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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
+[email protected]
+2 Microsoft Research, Redmond [email protected]
+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).
+⊓⊔
+

-dE1T4oBgHgl3EQfUgPr/content/tmp_files/2301.03092v1.pdf.txt

@@ -0,0 +1,1059 @@
+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:
+[email protected]).
+Sepehr Eskandari is with Microwave Systems, Sensors, and Imaging Lab
+(MiXIL), University of Southern California, Los Angeles, CA 90089 USA
+(e-mail: [email protected]).
+Vahid Khorashadi-Zadeh is with School of Electric and Computer
+Engineering, University of Tehran, Tehran 14399-57131, Iran (e-mail:
+[email protected]).
+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:
+[email protected]).
+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)
+<latexit sha1_base64="C4QjJmxveAJDFx/9MTdnNEoO
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+v1MWmds9KZHfQH1ucPKAKXSg=</latexit>
+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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+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: [email protected]
+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; �� ≥ 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).
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+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.
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+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.
+

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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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+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:
+[email protected]
+2Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland.
+email: [email protected]
+3Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Poland.
+email: [email protected]
+4Universit´e Cˆote d’Azur, INRIA, Sophia Antipolis, France. email:
+[email protected]
+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 �� > 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.
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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 ([email protected]) 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
+Chinese
+1.0
+1.0
+1.0
+1.0
+1.0
+0.837
+0.783
+0.8 -
+0.727
+0.8
+0.756
+0.751
+0.8
+0.8
+0.8
+0.692
+0.683
+0.689
+0.654
+0.626
+0.662
+0.666
+0.615
+0.594
+0.410
+0.2 
+0.2 
+0.2 
+0.2
+0.2 
+0.0
+0.0 -
+0.0
+0.0
+0.0
+Croatian
+English
+German
+Hindi
+Hungarian
+1.0
+1.0
+1.0
+1.0
+1.0
+0.892
+0.794
+0.788
+0.813
+0.8 
+0.742
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+0.705
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+0.742
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+0.665
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+0.2
+0.2 
+0.2 
+0.2
+0.2 -
+0.000
+0.000
+0.0
+0.0
+0.0
+0.0
+0.0
+Polish
+Portuguese
+Russian
+Slovak
+Slovenian
+1.0 -
+1.0
+1.0
+1.0
+1.0
+0.8
+0.8 -
+0.8
+0.8
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+0.720
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+0.2 
+0.2 -
+0.2 -
+0.2 
+0.2
+0.0
+0.0
+0.0
+0.0
+0.0
+Swedish
+Spanish
+1.0 -
+1.0
+0.8-
+0.761
+English Translated w/o Neutrals
+0.8 -
+0.594
+0.611
+0.548
+0.502
+0.467
+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
+0.817
+0.831
+0.827
+0.786
+0.760
+0.764
+0.748
+0.740
+0.670
+0.636
+0.610
+0.590
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+0.537
+0.514
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+0.514
+0.506
+0.501
+0.502
+0.501
+0.505
+0.512
+0.4980.868
+0.785
+0.794
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+0.684
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+0.596
+0.570
+0.538
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+0.474
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+0.293
+0.229
+0.198
+0.170
+0.118
+0.094
+0.097
+0.093
+0.054
+0.063
+0.056
+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
+0.428
+0.416
+0.408
+0.393
+0.399
+0.380
+0.352
+0.343
+0.317
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+0.094
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+0.099
+0.095
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+0.085
+0.101
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+0.078
+0.059
+0.060
+0.097
+0.091
+0.100
+0.065
+0.085
+0.053
+0.089
+0.076
+0.085
+0.065
+0.067
+0.073
+0.059
+0.028
+0.041
+0.044References
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+

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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: [email protected]).
+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:
+[email protected], [email protected]).
+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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+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)
+��������
+����������������
+�������
+�
+���
+�
+���
+�
+���
+���������
+�������
+�
+���
+�
+���
+�
+���
+��������
+�������
+�
+���
+�
+���
+�
+���
+���������
+�������
+�
+���
+�
+���
+�
+���
+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
+����
+����
+����
+����
+����
+����
+����
+����
+�
+�
+��
+��
+��
+����
+����
+����
+����
+����
+����
+����
+����
+��
+�
+��
+��
+��
+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.
+∗ [email protected]
+† [email protected]
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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: [email protected]
+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: [email protected] ( 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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+21
+

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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 
+ 
+[email protected] 
+[email protected]  
+ 
+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)

AtE2T4oBgHgl3EQfRQeF/content/tmp_files/load_file.txt

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+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  (���𝑚𝑎𝑥  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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