K )  =⇒  Reg[j](T)  ln T  ≥ 2K ¯σ2−ϵv  K  (ϵµ)2  (∆ + ϵµ) ≥ Kσ2/2  (ϵµ)2 · (∆ + ϵµ)  =⇒  Reg[j](T) = Ω  �  K  ∆2  Pj  (∆Pj + ∆) ln T  �  (36)  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  If T(ν(0)  Pj ) ≥ Tsafe(ν(0)  Pj ), it indicates the suboptimality of Pj is identified after the safeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus, whenever a solution  S ⊂ Pj is sampled, S can have most K − 1 items before the safeness checking is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' So the regret is lower bounded by  Reg[j](T)  ln T  ≥  Tsafe(ν(0)  Pj )  K − 1  (ϵµ + ∆ − ϵµ  K ) +  T(ν(0)  Pj ) − Tsafe(ν(0)  Pj )  K  ϵµ  ≥  T(ν(0)  Pj )  K  ϵµ + (∆ − ϵµ  K ) ·  Tsafe(ν(0)  Pj )  K − 1  ≥ 2K ¯σ2−ϵv  K  (ϵµ)2  ϵµ + (∆ − ϵµ  K ) ·  1  K − 1 · 4K2(¯σ2/K)2  (ϵv)2  ln  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4δT  ln T  ≥ Kσ2/2  ϵµ  + (∆ − ϵµ  K ) ·  1  K − 1 · K2(σ2)2  (ϵv)2  ln  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4δT  ln T  =⇒  Reg[j](T) = Ω  �  K  ∆Pj  ln T + (∆ − ∆Pj  K )  K  (∆v  Pj)2 ln 1  δT  �  (37)  Based on (36) and (37), the regret is  Reg[j](T) = Ω  �  K  ∆Pj  ln T + ∆ · min  �  K  ∆2  Pj  ln T,  K  (∆v  Pj)2 ln 1  δT  ��  Case 3: the risky paths  For any risky path Pj (L1 + 2 ≤ j ≤ L2 + 1, we define instance Λj = (E, AK, ν(j), ¯σ2) with  ν(j)  i  =  �  �  �  �  �  ν(j)  Pj = N(∆ + ϵµ  K , ¯σ2 − ϵv  j  K  ),  i ∈ Pj  ν(0)  i  ,  i /∈ Pj  where ϵv  j is an arbitrarily small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' So Pj is the optimal safe solution under instance j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Fix any item i ∈ Pj, consider the event Ej = {Ni(T) ≥ T  2 }, under instance 1, Ej indicates the optimal safe solution P1  is sampled less than T  2 times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' under instance j, Ec  j indicates the optimal safe solution Pj is sampled less than T  2 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Therefore,  RegΛ0(T) + RegΛj(T) ≥ T  2  �  ∆ − K − 1  K  ϵµ  �  P0[Ej] + T  2 ϵµPj[Ec  j ]  ≥ T  2 ϵµ ·  �  P0[Ej] + Pj[Ec  j ]  �  According to Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 and Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we have  P0[Ej] + Pj[Ec  j ] ≥ 1  2 exp{−KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj)}  KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj) =  L  �  i=1  E0[Ni(T)] · KL  �  ν(0)  i  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ν(j)  i  �  =  �  i∈Pj  E0[Ni(T)] · KL  �  ν(0)  i  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ν(j)  i  �  Thus  RegΛ0(T) + RegΛj(T) ≥ T  2 ϵµ · 1  2 exp{−KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj)}  ⇐⇒  ln  �  RegΛ0(T) + RegΛj(T)  �  ln T  ≥ 1 +  min{ϵµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ϵµ  j }/4  ln T  − KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj)  ln T  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  (∗)  =⇒  KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj)  ln T  ≥ 1  ⇐⇒  �  i∈Pj E0[Ni(T)]  ln T  ≥  1  KL  �  ν(0)  Pj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ν(j)  Pj  � = K2(σ2)2  (ϵv)2  =: T(ν(0)  Pj )  where in (∗) we let T → ∞ and note that both RegΛ0(T) and RegΛj(T) are of order o(T a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∀a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Note that Pj is a risky path and if any solution S ⊂ Pj is sampled, |S| ≤ K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus, E[Mj(T)] ≥  T (ν(0)  Pj )  K−1 ln T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The  regret is lower bounded by  Reg[j](T) = Ω  ��  ∆ − K − 1  K  ϵµ  �  1  K − 1  K2(σ2)2  (ϵv)2  ln T  �  = Ω  �  K∆  (∆v  Pj)2 ln T  �  Case 4: the unsafe and suboptimal paths  For any unsafe and suboptimal path Pj(L2 + 2 ≤ j ≤ L0), we define instance Λj = (E, AK, ν(j), ¯σ2) with  ν(j)  i  =  �  �  �  �  �  ν(j)  Pj = N(∆ +  ϵµ  j  K , ¯σ2 − ϵv  j  K  ),  i ∈ Pj  ν(0)  i  ,  i /∈ Pj  where ϵµ  j < ∆, ϵv  j < ¯σ2  K are arbitrarily small constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' So Pj is the optimal safe solution under instance j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Fix any item i ∈ Pj, consider the event Ej = {Ni(T) ≥ T  2 }, under instance 1, Ej indicates the optimal safe solution P1  is sampled less than T  2 times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' under instance j, Ec  j indicates the optimal safe solution Pj is sampled less than T  2 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Therefore,  RegΛ0(T) + RegΛj(T) ≥ T  2 ϵµP0[Ej] + T  2 ϵµ  j Pj[Ec  j ]  ≥ T  2 min{ϵµ, ϵµ  j }  �  P0[Ej] + Pj[Ec  j ]  �  According to Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 and Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we have  P0[Ej] + Pj[Ec  j ] ≥ 1  2 exp{−KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj)}  KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj) =  L  �  i=1  E0[Ni(T)] · KL  �  ν(0)  i  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ν(j)  i  �  =  �  i∈Pj  E0[Ni(T)] · KL  �  ν(0)  i  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ν(j)  i  �  Thus  RegΛ0(T) + RegΛj(T) ≥ T  2 min{ϵµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ϵµ  j } · 1  2 exp{−KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj)}  ⇐⇒  ln  �  RegΛ0(T) + RegΛj(T)  �  ln T  ≥ 1 +  min{ϵµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ϵµ  j }/4  ln T  − KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj)  ln T  (∗)  =⇒  KL(P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj)  ln T  ≥ 1  ⇐⇒  �  i∈Pj E0[Ni(T)]  ln T  ≥  1  KL  �  ν(0)  Pj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ν(j)  Pj  � = 4K2  �  �  �  ϵv  j + ϵv  (¯σ2 − ϵv  j )/K  �2  +  2(ϵµ  j + ϵµ)2  (¯σ2 − ϵv  j )/K  �  �  −1  (∗∗)  =⇒  �  i∈Pj E0[Ni(T)]  ln T  ≥ 4K2  �� ϵv  σ2/2  �2  + 2(ϵµ)2  σ2/2  �−1  =: T(ν(0)  Pj )  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  where in (∗) we let T → ∞ and note that both RegΛ0(T) and RegΛj(T) are of order o(T a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ∀a > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' in (∗∗) we take the  supremum over ϵµ  j > 0, ϵv  j > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Note that Pj is an unsafe path and if any solution S ⊂ Pj is sampled, |S| ≤ K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Thus, E[Mj(T)] ≥  T (ν(0)  Pj )  K−1 ln T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' The  regret is lower bounded by  Reg[j](T) ≥  �  ∆ + K − 1  K  ϵµ  � 4K2  K − 1  �� ϵv  σ2/2  �2  + 2(ϵµ)2  σ2/2  �−1  ln T  = Ω  � K∆ + Kϵµ  max{ϵµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ϵv}2 ln T  �  = Ω  �  min  �  K∆  ∆2  Pj  ln T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  K∆  (∆v  Pj)2 ln T  ��  In conclusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' the regret yielded from these paths is lower bounded by  Reg(T)  1 − δT  ≥ Reg[1](T) +  L1+1  �  j=2  Reg[j](T) +  L2+1  �  j=L1+2  Reg[j](T) +  L0  �  j=L2+2  Reg[j](T)  ≥ Ω  � K∆  (ϵv)2 ln 1  δT  �  +  L1+1  �  j=2  Ω  �K  ϵµ ln T + ∆ · min  � K  (ϵµ)2 ln T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  K  (ϵv)2 ln 1  δT  ��  +  L2+1  �  j=L1+2  Ω  � K∆  (ϵv)2 ln T  �  +  L0  �  j=L2+2  Ω  �  min  � K∆  (ϵµ)2 ln T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' K∆  (ϵv)2 ln T  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Note  L1 = L2 − L1 = L0 − L2 − 1 = L0−1  3  , L0 = L  K and µ⋆ = K∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  for S⋆, ϵv = ∆v  S⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  for S ∈ S ∩ B and i ∈ S, we check that if S ⊂ Pj, 2 ≤ j ≤ L1 + 1 , ∆i,S∩B,min = ϵµ and  Ψ′  i,S∩B ≥ min  �ln T  ∆2  S  , 9 ln(1/δT )  (∆v  S)2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  where the equality holds when S = Pj, 2 ≤ j ≤ L1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  for S ∈ R and i ∈ S, ϵv = ∆v  i,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  S ∈ Sc ∩ B and i ∈ S, we can easily check S = Pj for some L2 + 2 ≤ j ≤ L0, thus ∆i,Sc∩B,min = ϵµ and  Φi,Sc∩B = min  �ln T  ∆2  S  , 9 ln T  (∆v  S)2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Therefore,  Reg(T) ≥ Ω  �  L ln T  ∆i,S∩B,min  �  + µ⋆  K · Ω  �  K · ln(1/δT )  (∆S⋆)2  +  �  i∈E  Ψ′  i,S∩B +  �  i∈E  ln T  (∆v  i,R)2 +  �  i∈E  Φi,Sc∩B  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3 (Problem-dependent lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let {δT }∞  T =1 ∈ o(1) be a sequence that satisfies ln(1/δT ) = o(T b) for  all b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' There exists an instance Λ such that for any {δT }T ∈N-variance-constrained consistent algorithm π, the regret is  lower bounded by  Ω  � �  i∈E  ln T  ∆i,S∩B,min  �  + µ⋆  K · Ω  �K ln(1/δT )  (∆v  S⋆)2  +  �  i∈E  �  Ψ′  i,S∩B +  ln T  (∆v  i,R)2 + Φi,Sc∩B  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We divide the whole ground set into several paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Meanwhile, we define four sets E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , E4 such that Ei ∩Ej = ∅  for any i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' P1 = E1, Pj ⊂ Ej for j = 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For any j > 4, there exists k ̸= 1 such that Pj ⊂ Ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' For any path Pj, if  Pj ⊂ E4, |Pj| = K2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' otherwise, |Pj| = K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Pj consists of arms  j−1  �  i=1  |Pi| + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ,  j−1  �  i=1  |Pi| + K1 · 1{Pj ̸⊂ E4} + K2 · 1{Pj ⊂ E4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The feasible solution set AK consists of all subsets of each path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We let Bern(a) denote the Bernoulli distribution with parameter a(a ∈ (0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Note that the variance of Bern(a) is a(1−a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We construct an instance with νi = Bern(µi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' With ε1, ε2 > 0, we set  µi = ∆  ∀i ∈ E1,  µi = ∆ − ε1  ∀i ∈ E2,  µi = ∆ + ε2  ∀i ∈ E3,  µi = ∆ − ε3  ∀i ∈ E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We let 2 ≤ K1 < K2 ≤ K, ∆ < 1/2 and  K1∆(1 − ∆) < ¯σ2 (paths in E1 and E2 are safe),  K1(∆ + ε2)(1 − ∆ − ε2) > ¯σ2 (paths in E3 are unsafe),  K2(∆ − ε3)(1 − ∆ + ε3) > ¯σ (paths in E4 are unsafe),  K2(∆ − ε3) < K1∆ (paths in E4 are suboptimal),  (K1 − 1) · (∆ + ε2) < K1 · ∆ ⇔ (K1 − 1) · ε2 < ∆ ⇔ ε2 <  ∆  K1 − 1 (paths in E3 are suboptimal or unsafe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  The conditions above indicate that  P1 is the unique optimal safe set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  if Pj ⊂ E2, Pj and its subsets are safe but suboptimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  if Pj ⊂ E3, Pj is risky, and its proper subsets are suboptimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  if Pj ⊂ E4, Pj and its subsets are suboptimal, and Pj is unsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Let  p1 := 1 −  �  1 − ¯σ2/K1  2  and p2 := 1 −  �  1 − ¯σ2/K2  2  ,  The relations between µi’s are as in the following figure:  With a similar proof to that of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4, we have, the accumulative regret from the optimal P1 is lower bounded by  Reg[1](T) ≥ Ω( K1∆  (∆v  P1)2 ln 1  δT  );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  the accumulative regret from a safe and suboptimal path in E2 is lower bounded by  Reg[j](T) ≥ Ω  �  K1  ∆Pj  ln T + ∆ · min  �  K1  ∆2  Pj  ln T,  K1  (∆v  Pj)2 ln 1  δT  ��  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  the accumulative regret from a risky path in E3 is lower bounded by  Reg[j](T) ≥ Ω  �  K1∆  (∆v  Pj)2 ln T  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  the accumulative regret from a unsafe and suboptimal path in E4 is lower bounded by  Reg[j](T) ≥ Ω  �  min  �  K2∆  ∆2  Pj  ln T, K2∆  (∆v  Pj)2 ln T  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We let  ϵµ = min  j {∆Pj},  ϵv = min  j {∆v  Pj};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  and set K1, K2, ε1, ε2, ε3 such that  K1 > 3  4 · K2,  K2 = K,  min  j {∆Pj} < 2ϵµ,  min  j {∆v  Pj} < 2ϵv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In conclusion, the regret yielded from these paths is lower bounded by  Reg(T)  1 − δT  ≥ Reg[1](T) +  �  Pj∈E2  Reg[j](T) +  �  Pj∈E3  Reg[j](T) +  �  Pj∈E4  Reg[j](T)  ≥ Ω  � K∆  (ϵv)2 ln 1  δT  �  + |E2| · Ω  �K  ϵµ ln T + ∆ · min  � K  (ϵµ)2 ln T,  K  (ϵv)2 ln 1  δT  ��  + |E3| · Ω  � K∆  (ϵv)2 ln T  �  + |E4| · Ω  �  min  � K∆  (ϵµ)2 ln T, K∆  (ϵv)2 ln T  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Note that  µ⋆ = K1∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  for S⋆, ϵv = ∆v  S⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  for S ∈ S ∩ B and i ∈ S, we check that if S ⊂ Pj, j ∈ E2 , ∆i,S∩B,min = ϵµ and  Ψ′  i,S∩B ≥ min  �ln T  ∆2  S  , 9 ln(1/δT )  (∆v  S)2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  where the equality holds when S = Pj, Pj ∈ E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  for S ∈ R and i ∈ S, ϵv = ∆v  i,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  S ∈ Sc ∩ B and i ∈ S, we can easily check S = Pj for some Pj ∈ E4, thus ∆i,Sc∩B,min = ϵµ and  Φi,Sc∩B = min  �ln T  ∆2  S  , 9 ln T  (∆v  S)2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Therefore,  Reg(T) ≥ Ω  �  L ln T  ∆i,S∩B,min  �  + µ⋆  K · Ω  �  K · ln(1/δT )  (∆S⋆)2  +  �  i∈E  Ψ′  i,S∩B +  �  i∈E  ln T  (∆v  i,R)2 +  �  i∈E  Φi,Sc∩B  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Problem-independent Lower bound  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 (Problem-independent lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let the minimum variance gap be ∆v := minS∈AK ∆v  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When  K3 ≥ L2, we have  Reg(T) = Ω  �√  KLT + min  �  L  (∆v)2 ln  � 1  δT  �  , T  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Since the rewards of items are bounded in [0, 1], the variance of each arm is at most 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Therefore, when  ¯σ2 ∈ [K/4, ∞), any solution in AK is safe and there exists a generic lower bound (Kveton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When  ¯σ2 ∈ (0, K/4), let  ¯a := 1 +  �  1 − ¯σ2/K  2  and a := 1 −  �  1 − ¯σ2/K  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We consider the instances containing items with Bernoulli reward distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' We let Bern(a) denote the Bernoulli  distribution with mean a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' An item i ∈ [L] with reward distribution Bern(µi) is with variance µi(1 − µi), which is smaller  than ¯σ2/K if and only if µi ∈ (0, a) ∪ (¯a, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We will construct 3 instances such that under instance k (0 ≤ k ≤ 2), the stochastic reward of an item in path j (1 ≤ j ≤ L0)  is drawn from distribution ν(k)  j  = Bern(µ(k)  j ), where µ(k)  j  will be specified in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Under instance 0, with µ0 < a, we let µ(0)  j  = µ0 for all j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', the reward distribution of each item is Bern(µ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Since  µPj = Kµ0 and σ2  Pj = Kµ0(1 − µ0) < ¯σ2 for all j, each path is an identical safe and optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Since all paths  are equivalent under instance 0, we have E0[Mj(t)] = T/L0 for all j ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , L0, where E0 denote the expectation under  instance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We next construct instance 1 such that  µ(1)  1  = µ1,  µ(1)  j  = µ0  j ̸= 1,  where µ0 < µ1 < a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='2 Hence, P1 is the unique optimal safe solution under instance 1 while all other solutions are safe but  suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  With an analysis similar to that of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4 in Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' (2021), we can show that  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let the reward distribution of item i be ν(j)  i  under instance j(j = 1, 2), then  KL  �  H(1)  T , H(2)  T  �  =  L  �  i=1  E0[Ni(T)] · KL  �  ν(1)  i  , ν(2)  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Hence, we have  KL  �  H(0)  T , H(1)  T  �  =  �  i∈P1  E0[Ni(T)] ·  �  ν(0)  i  , ν(1)  i  � (a)  ≤ K · E0[M1(t)] · d(µ0, µ1) = K · T  L0  d(µ0, µ1),  where (a) follows from the fact that at most K items are selected at one time step and the definition of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Next, we apply Pinsker’s Inequality to bound E1[M1(T)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 indicates that  |E0[M1(T)] − E1[M1(T)]| ≤  �  1  2KL  �  H(0)  T , H(1)  T  �  ,  ⇒  ����E1[M1(T)] − T  L0  ���� ≤  �  KT  2L0  d(µ0, µ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  2In this proof, µ1 are µ2 are redefined to minimize clutter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' the previous definitions of them not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  Moreover, since the paths Pj for j ̸= 1 are identical under instance 1, we have  E1[Mj(T)] =  1  L0 − 1  �  T − E1[M1(T)]  �  ≥  1  L0 − 1  �  T − T  L0  −  �  KT  2L0  d(µ0, µ1)  �  := M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  To learn the regret incurred by P2 under instance 1, we need to take the effects of the safety constraint into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='3 indicates that  if M < T(ν0), at each of the first M time steps in S2(T), at most K − 1 items are pulled and regret of at least  [K(µ1 − µ2) + µ2] · M is incurred, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=',  Reg[2] ≥ [K(µ1 − µ2) + µ2] · M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  if M ≥ T(ν0), at each of the first T(ν0) time steps in S2(T), at most K − 1 items are pulled and regret of at least  [K(µ1 − µ2) + µ2] · T(ν0) is incurred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' besides, at the subsequent time steps in S2(T), regret of at least K(µ1 − µ2) ·  [M − T(ν0)] is incurred, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=',  Reg[2] ≥ [K(µ1 − µ2) + µ2] · T(ν0) + K(µ1 − µ2) · [M − T(ν0)] = K(µ1 − µ2) · M + µ2 · T(ν0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  In short, we have  Reg[2] ≥ K(µ1 − µ2) · M + µ2 · min{T(ν0), M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We can lower bound Reg(j) for all j = 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' , L0 with the same method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Besides, since  E1[M1(T)] ≥ T −  �  KT  2L0  d(µ0, µ1)  and at most K − 1 items are selected at each of the first T(ν1) time steps in S1(T), we have  Reg[1] ≥ µ1 · min  �  T(ν1), T −  �  KT  2L0  d(µ0, µ1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' under instance 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' we have  Reg(T)  1 − δ  ≥  L0  �  j=1  Reg[j]  ≥ µ1 · min  �  T(ν1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' T −  �  KT  2L0  d(µ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' µ1)  �  + (L0 − 1) · [K(µ1 − µ2) · M + µ2 · min{T(ν0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' M}]  = µ1 · min  �  sup  ν′  1∈E(ν1)  1  K − 1 · d(δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 1 − δ)  KL(ν1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ν′  1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' T −  �  KT  2L0  d(µ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' µ1)  �  + (L0 − 1) ·  �  K(µ1 − µ2) · M + µ2 · min  �  sup  ν′  0∈E(ν0)  1  K − 1 · d(δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' 1 − δ)  KL(ν0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' ν′  0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' M  ��  where  E(ν0) = {ν(0′) : the variance related to ν(0′) is larger than ¯σ2/K},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  E(ν1) = {ν(1′) : the variance related to ν(1′) is larger than ¯σ2/K},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  M =  1  L0 − 1  �  T − T  L0  −  �  KT  2L0  d(µ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' µ1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits  By Pinsker’s inequality (see Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1), for µi ≥ 1/2,  d(µi, ¯a) ≤ (µi − ¯a)2 · (1 − µi − ¯a)2  a(1 − µi − ¯a)2  = [µi(1 − µi) − ¯σ2/K]2  a(1 − µi − ¯a)2  ≤ [µi(1 − µi) − ¯σ2/K]2  a(¯a − 1/2)2  = [µi(1 − µi) − ¯σ2/K]2  a(1/2 − a)2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  for µi < 1/2,  d(µi, a) ≤ (µi − a)2 · (1 − µi − a)2  a(1 − µi − a)2  = [µi(1 − µi) − ¯σ2/K]2  a(1 − µi − a)2  ≤ [µi(1 − µi) − ¯σ2/K]2  a(1/2 − a)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Since ν0 = Bern(µ0), ν1 = Bern(µ1), and 0 < µ0 < µ1 < a < 1/2, we have  sup  ν′  0∈E(ν0)  1  K − 1 · d(δ, 1 − δ)  KL(ν0, ν′  0) = d(δ, 1 − δ)  K − 1 (1/2 − a)2  [µ0(0 − µ0) − ¯σ2/K]2 ,  sup  ν′  1∈E(ν1)  1  K − 1 · d(δ, 1 − δ)  KL(ν1, ν′  1) = d(δ, 1 − δ)  K − 1 (1/2 − a)2  [µ1(1 − µ1) − ¯σ2/K]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  We define the minimum variance gap ∆v := minS∈A ∆v  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' When K3 ≤ L2 and LK/T ≤ a2, we can let µ1 − µ2 =  �  L/KT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Then we have  Reg(T) = Ω  �  min  �  1  K − 1 · d(δ, 1 − δ)  (∆v/K)2 , T  �  +  √  KLT + L0 · min  �  1  K − 1 · d(δ, 1 − δ)  (∆v/K)2 , TK  L  ��  = Ω  �  min  �  K · d(δ, 1 − δ)  (∆v/)2  , T  �  +  √  KLT + L · min  �d(δ, 1 − δ)  (∆v)2  , T  L  ��  = Ω  �√  KLT + min  �  L · d(δ, 1 − δ)  (∆v)2  , T  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  Moreover, we complete the proof with d(δ, 1 − δ) ≥ − ln(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='4δ) and δT = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Tightness of the Upper bound  Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='1 (Tightness of problem-dependent bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=' Let {δT }∞  T =1 ∈ o(1) be a sequence that satisfies ln(1/δT ) =  o(T b) for all b > 0,  if ln(1/δT ) ∈ o(ln T), in particular, if δT = δ0 > 0 for all T, the regret Reg(T) is  Ω  ��  i∈E  ln T  ∆i,S∩B,min  + µ⋆ ln T/K  (∆v  i,R)2  + µ⋆  K Φi,Sc∩B  �  ∩ O  ��  i∈E  K ln T  ∆i,S∩B,min  + µ⋆K ln T  (∆v  i,R)2 + µ⋆KΦi,Sc∩B  �  if ln(1/δT ) = λ ln T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content=', δT = T −λ with a fixed λ > 0, the regret Reg(T) is  Ω  ��  i∈E  ln T  ∆i,S∩B,min  + λµ⋆ ln T  (∆v  S⋆)2 + µ⋆  K  �  i∈E  �  Ψ′  i,S∩B +  ln T  (∆v  i,R)2 + Φi,Sc∩B  ��  ∩ O  ��  i∈E  K ln T  ∆i,S∩B,min  + λµ⋆K2 ln T  (∆v  S⋆)2  + Kµ⋆ �  i∈E  �  Ψi,S∩B +  ln T  (∆v  i,R)2 + Φi,Sc∩B  ��  where ln(1/δT ) in the Ψ and Ψ′ functions should be replaced by λ ln T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
+page_content='  if ln(1/δT ) ∈ ω(ln T), the regret Reg(T) is  Reg(T) ∈ Ω  �µ⋆ ln(1/δT )  (∆v  S⋆)2  �  ∩ O  �µ⋆K2 ln(1/δT )  (∆v  S⋆)2  �  The upper bounds above are achieved by PASCOMBUCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNFQT4oBgHgl3EQftzab/content/2301.13393v1.pdf'}
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