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+arXiv:2301.02138v1  [math.CO]  5 Jan 2023
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS
+VIII. EXCLUDING A FOREST IN (THETA, PRISM)-FREE GRAPHS
+TARA ABRISHAMI∗†, BOGDAN ALECU∗∗¶, MARIA CHUDNOVSKY∗∐, SEPEHR HAJEBI §,
+AND SOPHIE SPIRKL§∥
+Abstract. Given a graph H, we prove that every (theta, prism)-free graph of sufficiently large
+treewidth contains either a large clique or an induced subgraph isomorphic to H, if and only if
+H is a forest.
+1. Introduction
+All graphs in this paper are finite and simple unless specified otherwise. Let G, H be graphs.
+We say that G contains H if G has an induced subgraph isomorphic to H, and we say G is
+H-free if G does not contain H. For a family H of graphs we say G is H-free if G is H-free
+for every H ∈ H. A class of graphs is hereditary if it is closed under isomorphism and taking
+induced subgraphs, or equivalently, if it is the class of all H-free graphs for some other family
+H of graphs.
+For a graph G = (V (G), E(G)), a tree decomposition (T, χ) of G consists of a tree T and a
+map χ : V (T) → 2V (G) with the following properties:
+(i) For every v ∈ V (G), there exists t ∈ V (T) such that v ∈ χ(t).
+(ii) For every v1v2 ∈ E(G), there exists t ∈ V (T) such that v1, v2 ∈ χ(t).
+(iii) For every v ∈ V (G), the subgraph of T induced by {t ∈ V (T) | v ∈ χ(t)} is connected.
+For each t ∈ V (T), we refer to χ(t) as a bag of (T, χ). The width of a tree decomposition
+(T, χ), denoted by width(T, χ), is maxt∈V (T) |χ(t)| − 1. The treewidth of G, denoted by tw(G),
+is the minimum width of a tree decomposition of G.
+Treewidth was first popularized by Robertson and Seymour in their graph minors project,
+and has attracted a great deal of interest over the past three decades. Particularly, graphs of
+bounded treewidth have been shown to be well-behaved from structural [19] and algorithmic [6]
+viewpoints.
+This motivates investigating the structure of graphs with large treewidth, and especially,
+the substructures emerging in them. The canonical result in this realm is the Grid Theorem
+of Robertson and Seymour [19], the following, which describes the unavoidable subgraphs of
+graphs with large treewidth. For a positive integer t, the (t × t)-wall, denoted by Wt×t, is a
+planar graph with maximum degree three and treewidth t (see Figure 1; a formal definition can
+be found in [3]).
+∗Princeton University, Princeton, NJ, USA
+∗∗School of Computing, University of Leeds, Leeds, UK
+§Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario,
+Canada
+† Supported by NSF-EPSRC Grant DMS-2120644.
+∐ Supported by NSF-EPSRC Grant DMS-2120644 and by AFOSR grant FA9550-22-1-0083.
+¶ Supported by DMS-EPSRC Grant EP/V002813/1.
+∥ We acknowledge the support of the Natural Sciences and Engineering Research Council of
+Canada (NSERC), [funding reference number RGPIN-2020-03912]. Cette recherche a été financée
+par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de
+référence RGPIN-2020-03912]. This project was funded in part by the Government of Ontario.
+Date: January 6, 2023.
+1
+
+2
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+Figure 1. W5×5
+Theorem 1.1 (Robertson and Seymour [19]). For every integer t ≥ 1 there exists w = w(t) ≥ 1
+such that every graph of treewidth more than w contains a subdivision of Wt×t as a subgraph.
+Theorem 1.1 can also be reformulated into a full characterization of unavoidable minors in
+graphs of large treewidth, that every graph of sufficiently large treewidth contains any given
+planar graph as a minor (and no non-planar graph has this property). In contrast, unavoidable
+induced subgraphs of graphs with large treewidth are far from completely understood. There
+are some natural candidates though, which we refer to as the “basic obstructions”: complete
+graphs and complete bipartite graphs, subdivided walls mentioned above, and line graphs of
+subdivided walls, where the line graph L(F) of a graph F is the graph with vertex set E(F),
+such that two vertices of L(F) are adjacent if and only if the corresponding edges of F share an
+end. Note that the complete graph Kt+1, the complete bipartite graph Kt,t, and the line graph
+of every subdivision of Wt×t all have treewidth t. For a positive integer t, let us say a graph H
+is a t-basic obstruction if H is one of the following graphs: Kt, Kt,t, a subdivision of Wt×t, or
+the line graph of a subdivision of Wt×t. We say a graph G is t-clean if G does not contain a
+t-basic obstruction.
+The basic obstructions do not form a comprehensive list of induced subgraph obstructions
+for bounded treewidth. Equivalently, there are t-clean graphs of arbitrarily large treewidth for
+small values of t. A well-known hereditary class of graphs evidencing this fact is the class of
+even-hole-free graphs, where a hole is an induced cycle on at least four vertices, the length of
+a hole is its number of edges and an even hole is a hole with even length. In fact, for every
+positive integer t ≥ 1, one may observe that an even-hole-free graph is t-clean if and only if it is
+Kt-free. It is therefore tempting to ask whether even-hole-free graphs excluding a fixed complete
+graph have bounded treewidth. Sintiari and Trotignon [20] answered this with a vehement no,
+providing a construction of (even-hole, K4)-free graphs with arbitrarily large treewidth, hence
+proving that there are t-clean (even-hole-free) graphs of arbitrarily large treewidth for every
+fixed t ≥ 4. In addition, graphs from this construction are rather sparse, in the sense that they
+exclude short holes.
+Theorem 1.2 (Sintiari and Trotignon [20]). For all integers w, l ≥ 1, there exists an (even-hole,
+K4)-free graph Gw,l of treewidth more than w and with no hole of length at most l.
+Note that t-clean graphs for t ≤ 2 have empty vertex set or edge set. But one might still
+hope for 3-clean graphs to have bounded treewidth. This is in fact supported by a result from
+[7] asserting that 3-clean even-hole-free graphs have treewidth at most five. However, another
+construction by Sintiari and Trotignon [20] shows that being 3-clean fails to guarantee bounded
+treewidth in the more general class of theta-free graphs (see the next section for the definition of
+a theta; one may check that the every t-basic obstruction for t ≥ 3 contains either a theta or a
+triangle). Indeed, the treewidth of theta-free graphs remains unbounded even when forbidding
+short cycles.
+Theorem 1.3 (Sintiari and Trotignon [20]). For all integers w, g ≥ 1, there exists a theta-free
+graph Gw,g of treewidth more than w and girth more than g.
+A natural question to ask then is what further conditions must be imposed to force bounded
+treewidth in even-hole-free graphs. For instance, graphs from both Theorems 1.2 and 1.3 have
+
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+3
+vertices of arbitrarily large degree, and so it was conjectured in [1] that (theta, triangle)-
+free graphs of bounded maximum degree have bounded treewidth and even-hole-free graphs
+of bounded maximum degree have bounded treewidth. These were proved in [3] and [4], respec-
+tively. In the same paper [1], a stronger conjecture was made, asserting that basic obstructions
+are in fact the only obstructions to bounded treewidth in graphs of bounded maximum degree.
+This was later proved in [16], which closed the line of inquiry into graph classes of bounded
+maximum degree.
+Theorem 1.4 (Korhonen [16]). For all integers t, δ ≥ 1, there exists w = w(t, δ) such that every
+t-clean graph of maximum degree at most δ has treewidth at most w.
+Despite its generality, the proof of Theorem 1.4 is surprisingly short. However, the case of
+proper hereditary classes containing graphs of unbounded maximum degree seems to be much
+harder. For graph classes G and H, let us say H modulates G if for every positive integer t,
+there exists a positive integer w(t) (depending on G and H) such that every t-clean H-free graph
+in G has treewidth at most w(t). An induced-subgraph analogue to Theorem 1.1 is therefore
+equivalent to a full characterization of graph classes H which modulate the class of all graphs.
+This remains out of reach, but the special case where |H| = 1 turns out to be more approachable.
+For a graph H and a graph class G, let us say H modulates G if {H} modulates G. Building
+on a method from [17], recently we characterized all graphs H which modulate the class of all
+graphs:
+Theorem 1.5 (Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl [2]). Let H be a graph. Then
+H modulates the class of all graphs if and only if H is a subdivided star forest, that is, a forest
+in which every component has at most one vertex of degree more than two.
+In general, for a hereditary class G containing t-clean graphs of arbitrarily large treewidth for
+small t, one may ask for a characterization of graphs H modulating G. Given Theorem 1.2, a
+natural class G to consider is the class of even-hole-free graphs. Note that Theorem 1.2 shows
+that a graph H modulates even-hole-free graphs only if H is a chordal graph (that is, a graph
+with no hole) of clique number at most three. As far as we know, the converse may also be
+true, that every chordal graph of clique number at most three modulates even-hole-free graphs.
+In fact, in this paper we narrow the gap, showing that every chordal graph of clique number at
+most two, that is, every forest, modulates the class of even-hole-free graphs.
+Theorem 1.6. For every forest H and every integer t ≥ 1, every even-hole-free graph of suffi-
+ciently large treewidth contains either H or a clique of cardinality t.
+This aligns with the observation [21] that every forest is contained in some graph Gw,l from
+Theorem 1.2. As mentioned above, one way to improve on Theorem 1.6 is to push H towards
+being an arbitrary chordal graph of clique number three. Another way to strengthen Theorem 1.6
+is to find a superclass G of even-hole-free graphs for which forests are the only graphs modulating
+G. While the former remains open, we provide an appealing answer to the latter: our main result
+shows that forests are exactly the graphs which modulate the class of (theta, prism)-free graphs
+(see the next section for the definition of a prism; again one may check that in (theta, prism)-free
+graphs, being t-clean is equivalent to being Kt-free for every positive integer t).
+Theorem 1.7. Let H be a graph. Then H modulates (theta, prism)-free graphs if and only if
+H is a forest. In other words, given a graph H, for every integer t ≥ 1, every (theta, prism)-free
+graph of sufficiently large treewidth contains either H or a clique of cardinality t, if and only if
+H is a forest.
+Let C be the class of all (theta, prism)-free graphs. It is easily seen that C contains all even-
+hole-free graphs, and so Theorem 1.7 implies Theorem 1.6. Note that the “only if” direction
+of Theorem 1.7 follows immediately from Theorem 1.3 as prisms contain triangles. Since every
+forest is an induced subgraph of a tree, in order to prove Theorem 1.7, it suffices to prove
+
+4
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+Theorem 1.8 below, which we do in Section 7. For a positive integer t and a tree F, we denote
+by Ct the class of all graphs in C with no clique of cardinality t (that is, t-clean graph in C), and
+by Ct(F) the class of all F-free graphs in Ct.
+Theorem 1.8. For every tree F and every integer t ≥ 1, there exists an integer τ(F, t) ≥ 1
+such that every graph in Ct(F) has treewidth at most τ(F, t).
+We conclude this introduction by sketching our proofs (the terms we use here are defined in
+later sections). The proof of Theorem 1.8 begins with a two-step preparation which culminates
+in the proof of Theorem 6.2, a result we will also use in subsequent papers in this series. As the
+first step, inspired by a result from [9], we show that for every graph G ∈ C which contains a
+pyramid with certain conditions on the apex and its neighbors, G admits a construction which we
+call a “(T, a)-strip-structure,” where a is the apex of the pyramid and T is an optimally chosen
+tree. Roughly speaking, we show that G\{a} can be partitioned into two induced subgraphs H
+and J where H is more or less similar to the line graph of the tree T and every vertex in J with a
+neighbor in H attaches at a pyramid lurking in H in a restricted way; we call the latter vertices
+“jewels”. The proof of this theorem occupies Sections 3 and 4. The second step is to employ the
+previous result to show that if G ∈ Ct admits a (C, a)-strip-structure where C is a caterpillar,
+then every vertex in G \ NG[a] can be separated from a by removing a few vertices (our proof
+works more generally when C is any tree of bounded maximum degree, but the caterpillar case
+suffices for our application). We prove this in Section 6. The central difficulty in the proof is to
+deal with the jewels separately. This is surmounted in Section 5 where we prove several results
+concerning the properties of jewels. Most notably, we show that jewels only attach at “local
+areas of the line-graph-like part” of G, and that only a few jewels attach at each local area. This
+concludes the preparation for proving Theorem 1.8.
+Next, we embark on the proof of Theorem 1.8. We assume that G ∈ Ct has large treewidth,
+which together with results from Section 2 implies that G contains two vertices x, y joined by
+many pairwise internally disjoint induced paths P1, . . . , Pm. Now we analyze the structure of
+the graph G[P1 ∪ · · · ∪ Pm]. It turns out that, if m is large enough, then either
+• there are many paths among Pi’s whose union H admits a (C, x)-strip-structure for some
+caterpillar C, or
+• for some large value of d, G[P1 ∪ · · · ∪ Pm] contains a tree S isomorphic to the complete
+bipartite graph K1,d, such that x is the vertex of degree d in S, and for every leaf l of S,
+there are many pairwise internally disjoint induced paths between l and y, such that in
+addition, paths corresponding to distinct leaves of S are also pairwise internally disjoint.
+The former case implies that y can be separated from x by removing few vertices, which using
+a result from Section 6, yields a contradiction with Menger’s theorem. The latter case is the first
+step towards building the large tree in G as a subgraph. We now iterate the argument we just
+described, applying it to each leaf l of S and y, obtaining larger and larger trees. The process is
+stopped once we reach a sufficiently large tree as a subgraph of G. This, combined with the fact
+that G ∈ Ct and a result of Kierstead and Penrice [15], yields the desired tree F as an induced
+subgraph of G.
+This paper is organized as follows. Section 2 covers preliminary definitions as well as some
+results from the literature used in our proofs. Section 3 investigates the behavior of pyramids in
+graphs from C. Section 4 is devoted to defining strip-structures and jewels, and showing how they
+arise from pyramids in graphs in C. Section 5 takes a closer look at jewels for the strip-structures
+obtained in Section 4. In Section 6 we show that admitting certain strip-structures weakens the
+connectivity of most vertices to the apex. Finally, in Section 7, we prove Theorem 1.8.
+2. Preliminaries and results from the literature
+Let G = (V (G), E(G)) be a graph. For a set X ⊆ V (G) we denote by G[X] the subgraph of
+G induced by X. For X ⊆ V (G)∪E(G), G\X denotes the subgraph of G obtained by removing
+
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+5
+X. Note that if X ⊆ V (G), then G \ X denotes the subgraph of G induced by V (G) \ X. In
+this paper, we use induced subgraphs and their vertex sets interchangeably.
+Let x ∈ G and d be a positive integer. We denote by N d
+G(x) the set of all vertices in G at
+distance d from some x, and by N d
+G[x] the set of all vertices in G at distance at most d from x.
+We write NG(x) for N 1
+G(x) and NG[x] for N 1
+G[x]. For an induced subgraph H of G, we define
+NH(x) = NG(x) ∩ H, NH[x] = NG[x] ∩ H. Also, for X ⊆ G, we denote by NG(X) the set of all
+vertices in G \ X with at least one neighbor in X, and define NG[X] = NG(X) ∪ X.
+Let X, Y ⊆ G be disjoint. We say X is complete to Y if all edges with an end in X and an
+end in Y are present in G, and X is anticomplete to Y if there are no edges between X and Y .
+A path in G is an induced subgraph of G that is a path.
+If P is a path in G, we write
+P = p1- · · · -pk to mean that V (P) = {p1, . . . , pk} and pi is adjacent to pj if and only if |i−j| = 1.
+We call the vertices p1 and pk the ends of P, and say that P is from p1 to pk. The interior of
+P, denoted by P ∗, is the set P \ {p1, pk}. The length of a path is its number of edges (so a path
+of length at most one has empty interior). Similarly, if C is a cycle, we write C = c1- · · · -ck-c1
+to mean that V (C) = {c1, . . . , ck} and ci is adjacent to cj if and only if |i − j| ∈ {1, k − 1}. The
+length of a cycle is its number edges (or equivalently, vertices.)
+A theta is a graph Θ consisting of two non-adjacent vertices a, b, called the ends of Θ, and
+three pairwise internally disjoint paths P1, P2, P3 from a to b of length at least two, called the
+paths of Θ, such that P ∗
+1 , P ∗
+2 , P ∗
+3 are pairwise anticomplete to each other. For a graph G, by a
+theta in G we mean an induced subgraph of G which is a theta.
+A prism is a graph Π consisting of two disjoint triangles {a1, a2, a3}, {b1, b2, b3} called the
+triangles of Π, and three pairwise disjoint paths P1, P2, P3 called the paths of Π, where Pi has
+ends ai, bi for each i ∈ {1, 2, 3}, and for distinct i, j ∈ {1, 2, 3}, aiaj and bibj are the only edges
+between Pi and Pj. For a graph G, by a prism in G we mean an induced subgraph of G which
+is a prism.
+A pyramid is a graph Σ consisting of a vertex a, a triangle {b1, b2, b3} and three paths P1, P2, P3
+of length at least one with Pi from a to bi for each i ∈ {1, 2, 3} and otherwise pairwise disjoint,
+such that for distinct i, j ∈ {1, 2, 3}, bibj is the only edge between Pi \ {a} and Pj \ {a}, and
+at most one of P1, P2, P3 has length exactly one. We say that a is the apex of the pyramid and
+b1b2b3 is the base of the pyramid. The pyramid Σ is said to be long if Pi has length more than
+one for every i ∈ {1, 2, 3}. For a graph G, by a pyramid in G we mean an induced subgraph of
+G which is a pyramid.
+Figure 2. Theta, pyramid and prism. The dotted lines represent paths of
+length at least one.
+Let us now mention a few results from the literature which we will use in this paper. Let
+G be a graph. By a separation in G we mean a triple (L, M, R) of pairwise disjoint subsets of
+vertices in G with L ∪ M ∪ R = G, such that neither L nor R is empty and L is anticomplete
+to R in G. Let x, y ∈ G be distinct. We say a set M ⊆ G \ {x, y} separates x and y if there
+exists a separation (L, M, R) in G with x ∈ L and y ∈ R. Also, for disjoint sets X, Y ⊆ G, we
+say a set M ⊆ G \ (X ∪ Y ) separates X and Y if there exists a separation (L, M, R) in G with
+X ⊆ L and Y ⊆ R. If X = {x}, we say that M separates x and Y to mean M separates X and
+Y . Recall the following well-known theorem of Menger [18]:
+
+6
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+Theorem 2.1 (Menger [18]). Let k ≥ 1 be an integer, let G be a graph and let x, y ∈ G be
+distinct and non-adjacent. Then either there exists a set M ⊆ G \ {x, y} with |M| < k such that
+M separates x and y, or there are k pairwise internally disjoint paths in G from x to y.
+Let k be a positive integer and let G be a graph. A strong k-block in G is a set B of at least k
+vertices in G such that for every 2-subset {x, y} of B, there exists a collection P{x,y} of at least
+k distinct and pairwise internally disjoint paths in G from x to y, where for every two distinct
+2-subsets {x, y}, {x′, y′} ⊆ B of G, and every choice of paths P ∈ P{x,y} and P ′ ∈ P{x′,y′}, we
+have P ∩ P ′ = {x, y} ∩ {x′, y′}.
+For a tree T and xy ∈ E(T), we denote by Tx,y the component of T − xy containing x. Let G
+be a graph and (T, χ) be a tree decomposition for G. For every S ⊆ T, let χ(S) = �
+x∈S χ(x).
+By an adhesion of (T, χ) we mean the set χ(x) ∩ χ(y) = χ(Tx,y) ∩ χ(Ty,x) for some xy ∈ E(T).
+For every x ∈ V (T), by the torso at x, denoted by ˆχ(x), we mean the graph obtained from
+the bag χ(x) by, for each y ∈ NT (x), adding an edge between every two non-adjacent vertices
+u, v ∈ χ(x, y). In [2], we used Theorem 1.4 and the following result from [13]:
+Theorem 2.2 (Erde and Weißauer [13], see also [14]). Let r be a positive integer, and let G
+be a graph containing no subdivision of Kr as a subgraph. Then G admits a tree decomposition
+(T, χ) for which the following hold.
+• Every adhesion of (T, χ) has cardinality less than r2.
+• For every x ∈ V (T), either ˆχ(x) has fewer than r2 vertices of degree at least 2r4, or ˆχ(x)
+has no minor isomorphic to K2r2.
+to prove the following.
+Theorem 2.3 (Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl [2]). Let k, t ≥ 1 be integers.
+Then there exists an integer w = w(k, t) ≥ 1 such that every t-clean graph with no strong k-block
+has treewidth at most w.
+Note that for every t ≥ 3, every subdivision of Wt×t contains a theta and the line graph of
+every subdivision of Wt×t contains a prism. It follows that for every t ≥ 1, every graph in Ct is
+t-clean, and so the following is immediate from Theorem 2.3:
+Corollary 2.4. For all integers k, t ≥ 1, there exists an integer β = β(k, t) such that every
+graph in Ct with no strong k-block has treewidth at most β(k, t).
+A vertex v in a graph G is said to be a branch vertex if v has degree more than two. By
+a caterpillar we mean a tree C with maximum degree three such that there is a path P in
+C containing all branch vertices of C (our definition of a caterpillar is non-standard for two
+reasons: a caterpillar is often allowed to be of arbitrary maximum degree, and the path P from
+the definition often contains all vertices of degree more than one). By a subdivided star we mean
+a graph isomorphic to a subdivision of the complete bipartite graph K1,δ for some δ ≥ 3. In
+other words, a subdivided star is a tree with exactly one branch vertex, which we call its root.
+For every graph H, a vertex v of H is said to be simplicial if NH(v) is a clique. We denote by
+Z(H) the set of all simplicial vertices of H. Note that for every tree T, Z(T) is the set of all
+leaves of T. An edge e of a tree T is said to be a leaf-edge of T if e is incident with a leaf of
+T. It follows that if H is the line graph of a tree T, then Z(H) is the set of all vertices in H
+corresponding to the leaf-edges of T. The following is proved in [2] based on (and refining) a
+result from [11].
+Theorem 2.5 (Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl [2]). For every integer h ≥ 1,
+there exists an integer µ = µ(h) ≥ 1 with the following property. Let G be a connected graph
+with no clique of cardinality h and let S ⊆ G such that |S| ≥ µ. Then either some path in G
+contains h vertices from S, or there is an induced subgraph H of G with |H ∩ S| = h for which
+one of the following holds.
+• H is either a caterpillar or the line graph of a caterpillar with H ∩ S = Z(H).
+• H is a subdivided star with root r such that Z(H) ⊆ H ∩ S ⊆ Z(H) ∪ {r}.
+
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+7
+3. Jumps and jewels on pyramids with trapped apices
+For a graph G, an induced subgraph H of G and a vertex a ∈ H, we say a is trapped in H if
+• we have N 2
+G[a] ⊆ H, and;
+• every vertex in NH(a) = NG(a) has degree two in H (and so in G).
+The goal of this section is, for a graph G ∈ C, H ⊆ G and a pyramid Σ in H, to investigate the
+adjacency between Σ and a path in G \ H, assuming that the apex of Σ is trapped in H. This
+will be of essential use in the next section.
+We begin with a few definitions. Let G be a graph and let Σ be a pyramid in G with apex
+a, base b1b2b3 and paths P1, P2, P3. A set X ⊆ Σ is said to be local (in Σ) if either X ⊆ Pi for
+some i ∈ {1, 2, 3} or X ⊆ {b1, b2, b3}. Let P be a path in G \ Σ with (not necessarily distinct)
+ends p1, p2. For i ∈ {1, 2, 3}, we say P is a corner path for Σ at bi if
+• p1 has at least one neighbor in Pi \ {bi};
+• p2 is complete to {b1, b2, b3} \ {bi}, and;
+• except for the edges between {p1, p2} and Σ described in the above two bullets, there is
+no edge with an end in P and an end in Σ \ {bi}.
+By a corner path for Σ we mean a corner path for Σ at one of b1, b2 or b3.
+Let p ∈ G \ Σ. Then p is said to be narrow for Σ if NΣ(p) is local in Σ. Otherwise, we say
+p is wide for Σ. For i ∈ {1, 2, 3}, we say p is a jewel for Σ at bi if p is anticomplete to Pi (in
+particular, p is anticomplete to a), and for every j ∈ {1, 2, 3} \ {i}, we have NPj(p) = NPj[bj].
+By a jewel for Σ we mean a jewel for Σ at one of b1, b2 or b3. Note that if p is either a corner
+path or a jewel for Σ, then p is wide for Σ. The following lemma establishes a converse to this
+fact for graphs in C and pyramids with a trapped apex.
+Lemma 3.1. Let G ∈ C be graph, let H ⊆ G and let a ∈ H be trapped in H. Let Σ be a pyramid
+in H with apex a, base b1b2b3 and paths P1, P2, P3. Let p ∈ G \ H. Then p is wide for Σ if and
+only if p is either a corner path for Σ or a jewel for Σ.
+Proof. We only need to prove the “only if” direction. Assume that p ∈ G \ H is wide for Σ and
+p is not a corner path for Σ. Since a is trapped in H and p ∈ G \ H, it follows that Σ is long
+and p is anticomplete to NΣ[a]. First, we show that:
+(1) There exists i ∈ {1, 2, 3} for which p is anticomplete to Pi.
+Suppose for a contradiction that p has a neighbor in each of P1, P2, P3. Since p is wide for Σ
+and p is not a corner path for Σ, we may assume without loss of generality that p has a neighbor
+in P ∗
+1 and a neighbor in P ∗
+2 . For each i ∈ {1, 2, 3}, traversing Pi from a to bi, let xi be the first
+neighbor of p in Pi. Since a is trapped, it follows that x1 ∈ P ∗
+1 , x2 ∈ P ∗
+2 and x3 ∈ P3\NΣ[a]. But
+then there is a theta in G with ends a, p and paths a-Pi-xi-p for i ∈ {1, 2, 3}, a contradiction.
+This proves (1).
+By (1) and without loss of generality, we may assume that p is anticomplete to P3. Note that
+since p is wide for Σ, it follows that for every j ∈ {1, 2}, p has a neighbor in Pj, and there exists
+j ∈ {1, 2} for which p has a neighbor in P ∗
+j . For each j ∈ {1, 2}, traversing Pj from a to bj, let xj
+and yj be the first and the last neighbor of p in Pj, respectively. Then we have xj ∈ P ∗
+j \NPj(a)
+for some j ∈ {1, 2}. In fact, the following holds.
+(2) For every j ∈ {1, 2}, we have xj ∈ P ∗
+j \ NPj(a).
+Suppose not. Then since p is wide for Σ, we may assume without loss of generality that p has
+a neighbor in P ∗
+1 and we have x2 = y2 = b2. But now there is a theta in G with ends a, b2 and
+paths a-P1-x1-p-b2, a-P2-b2 and a-P3-b3-b2, a contradiction. This proves (2).
+(3) For every j ∈ {1, 2}, NPj(p) is a clique of cardinal ity two.
+Suppose not. Then we may assume without loss of generality that either x1 = y1 or x1 and y1
+are distinct and non-adjacent. By (2), for every j ∈ {1, 2}, we have xj ∈ P ∗
+j \NPj(a). Therefore,
+
+8
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+if x1 = y1, then there is a theta in G with ends a, x1 and paths a-P1-x1, a-P2-x2-p-x1 and
+a-P3-b3-b1-P1-x1, which is impossible. Thus, x1 and y1 are distinct and non-adjacent. But now
+there is a theta in G with ends a, p and paths a-P1-x1-p, a-P2-x2-p and a-P3-b3-b1-P1-y1-p, a
+contradiction. This proves (3).
+The proof is almost concluded. By (3), for every j ∈ {1, 2}, we have NPj(p) = {xj, yj} and xj
+is adjacent to yj. If yj ∈ P ∗
+j for some j ∈ {1, 2}, then there is a prism in G with triangles xjyjp
+and b1b2b3 and paths xj-Pj-a-P3-b3, yj-Pj-bj and p-y3−j-P3−j-b3−j, a contradiction. Hence, we
+have yj = bj for every j ∈ {1, 2}, and so p is a jewel corner for Σ at bi. This completes the proof
+of Lemma 3.1.
+■
+We can now prove the main result of this section.
+Theorem 3.2. Let G ∈ C be a graph, let H ⊆ G and let a ∈ H be trapped in H. Let Σ be a
+pyramid in H with apex a, base b1b2b3 and paths P1, P2, P3. Let P be a path in G \ H. Then
+one of the following holds.
+• NΣ(P) is local in Σ.
+• P contains a corner path for Σ.
+• P contains a jewel for Σ.
+Proof. Suppose for a contradiction that there exists a path P in G \ H for which none of the
+outcomes of Theorem 3.2 hold. We choose such a path P with |P| as small as possible. It follows
+that NΣ(P) is not local in Σ, NΣ(X) is local in Σ for every connected set X ⊊ P, P contains
+no corner path for Σ and P contains no jewel for Σ. Therefore, by Lemma 3.1, we have |P| > 1.
+Since a is trapped in H and P ⊆ G\H, it follows that Σ is long and P is anticomplete to NΣ[a].
+For every i ∈ {1, 2, 3}, let P ′
+i = Pi \ NPi[a]. Since NΣ(P) is not local and P is minimal subject
+to this property, we may assume without loss of generality that
+• NΣ(p1) ⊆ P ′
+1 and p1 has a neighbor in P ′
+1 \ {b1}, and;
+• p2 has a neighbor in P ′
+2, and either NΣ(p2) ⊆ P ′
+2, or NΣ(p2) ⊆ {b1, b2, b3}.
+It follows from the choice of P that P ∗ is anticomplete to Σ\{b1}. For each i ∈ {1, 2}, traversing
+Pi from a to bi, let xi and yi be the first and the last neighbor of pi in Pi, respectively. So we
+have x1 ∈ P ′
+1 \ {b1}, y1 ∈ P ′
+1 and x2, y2 ∈ P ′
+2. In fact, the following holds.
+(4) We have x2 ∈ P ′
+2 \ {b2}.
+Suppose not. Then we have x2 = y2 = b2, and so b2 ∈ NΣ(p2) ⊆ {b1, b2, b3}. Note that if p2
+is adjacent to b3, then P is a corner path for Σ at b1, which is impossible. So p2 is not adjacent
+to b3. But now there is a theta in G with ends a, b2 and paths a-P1-x1-p1-P-p2-b2, a-P2-b2 and
+a-P3-b3-b2, a contradiction. This proves (4).
+In view of (4) and the choice of P, we conclude that P ∗ is anticomplete to Σ, and for every
+i ∈ {1, 2}, we have NΣ(pi) = NP ′
+i (pi), xi ∈ P ′
+i \ {bi} and yi ∈ P ′
+i.
+(5) For every i ∈ {1, 2}, xi and yi are distinct and adjacent.
+Suppose not. Then we may assume without loss of generality that either x1 = y1 or x1 and
+y1 are distinct and non-adjacent. In the former case, there is a theta in G with ends a, x1 and
+paths a-P1-x1, a-P2-x2-p2-P-p1-x1 and a-P3-b3-b1-P1-x1, which is impossible. It follows that x1
+and y1 are distinct and non-adjacent. But then there is a theta in G with ends a, p1 and paths
+a-P1-x1-p1, a-P2-x2-p2-P-p1 and a-P3-b3-b1-P1-y1-p1, a contradiction. This proves (5).
+By (5), for every i ∈ {1, 2}, we have NPi(p) = {xi, yi} and xi is adjacent to yi. But now there is
+a prism in G with triangles p1x1y1 and p2x2y2 and paths P, x1-P1-a-P2-x2 and y1-P1-b1-b2-P2-y2,
+a contradiction. This completes the proof of Theorem 3.2.
+■
+
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+9
+4. Strip structures with an ornament of jewels
+The main result of this section, Theorem 4.2, provides a description of the structure of graphs
+in C which have an induced subgraph containing a pyramid with a trapped apex.
+We first set up a framework that allows us to think of a pyramid with apex a as a special case
+of a construction similar to the line graph of a tree T, which we call a “(T, a)-strip-structure.”
+We start with an induced subgraph W of G that admits an “optimal” (T, a)-strip-structure in
+G in a certain sense, and show that the rest of the graph fits into the same construction, except
+for vertices which are jewels for certain canonically positioned pyramids in W.
+First, we need to properly define a strip-structure (this is similar to [8], [9] and [10]). A
+tree T is said to be smooth if T has at least three vertices and every vertex of T is either
+a branch vertex or a leaf.
+Let G be a graph, let a ∈ G, let T be a smooth tree, and let
+η : V (T) ∪ E(T) ∪ (E(T) × V (T)) → 2G\{a} be a function. For every S ⊆ V (T), we define
+η(S) = �
+v∈S,e∈E(T[S])(η(v) ∪ η(e)) and η+(S) = η(S) ∪ {a}. For every vertex v ∈ V (T), we
+define Bη(v) to be the union of all sets η(e, v) taken over all edges e ∈ E(T) incident with v (we
+often omit the subscript η unless there is ambiguity).
+The function η is said to be a (T, a)-strip-structure in G if the following conditions are satisfied.
+(S1) For all distinct o, o′ ∈ V (T) ∪ E(T), we have η(o) ∩ η(o′) = ∅.
+(S2) If l ∈ V (T) is a leaf of T, then η(l) is empty.
+(S3) For all e ∈ E(T) and v ∈ V (T), we have η(e, v) ⊆ η(e) and η(e, v) ̸= ∅ if and only if e is
+incident with v.
+(S4) For all distinct edges e, f ∈ E(T) and every vertex v ∈ V (T), η(e, v) is complete to η(f, v),
+and there are no other edges between η(e) and η(f). In particular, if e and f share no end,
+the η(e) is anticomplete to η(f).
+(S5) For every e ∈ E(T) with ends u, v, define η◦(e) = η(e) \ (η(e, u) ∪ η(e, v)). Then for every
+vertex x ∈ η(e), there is a path in η(e) from x to a vertex in η(e, u) with interior contained
+in η◦(e), and there is a path in η(e) from x to a vertex in η(e, v) with interior contained in
+η◦(e).
+(S6) For all v ∈ V (T) and e ∈ E(T), η(v) is anticomplete to η(e) \ η(e, v). In other words, we
+have Nη(T)(η(v)) ⊆ Bη(v).
+(S7) For every v ∈ V (T) and every connected component D of η(v), we have NBη(v)(D) ̸= ∅.
+(S8) For every leaf l ∈ V (T) of T, assuming e ∈ E(T) to be the leaf-edge of T incident with l,
+a is complete to η(e, l). Also, a has no other neighbors in η(T).
+Let S ⊆ η(T). We say that S is local in η if S ⊆ η(e) for some e ∈ E(T) or S ⊆ Bη(v) ∪ η(v)
+for some v ∈ V (T). The following lemma shows that every non-local subset contains a 2-subset
+(that is, a subset of cardinality two) which is non-local.
+Lemma 4.1. Let G be a graph and a ∈ G. Let T be a smooth tree and η be a (T, a)-strip-
+structure in G. Assume also that C ⊆ η(T) is not local in η. Then there is a 2-subset of C
+which is not local in η.
+Proof. First, suppose there exists a vertex x ∈ C ∩ η◦(e) for some e ∈ E(T). Since C is not
+local, there exists y ∈ C \ η(e). Now {x, y} is a 2-subset of C which is not local in η, as desired.
+Therefore, we may assume that C ⊆ �
+v∈V (T)(B(v) ∪ η(v)). Since the empty set is local in η,
+we have C ̸= ∅; thus, we may pick x ∈ C, v ∈ V (T) and e ∈ E(T) such that x ∈ η(e, v) ∪ η(v).
+If there exists a vertex y ∈ C \ (η(e) ∪ B(v) ∪ η(v)), then {x, y} is a 2-subset of C which is not
+local in η, and so we are done. Consequently, we may assume that C ⊆ η(e) ∪ B(v) ∪ η(v).
+Since C is not local, there exist x′ ∈ η(e) \ (B(v) ∪ η(v))) and y′ ∈ (B(v) ∪ η(v)) \ η(e) such that
+{x′, y′} ⊆ C. Now {x′, y′} is a 2-subset of C which is not local in η, as required. This completes
+the proof of Lemma 4.1.
+■
+In order to state and prove the main result of this section, we need to define several notions
+related to strip-structures. From here until the statement of Theorem 4.2, let us fix a graph G,
+a vertex a ∈ G, a smooth tree T and a (T, a)-strip-structure η in G.
+
+10
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+For every edge e ∈ E(T) with ends u, v, by an η(e)-rung, we mean a path P in η(e) ⊆ η(T)
+for which either |P| = 1 and P ⊆ η(e, u) ∩ η(e, v), or P has an end in η(e, u) \ η(e, v) and an
+end in η(e, v) \ η(e, u) and we have P ∗ ⊆ η◦(e). Equivalently, a path P in η(e) is an η(e)-rung
+if P has an end in η(e, u) and an end in η(e, v) and we have |P ∩ η(e, u)| = |P ∩ η(e, v)| = 1. It
+follows from (S5) that every vertex in η(e) \ η◦(e) is contained in an η(e)-rung. In particular,
+if either η(e, u) ⊆ η(e, v) or η(e, v) ⊆ η(e, u), then η(e, u) = η(e, v). An η(e)-rung is said to be
+long if it is of non-zero length.
+For every edge e ∈ E(T), let ˜η(e) be the set of vertices in η(e) that are not in any η(e)-rung
+(so ˜η(e) ⊆ η◦(e).) We say that η is tame if
+• η(v) = ∅ for every v ∈ V (T), and;
+• ˜η(e) = ∅ for every e ∈ E(T).
+In other words, η is tame if and only if every vertex in η(T) is in an η(e)-rung for some e ∈ E(T).
+For a (T, a)-strip-structure η′ in G, we write η ≤ η′ to mean that for every o ∈ V (T)∪E(T)∪
+(E(T)×V (T)), we have η(o) ⊆ η′(o). We say a (T, a)-strip-structure η is substantial if for every
+e ∈ E(T), there exists a long η(e)-rung in G. Equivalently, η is substantial if for every edge
+e ∈ E(T) with ends u, v, we have η(e, u) ̸= η(e, v), and so η(e, u) \ η(e, v), η(e, v) \ η(e, u) ̸= ∅.
+One may observe that since T has at least three vertices, if η is substantial and η ≤ η′, then η′
+is substantial too.
+We say η is rich if
+• a is trapped in η+(T), and;
+• for every leaf l ∈ V (T) of T, assuming e ∈ E(T) to be the leaf-edge of T incident with
+l, we have |η(e, l)| = 1.
+It follows that if there exists a rich (T, a)-strip-structure η in G, then T has exactly |NG(a)|
+leaves, and for every leaf l ∈ V (T) of T, assuming e ∈ E(T) to be the leaf-edge of T incident
+with l and v ∈ V (T) to be the unique neighbor of l in T, we have η(e, v) ∩ η(e, l) = ∅.
+By a seagull in T we mean a triple (v, e1, e2) where v ∈ V (T) and e1, e2 are two distinct
+edges of T incident with v. By a claw in T we mean a 4-tuple (v, e1, e2, e3) where v ∈ V (T) and
+e1, e2, e3 are three distinct edges of T incident with v.
+Let (v, e1, e2, e3) be a claw in T. By an η-pyramid at (v, e1, e2, e3), we mean a pyramid Σ with
+apex a, base b1b2b3 and paths P1, P2, P3, satisfying the following for each i ∈ {1, 2, 3}.
+• bi ∈ η(ei, v).
+• There exists a leaf li of T with the following properties:
+(1) li belongs to the component of T \ {ei} not containing v.
+(2) Let Λi be the unique path in T from v to li (so ei ∈ E(Λi)). Then Pi = Γi ∪ {a},
+where Γi is a path in �
+e∈E(Λi) η(e) such that Ri = Γi ∩ η(ei) is a long η(ei)-rung
+and Γi ∩ η(e) is a η(e)-rung for each e ∈ E(Λi) \ {ei}.
+In particular, assuming ui to be the ends of ei distinct from v and ci to be the unique vertex in
+NRi(bi) = NPi(bi) for each i ∈ {1, 2, 3}, we have bi ∈ η(ei, v) \ η(ei, ui) and ci ∈ η(ei) \ η(ei, v).
+For a branch vertex v ∈ V (T), by an η-pyramid at v we mean an η-pyramid at (v, e1, e2, e3) for
+some claw (v, e1, e2, e3) in T. Also, by an η-pyramid we mean an η-pyramid at v for some branch
+vertex v ∈ V (T). It follows that every η-pyramid is a long pyramid. Also, if η is substantial,
+then for every claw (v, e1, e2, e3) in T there is a η-pyramid at (v, e1, e2, e3).
+Let (v, e1, e2) be a seagull in T. A vertex p ∈ G\η+(T) is said to be a jewel for η at (v, e1, e2)
+if for some edge e3 ∈ E(T)\{e1, e2} incident with v, there exists an η-pyramid Σ at (v, e1, e2, e3)
+with base b1b2b3 where bi ∈ η(ei, v) for each i ∈ {1, 2, 3}, such that p is a jewel for Σ at b3. In
+particular, for each i ∈ {1, 2}, p is adjacent to bi and the unique vertex ci in NPi(bi). Therefore,
+since Σ is an η-pyramid at (v, e1, e2, e3), assuming ui to be the end of ei distinct from v, it
+follows that p has a neighbor bi ∈ η(ei, v) \ η(ei, ui) and a neighbor ci ∈ η(ei) \ η(ei, v).
+For a vertex v ∈ V (T), by a jewel for η at v we mean a jewel for η at (v, e1, e2) for some
+seagull (v, e1, e2) in T. Also, by a jewel for η we mean a jewel for η at v for some branch vertex
+v ∈ V (T). We denote by Jη the set of all jewels for η. It follows that Jη ⊆ G \ η+(T).
+
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+11
+We are now in a position to prove the main result of this section:
+Theorem 4.2. Let G ∈ C, let a ∈ G and let T be a smooth tree. Suppose that there exists
+a tame, substantial and rich (T, a)-strip-structure in G. Then there is a substantial and rich
+(T, a)-strip-structure ζ in G for which G \ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T).
+Proof. Let η be a tame, substantial and rich (T, a)-strip-structure in G such that η(T) is maximal
+with respect to inclusion. Let M = G \ (η+(T) ∪ Jη).
+(6)
+Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and
+x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible
+subject to this property. Then there exists {j1, j2} = {1, 2} and f = v1v2 ∈ E(T) such that
+xj1 ∈ B(vj1) \ η(f) and xj2 ∈ (B(vj2) ∪ η(f)) \ B(vj1).
+Suppose not. For each i ∈ {1, 2}, let ei ∈ E(T) such that xi ∈ η(ei) (hence e1 ̸= e2) and
+si be an end of ei such that there exists a path Λ0 (possibly of length zero) from s1 to s2 in
+T \ {e1, e2}. We claim that there is a vertex v ∈ Λ0 such that B(v) ∩ {x1, x2} = ∅. Suppose first
+that s1 ̸= s2; let v1 be unique neighbor of s1 in Λ0. Then we have x1 /∈ B(v1) and x2 /∈ B(s1).
+Also, since f = s1v1 does not satisfy (6), we have either x1 /∈ B(s1) or x2 /∈ B(v1). But then
+either v = s1 or v = v1 satisfies the claim. Thus, we may assume that v = s1 = s2. Note
+that since neither e1 nor e2 satisfies (6), we have x1 /∈ B(s1) and x2 /∈ B(s2). In other words,
+we have B(v) ∩ {x1, x2} = ∅, and the claim follows. Henceforth, let v be as promised by the
+above claim. For each i ∈ {1, 2}, let ui be the end of ei distinct from si (hence u1 ̸= u2). Let
+Λ = u1-s1-Λ0-s2-u2 and let u′
+1, u′
+2 be the neighbors of v in Λ such that Λ traverses u1, u′
+1, v, u′
+2, u2
+in this order (so either of u1 = u′
+1 and u2 = u′
+2 is possible). Let e′
+i = u′
+iv for each i ∈ {1, 2}. Since
+T is smooth, there exists a vertex u′
+3 ∈ NT (v) \ Λ; let e′
+3 = u′
+3v. For each i ∈ {1, 2, 3}, let Ti be
+the component of T \(NT (v)\{u′
+i}) containing v (so e′
+i ∈ E(Ti)). Then since B(v)∩{x1, x2} = ∅
+and since η is tame and substantial, there exists an η-pyramid Σ at (v, e′
+1, e′
+2, e′
+3) with apex a,
+base b1b2b3 and paths P1, P2, P3 such that we have
+• bi ∈ η(e′
+i, v) and Pi \ {a, bi} ⊆ η(Ti) \ B(v) for each i ∈ {1, 2, 3}, and;
+• xi ∈ P ∗
+i for each i ∈ {1, 2}.
+In particular, the second bullet above implies that NΣ(P) is not local in Σ and P is not a corner
+path for Σ. Since P ⊆ M, we have P ∩ Jη = ∅. Thus, Σ being an η-pyramid, it follows that
+P contains no jewel for Σ. Also, since η is rich, a is trapped in η+(T). Therefore, applying
+Theorem 3.2 to G, H = η+(T), a, Σ and P, we deduce that P contains a corner path for Σ.
+On the other hand, note that by the second bullet above, for every vertex x ∈ Σ \ {a}, either
+{x, x1} or {x, x2} is not local in η. From this, the minimality of |P| and the fact that η is rich,
+it follows that P ∗ is anticomplete to Σ. But then P is a corner path for Σ, a contradiction. This
+proves (6).
+(7) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and
+x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible
+subject to this property. Let f = v1v2 ∈ E(T) and {j1, j2} = {1, 2} be as guaranteed by (6) applied
+to P, x1 and x2. Then we have Nη(T)(P ∗) ⊆ η(f, vj1) and Nη(T)({p1, p2}) ⊆ η(f)∪B(v1)∪B(v2).
+Suppose not. Without loss of generality, we may assume that j1 = 1 and j2 = 2. Note that by
+the minimality of |P|, we have Nη(T)(P ∗) ⊆ η(f, v1). Therefore, one of p1 and p2 has a neighbor
+in η(T) \ (η(f) ∪ B(v1) ∪ B(v2)); say p1 is adjacent to x′
+1 ∈ η(T) \ (η(f) ∪ B(v1) ∪ B(v2)). For
+each i ∈ {1, 2}, let Ti be the component of T \ {f} containing vi. It follows that there exists
+j ∈ {1, 2} such that x′
+1 ∈ η(Tj) \ B(vj). Assume that |P| > 1. By the minimality of |P|, we
+have j = 1. But then P, x′
+1 and x2 violate (6). We deduce that |P| = 1. But now P, x′
+1 and x3−j
+violate (6). This proves (7).
+
+12
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+(8) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and
+x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as
+possible subject to this property. Suppose that there exist {k1, k2} = {1, 2}, f = v1v2 ∈ E(T)
+and e1 ∈ E(T) \ {f} incident with vk1 such that pk1 has a neighbor in η(e1, vk1) and pk2 has a
+neighbor in (B(vk2) ∪ η(f)) \ B(vk1). Then pk1 is complete to B(vk1) \ (η(e1, vk1) ∪ η(f)).
+Due to symmetry, we may assume that k1 = 1 and k2 = 2. Let e3 ∈ E(T)\{e1, f} be incident
+with v1 and let b3 ∈ η(e3, v1) be arbitrary. We need to show that p1 is adjacent to b3. Suppose
+for a contradiction that p1 and b3 are non-adjacent. Let b1 ∈ η(e1, v1) be adjacent to p1 and let
+x ∈ (B(v2) ∪ η(f)) \ B(v1) be adjacent to p2. Let T2 be the component of T \ (NT (v1) \ {v2})
+containing v1 (so f ∈ E(T2)). Also, for each i ∈ {1, 3}, let ui be the end of ei distinct from v1
+and let Ti be the component of T \ (NT (v1) \ {ui}) containing v1 (so ei ∈ E(Ti)). By (6) and
+(7), there exists an edge f ′ = v′
+1v′
+2 ∈ E(T) such that Nη(T)({p1, p2}) ⊆ η(f ′) ∪ B(v′
+1) ∪ B(v′
+2).
+This, along with the minimality of |P|, implies that p1 is anticomplete to (η(T1)∪η(T3))\B(v1),
+P \ {p1} is anticomplete to η(T1) ∪ η(T3) and P \ {p2} is anticomplete to η(T2) \ B(v1). Since
+p2 has a neighbor x ∈ (B(v2) ∪ η(f)) \ B(v1) and since η is tame, there exists a path P2 in G
+from a to p2 with P ∗
+2 ⊆ η(T2) \ B(v1). Also, for each i ∈ {1, 3}, there exists a path Pi in G
+from a to bi with P ∗
+i ⊆ η(Ti) \ B(v1). Note that since η is rich, P anticomplete to NG[a]; in
+particular, P1 has length at least two. But now there is a theta in G with ends a and b1 and
+paths P1, a-P2-p2-P-p1-b1 and b1-b3-P3-a, a contradiction. This proves (8).
+The following is immediate from (8) and the fact that T is smooth.
+(9) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and
+x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible
+subject to this property. Suppose that there exist {k1, k2} = {1, 2} and f = v1v2 ∈ E(T) such that
+xk1 ∈ B(vk1) \ (η(f)) and xk2 ∈ (B(vk2) ∪ η(f)) \ B(vk1). Then pk1 is complete to B(vk1) \ η(f).
+We now deduce:
+(10) Let D be a component of M. Then Nη(T)(D) is local in η.
+Suppose not. By Lemma 4.1, there exist x1, x2 ∈ Nη(T)(D) such that {x1, x2} is not local in η.
+For each i ∈ {1, 2}, let pi be a neighbor of xi in D. Since D is connected, there exists a path P
+in D ⊆ M from p1 to p2. In other words, there exists a path P in M with ends p1, p2 along with
+x1 ∈ Nη(T)(p1) and x2 ∈ Nη(T)(p2) such that {x1, x2} is not local in η. Now, let P be a path in M
+with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and x2 ∈ Nη(T)(p2) for which {x1, x2}
+is not local in η, and such that |P| ≥ 1 is as small as possible subject to this property. So we can
+apply (6) to P, x1 and x2; let {j1, j2} = {1, 2} and f = v1v2 ∈ E(T) be as in (6). We may assume
+without loss of generality that j1 = 1 and j2 = 2; thus, v1 is a branch vertex of T. It follows from
+(7) that Nη(T)(P ∗) ⊆ η(f, v1) and Nη(T)({p1, p2}) ⊆ η(f) ∪ B(v1) ∪ B(v2). By (9) applied to
+k1 = 1 and k2 = 2, p1 is complete to B(v1) \ η(f). Also, from (9) applied to k1 = 2 and k2 = 1,
+it follows that either p2 is complete to B(v2) \ η(f) and B(v2) \ η(f) ̸= ∅, or p2 is anticomplete
+to B(v2) \ η(f). Note that if |P| > 1, then by the minimality of |P|, we have Nη(T)(p1) ⊆ B(v1)
+and Nη(T)(p2) ⊆ (B(v2)∪η(f))\B(v1). Let us define η′ : V (T)∪E(T)∪(E(T)×V (T)) ⊆ 2G\{a}
+as follows. Let η′(f) = η(f) ∪ P and let η′(f, v1) = η(f, v1) ∪ {p1}. Let
+• η′(f, v2) = η(f, v2) ∪ {p2} if p2 is complete to B(v2) \ η(f) and B(v2) \ η(f) ̸= ∅, and;
+• η′(f, v2) = η(f, v2) if p2 is anticomplete to B(v2) \ η(f).
+Let η′ = η elsewhere on V (T) ∪ E(T) ∪ (E(T) × V (T)). Then since η is tame, substantial and
+rich, and p2 is adjacent to x2 ∈ B(v2) ∪ η(f)) \ B(v1), it is straightforward to check that η′
+is also a tame, substantial and rich (T, a)-strip-structure. But we have η′(T) = η(T) ∪ P, a
+contradiction with the maximality of η(T). This proves (10).
+The proof is almost concluded. Let X be the union of all the components D of M such that
+D is anticomplete to η+(T). Since η is rich, it follows that for every component D of M \ X, a
+
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+13
+is anticomplete to X and Nη+(T)(D) = Nη(T)(D) is non-empty. By (10), for every component D
+of M \ X, Nη(T)(D) is local in η. Let D be the set of all components D of M \ X for which we
+have Nη+(T)(D) ⊆ Bη(v) for some v ∈ V (T). Breaking the ties arbitrarily and by the definition
+of X, we may write D = �
+v∈V (T) Dv, where
+• for all distinct u, v ∈ V (T), we have Du ∩ Dv = ∅, and;
+• for all v ∈ V (T) and every D ∈ Dv, we have Nη+(T)(D) ⊆ Bη(v) and Nη+(T)(D) ̸= ∅.
+Also, for every e = uv ∈ E(T), let De be the set of all components D of M \ X for which we
+have Nη+(T)(D) ⊆ η(e) and
+• either Nη(T)(D) ∩ η◦(e) ̸= ∅, or;
+• Nη(T)(D) ∩ (η(e, u) \ η(e, v)) ̸= ∅ and Nη(T)(D) ∩ (η(e, v) \ η(e, v)) ̸= ∅.
+From the definition of X, it follows that every component of M \ X belongs to exactly one of
+the sets {Dv, De : v ∈ V (T), e ∈ E(T)} (note that since η is rich, a is anticomplete to each such
+component).
+Let ζ : V (T) ∪ E(T) ∪ (E(T) × V (T)) ⊆ 2G\{a} be defined as follows. For all v ∈ V (T) and
+e ∈ E(T), let
+• ζ(v) = �
+D∈Dv D;
+• ζ(e) = η(e) ∪ (�
+D∈De D), and;
+• ζ(e, v) = η(e, v).
+It is easily seen that ζ satisfies the conditions (S1-S8) from the definition of a (T, a)-strip-
+structure. In particular, since η is rich, ζ satisfies (S2), and from the definitions of X, Dv’s and
+De’s, it follows that ζ satisfies (S5) and (S7). Also, we have η ≤ ζ.
+Now, since η is substantial and rich, since η ≤ ζ and from the definitions of X and ζ, it follows
+that ζ is a substantial and rich (T, a)-strip-structure with Jζ = Jη. Moreover, note that we have
+ζ+(T) = η(T)+ ∪ (M \ X), and so G \ (ζ+(T) ∪ Jζ) = G \ (ζ+(T) ∪ Jη) = X is anticomplete to
+ζ+(T). This completes the proof of Theorem 4.2.
+■
+5. Jewels under the loupe
+Here we revisit jewels for strip-structures, establishing several results about their proper-
+ties in various settings. This will help attune Theorem 4.2 for its application in the proof of
+Theorem 6.1.
+First we need to introduce some notation. Let G be a graph and let a ∈ G. Let T be a smooth
+tree and let ζ be a (T, a)-strip-structure in G. Let v ∈ V (T) and let e ∈ E(T) be incident with
+v. We denote by ζe(v) the set of all components D of ζ(v) for which we have NB(v)(D) ⊆ η(e, v),
+or equivalently, Nζ(T)\ζ(e,v)(D) = ∅.
+Let (v, e1, e2) be a seagull in T and let ui be the end of ei distinct from v for each i ∈ {1, 2}.
+We define
+ζ(v, e1, e2) = ζ(e1) ∪ ζ(e2) ∪ ζe1(u1) ∪ ζe2(u2) ∪ ζ(v).
+We denote by Jζ,(v,e1,e2) the set of all jewels for ζ at (v, e1, e2), and for every vertex v ∈ V (T),
+Jζ,v stands for the set of all jewels for η at v. It follows that Jζ,v = ∅ if v is a leaf of T.
+The first result in this section describes, for a (T, a)-strip-structure in a theta-free graph, the
+attachments of jewels at a vertex of T.
+Theorem 5.1. Let G be a theta-free graph and let a ∈ G. Let T be a smooth tree and let ζ be
+a (T, a)-strip-structure in G. Let (v, e1, e2) be a seagull in T and let x ∈ Jζ,(v,e1,e2). Then the
+following hold.
+• We have Nζ+(T)(x) ⊆ ζ(v, e1, e2), and so Nζ+(T)(Jζ,(v,e1,e2)) ⊆ ζ(v, e1, e2). Consequently,
+for every vertex v ∈ V (T), we have Nζ+(T)(Jζ,v) ⊆ ζ(NT [v]), and for every two distinct
+vertices v, v′ ∈ V (T), we have Jζ,v ∩ Jζ,v′ = ∅.
+
+14
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+• Assume that ζ is rich. Let i ∈ {1, 2} and let R be a long ζ(ei)-rung, let r be the end of
+R in ζ(ei, v) and let r′ be the unique neighbor of r in R. Then either x is anticomplete
+to R or NR(x) = {r, r′}.
+Proof. Note that v is a branch vertex of T. For each i ∈ {1, 2}, let ui be the end of ei distinct
+from v and let Ti be the component of T \(NT (v)\{ui}) containing v. Let T ′ be the component
+of T \{u1, u2} containing v. Let x ∈ Jζ,(v,e1,e2). Since x ∈ Jζ,(v,e1,e2) is a jewel for ζ, there exists
+an edge e3 ∈ E(T)\{e1, e2} incident with v and a ζ-pyramid Σ at (v, e1, e2, e3) with apex a, base
+b1b2b3 and paths P1, P2, P3 such that x is a jewel for Σ at b3. In particular, for each j ∈ {1, 2, 3},
+Pj ∩ ζ(ej) is a long ζ(ej)-rung Rj with bj as its end in ζ(ej, v).
+Also, x is anticomplete to
+P3 (and so x is not adjacent to a), and for each j ∈ {1, 2}, assuming cj to be the unique
+vertex in NRj(bj) = NPj(bj), x is adjacent to bj ∈ ζ(ej, v) \ ζ(ej, uj) and cj ∈ ζ(ej) \ ζ(ej, v).
+Therefore, there exist paths Qi, Si of length more than one in G from a to x for which we have
+bi ∈ Q∗
+i ⊆ (ζ(T ′) \ ζ(v)) ∪ (ζ(ei, v) \ ζ(ei, ui)) and ci ∈ S∗
+i ⊆ ζ(Ti) \ (B(v) ∪ ζ(ui) ∪ ζ(v)).
+To prove the first assertion of the Theorem 5.1, assume for a contradiction that x has a
+neighbor y ∈ ζ+(T) \ ζ(v, e1, e2). Since x is not adjacent to a, we have y ∈ ζ(T) \ ζ(v, e1, e2).
+First, assume that y ∈ ζ(T ′)\ζ(v). Then by (S5) and (S7) from the definition of a strip-structure,
+there exists a path Q′ of length more than one in G from a to x with Q′∗ ⊆ ζ(T ′) \ ζ(v). But
+now there is a theta in G with ends a, x and paths a-S1-x, a-S2-x and a-Q′-x, a contradiction.
+It follows that y ∈ ζ(T1 ∪ T2) \ ζ(v, e1, e2).
+In other words, for some i ∈ {1, 2}, we have
+y ∈ ζ(Ti) \ (ζ(ei) ∪ ζei(ui) ∪ ζ(v)). As a result, by (S5) and (S7) from the definition a strip-
+structure, and by the definition of ζei(ui), there exists a path S′
+i of length more than one in G
+from a to x with S′∗
+i ⊆ ζ(Ti)\(ζ(ei)∪ζei(ui)∪ζ(v)). But now assuming i′ ∈ {1, 2} to be distinct
+from i, there is a theta in G with ends a, x and paths a-Qi-x, a-S′
+i-x and a-Si′-x, a contradiction.
+This proves the the first assertion.
+Next we prove the second assertion of Theorem 5.1. By symmetry, we may assume that i = 1.
+Assume that x has a neighbor y ∈ R. Let P ′
+1 = (P1 \ R1) ∪ R. Let Σ′ be the pyramid with
+apex a, base rb2b3 and paths P ′
+1, P2 and P3. Recall that since ζ is rich, a is trapped in ζ+(T).
+Also, Σ′ is a pyramid in ζ+(T), x is adjacent to y ∈ P ′
+1, x is adjacent to b2, c2 ∈ P2 and x is
+anticomplete to P3. It follows that x is a wide vertex for Σ′ which is not a corner path for Σ′.
+Now applying Lemma 3.1 to G, a, H = ζ+(T), Σ′ and p = x, we deduce that x is a jewel for Σ′
+at b3, and so NR(x) = NP ′
+1(x) = {r, r′}. This completes the proof of Theorem 5.1.
+■
+Our next goal is to show that for every rich (T, a)-strip-structure in a graph G ∈ Ct, there
+are only a few jewels at each vertex of T. Let us begin with a lemma, asserting that for a rich
+(T, a)-strip-structure ζ in a theta-free graph, each set Bζ(v) is almost a clique.
+Lemma 5.2. Let G be a theta-free graph and a ∈ G. Let T be a smooth tree and ζ be a rich
+(T, a)-strip-structure in G. Then for every v ∈ V (T), there exists at most one edge f ∈ E(T)
+such that η(f, v) is not a clique.
+Proof. Suppose for a contradiction that there are two distinct edges f1, f2 ∈ E(T) incident with
+v, and for each i ∈ {1, 2}, there exist xi, yi ∈ ζ(fi, v) such that xi is not adjacent to yi. Then
+v is not a leaf of T and H = x1-x2-y1-y2-x1 is a hole of length four in G. Since ζ is rich, a
+is anticomplete to H. Let f1 = u1v. Let l1 be a leaf of T which belongs to the component of
+T \ {v} containing u1, and let Λ1 be the unique path in T from v to l1 (so f1 ∈ E(Λ1)). Let
+Rx1 be an ζ(f)-rung containing x1 and let Ry1 be an ζ(f)-rung containing y1. Since ζ is rich,
+H1 = Rx1 ∪ Rx2 ∪ B(u1) is a connected induced subgraph of G, and so there is a path Q in H1
+from x1 to y1. It follows that Q has length more than one and Q∗ ⊆ (B(u1) ∪ ζ(f1))\B(v). But
+now there is a theta in G with ends x1, y1 and paths Q, x1-x2-y1 and x1-y2-y1, a contradiction.
+This completes the proof of Lemma 5.2.
+■
+Recall the following classical result of Ramsey (see, for instance, [5] for an explicit bound.)
+
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+15
+Theorem 5.3 (See [5]). For all integers a, b ≥ 1, there exists an integer R = R(a, b) ≥ 1 such
+that every graph G on at least R(a, b) vertices contains either a clique of cardinality a or a stable
+set of cardinality b.
+We can now prove the second main result of this section.
+Theorem 5.4. For all positive integers t, δ, there exists a positive integer j = j(t, δ) with the
+following property. Let G ∈ Ct be a graph and let a ∈ G and let T be a smooth tree of maximum
+degree δ. Let ζ be a rich (T, a)-strip-structure in G. Then for every vertex v ∈ V (T), we have
+|Jζ,v| < j.
+Proof. Let j = j(t, δ) =
+�δ
+2
+�R(t, 3) with R(·, ·) as in Theorem 5.3.
+Then in order to prove
+|Jζ,v| < j, it is enough to show that |Jζ,(v,e1,e2)| < R(t, 3) for every seagull (v, e1, e2) in T.
+Suppose for a contradiction that |Jζ,(v,e1,e2)| ≥ R(t, 3) for some seagull (v, e1, e2) in T. Then v is
+a branch vertex of T. For each i ∈ {1, 2}, let ui be the end of ei different from v. Since G ∈ Ct,
+it follows from Theorem 5.3 that Jζ,(v,e1,e2) contains a stable set X of cardinality three. For
+every x ∈ X, since x is a jewel for ζ at (v, e1, e2), it follows that for every i ∈ {1, 2}, there exists
+a long ζ(ei)-rung Rx
+i such that Qx
+i = Rx
+i \ ζ(ei, v) is a path in ζ(ei) \ ζ(ei, v) from a neighbor
+of x to a vertex in ζ(ei, ui) \ ζ(ei, v); in particular, Rx
+i contains a neighbor of x. Therefore, for
+each i ∈ {1, 2}, we may pick a non-empty set Ri of long ζ(ei)-rungs such that every vertex in X
+has a neighbor in at least one rung in Ri, and with Ri minimal with respect to inclusion. We
+deduce:
+(11) There exists i ∈ {1, 2} with |Ri| > 1.
+Suppose not. Then for every i ∈ {1, 2}, there exists a long ζ(ei)-rung Si such that every vertex
+in X has a neighbor in Si. Let si be the end of Si in ζ(ei, v) and s′
+i be unique neighbor of si in
+Si. By the second assertion of Theorem 5.1, X is complete to {s′
+1, s′
+2}. But now X ∪ {s′
+1, s′
+2} is
+a theta in G with ends s′
+1, s′
+2, a contradiction. This proves (11).
+By (11) and due to symmetry, we may assume that |R1| > 1.
+This, together with the
+minimality of R1, implies that there exist distinct vertices x, y ∈ X as well as distinct long
+ζ(e1)-rungs Rx, Ry ∈ R1 such that x has a neighbor in Rx, y has a neighbor in Ry, x is
+anticomplete to Ry, and y anticomplete to Rx. Let rx and ry be the ends of Rx and Ry in
+ζ(e1, v), respectively. Let r′
+x be the unique neighbor of rx in Rx and r′
+y be the unique neighbor
+of ry in Ry; so we have r′
+x, r′
+y ∈ ζ(e1) \ ζ(e1, v). By the second assertion of Theorem 5.1, we
+have NRx∪Ry(x) = {rx, r′
+x} and NRx∪Ry(y) = {ry, r′
+y}. It follows that rx, r′
+x ∈ Rx \ Ry and
+ry, r′
+y ∈ Ry \ Rx. Also, rx is anticomplete to Ry \ {ry}, as otherwise (Ry \ {ry}) ∪ {rx} contains
+a long ζ(e1)-rung R with NR(x) = {rx}, which violates the second assertion of Theorem 5.1.
+Similarly, ry is anticomplete to Rx \ {rx}.
+Now, let G1 = G[(B(u1)\ζ(e1, u1))∪((Rx∪Ry)\{rx, ry})] and let G2 = G[(B(u2)\ζ(e2, u2))∪
+Qx
+2 ∪ Qy
+2]. Since ζ is rich, the second bullet in the definition of a rich strip-structure implies that
+G1 and G2 are connected. Consequently, there exists a path Q1 in G1 from r′
+x to r′
+y, and there
+exists a path Q2 from x to y with Q∗
+2 ⊆ G2. Also, since v is a branch vertex of T, we may choose
+an edge e3 ∈ E(T) \ {e1, e2} incident with v. By the first assertion of Theorem 5.1, {x, y} is
+anticomplete to ζ(e3, v). Let Q3 be a path from rx to ry with Q∗
+3 ⊆ ζ(e3, v) (thus |Q3| ∈ {2, 3}).
+But now there is a prism with triangles xrxr′
+x and yryr′
+y and paths Q1, Q2, Q3, a contradiction.
+This completes the proof of Theorem 5.4.
+■
+Our last theorem in this section examines the connectivity within G \ ζ+(T) for a (T, a)-
+strip-structure ζ arising from Theorem 4.2. We need the following lemma, the proof of which is
+similar to that of Theorem 5.1.
+Lemma 5.5. Let G be a theta-free graph and let a ∈ G. Let T be a smooth tree and let ζ be a
+(T, a)-strip-structure in G. Let v, v′ ∈ V (T) be distinct and let P be a path in G \ ζ+(T) with
+
+16
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+ends x, x′ such that x ∈ Jζ,v, x′ ∈ Jζ,v′ and P ∗ is anticomplete to ζ+(T). Then v and v′ are
+adjacent in T.
+Proof. Suppose not. Note that by Theorem 5.1, x and x′ are distinct. Let Λ be the path in T
+from v to v′. Then Λ has length more than one, and so there are two distinct edges f, f ′ ∈ E(Λ)
+such that f is incident with v and f ′ is incident with v′. Let u be the end of f distinct from
+v and u′ be the end of f ′ distinct from v′. Let (v, e1, e2) and (v′, e′
+1, e′
+2) be two seagulls in G
+such that x ∈ Jζ,(v,e1,e2) and x′ ∈ Jζ,(v′,e′
+1,e′
+2). For each i ∈ {1, 2}, let ui be the end of ei distinct
+from v and let u′
+i be the end of e′
+i distinct from v′. Without loss of generality, we may assume
+that u2, u′
+2 /∈ Λ. Let T2 be the component of T \ (NT (v) \ {u2}) containing v and let T ′
+2 be the
+component of T \(NT (v′)\{u′
+2}) containing v′. Let T ′ be the component of T \{u′, u′
+2} containing
+v′. Since x is a jewel for ζ at (v, e1, e2), it follows that x is not adjacent to a, and x has a neighbor
+c ∈ ζ(e2) \ ζ(e2, v) ⊆ ζ(T2) \ (B(v) ∪ ζ(u2) ∪ ζ(v)). Therefore, there exists a path Q of length
+more than one in G from a to x for which we have c ∈ Q∗ ⊆ ζ(T2) \ (B(v) ∪ ζ(u2) ∪ ζ(v)). Also,
+since x′ is a jewel for ζ at (v′, e′
+1, e′
+2), it follows that x′ is not adjacent to a, and x′ has a neighbor
+b′ ∈ B(v′)\(ζ(f ′, u′)∪ζ(e′
+2, v′)) and a neighbor c′ ∈ ζ(e′
+2)\ζ(e′
+2, v′) ⊆ ζ(T ′
+2)\(B(v′)∪ζ(u′
+2)∪ζ(v′)).
+Therefore, there exist paths P ′, Q′ of length more than one in G from a to x′ for which we have
+b′ ∈ P ′∗ ⊆ (ζ(T ′) \ ζ(v′)) ∪ (ζ(f ′, v′) \ ζ(f ′, u′)) and c′ ∈ Q′∗ ⊆ ζ(T ′
+2) \ (B(v′) ∪ ζ(u2) ∪ ζ(v′)).
+But now there is a theta in G with ends a, x′ and paths a-P ′-x′, a-Q′-x′ and a-Q-x-P-x′, a
+contradiction. This proves Lemma 5.5.
+■
+Theorem 5.6. Let t, δ ≥ 1 be integers and let j(t, δ) be as in Theorem 5.4. Let G ∈ Ct be a
+graph and let a ∈ G. Let T be a smooth tree of maximum degree δ and let v ∈ V (T). Let ζ
+be a rich (T, a)-strip-structure in G such that G \ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T). Let
+x ∈ G \ (ζ+(T) ∪ Jζ). Then there exists Sx ⊆ G \ (ζ+(T) ∪ {x}) such that |Sx| < 2j(t, δ) and Sx
+separates x and Jζ \ ({x} ∪ Sx) in G \ ζ+(T). Consequently, Sx separates x and ζ+(T) in G.
+Proof. By Theorem 5.1, {Jζ,v : v ∈ V (T)} is a partition of Jζ. Let G′ be the graph obtained
+from G\ζ+(T) by contracting the set Jζ,v into a vertex zv for each v ∈ V (T) with Jζ,v ̸= ∅, and
+then adding a new vertex z such that NG′(z) = {zv : v ∈ V (T), Jζ,v ̸= ∅}. We claim that there
+is a set Y ⊆ G′ \ {x, z} of cardinality at most two which separates x and z in G′. Suppose not.
+By Theorem 2.1, there are three pairwise internally disjoint paths in G′ from x to z. Thus, there
+exist S ⊆ T with |S| = 3 as well as three paths {Pv : v ∈ S} in G \ ζ+(T) all having x as an end
+and otherwise disjoint, such that for each v ∈ S, Pv has an end yv ∈ Jζ,v distinct from x, and
+we have P ∗
+v ⊆ G\(ζ+(T) ∪ Jζ). As a result, for all distinct v, v′ ∈ S, Pv,v′ = yv-Pv-x-Pv′-yv′ is a
+path in G\ζ+(T) from yv ∈ Jζ,v to yv′ ∈ Jζ,v′ such that P ∗
+v,v′ ⊆ G\(ζ+(T)∪Jζ). In particular,
+P ∗
+v,v′ is anticomplete to ζ+(T). But then by Lemma 5.5, S is a clique in T, which is impossible.
+The claim follows.
+Let Y be as in the above claim. For each y ∈ Y , if y = zv for some v ∈ V (T), then let
+Ay = Jζ,v. Otherwise, let Ay = {y}. Let Sx = �
+y∈Y Ay. Then Sx ⊆ G\(ζ+(T)∪{x}) separates
+x and Jζ \({x}∪Sx) in G\ζ+(T). Also, by Theorem 5.4, we have |Sx| < 2j(t, δ). This completes
+the proof of Theorem 5.6.
+■
+6. Strip structures and connectivity
+In this section, we investigate the connectivity implications of the presence of certain (T, a)-
+strip-structures in graphs from Ct. The main result is the following.
+Theorem 6.1. For all integers t, δ ≥ 1, there exists an integer σ = σ(t, δ) ≥ 1 with the following
+property. Let G ∈ Ct be a graph and let a ∈ G. Let T be a smooth tree of maximum degree δ and
+let ζ be a rich (T, a)-strip-structure in G such that G \ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T).
+Then for every vertex x ∈ G \ NG[a], there exists a set Sx ⊆ G \ {a, x} with |Sx| < σ such that
+S separates a and x in G.
+
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+17
+Proof. Let j(t, δ) be as in Theorem 5.4. We claim that
+σ = σ(t, δ) = 2δ(j(t, δ) + t)
+satisfies Theorem 6.1. For every vertex v ∈ V (T), we define Cv = B(v) if v is a leaf of T and
+Cv = ∅ otherwise. Also, for every vertex v ∈ V (T), let Kv be a maximal clique of G contained
+in B(v). Thus, we have |Kv| < t. Moreover, Lemma 5.2 along with the assumption that ζ is
+rich implies if v is a leaf of T, then we have Kv = B(v) = Cv (and so |Kv| = 1), and if v is a
+branch vertex of T, then Kv contains all but possibly one of the sets η(f, v) for f ∈ E(T). For
+every S ⊆ T, we define
+MS =
+�
+w∈NT (S)
+Jη,w,
+NS =
+�
+w∈NT (S)
+Kw.
+Also, we write Mv for M{v} and Nv for N{v}. For every v ∈ V (T), let Ov = Mv ∪ Nv. The
+following is immediate from Theorems 5.1 and 5.4 and Lemma 5.5.
+(12) For every v ∈ V (T), we have
+• Ov ⊆ G \ (Jζ,v ∪ {a});
+• |Ov| < δ(j(t, δ) + t) ≤ σ, and;
+• Ov separates a and Jζ,v in G.
+Now, for every x ∈ G \ NG[a], we define Sx as follows. First, assume that x ∈ ζ(T) \ NG[a].
+Then either x ∈ ζ(e) for some edge e = uv ∈ E(T), or x ∈ ζ(v) for some branch vertex v ∈ V (T).
+In the former case, let
+Ex = Mu ∪ Mv,
+Ix = N{u,v} ∪ Cu ∪ Cv.
+In the latter case, let
+Ex = Mv ∪ Jζ,v
+Ix = Nv.
+Let Sx = Ex ∪ Ix.
+Observe that since x ∈ G \ NG[a], we have Sx ⊆ G \ {a, x}.
+Also, by
+Theorem 5.4, we have |Ex| ≤ 2δj(t, δ) and so |Sx| < 2δ(j(t, δ) + t) = σ.
+Moreover, from
+Theorem 5.1 and the fact that ζ is rich, it is easy to check that for every path P in G from a
+to x, if P ⊆ ζ+(T), then P contains a vertex from Ix, and otherwise P contains a vertex from
+either Ix or Ex. Therefore, Sx separates a and x in G.
+Next, assume that x ∈ Jζ. Then by Theorem 5.1, there exists a unique vertex v ∈ V (T) such
+that x ∈ Jζ,v. Let Sx = Ov. Then by (12), we have Sx ⊆ G \ {a, x}, |Sx| < σ and Sx separates
+a and x in G.
+Finally, assume that x ∈ G\(ζ+(T)∪Jζ). Then letting Sx to be as in Theorem 5.6, it follows
+from Theorem 5.6 that Sx ⊆ G \ {a, x}, |X| < 2j(t, δ) ≤ σ and Sx separates a and x in G. This
+completes the proof of Theorem 6.1.
+■
+Our application of Theorem 6.1 though is confined to the case where T is a caterpillar. More
+precisely, for a graph G and a vertex a ∈ G, an induced subgraph H ⊆ G \ {a} is said to be an
+a-seed in G if the following hold.
+• There exists a caterpillar C such that H is the line graph of a 1-subdivision of C and
+NG(a) = Z(H).
+• The vertex a is trapped in H ∪ {a}.
+It follows that Z(H) is the set of all degree-one vertices of H. We now combine Theorems 4.2
+and 6.1 to deduce the following.
+
+18
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+Theorem 6.2. For every integer t ≥ 1, there exists an integer s = s(t) ≥ 1 with the following
+property. Let G ∈ Ct be a graph and a ∈ G. Assume that there is an a-seed in G. Then for
+every vertex x ∈ G \ NG[a], there exists Sx ⊆ G \ {a, x} with |Sx| < s such that Sx separates a
+and x in G.
+Proof. Let σ(·, ·) be as in Theorem 6.1. We show that s = s(t) = σ(t, 3) satisfies Theorem 6.2.
+Pick an a-seed H in G. Let T be the unique smooth caterpillar with |NG(a)| leaves. Then T has
+maximum degree three. Also, one may immediately observe that there is a tame, substantial
+and rich (T, a)-strip-structure η in G with η(T) = H. Now we can apply Theorem 4.2 to G, a
+and T, deducing that there exists a substantial and rich (T, a)-strip-structure ζ in G such that
+G \ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T). Hence, by Theorem 6.1 applied to G, a, T and ζ,
+for every vertex x ∈ G \ NG[a], there exists Sx ⊆ G \ {a, x} with |Sx| < s such that Sx separates
+a and x in G. This completes the proof of Theorem 6.2.
+■
+7. From blocks to trees
+In this section, we prove Theorem 1.8. We begin with a result which captures the use of
+Theorem 6.2 in the proof of Theorem 1.8. For a positive integer n, we write [n] = {1, . . . , n}.
+Theorem 7.1. For all integers ν, t ≥ 1, there exists an integer ψ = ψ(t, ν) ≥ 1 with the
+following property. Let G ∈ Ct, let a, b ∈ G be distinct and non-adjacent and let {Pi : i ∈ [ψ]}
+be a collection of ψ pairwise internally disjoint paths in G from a to b. For each i ∈ [ψ], let
+ai be the neighbor of a in Pi (so ai ̸= b). Then there exists I ⊆ [ψ] with |I| = ν for which the
+following holds.
+• {ai : i ∈ I} ∪ {b} is a stable set in G.
+• For all i, j ∈ I with i < j, ai has a neighbor in P ∗
+j \ {aj}.
+Proof. Let s = s(t) be as in Theorem 6.2 and let µ = µ(max{2s + 1, t}), where µ(·) is as in
+Theorem 2.5. Let R(·, ·) be as in Theorem 5.3. For every integer p ≥ 1, let Rtourn(p) be the
+smallest positive integer n such that every tournament on at least n vertices contains a transitive
+tournament on p vertices; the existence of Rtourn(p) follows easily from Theorem 5.3 (in fact,
+one may observe that Rtourn(p) ≤ R(p, p)). Let γ = R(Rtourn(ν + 1), µ). We prove that
+ψ = ψ(t, ν) = R(γ, t)
+satisfies Theorem 7.1. Let P1, . . . , Pψ be ψ pairwise internally disjoint paths in G from a to
+b. Since G is Kt-free, it follows from Theorem 5.3 and the definition of ψ that there exists a
+stable set N ⊆ {ai : i ∈ [ψ]} in G with |N| = γ; we may assume without loss of generality that
+N = {ai : i ∈ [γ]}.
+Let D be a directed graph with V (D) = N such that for distinct i, j ∈ [γ], there is an arc
+from ai to aj in D if and only if xi has a neighbor in P ∗
+j \ {aj}. Note that D may contain both
+arcs (ai, aj) and (aj, ai), and so the undirected underlying graph of D might not be simple. Let
+D− be the simple graph obtained from the undirected underlying graph of D by removing one
+of every two parallel edges.
+(13) D− contains no stable set of cardinality µ.
+Suppose for a contradiction that D− contains a stable set S of cardinality µ. We may assume
+without loss of generality that S = {a1, . . . , aµ}.
+Let G1 = G[(�µ
+j=1 Pj) \ {a}].
+Note that
+by the definition of D, for every i ∈ [µ], we have NG1(ai) = NPi(ai) \ {a}, and in particular
+|NG1(ai)| = 1. Since G1 is connected and Kt-free, and since and |S| = µ = µ(max{2s + 1, t}),
+we can apply Theorem 2.5 to G1 and S. Note that every vertex in S has a unique neighbor in
+G1, and so no path in G1 contains max{2s + 1, t} ≥ 3 vertices from S. Consequently, there is
+an induced subgraph H1 of G1 with |H1 ∩ S| = 2s + 1 for which one of the following holds.
+• H1 is either a caterpillar or the line graph of a caterpillar with H1 ∩ S = Z(H1).
+• H1 is a subdivided star with root r1 such that Z(H1) ⊆ H1 ∩ S ⊆ Z(H1) ∪ {r1}.
+
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+19
+If H1 is a caterpillar, then G[H1∪{a}] contains a theta with ends a and a′ for every vertex a′ ∈ H1
+of degree more than two, a contradiction. Also, if the second bullet above holds, then since every
+vertex in S is of degree one in G1, we have H1 ∩ S = Z(H1), and so r1 is not adjacent to a. But
+then G[H1 ∪ {a}] contains a theta with ends x and r1, a contradiction. It follows that H1 is the
+line graph of a caterpillar with |H1 ∩ S| = 2s + 1 and H1 ∩ S = Z(H1). This, together with the
+fact that every vertex in H1 ∩ S ⊆ S has a unique neighbor in H1 ⊆ G, implies that H1 contains
+the line graph H2 of a 1-subdivision of a caterpillar with |H2 ∩ S| = s and H2 ∩ S = Z(H2).
+Let S2 = H2 ∩ S = Z(H2); then S2 is the set of all vertices of degree one in H2, and we may
+assume without loss of generality that S2 = {a1, . . . , as}. Let G2 = G[H2 ∪(�s
+j=1 Pj)]. It follows
+that G2 ∈ Ct, NG2(a) = S2 = Z(H2) and a is trapped in H2 ∪ {a}. Therefore, H2 is an a-seed
+in G2. Since b ∈ G2 \ NG2[a], applying Theorem 6.2 to G2 and a, we deduce that there exists
+Sb ⊆ G2 \{a, b} such that |Sb| < s and Sb separates a and b in G2. But P1, . . . , Ps are s pairwise
+internally disjoint paths in G2 from a to b, a contradiction with Theorem 2.1. This proves (13).
+By (13), Theorem 5.3 and the definition γ, D− contains a clique of cardinality Rtourn(ν + 1).
+This, along with the definition of Rtourn(·), implies that D contains (as a subdigraph) a transitive
+tournament K on ν + 1 vertices.
+We may assume without loss of generality that V (K) =
+{a1, . . . , aν+1} such that for distinct i, j ∈ [ν + 1], (ai, aj) is an arc in K if i < j. From the
+definition of D, it follows that {a2, . . . , aν+1, b} is a stable set in G, and for all i, j ∈ {2, . . . , ν+1}
+with i < j, ai has a neighbor in P ∗
+j \{aj}. Hence, I = {2, . . . , ν + 1} satisfies Theorem 7.1. This
+completes the proof.
+■
+For positive integers d and r, let T r
+d denote the rooted tree in which every leaf is at distance
+r from the root, the root has degree d, and every vertex that is neither a leaf nor the root has
+degree d + 1. We need a result from [15]:
+Theorem 7.2 (Kierstead and Penrice [15]). For all integers d, r, s, t ≥ 1, there exists an integer
+f = f(d, r, s, t) ≥ 1 such that if G contains T f
+f as a subgraph, then G contains one of Ks,s, Kt
+and T r
+d as an induced subgraph.
+The following lemma is the penultimate step in the proof of Theorem 1.8.
+Lemma 7.3. For all integers d, r, t ≥ 1, there exists an integer m = m(d, r, t) with the following
+property. Let G ∈ Ct be a graph, let a, b ∈ G be non-adjacent and let {Pi : i ∈ [m]} be a collection
+of m pairwise internally disjoint paths in G from a to b. Then G[�m
+j=1 Pj] contains a subgraph
+J isomorphic to T r
+d such that a ∈ J and a has degree d in J (that is, a is the root of J), and we
+have b /∈ J.
+Proof. Let d, t ≥ 1 be fixed. Let m1 = d. For every integer r > 1, let mr = ψ(t, (mr−1 + 1)d)
+where ψ(·, ·) is as in Theorem 7.1. We prove by induction on r ≥ 1 that m(d, r, t) = mr satisfies
+Lemma 7.3. Let P1, . . . , Pmr be mr pairwise internally disjoint paths in G from a to b. Since a
+and b are not adjacent, it follows that for each i ∈ [mr], we have P ∗
+i ̸= ∅; let ai be the neighbor
+of a in Pi. In particular, we have b /∈ {ai : i ∈ [mr]}. Suppose first that r = 1. Then we have
+|{ai : i ∈ [m1]}| = m1 = d, and so G[{ai : i ∈ [mr]} ∪ {a}] contains a (spanning) subgraph
+J isomorphic to T 1
+d such that a ∈ J and a has degree d in J, and we have b /∈ J, as desired.
+Therefore, we may assume that r ≥ 2. Since mr = ψ(t, (mr−1 + 1)d), we can apply Theorem 7.1
+to a, b and {Pi : i ∈ [mr]}, obtaining I ⊆ [mr] with |I| = (mr−1 + 1)d which satisfies the two
+outcomes of Theorem 7.1. Without loss of generality, we may assume that I = [(mr−1 + 1)d].
+It follows that {a1, · · · , a(mr−1+1)d, b} is a stable set in G, and for all i, j ∈ [(mr−1 + 1)d] with
+i < j, ai has a neighbor in P ∗
+j \ {aj}. For every i ∈ [d], let a′
+i = a(i−1)mr−1+i and let
+Ai = {(i − 1)mr−1 + i + 1, . . . , (i − 1)mr−1 + i + mr−1}.
+In particular, we have |Ai| = mr−1. Then for each i ∈ [d] and each j ∈ Ai, a′
+i has a neighbor in
+P ∗
+j \ {aj}, and so there exists a path Qj in G from a′
+i to b with Q∗
+j ⊆ P ∗
+j . Now, for every i ∈ [d],
+a′
+i and b are non-adjacent, and {Qj : j ∈ Ai} is a collection of mr−1 pairwise internally disjoint
+
+20
+INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.
+paths in G from a′
+i to b. It follows from the induction hypothesis that G[�
+j∈Ai Qj] contains a
+subgraph Ji isomorphic to T r−1
+d
+such that a′
+i ∈ Ji and a′
+i has degree d in Ji, and we have b /∈ Ji.
+But now G[(�d
+i=1 V (Ji)) ∪ {a}] ⊆ G[�mr
+j=1 Pj] contains a (spanning) subgraph J isomorphic to
+T r
+d such that a ∈ J and a has degree d in J, and we have b /∈ J. This completes the proof of
+Lemma 7.3.
+■
+Finally, we prove Theorem 1.8, which we restate:
+Theorem 1.8. For every tree F and every integer t ≥ 1, there exists an integer τ(F, t) ≥ 1
+such that every graph in Ct(F) has treewidth at most τ(F, t).
+Proof. Let d and r be the maximum degree and the radius of F, respectively. It follows that
+T r
+d contains F as an induced subgraph.
+Let f = f(d, r, 3, t) be as in Theorem 7.2 and let
+m = m(f, f, t) be as in Lemma 7.3. Let β(·, ·) be as in Corollary 2.4. We claim that τ(F, t) =
+β(max{m, t + 1}, t) satisfies Theorem 1.8. Suppose for a contradiction that tw(G) > τ for some
+G ∈ Ct(F). By Corollary 2.4, G contains a max{m, t + 1}-block B. Consequently, since G is
+Kt-free, there are two distinct and non-adjacent vertices a, b ∈ B, and m pairwise internally
+disjoint paths P1, . . . , Pm in G from a to b. It follows from Lemma 7.3 that G contains T f
+f as a
+subgraph. Also, since G ∈ Ct(F) ⊆ Ct, G is (K3,3, Kt)-free. But now by Theorem 7.2, G contains
+T r
+d , and so F, as an induced subgraph, a contradiction. This completes the proof.
+■
+References
+[1] P. Aboulker, I. Adler, E. J. Kim, N. L. D. Sintiari, and N. Trotignon. “On the treewidth of even-hole-free
+graphs.” European Journal of Combinatorics 98, (2021), 103394.
+[2] T. Abrishami, B. Alecu, M. Chudnovsky, S. Hajebi, and S. Spirkl, “Induced subgraphs and tree decomposi-
+tions VII. Basic obstructions in H-free graphs.” arXiv:2212.02737, (2022).
+[3] T. Abrishami, M. Chudnovsky, C. Dibek, S. Hajebi, P. Rzążewski, S. Spirkl, and K. Vušković, “Induced
+subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree.”
+arXiv:2108.01162, (2021).
+[4] T. Abrishami, M. Chudnovsky and K. Vušković, “Induced subgraphs and tree decompositions I. Even-hole-
+free graphs of bounded degree.” J. Combin. Theory Ser. B, 157 (2022), 144-175.
+[5] M. Ajtai, J. Komlós and E. Szemerédi. “A note on Ramsey numbers.” J. Combinatorial Theory, Ser. A 29
+(1980), 354–360.
+[6] H. L. Bodlaender. “Dynamic programming on graphs with bounded treewidth.” Springer, Berlin, Heidelberg,
+(1988), pp. 105–118.
+[7] K. Cameron, M.V. da Silva, S. Huang, and K. Vušković, “Structure and algorithms for (cap, even hole)-free
+graphs.” Discrete Mathematics 341, 2 (2018), 463-473.
+[8] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, “The strong perfect graph theorem.” Annals of
+Math 164 (2006), 51-229.
+[9] M. Chudnovsky and P. Seymour, “Even-hole-free graphs still have bisimplicial vertices.” arXiv:1909.10967,
+(2019).
+[10] M. Chudnovsky and P. Seymour, “The three-in-a-tree problem.” Combinatorica 30, 4 (2010): 387-417.
+[11] J. Davies, “Vertex-minor-closed classes are χ-bounded.” arXiv:2008.05069, (2020).
+[12] J. Davies, appeared in an Oberwolfach technical report DOI:10.4171/OWR/2022/1.
+[13] J. Erde and D. Weißauer. “A short derivation of the structure theorem for graphs with excluded topological
+minors.” SIAM Journal of Discrete Mathematics 33, 3 (2019), 1654–1661.
+[14] M. Grohe and D. Marx. “Structure theorem and isomorphism test for graphs with excluded topological
+subgraphs,” SIAM Journal on Computing 44, 1 (2015), 114–159.
+[15] H.A. Kierstead and S. G. Penrice, “Radius two trees specify χ-bounded classes.” J. Graph Theory 18, 2
+(1994): 119–129.
+[16] T. Korhonen, “Grid Induced Minor Theorem for Graphs of Small degree.” arXiv:2203.13233, (2022).
+[17] V. Lozin and I. Razgon. “Tree-width dichotomy.” European J. Combinatorics 103 (2022): 103517.
+[18] K. Menger, “Zur allgemeinen Kurventheorie.” Fund. Math. 10, 1927, 96–115.
+[19] N. Robertson and P. Seymour. “Graph minors. V. Excluding a planar graph.” J. Combin. Theory Ser. B, 41
+(1) (1996), 92–114.
+[20] N.L.D. Sintiari and N. Trotignon. “(Theta, triangle)-free and (even-hole, K4)-free graphs. Part 1: Layered
+wheels.” J. Graph Theory 97 (4) (2021), 475-509.
+[21] N. Trotignon, private communication, 2021.
+

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+arXiv:2301.02137v1  [nucl-th]  5 Jan 2023
+Reduced nuclear helicity amplitudes for deuteron
+electrodisintegration and other processes
+J. Flores and S. S. Chabysheva
+Department of Physics, University of Idaho, Moscow ID 83844 USA
+J. R. Hiller
+Department of Physics, University of Idaho, Moscow ID 83844 USA and
+Department of Physics and Astronomy,
+University of Minnesota-Duluth, Duluth, Minnesota 55812 USA
+(Dated: January 6, 2023)
+1
+
+Abstract
+We extend the original idea of reduced nuclear amplitudes to capture individual helicity ampli-
+tudes and discuss various applications to exclusive processes involving the deuteron. Specifically,
+we consider deuteron form factors, structure functions, tensor polarization observables, photodisin-
+tegration, and electrodisintegration. The basic premise is that nuclear processes at high momentum
+transfer can be approximated by tree graphs for point-like nucleons supplemented by empirical form
+factors for each nucleon. The latter represent the internal structure of the nucleon, and incorporate
+nonperturbative physics, which can allow for early onset of scaling behavior. The nucleon form
+factors are evaluated at the net momentum transfer experienced by the given nucleon, with use of
+GE for a no-flip contribution and GM for a helicity-flip contribution. Results are compared with
+data where available. The deuteron photodisintegration asymmetry Σ is obtained with a value of
+Σ(90◦) ≃ −0.06, which is much closer to experiment than the value of -1 originally expected. The
+method also provides an estimate of the momentum transfer values required for scaling onset. We
+find that the deuteron structure function B is a good place to look, above momentum transfers of
+10 GeV2.
+I.
+INTRODUCTION
+With the advent of the upgraded electron accelerator at the Thomas Jefferson National
+Accelerator Facility, scattering experiments with polarized beams and targets at high energy
+and high momentum transfer become possible. In the regime of high momentum transfer to
+all relevant nucleons, quantum chromodynamics (QCD) implies that the internal structure of
+every nucleon is important. Until ab initio QCD (lattice) calculations for nuclear scattering
+processes are available for more than very simple processes, one is led to consider models
+that can represent the basic physics.
+One such approach is the reduced nuclear amplitude (RNA) analysis pioneered by Brod-
+sky and Chertok [1]. In addition to their application to a generic deuteron form factor,
+the approach has been applied to deuteron disintegration [2], pion photoproduction [3], and
+photodisintegration of 3He [4]. As originally developed, a nuclear process was modeled as
+a tree-level amplitude multiplied by a generic form factor for each nucleon, with each form
+factor evaluated at the net momentum transferred to that nucleon. In order to model the
+behavior of polarization observables [5–15], we extend this approach to a reduced nuclear
+helicity amplitude (RNHA) method to combine a tree-level helicity amplitude for point-like
+nucleons with the appropriate form factor for each nucleon. When the nucleon does (not)
+flip its helicity, we use the electric (magnetic) form factor GEN (GMN). As a check on the
+procedure, virtual photon absorption by a single nucleon in the RNHA approach is consistent
+with the definitions of GEN and GMN.
+A caveat in applications of the RNA approach is that the normalization is not determined
+by the model and is fixed to data at infinite momentum transfer by the coefficient of the
+leading power-law behavior. This means that the normalization cannot be determined in
+practice; fitting to a data point at some intermediate kinematics will give the wrong nor-
+malization and the wrong magnitude at higher momentum transfer. Instead, ratios need to
+be considered, so that the normalization becomes irrelevant.
+The primary criterion for the asymptotic region is in the momentum transfer to each
+nucleon. For every nucleon in the process, the momentum transfer must be above some
+common threshold, which is at least 1 GeV2. For example, for deuteron photodisintegration,
+2
+
+the momentum transfer to a nucleon is −tN = −(pN − p/2)2, where pN is the final four-
+momentum of the nucleon and p is the initial deuteron momentum. When expressed in
+terms of the photon energy Eγ and the final nucleon angle θ, the constraint to be above 1
+GeV2 becomes [16]
+mNEγ
+�
+1 −
+�
+Eγ
+mN + Eγ
+| cos θ|
+�
+≥ 1 GeV2.
+(1.1)
+This relationship is illustrated in Fig. 1.
+Notice that away from 90◦, the lower limit is
+quite high. For electrodisintegration, only the most recent data [17, 18] begins to reach this
+θ (deg)
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Eγ (GeV)
+0
+2
+4
+6
+8
+10
+FIG. 1. Angular dependence of the scale for large momentum transfer in deuteron photodisinte-
+gration.
+threshold.
+Here we will focus on deuteron processes, including photodisintegration and electrodisin-
+tegration. For recent reviews of deuteron studies at high momentum transfer, see [16, 19, 20].
+Elastic electron scattering data at high momentum transfer is presented in [21–25]. Recent
+photodisintegration data can be found in [26–29], and for electrodisintegration data, in
+[17, 18, 30–33]. Other analyses of deuteron processes include hidden-color contributions to
+deuteron form factors [34], the hard rescattering mechanism [35], quark-gluon strings [36],
+the Moscow NN potential [37], and AdS/QCD models [38, 39].
+One recent experiment [17] used the 10.6 GeV electron beam at JLab and the Hall
+C spectrometers to measure electron scattering from a liquid deuterium target. The final
+electron and the proton were detected, with the kinematics restricted to the exclusive process
+ed → e′pn. One spectrometer measured the final electron at a nominal 12.2◦ degrees from
+the beam direction, with a momentum of 8.5-9.1 GeV such that the recorded events had a
+distribution of momentum transfer squared reaching 5 GeV2. The events studied were taken
+from a bin of 4.5±0.5 GeV2 in the tail of the distribution; however, the nominal transfer
+was 4.2 GeV2, because the majority of the events were in the lower half of the bin.
+A second spectrometer measured the proton momentum at a range of angles to the
+beam direction, tuned to select events where the (missing) neutron had an angle relative to
+the direction of the momentum transfer that fell within a chosen bin. In the one-photon
+exchange approximation, which we assume, the momentum transferred is, of course, the
+photon momentum. The published neutron angles are binned at 35◦, 45◦ and 75◦, with the
+first two selected to minimize final-state interactions. For our purposes, the importance of
+3
+
+these two angles is that the momentum transferred to the neutron reaches 1 GeV in a zero-
+binding approximation, so that, rather than focus on the internal structure of the deuteron,
+we can consider the response to a large momentum transfer to all the nucleons involved and
+we can see that experiments may be approaching the threshold where our model can be
+applied.
+The RNHA model is constructed in detail in Sec. II for two-nucleon processes. In the
+remainder of the paper, we consider various processes for the deuteron. In Sec. III, the
+form factors,1 structure functions, and tensor polarization observables of elastic electron
+scattering from the deuteron are obtained. Photodisintegration and electrodisintegration
+are analyzed in Secs. IV and V. Within the zero-binding approximation, elastic scattering
+and photodisintegration live at edges of the kinematic range of electrodisintegration and
+are essentially special cases that provide introductory examples.
+Section VI contains a
+summary of the results and suggestions for additional applications. Many details of the
+electrodisintegration helicity amplitudes are left to an appendix.
+II.
+CONSTRUCTION OF THE MODEL
+The basic process for a two-nucleon system to absorb a photon and exchange momentum
+between the nucleons is illustrated in Fig. 2. These diagrams are modeled on the primitive
+process of γ∗ff → ff, with f representing a point-like nucleon.2 The structure of each
+nucleon is then introduced by combining the Feynman amplitude for each diagram with the
+appropriate form factor for each nucleon, evaluated at the net momentum transfer for that
+nucleon. For a deuteron process in the zero-binding limit, the initial nucleons share the
+initial deuteron momentum p equally, so that pp = pn = p/2. We also neglect the nucleon
+mass difference, setting mp = mn ≡ m. The distinction between different photon-absorption
+processes is then in the nature of the photon, being either real or virtual, and in the outcome
+for the final nucleons, bound as a deuteron or not.
+The tree-level amplitudes for the four diagrams in Fig. 2 are
+Mν
+a (λ′
+p, λ′
+n, λp, λn) = Aµν
+p (p/2 + q; λ′
+p, λp)
+1
+(p′
+n − p/2)2Bnµ(λ′
+n, λn),
+(2.1)
+Mν
+b (λ′
+p, λ′
+n, λp, λn) = Aνµ
+p (p′
+p − q; λ′
+p, λp)
+1
+(p′n − p/2)2Bnµ(λ′
+n, λn),
+(2.2)
+Mν
+c (λ′
+p, λ′
+n, λp, λn) = Aµν
+n (p/2 + q; λ′
+n, λn)
+1
+(p′p − p/2)2Bpµ(λ′
+p, λp),
+(2.3)
+Mν
+d (λ′
+p, λ′
+n, λp, λn) = Aνµ
+n (p′
+n − q; λ′
+n, λn)
+1
+(p′
+p − p/2)2Bpµ(λ′
+p, λp),
+(2.4)
+where
+Aµν
+N (p; λ′
+N, λN) = ¯u′
+Nγµ ̸p + m
+p2 − m2γνuN, Bµ
+N(λ′
+N, λN) = ¯u′
+NγµuN,
+(2.5)
+with uN (¯u′
+N) the initial (final) spinor for the nucleon N with helicity λN (λ′
+N). The sub-
+amplitude AN represents the fermion line that absorbs the photon, and BN represents the
+1 For discussion specifically in terms of perturbative QCD, see [34].
+2 In [2], the primitive process was γ∗q¯q → q¯q, with q corresponding to a point-like proton and ¯q to a point-
+like neutron, and direct interaction of the photon with the neutron was neglected. Here we amend and
+extend this, to retain information about helicity states of the fermions and include photon absorption by
+the neutron.
+4
+
+pp, λp
+pn, λn
+p′
+p, λ′
+p
+p′
+n, λ′
+n
+q, λγ
+pp, λp
+pn, λn
+p′
+p, λ′
+p
+p′
+n, λ′
+n
+q, λγ
+(a)
+(b)
+pp, λp
+pn, λn
+p′
+p, λ′
+p
+p′
+n, λ′
+n
+q, λγ
+pp, λp
+pn, λn
+p′
+p, λ′
+p
+p′
+n, λ′
+n
+q, λγ
+(c)
+(d)
+FIG. 2. Tree graphs for deuteron processes that absorb a photon of momentum q and helicity
+λγ. The initial (final) nucleon momentum and helicity are pN (p′
+N) and λN (λ′
+N), with N = p or
+n. The two nucleons exchange momentum via a vector particle. The four diagrams differ in the
+nature of the photon-absorbing nucleon and the order of this absorption and momentum transfer
+between nucleons.
+other fermion line. Calculation of these sub-amplitudes can be checked against the trace
+theorem for sums over helicities:
+�
+λN,λ′
+N
+Aν′µ′∗
+N
+(p; λ′
+N, λN)Aµν
+N (p; λ′
+N, λN) = Tr
+�
+γν′ ̸p + m
+p2 − m2γµ′(̸p′
+N + m)γµ ̸p + m
+p2 − m2γν(̸pN + m)
+�
+,
+(2.6)
+�
+λN,λ′
+N
+Bµ′∗
+N (λ′
+N, λN)Bµ
+N(λ′
+N, λN) = Tr
+�
+γµ′(̸p′
+N + m)γµ(̸pN + m)
+�
+.
+(2.7)
+The full amplitude is constructed from the MX by combining them with form factors for
+each nucleon. For a deuteron with initial helicity λd, we have
+Mν(λ′
+p, λ′
+n, λd) =
+�
+λp,λn
+Cλd
+λpλn
+
+
+�
+X=a,b,c,d
+Mν
+X(λ′
+p, λ′
+n, λp, λn)
+
+ Gpλ′pλp(Q2
+p)Gnλ′nλn(Q2
+n),
+(2.8)
+5
+
+where Q2
+N = −(p′
+N − pN)2,
+Cλd
+λpλn =
+
+
+
+δλp± 1
+2δλn± 1
+2,
+λd = ±1
+1
+√
+2
+�
+δλp 1
+2δλn− 1
+2 + δλp− 1
+2δλn+ 1
+2
+�
+, λd = 0,
+(2.9)
+and
+GNλ′λ =
+�
+GEN, λ′ = λ
+GMN, λ′ = −λ.
+(2.10)
+The form factors GEN and GMN represent the internal structure of the nucleons. They can
+be represented by data or empirical fits. For simplicity, we use the fits [40]
+GEp ≃
+�
+1 + Q2
+N
+m2
+0
+�−2
+, GMp ≃ µpGEp, GMn ≃ µnGEp, GEn ≃ −
+µnτ
+1 + 5.6τ GEp,
+(2.11)
+where m2
+0 = 0.71 GeV2, τ =
+Q2
+N
+4m2 , µp = 2.79, and µn = −1.91. To limit the analysis to a
+single mass scale, we take the parameter m0 to be proportional to the nuclear mass, with
+m2
+0 = 0.80 m2. We do not attempt to compute or assign an overall normalization to Mν,
+and the running of the strong coupling constant is not included.
+The initial nucleon spinor, for a deuteron traveling along the negative z direction, is [41]
+uN =
+̸p/2 + m
+�
+Ed/2 + m
+�
+φ(λN)(−ˆz)
+0
+�
+,
+(2.12)
+with
+φ(1/2)(−ˆz) =
+�
+0
+1
+�
+, φ(−1/2)(−ˆz) =
+�
+1
+0
+�
+.
+(2.13)
+The final nucleon spinor is
+u′
+N = ̸p′
+N + m
+�
+E′
+N + m
+�
+φ(λ′
+N)(ˆp′
+N)
+0
+�
+,
+(2.14)
+with
+φ(1/2)(ˆp′
+N) =
+�
+cos(θN/2)
+eiφN sin(θN/2)
+�
+, φ(−1/2)(ˆp′
+N) =
+�
+−e−iφN sin(θN/2)
+cos(θN/2)
+�
+,
+(2.15)
+where θN and φN are the polar and azimuthal angles of the outgoing momentum of the
+particular nucleon.
+As discussed in the Introduction, the overall normalization of the RNHA amplitude is
+unknown. For comparison with data, we consider quantities which are themselves ratios or
+a ratio of the model to data.
+III.
+ELASTIC ELECTRON SCATTERING
+A.
+Form factors
+The three deuteron form factors, GC, GM, and GQ, are readily obtained from the hadronic
+helicity amplitudes of elastic electron-deuteron scattering in the Breit frame [42]. The kine-
+matics are shown in Fig. 3. The photon four-momentum is q = (0, 0, 0, qz) and the initial
+6
+
+(final) deuteron four-momentum is p = (Ed, 0, 0, −qz/2) (p′ = (Ed, 0, 0, qz/2)), with q2
+z = Q2
+and Ed =
+�
+Q2/4 + m2
+d. In the zero-binding limit,3 md = 2m and the individual nucleon
+four-momenta are pp = pn = p/2 and p′
+p = p′
+n = p′/2. The hadronic matrix elements are
+given by
+Gµ
+λ′
+d,λd =
+�
+λ′p,λ′n
+C
+λ′
+d
+λ′pλ′nMµ(λ′
+p, λ′
+n, λd).
+(3.1)
+The initial spinors are as in (2.12); the final spinors are specified by
+u′
+N =
+̸p′/2 + m
+�
+Ed/2 + m
+�
+φ(λ′
+N)(ˆz)
+0
+�
+.
+(3.2)
+⃗q, λγ
+−⃗q/2, λd
+⃗q/2, λ′
+d
+z
+⃗pe, λe
+⃗p ′
+e, λ′
+e
+FIG. 3. Kinematics for elastic electron-deuteron scattering in the Breit frame. The photon travels
+along the positive z direction, and the deuteron comes from the right, along the negative z direction.
+The three form factors are then extracted as [42, 43]
+GC =
+−1
+2md
+√1 + η
+G+
+00 − 2G+
++−
+3
+, GM =
+2
+2md
+√1 + η
+Gx
++0
+√2η, GQ =
+−1
+2md
+√1 + η
+G+
+00 + G+
++−
+2η
+,
+(3.3)
+with η ≡
+Q2
+4m2
+d and the + superscript denoting the light-front sum of the 0 and z components.
+For the helicity matrix elements, the model yields the following Q2 dependence:
+G+
+00 = 0.5588N m
+�m
+Q
+�9 �
+1 + 129.1m2
+Q2 + O(m4
+Q4 )
+�
+,
+(3.4)
+G+
++− = −69.85N m
+�m
+Q
+�11 �
+1 + 4.8m2
+Q2 + O(m4
+Q4 )
+�
+,
+(3.5)
+Gx
++0 = 8.851N m
+�m
+Q
+�10 �
+1 + 4.8m2
+Q2 + O(m4
+Q4 )
+�
+,
+(3.6)
+with N the unknown normalization. The factor of m/Q associated with each helicity flip
+[44] is clearly evident. For the form factors, we find
+GC = − 0.5588
+√1 + η
+N
+12
+�m
+Q
+�9 �
+1 + 379.1m2
+Q2 + O(m4
+Q4 )
+�
+,
+(3.7)
+3 The difference between the proton and neutron masses is neglected in addition to the deuteron binding
+energy, the two being of the same order.
+7
+
+GM =
+8.851
+�
+η(1 + η)
+N
+2
+√
+2
+�m
+Q
+�10 �
+1 + 4.8m2
+Q2 + O(m4
+Q4 )
+�
+,
+(3.8)
+GQ = − 0.5588
+η√1 + η
+N
+8
+�m
+Q
+�9 �
+1 + 4.086m2
+Q2 + O(m4
+Q4 )
+�
+.
+(3.9)
+The leading ± signs are as expected for large Q2.
+We have left the kinematic factor η = Q2/16m2 without substitution, because there can
+be three regimes for Q2. In addition to Q2 large or small, there can be an intermediate
+region where Q2 is large but η is not. Such an intermediate regime does exist for GM and
+GQ, where the coefficients of the nonleading terms are small enough for this correction to
+be small while η is also small. For GC, this is not the case, because the coefficient of the
+nonleading term is large enough to require a Q2 value for which η is also large. In the
+intermediate regime, we obtain
+GM ∼
+�m
+Q
+�11
+, GQ ∼
+�m
+Q
+�11
+,
+(3.10)
+and for the large-η regime
+GC ∼
+�m
+Q
+�10
+, GM ∼
+�m
+Q
+�12
+, GQ ∼
+�m
+Q
+�12
+.
+(3.11)
+Ratios of these form factors at very large Q2 can be compared with the tree-level ratios
+for a point-like spin-one particle, such as the W +, which are [43]
+GC
+GQ
+= 2
+3η − 1,
+GM
+GQ
+= −2.
+(3.12)
+Such behavior is immediately reproduced for form factors separated according to a Drell–
+Yan frame [45], with the assumption of strict G+
+00 dominance [43]. In terms of our hadronic
+matrix elements, we have
+GC
+GQ
+= 2
+3η − 2η
+G+
++−
+G+
+00 + G+
++−
+,
+GM
+GQ
+= −2
+�
+2η
+Gx
++0
+G+
+00 + G+
++−
+.
+(3.13)
+As already observed in [43], these Breit-frame ratios cannot both be resolved by simply
+assuming G+
+00 dominance. From our model, we obtain
+GC
+GQ
+= 2
+3η + 15.6 + O(m2
+Q2 ),
+GM
+GQ
+= −11.2 + O(m2
+Q2 ).
+(3.14)
+The leading 2
+3η is just kinematic. The deviations of 15.6 and -11.2 from -1 and -2, respec-
+tively, are due to nonleading contributions multiplied by powers of η. Similar deviations will
+arise for calculations done in the Drell–Yan frame, because η factors again interfere with
+strict G+
+00 dominance. Plots of these ratios are shown in Fig. 4.
+8
+
+�����V����e/�����
+V��
+V��
+�
+��
+��
+��
+��/M����0
+�
+��
+���
+FIG. 4. Ratios GC/GQ − 2η/3 (dashed) and GM/GQ (solid) for the model deuteron form factors.
+B.
+Structure functions
+Experiments designed to extract these form factors measure cross sections and polariza-
+tion observables in elastic electron-deuteron scattering. The unpolarized cross section
+dσ
+dΩ ∝ S, S ≡ A(Q2) + B(Q2) tan2(θe/2)
+(3.15)
+depends on the electron scattering angle θe and two structure functions
+A(Q2) ≡ G2
+C + 8
+9η2G2
+Q + 2
+3ηG2
+M,
+(3.16)
+B(Q2) ≡ 4
+3η(1 + η)G2
+M.
+(3.17)
+These have been measured at the highest Q2 yet attained at JLab [22–24], and A has been
+measured at comparable Q2 at SLAC [21]. However, these do not yet reach the Q2 values
+needed for a definitive comparison. Figures 5 and 6 show plots of the data divided by the
+model, including an arbitrary normalization factor.
+In our model, expansions of these functions in inverse powers of Q2 are
+A(Q2) = 0.1041N 2
+�m
+Q
+�20 �
+1 + 1246m2
+Q2 + O(m4
+Q4 )
+�
+,
+(3.18)
+B(Q2) = 13.06N 2
+�m
+Q
+�20 �
+1 + 9.6m2
+Q2 + O(m4
+Q4 )
+�
+.
+(3.19)
+Because the expansion for GC is valid only for large η, we have used the explicit form of η in
+constructing the expansion for A. The function B is independent of η; the leading factor of
+9
+
+�
+�l�70V�����
+5
+6
+7
+�
+�
+�7Ql���70
+5
+6
+7
+�
+�
+�
+�
+�
+FIG. 5. Data for the deuteron structure function A(Q2) divided by the model function, including
+an arbitrary normalization. Experimental values are taken from [23] (circles) and [24] (squares).
+η(1+η) in its definition exactly cancels against factors in the relationship of GM to hadronic
+matrix elements. The expansion for B converges much faster than the expansion for A, and
+the leading Q2 behavior is dominant for Q2 ≫ 10 GeV2 only for B. For A, one must wait
+until impossibly large Q2, which enters a regime where the collective quark substructure is
+important, including hidden-color effects [34], and the point-like approximation used in our
+model is invalid.
+In [22] the large Q2 behavior of B is quoted as being Q−24 from perturbative QCD.
+This faster fall off compared to A is attributed to the extra suppression of the helicity flip
+involved in GM. However, there are other compensating factors, and, just as in our model,
+the behavior of B should be Q−20, which is the same as A. In Fig. 7 we plot the ratio of
+B to A for a large range of Q2. This ratio becomes constant at very large Q2. Although
+the plots begin at low Q2, there is nothing in the model that could reproduce diffractive
+minima, hence the smooth appearance.
+C.
+Tensor polarization observables
+Experiments can also extract tensor polarization observables [20, 25]
+t20 ≡ −
+1
+√
+2S
+�8
+3ηGCGQ + 8
+9η2G2
+Q + 1
+3η
+�
+1 + 2(1 + η) tan2(θe/2)
+�
+G2
+M
+�
+,
+(3.20)
+t21 ≡
+2η
+√
+3S cos(θe/2)
+�
+η + η2 sin2(θe/2)GMGQ,
+(3.21)
+t22 ≡ −
+η
+2
+√
+3S G2
+M.
+(3.22)
+10
+
+�
+�l��-G�����
+1V �
+V
+V �
+3
+3 �
+�
+� �
+��Ql����-
+3
+3 �
+�
+� �
+�
+FIG. 6. Data for the deuteron structure function B(Q2) divided by the model function, including
+an arbitrary normalization. Experimental values are taken from [22].
+�
+�8��15�8��1
+�
+��
+��
+��
+��
+���
+���
+���
+��Q8����1
+���
+���
+���
+���
+���
+���
+���
+FIG. 7. Ratio of B to A for the model deuteron structure functions.
+The highest Q2 measurements of these were also done at JLab [25]. When η is held explicit,
+expansions in m/Q are
+t20 = −
+√
+2 + 1064[1 + 2(1 + η) tan2(θe/2)]
+�m
+Q
+�2
++ O(m4
+Q4 ),
+(3.23)
+11
+
+t21 = 38.8 sec(θe/2)
+�
+η + sin2(θe/2)m
+Q
+(3.24)
++ sec(θe/2)
+�
+η + sin2(θe/2)[48606 + 77869(1 + η) tan2(θe/2)]
+�m
+Q
+�3
++ O(m4
+Q4 ),
+t22 = −434.5
+�m
+Q
+�2
++ [544703 + 872133(1 + η) tan2(θe/2)]
+�m
+Q
+�4
++ O(m6
+Q6 ).
+(3.25)
+While at very large Q2, they are
+t20 = −
+√
+2 + 133 tan2(θe/2) + 1064[1 + 2 tan2(θe/2)]
+�m
+Q
+�2
++ O(m4
+Q4 ),
+(3.26)
+t21 = 1217 sec(θe/2) sin(θe/2)[tan2(θe/2) − 0.007972] + []
+�m
+Q
+�2
++ O(m4
+Q4 ),
+(3.27)
+t22 = −[434.5 − 54508 tan2(θe/2)]
+�m
+Q
+�2
+(3.28)
++[544703 + 872133 tan2(θe/2)]
+�m
+Q
+�4
++ O(m6
+Q6 ).
+The coefficients of nonleading terms are quite large. Thus, very large Q2 is required for
+the leading term to be dominant, well beyond any available data. The limit of −
+√
+2 for
+t20 at θe = 0◦ was an early prediction of perturbative QCD [44, 46]. However, as argued
+elsewhere [43], this value is obtained only at very large Q2, and the value is quite different for
+small η. Figures 8 and 9 show plots of these observables at angles of 0◦ and 30◦, respectively.
+We also compare with data [25] in Figs. 10, 11, and 12. At these ‘small’ values of Q2, only
+t22 is consistent with data, something which is likely accidental with both data and model
+values near zero.
+IV.
+PHOTODISINTEGRATION
+In the photodisintegration of a deuteron, a real photon is absorbed and the two constituent
+nucleons emitted. This process is depicted in Fig. 13. The initial deuteron and photon four-
+momenta in the center-of-mass (c.m.) frame are p = (Ed, 0, 0, −qz) and q = (qz, 0, 0, qz),
+where the incident photon is taken along the positive z axis. The final proton and neutron
+four-momenta are p′
+p = (E′
+p, ⃗p ′
+p) and p′
+n = (E′
+n, ⃗p ′
+n), with θp and φp the polar and azimuthal
+angles of the final proton.
+By ignoring the nucleon mass difference, we have E′
+p = E′
+n,
+because momentum conservation guarantees ⃗p ′
+n = −⃗p ′
+p in the c.m. frame.
+In terms of the Mandelstam variable s, the c.m. energies and momenta are
+Ed = (s + 4m2)/(2√s), qz = (s − 4m2)/(2√s), E′
+p = √s/2, |⃗p ′
+p| =
+√
+s − 4m2/2.
+(4.1)
+The photon energy in the lab frame is Eγ = (s − 4m2)/4m. We will work at large s, so that
+momentum transfers are large.
+The standard definition of helicity amplitudes for photodisintegration is [47]
+Fi± ≡ ǫν(λγ)Mν(λ′
+p, λ′
+n, λd)
+(4.2)
+12
+
+�
+��5e0���e0���
+V�)�
+V�)�
+V�)�
+V�
+V5)�
+V5)�
+V5)�
+V5)�
+5
+��08����Q
+�55
+�5�
+�5�
+�5�
+�5�
+�5�
+�5�
+FIG. 8.
+Deuteron tensor polarization observables t20 (solid), t21 (dashed), and t22 (dotted) as
+computed in the model at an angle of θe = 0◦.
+The asymptotic value of t20(0◦) is −
+√
+2, as
+predicted by perturbative QCD [44, 46].
+with ǫ the polarization vector for a photon with helicity λγ and Mν given in (2.8). The
+index i is associated with particular helicity combinations as follows:
+F1± = ǫν(1)Mν(±1
+2, ±1
+2, 1), F2± = ǫν(1)Mν(±1
+2, ±1
+2, 0),
+(4.3)
+F3± = ǫν(1)Mν(±1
+2, ±1
+2, −1), F4± = ǫν(1)Mν(±1
+2, ∓1
+2, 1),
+(4.4)
+F5± = ǫν(1)Mν(±1
+2, ∓1
+2, 0), F6± = ǫν(1)Mν(±1
+2, ∓1
+2, −1).
+(4.5)
+The other helicity combinations are related to these by parity.
+The helicity amplitudes can be used to compute various polarization observables. The
+recoil-proton polarization Py measures the asymmetry parallel/antiparallel to the normal
+ˆy ∝ ⃗q × ⃗p ′
+p to the scattering plane:
+Py = 2Im
+3
+�
+i=1
+[F †
+i+Fi+3,− + F †
+i+3,+Fi−]/f(θ),
+(4.6)
+where f(θ) =
+�6
+i=1[|Fi+|2 + |Fi−|2] is the sum of all the helicity amplitudes squared. The
+transferred polarizations Cx′ and Cz′ measure asymmetries parallel/antiparallel to the ˆx′ ∝
+⃗p ′
+p × ˆy and ˆz′ = ˆp ′
+p directions:
+Cx′ = 2Re
+3
+�
+i=1
+[F †
+i+Fi+3,− + F †
+i+3,+Fi−]/f(θ),
+(4.7)
+Cz′ =
+6
+�
+i=1
+[|Fi+|2 − |Fi−|2]/f(θ).
+(4.8)
+13
+
+�
+���e0���e0���
+V�)�
+V�)�
+V�)�
+V�)�
+V�)�
+�
+��03����Q
+���
+���
+���
+���
+���
+���
+���
+FIG. 9. Same as Fig. 8 but for an angle of θe = 30◦.
+The asymmetry Σ for linearly polarized photons is given by
+Σ = −2Re
+��
+±
+(F †
+1±F3∓ − F †
+4±F6∓) − F †
+2+F2− + F †
+5+F5−
+�
+/f(θ).
+(4.9)
+Each observable is formed as a ratio, which sets aside questions of normalization.
+Because we only need to consider photons with helicity +1, the polarization vector is
+always ǫ = − 1
+√
+2(0, 1, i, 0), relative to the momentum in the positive z direction. The final
+Dirac spinors are
+u′
+N = ̸p′
+N + m
+�
+E′
+N + m
+�
+φ(λ′
+N)(ˆp′
+N)
+0
+�
+,
+(4.10)
+with θn = π − θp, φn = φp + π = π, and
+φ(1/2)(ˆp′
+N) =
+�
+cos(θN/2)
+eiφN sin(θN/2)
+�
+, φ(−1/2)(ˆp′
+N) =
+�
+−e−iφN sin(θN/2)
+cos(θN/2)
+�
+.
+(4.11)
+With these spinors as input, the amplitudes ǫνMν
+X can be evaluated in terms of Dirac
+matrix and spinor products and then combined to construct the predefined amplitudes Fi±.
+At large s, these RNHA predictions for the helicity amplitudes reduce to
+F1+ ∼ 4
+√
+2
+√scsc2(θp
+2 )GEn(θp)GEp(θp), F1− ∼ 0,
+(4.12)
+F2+ ∼ 2m
+s cot3(θp
+2 )[GEn(θp)GMp(θp) − GMn(θp)GEp(θp)],
+(4.13)
+F2− ∼ 2m
+s cot(θp
+2 )[GMn(θp)GEp(θp) − GEn(θp)GMp(θp)],
+F3+ ∼ 0, F3− ∼ 0,
+(4.14)
+14
+
+�
+���
+1���
+1���
+1���
+1���
+�
+���
+���
+���
+���
+��0(����G
+���
+���
+���
+���
+���
+���
+���
+FIG. 10. Plots of the tensor polarization observable t20 of the deuteron from both data [25] (circles)
+and the model (squares) considered in the text. The angle θe varies and is as follows in order of
+increasing Q2: 35.6◦, 33.4◦, 29.8◦, 27.3◦, 23.0◦, and 19.8◦.
+F4+ ∼ −4
+√
+2m
+s cot(θp
+2 )GMn(θp)GEp(θp), F4− ∼ 4
+√
+2m
+s cot(θp
+2 )GEn(θp)GMp(θp), (4.15)
+F5+ ∼ 2
+√s cot2(θp
+2 )GEn(θp)GEp(θp), F5− ∼ 2
+√sGEn(θp)GEp(θp),
+(4.16)
+F6+ ∼ 4
+√
+2m
+s cot3(θp
+2 )GEn(θp)GMp(θp), F6− ∼ −4
+√
+2m
+s cot(θp
+2 )GMn(θp)GEp(θp).(4.17)
+From these we can calculate the various observables. Plots of the results and recent data [48–
+50] are given in Figs. 14, 15, 16, and 17. Because the tree-level amplitudes are real, Py is
+automatically zero. That Cx′ is of order m/√s, rather than zero, is a correction to hadron
+helicity conservation [51]. Also, we find the asymmetry Σ(90◦) to be approximately -0.06,
+rather than the nominal expectation [52] of -1. In general, the trends with photon energy
+seem to be modestly consistent with data.
+V.
+ELECTRODISINTEGRATION
+The kinematics of the electrodisintegration process are shown in Fig. 18. The initial (final)
+momentum and helicity of the electron are pe (p′
+e) and λe (λ′
+e). The intermediate photon
+carries four-momentum q. The azimuthal angle φp of the proton measures the rotation of
+the hadronic reaction plane relative to the electron scattering plane.
+In the lab frame, with the z axis taken along the photon three-momentum and the electron
+mass neglected, the initial and final electron four-momenta are
+pe = (Ee, Ee sin θe, 0, Ee cos θe), p′
+e = (E′
+e, E′
+e sin θ′
+e, 0, E′
+e cos θ′
+e),
+(5.1)
+15
+
+�
+���
+V�)�
+V�)�
+�
+�)�
+�)�
+�)�
+�)�
+�)�
+�)�
+��1Q���� 
+�)�
+�)�
+�
+�)�
+�)�
+�)�
+�)�
+FIG. 11. Same as Fig. 10 but for t21.
+�
+���
+8�4��
+8�4�
+8�4��
+8�4�
+8�4��
+�
+�4��
+�4�
+�4��
+��-G����e
+�4�
+�4�
+�
+�4�
+�4�
+�4�
+�4�
+FIG. 12. Same as Fig. 10 but for t22.
+with E′
+e and ˜θ = θ′
+e − θe, the angle of the scattered electron to the beam direction, being
+measured. The photon four-momentum q = (Eγ, 0, 0, qz) is just pe − p′
+e, which yields
+Q2 ≡ −q2 = 2EeE′
+e(1 − cos ˜θ), Eγ = Ee − E′
+e, qz =
+�
+E2γ + Q2.
+(5.2)
+The deuteron four-momentum is p = (md = 2m, 0, 0, 0), and in the zero-binding limit, the
+initial proton and neutron four-momenta are pp = pn = (m, 0, 0, 0).
+The final nucleon
+16
+
+⃗q, λγ
+⃗p, λd
+⃗p ′
+p, λ′
+p
+⃗p ′
+n, λ′
+n
+θn = π − θp
+θp
+z
+FIG. 13. Kinematics for deuteron photodisintegration in the c.m. frame, with ⃗q the photon mo-
+mentum and ⃗p = −⃗q the deuteron momentum. The final proton and neutron momenta are ⃗p ′
+p and
+⃗p ′
+n. The λ’s are helicities. Coordinates are chosen such that the photon enters along the positive
+z direction and the azimuthal angle φp of the proton is zero.
+Eγ (GeV)
+0
+1
+2
+3
+4
+5
+6
+Py (90 deg)
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+θ (deg)
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Py (Eγ=2 GeV)
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a)
+(b)
+FIG. 14. Recoil proton polarization Py as a function of (a) photon energy Eγ and (b) proton angle
+θ. For the latter, the photon energy is 2 GeV. The solid line is the RNHA prediction; the data
+points are from [48, 49].
+four-momenta are
+p′
+p = (E′
+p =
+�
+⃗p ′2
+p + m2, |⃗p ′
+p| sin θp cos φp, |⃗p ′
+p| sin θp sin φp, |⃗p ′
+p| cos θp),
+(5.3)
+p′
+n = (E′
+n =
+�
+⃗p ′2
+n + m2, −|⃗p ′
+n| sin θn cos φp, −|⃗p ′
+n| sin θn sin φp, |⃗p ′
+n| cos θn).
+(5.4)
+Within the one-photon-exchange approximation, the scattering amplitude is proportional
+17
+
+Eγ (GeV)
+0
+1
+2
+3
+4
+5
+6
+Cx' (90 deg)
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+θ (deg)
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Cx' (Eγ=2 GeV)
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a)
+(b)
+FIG. 15. Same as Fig. 14 but for the transferred polarization Cx′.
+Eγ (GeV)
+0
+1
+2
+3
+4
+5
+6
+Cz' (90 deg)
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+θ (deg)
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Cz' (Eγ=2 GeV)
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a)
+(b)
+FIG. 16. Same as Fig. 15 but for Cz′.
+to
+Med(λ′
+p, λ′
+n, λ′
+e; λd, λe) = ¯u′
+eγµue
+Dµν
+q2 Mν(λ′
+p, λ′
+n, λd),
+(5.5)
+with ue (u′
+e) the initial (final) spinor of the electron and Mν given in (2.8). The numerator
+of the photon progator is the sum over photon polarizations
+Dµν =
+1
+�
+λ=−1
+(−1)λǫ∗
+µ(λ)ǫν(λ).
+(5.6)
+18
+
+Eγ (GeV)
+0
+1
+2
+3
+4
+5
+6
+Σ (90 deg)
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+θ (deg)
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Σ (Eγ=2 GeV)
+-0.10
+-0.08
+-0.06
+-0.04
+-0.02
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+(a)
+(b)
+FIG. 17. Same as Fig. 14 but for the asymmetry Σ. The data points are from [50].
+The polarization four-vectors are4
+ǫ(±1) = ∓ 1
+√
+2(0, 1, ±i, 0), ǫ(0) = (qz/Q, 0, 0, Eγ/Q)
+(5.7)
+relative to the photon four-momentum q = (Eγ, 0, 0, qz). Polarization observables [5–15] can
+then be computed from these helicity amplitudes.
+⃗pe, λe
+⃗p ′
+e, λ′
+e
+x
+⃗q
+⃗p ′
+p, λ′
+p
+⃗p ′
+n, λ′
+n
+y′
+x′
+z, z′
+y
+θp
+φp
+θe
+θ′
+e
+z
+θn
+˜θ
+FIG. 18. Kinematics for deuteron electrodisintegration. The unprimed axes are defined relative to
+the electron scattering plane, and the primed axes relative to the final nucleon momenta. The final
+proton momentum has polar angle θp and azimuthal angle φp relative to the unprimed frame.
+4 In the hadronic c.m. frame, the longitudinal polarization vector is ǫ(0) = (q′
+z/Q, 0, 0, E′
+γ/Q).
+19
+
+In keeping with the notation of [6, 7] and [15], the differential cross section for elec-
+trodisintegration, summed over the final electron and neutron helicities in the lab frame,
+is [14, 15]5
+dσ ≡
+dσ5
+dE′dΩ′
+edΩ′
+p
+(5.8)
+= mpmn|⃗p ′
+p|
+16π3md
+σMott
+frec
+[νLRL + νT RT + νTTRTT + νLTRLT + 2λeνLT ′TLT ′ + 2λeνT ′RT ′] ,
+where Ω′
+e (Ω′
+p) is the solid angle of the scattered electron (proton), σMott is the Mott cross
+section, frec = |1 + (Eγ|⃗p ′
+p| − E′
+pqz cos θp)/(md|⃗p ′
+p|)| is the lab recoil factor,
+νL = Q4
+q4z
+, νT = Q2
+2q2z
++ tan2 ˜θ
+2, νTT = Q2
+2q2z
+, νLT =
+Q2
+√
+2q2
+z
+�
+�
+�
+�Q2
+q2z
++ tan2 ˜θ
+2,
+(5.9)
+νLT ′ = − Q2
+√
+2q2z
+tan
+˜θ
+2, νT ′ = tan
+˜θ
+2
+�
+�
+�
+�Q2
+q2
+z
++ tan2 ˜θ
+2,
+and ˜θ = θ′
+e − θe is the angle between the incoming and outgoing electron. The response
+functions RX depend upon the hadronic helicity amplitudes and the azimuthal angle φp of
+the hadronic scattering plane. The subscripts refer to the polarization of the intermediate
+photon, which enters on substitution of the polarization expansion (5.6) for the numerator
+of the photon propagator in the hadronic amplitude (5.5). The amplitude then decomposes
+into separate leptonic and hadronic factors
+Med(λ′
+p, λ′
+n, λ′
+e; λd, λe) = −
+1
+�
+λ=−1
+¯u′
+e̸ ǫ∗(λ)ue
+(−1)λ
+Q2
+ǫν(λ)Mν(λ′
+p, λ′
+n, λd).
+(5.10)
+The leptonic factors give rise to the νX coefficients, and the hadronic factors to the response
+functions in the square of the amplitude used to construct the cross section [15].
+The
+subscript L(T) indicates a purely longitudinal (transverse) contribution, while LT is a cross
+term between longitudinal and transverse photon helicities. The TT subscript marks a cross
+term between different transverse helicities. A prime indicates a different combination of
+transverse helicities.
+The response functions are computed from components of the hadronic tensor
+wλ′,λ = 2
+3
+�
+λ′′p,λ′p,λ′n,λ′′
+d,λd
+ǫ∗
+ν(λ′)Mν∗(λ′′
+p, λ′
+n, λ′′
+d)ρp
+λ′′p,λ′pǫµ(λ)Mµ(λ′
+p, λ′
+n, λd)ρd
+λ′′
+d,λd,
+(5.11)
+with ρp(ρd) the density matrix for the proton (deuteron) helicity state. We construct these
+in the xyz coordinate system of the electron scattering plane. The particular components
+are [6]
+RL = w0,0, RT = w1,1 + w−1,−1, RT ′ = w1,1 − w−1,−1,
+(5.12)
+RTT = 2Rew1,−1, RLT = −2Re [w0,1 − w0,−1] , RLT ′ = −2Re [w0,1 + w0,−1] .
+5 In [6], h is 2λe but in [15], h is just λe, which leads to additional factors of 2.
+20
+
+For an unpolarized target, the deuteron density matrix is proportional to the identity,
+ρd =
+1
+3I; similarly, if the proton helicity is not detected, ρp =
+1
+2I.
+We then have the
+unpolarized cross section [6]
+dσunpol = mpmn|⃗p ′
+p|
+16π3md
+σMott
+frec
+σ0, σ0 ≡ νLRU
+L + νTRU
+T + νTTRU
+TT + νLTRU
+LT ,
+(5.13)
+where the RU
+X are computed with the simple density matrices. These are then computable
+in our model, with the basic computation being the evaluation of ǫ(λγ)µMµ, which differs
+from the photodisintegration calculation in only two ways: Q2 is not zero and λγ ranges
+over all three possibilities.
+The unpolarized response functions RU
+LT ′ and RU
+T ′ are identically zero. With ρd replaced
+by 1
+3I and the form (5.7) of the polarization vectors taken into account, Rew(0, 1) is just
+the negative of Rew(0, −1), and w1,1 is equal to w−1,−1. Thus, the inputs to RU
+LT ′ and RU
+T ′,
+as given in (5.12), immediately cancel.
+The recent ed → e′pn experiment at JLab [17] does not include polarization but does
+begin to reach momentum transfers sufficient to consider the RNHA approach. Once po-
+larization data is available, the expressions developed here and in the Appendix can be
+compared.
+VI.
+SUMMARY
+We have extended the reduced nuclear amplitude approach [1, 2] to helicity amplitudes
+and applied this model to analysis of elastic electron-deuteron scattering, deuteron photo-
+disintegration, and deuteron electrodisintegration. These are just examples of the approach,
+which is generally applicable to exclusive nuclear processes. The primary limitation is that,
+for any process, the net momentum transfer to every nucleon must be large; therefore,
+as the number of nucleons increases, the required beam energy can increase dramatically.
+The primary gain is precocious scaling in the dependence on momentum transfer. What the
+model (or the original RNA approach) does not provide, though, is an overall normalization;
+comparisons must be made in terms of ratios.
+By considering helicity amplitudes, many more quantities can be studied, including po-
+larization dependence. All three of the deuteron’s electromagnetic form factors can be cal-
+culated and from there various elastic scattering observables can be constructed. In Sec. III
+we considered the standard structure functions A and B as well as the tensor polarizations
+t2m. Generally, the model implies the need for momentum transfers larger than one would
+have hoped for seeing simple perturbative QCD scaling. However, our results do imply that
+the deuteron structure function B is a good place to look, above a transfer of 10 GeV2.
+The RNHA results for polarization observables in deuteron photodisintegration, consid-
+ered in Sec. IV, are somewhat consistent with experiment. In particular, our result for the
+asymmetry Σ, with a value of Σ(90◦) ≃ −0.06, is much better than the value of -1 originally
+expected [52]. Higher photon energies would, of course, be useful.
+We have also constructed the RNHA framework for analysis of deuteron electrodisinte-
+gration, in Sec. V. This stands ready for comparison with experiment when data is available
+at sufficient energies. One aspect that does remain is to consider polarization of the outgoing
+proton, in addition to polarization of the beam and target.
+Other processes that one might consider include deeply virtual Compton scattering on
+the deuteron, pion photoproduction on the deuteron [3], and photodisintegration of 3He [4].
+21
+
+In each case, our approach can provide not only information about helicity amplitudes but
+also an analysis of nonleading momentum transfer dependence with respect to the onset of
+perturbative QCD scaling. We look forward to experiments at larger momentum transfers
+for all of these processes.
+ACKNOWLEDGMENTS
+This work began in conversations with S.J. Brodsky and D.-S. Hwang. Some calculations
+were checked by W. Miller and C. Salveson. Diagrams were drawn with use of JaxoDraw [53].
+Appendix A: Electrodisintegration with polarization
+If we consider polarization for the beam and the target,6 the proton density matrix is
+still just ρp = 1
+2I, but the deuteron density matrix in the xyz frame is [6]
+ρd = 1
+3
+
+
+
+
+
+1 +
+�
+3
+2T10 +
+1
+√
+2T20 −
+�
+3
+2(T ∗
+11 + T ∗
+21)
+√
+3T ∗
+22
+−
+�
+3
+2(T11 + T21)
+1 −
+√
+2T20
+−
+�
+3
+2(T ∗
+11 − T ∗
+21)
+√
+3T22
+−
+�
+3
+2(T11 − T21) 1 −
+�
+3
+2T10 +
+1
+√
+2T20
+
+
+
+
+ .
+(A1)
+For a target polarization defined relative to the beam direction, rather than the xyz system
+used above, the tensor polarization coefficients TJM are related to the coefficients ˜TJM defined
+relative to the beam [6]. If only ˜T10 and ˜T20 are nonzero,7 the nonzero TJM are
+T10 = cos ˜θ ˜T10, T11 = − 1
+√
+2 sin ˜θ ˜T10,
+(A2)
+T20 = 1
+4(1 + 3 cos 2˜θ) ˜T20, T21 = −
+�
+3
+8 sin 2˜θ ˜T20, T22 =
+�
+3
+32(1 − cos 2˜θ) ˜T20.
+The density matrix can then be written as
+ρd =
+�1
+3I + ˜T10ρdV + ˜T20ρdT
+�
+,
+(A3)
+where
+ρdV = 1
+3
+
+
+
+
+
+�
+3
+2 cos ˜θ
+√
+3
+2 sin ˜θ
+0
+√
+3
+2 sin ˜θ
+0
+√
+3
+2 sin ˜θ
+0
+√
+3
+2 sin ˜θ −
+�
+3
+2 cos ˜θ
+
+
+
+
+
+(A4)
+and
+ρdT = 1
+3
+
+
+
+
+1
+4
+√
+2(1 + 3 cos 2˜θ)
+3
+4 sin 2˜θ
+3
+√
+32(1 − cos 2˜θ)
+3
+4 sin 2˜θ
+−
+1
+2
+√
+2(1 + 3 cos 2˜θ)
+−3
+4 sin 2˜θ
+3
+√
+32(1 − cos 2˜θ)
+−3
+4 sin 2˜θ
+1
+4
+√
+2(1 + 3 cos 2˜θ)
+
+
+
+ .
+(A5)
+6 For discussion of a polarized outgoing proton, see [7] and [15].
+7 The spherical tensor moments are related to the Cartesian tensor moments as ˜T10 =
+�
+3
+2Pz and ˜T20 =
+1
+√
+2Pzz.
+22
+
+The response functions can then be separated into unpolarized, vector, and tensor contri-
+butions as RX = RU
+X + ˜T10RV
+X + ˜T20RT
+X, with RU
+X, RV
+X, and RT
+X computed with ρd replaced
+by 1
+3I, ρdV , and ρdT , respectively.
+With dσunpol defined as the unpolarized cross section, given in (5.13), the full cross section
+can be written as
+dσ =
+�
+1 + ˜T10
+�
+AV
+d + 2λeAV
+ed
+�
++ ˜T20
+�
+AT
+d + 2λeAT
+ed
+��
+dσunpol,
+(A6)
+in terms of the single and double asymmetries
+AV
+d =
+�
+νLRV
+L + νTRV
+T + νTTRV
+TT + νLT RV
+LT
+�
+/σ0,
+(A7)
+AV
+ed =
+�
+νLT ′RV
+LT ′ + νT ′RV
+T ′
+�
+/σ0,
+(A8)
+AT
+d =
+�
+νLRT
+L + νTRT
+T + νTTRT
+TT + νLTRT
+LT
+�
+/σ0,
+(A9)
+AT
+ed =
+�
+νLT ′RT
+LT ′ + νT ′RT
+T ′
+�
+/σ0.
+(A10)
+For a recent summary of data, see [54].
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+

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+Recursive Fermi-operator expansion strategies to accelerate subspace
+diagonalization for large eigenvalue problems in density functional theory
+Sameer Khadatkar1 and Phani Motamarri1
+Indian Institute of Science, Bengaluru, India.
+(*Electronic mail: [email protected])
+Quantum mechanical calculations for material modeling using density functional theory (DFT) involves solving a
+large-scale nonlinear eigenvalue problem. These calculations are computationally demanding and have asymptotic
+cubic scaling complexity with the number of electrons in the material system. The efficient computational strategies
+used to solve these large nonlinear DFT eigenvalue problems rely on iterative orthogonal projection methods. The
+Rayleigh-Ritz projection step and the subspace diagonalization incur the dominant computational cost in these projec-
+tion methods. In this work, we explore scalable polynomial expansion based on recursive Fermi-operator expansion
+approaches using mixed-precision arithmetic as an alternative to subspace diagonalization of the projected Hamiltonian
+to reduce the computational cost. The performance and accuracy of these approaches have been thoroughly assessed by
+comparing them with the explicit diagonalization approach using the state-of-the-art ELPA library on both multinode
+CPUs and GPUs.
+I.
+INTRODUCTION
+Eigenvalue problems are frequently encountered in many
+scientific disciplines. For instance, the accurate and efficient
+computation of eigenvectors and eigenvalues is critical in the
+study of resonance, understanding the stability of fluid flows
+subjected to small perturbations, obtaining insights into vibra-
+tional modes of lattices, dimensionality reduction, and many
+more.
+Another well-known and challenging application of
+eigenvalue problems is in the area of quantum modeling of
+materials using Kohn-Sham density functional theory (DFT)1,
+which has been immensely successful in providing critical in-
+sights into various ground-state material properties. To com-
+pute the ground-state electronic structure in DFT, one is con-
+fronted with solving a large-scale nonlinear eigenvalue prob-
+lem using a self-consistent field iteration procedure (SCF) for
+N smallest eigenvalue/eigenvector pairs, with N being pro-
+portional to the number of electrons in the material system.
+This results in asymptotic cubic complexity O(N3) with the
+number of electrons for DFT, making these calculations com-
+putationally demanding and often restrictive in terms of sys-
+tem sizes that can be handled using widely used DFT codes.
+Many of these codes employ plane-wave basis sets, which re-
+strict simulation domains to periodic or atomic-orbital type
+basis sets, which are not systematically convergent, and these
+basis sets are not amenable for massive parallelization on par-
+allel computing architectures. To extend the range of system
+sizes to be studied, numerous past efforts have focused on de-
+veloping systematically convergent real-space computational
+methodologies 2–6 that have focused on reducing the prefac-
+tor associated with the cubic computational complexity along-
+side improving the parallel scalability, thereby enabling large-
+scale DFT calculations up to 100,000 electrons. These real-
+space DFT discretization approaches result in large sparse
+Hermitian eigenvalue problems of the form Hψψψi = εh
+i ψψψi to
+be solved for N smallest eigenvalue/eigenvector pairs, with N
+being proportional to M, the dimension of the sparse Hamil-
+tonian matrix H (M ≈ 105 −107). We note that N depends on
+the number of electrons in the material system and is usually
+0.1 − 0.5% of M, the degrees of freedom (DoFs) used in the
+simulation.
+In the electronic structure community, the most popular
+eigensolver strategies employed to solve these large DFT
+eigenvalue problems include the Davidson approach, Lo-
+cally Optimal Block Pre-conditioned Conjugate Gradient
+(LOBPCG) method, or the Chebyshev filtered subspace it-
+eration (ChFSI) approach. These eigensolvers belong to the
+category of iterative orthogonal projection methods (IOP)
+wherein the matrix H is orthogonally projected onto a care-
+fully constructed subspace rich in the wanted eigenvectors
+(Rayleigh-Ritz step), and subsequently, the resulting smaller
+dense projected Hamiltonian Hp is explicitly diagonalized
+(subspace diagonalization) to approximate the desired eigen-
+value/eigenvector pairs of the H matrix.
+The cubic scal-
+ing computational cost of this subspace diagonalization step
+dominates for medium to large-scale material systems (N >
+20,000) in comparison to the costs associated with subspace
+construction and Rayleigh-Ritz steps in IOP methods. For in-
+stance, the authors6 employing the ChFSI approach have re-
+ported that the subspace diagonalization constitutes roughly
+30% of the total ChFSI cost for N ≈ 30,000, whereas it ac-
+counts for around 56% of the total cost for N ≈ 50,000. To
+this end, the current work explores recursive polynomial ex-
+pansion approaches based on Fermi-operator expansion as an
+alternative to the subspace diagonalization procedure to im-
+prove the computational efficiency, thereby reducing the com-
+putational prefactor associated with the cubic complexity of
+the subspace diagonalization approach. Furthermore, the en-
+ergy efficiency and parallel scaling efficiency of these ap-
+proaches is examined on both multinode CPUs and multinode
+GPUs.
+Recursive polynomial expansion approaches (RPE) rely
+on the key idea that constructing a density matrix (projec-
+tor matrix corresponding to N smallest eigenvectors) suffices
+to compute ground-state electronic structure in DFT at zero-
+temperature without the necessity of knowing explicit eigen-
+values and eigenvectors. These RPE approaches 7–12 have
+been explored in the past for conducting ground-state DFT
+calculations using atomic-orbital basis. However, the compu-
+tational efficiency, scaling and energy efficiency of these ap-
+proaches have not been explored in comparison to subspace
+arXiv:2301.04642v1  [physics.comp-ph]  11 Jan 2023
+
+2
+diagonalization procedures for their use in iteration orthogo-
+nal projection methods on multinode CPU and GPU architec-
+tures. The evolving computing architectures in today’s ex-
+ascale era place a heavy emphasis on scalable methodolo-
+gies with a focus on reduced data movement and increased
+arithmetic intensity, with an equal emphasis on using energy-
+efficient algorithms. The current work assumes significance
+in this regard and is useful for solving large-scale eigenvalue
+problems arising from the discretization of DFT using sys-
+tematically convergent basis sets employing IOP methods. To
+this end, the key contributions of our current work, as de-
+scribed in the subsequent sections, include – (a) efficient im-
+plementation strategies of various recursive polynomial ex-
+pansion (RPE) techniques based on Fermi-operator expansion
+on both multinode CPU and GPU architectures for both zero-
+temperature case and the finite-temperature case of Fermi-
+dirac smearing of the occupancy function (b) design of mixed
+precision strategies in conjunction with RPE to reduce com-
+pute and data access costs (c) assessing accuracy, scaling effi-
+ciency and energy efficiency of the proposed implementation
+procedures by comparing it with explicit diagonalization al-
+gorithms provided by state-of-the-art ELPA library13.
+II.
+RELATED WORK AND BACKGROUND
+This section discusses key ideas central to recursive poly-
+nomial expansion approaches which are used to approximate
+the density matrix.
+A.
+Density matrix
+At zero electronic temperature, the density matrix (D) can
+be defined as a projector matrix corresponding to the lowest
+occupied (Nocc ≤ N) eigenvectors of the Kohn-Sham Hamil-
+tonian H matrix.
+Mathematically, it takes the form of a
+shifted Heaviside step function, θ(.), given by D = θ[µI−H].
+The density matrix (D) in the case of finite-temperature is a
+smeared version of zero-temperature density matrix and math-
+ematically represented by a Fermi-operator matrix function
+given by D = [eβ(H−µI) + I]−1, where, I denotes the identity
+matrix, β = 1/(kBTe) is the inverse electronic temperature, µ
+is the Fermi-energy, and H is the Hamiltonian matrix. Note
+that the eigenvalues fi of D are referred to as occupancies. fi
+is either 0 or 1 for a zero-temperature case whereas for the
+case of a finite-temperature case, fi ∈ [0,1].
+B.
+Recursive polynomial expansion techniques to
+approximate the density matrix
+Two types of polynomial expansion schemes can be used
+to approximate the density matrix – (a) Serial Fermi-operator
+expansion schemes (Chebyshev Fermi-operator expansion
+scheme14, Green’s function expansion scheme15, etc), (b)
+Recursive Fermi-operator expansion schemes 8–12.
+In this
+work, we employ the latter approach i.e., the recursive Fermi-
+operator expansion schemes as they are shown to be more ef-
+ficient and can be used to approximate the density matrix for
+both zero-temperature, and finite-temperature cases as well.
+1.
+Recursive Fermi-operator expansion for zero-temperature
+density matrix (Z-RFOE)
+The recursive Fermi-operator expansion8 involves succes-
+sive projections of a matrix Xn, where X0 = H and Xn+1 =
+F(Xn). The functions F(Xn) are chosen to project the eigen-
+value spectrum of Xn to eigenvalues closer either to 1 or to 0.
+Mathematically this can be represented as
+D = θ(µI−H) ≈ Fm(Fm−1(...F0(H)...))
+(1)
+One of the most efficient techniques in Z-RFOE is to use the
+second-order projection polynomials (SP2) 9 given by Xn+1 =
+Fn(Xn) = Xn ± (Xn − X2
+n). The SP2 here are continuously
+increasing and decreasing functions in the interval [0, 1]. The
+± sign is chosen to adjust the trace of Xn+1 in each projection
+such that it converges to Nocc.
+2.
+Accelerated recursive polynomial expansion for
+zero-temperature density matrix (A-Z-RFOE)
+This technique works on the concept of shifting and scaling.
+In Z-RFOE, we used SP2 polynomials, which either took the
+form F(X) = X2 or F(X) = 2X−X2. In A-Z-RFOE, instead
+of restricting ourselves to these fixed expansion functions, we
+give it some freedom to choose the expansion functions such
+that it moves the eigenvalues closer to either 1 or 0 faster. To
+optimize the convergence, we chose the polynomial such that
+each iteration gives the highest slope of projection around the
+eigenvalues, which are rescaled values of the HOMO (Highest
+Occupied Molecular Orbital) and LUMO (Lowest Unoccu-
+pied Molecular Orbital) eigenvalues and done such that there
+is no risk of eigenvalues switching the places between the oc-
+cupied and the unoccupied states10.
+3.
+Recursive Fermi-operator expansion scheme for
+finite-temperature cases (T-RFOE)
+Finite-temperature density matrix has occupancies fi ∈
+[0,1] and the SP2 recursion scheme discussed above is not
+well suited for approximating density matrix with fractional
+occupancies.
+To this end, an intermediate function gener-
+ated in Z-RFOE that is obtained before the convergence of
+the algorithm to the projector matrix is used. This serves as a
+smeared function to zero-temperature density matrix (Heavi-
+side step function). To this end, the truncated expansion for
+computing, the density matrix D can be given by the expres-
+sion in (2).
+Gm(H) = Fm(Fm−1(...F0(H)...))
+(2)
+Lower the electronic temperature Te higher will be the β value
+(refer to Sec. IIA), and more recursion steps m will be re-
+quired to approximate the density matrix11,12.
+C.
+Accuracy of the polynomial expansion procedures
+The accuracy of the aforementioned polynomial expansion
+procedures is given in terms of the degree npl of the polyno-
+mial needed to approximate the density matrix and is given
+by npl ∝ (εN −ε1) with ε1,εN being spectral bound estimates
+
+3
+of H16. It is well known that DFT discretized Hamiltonian H
+using real-space approaches has large spectral width εN − ε1
+resulting in higher npl for approximating the density matrix.
+Often this leads to an inefficient computational procedure to
+approximate the density matrix since the dimension of H can
+be of O(105 −107).
+III.
+COMPUTATIONAL METHODOLOGY AND
+IMPLEMENTATION
+A.
+Proposed methodology
+Due to the aforementioned limitations of employing the re-
+cursive polynomial expansion procedures on the real-space
+discretized DFT Hamiltonian (H), we resort to iterative or-
+thogonal projection (IOP) methods of solving a large sparse
+eigenvalue problem and choose to work with the smaller dense
+projected Hamiltonian Hp in the subspace rich with eigenvec-
+tors of H. To this end, we employ the recursive polynomial
+expansion procedures on Hp to approximate the density ma-
+trix in the subspace as an alternative to explicit subspace diag-
+onalization. Since the spectral width of Hp is commensurate
+with spectral width corresponding to occupied eigenstates, it
+is small and the proposed approach is computationally effi-
+cient as demonstrated subsequently.
+B.
+Algorithmic details
+Using Hp, Z-RFOE and A-Z-RFOE schemes employing
+SP2 polynomials for approximating zero-temperature density
+matrix and the T-RFOE scheme for the finite-temperature den-
+sity matrix have been implemented in a distributed setting.
+Figure 1 shows the schematic of the RFOE algorithm imple-
+mented in the current work.
+Furthermore, we also explored mixed-precision strategies
+in conjunction with the RFOE schemes implemented in this
+work. To this end, we rely on the fact that far away from
+RFOE convergence to the appropriate density matrix, the
+floating point operations involved in the initial RFOE itera-
+tions can be performed in single precision (FP32) and switch-
+ing to FP64 operations thereafter. The criteria to decide the
+number of initial FP32 iterations is linked to relative trace
+change of Xn (εtr) of two successive RFOE iterations. An es-
+timate of εtr could be obtained by examining the dependence
+of εtr on the relative change in occupied eigensubspace be-
+tween the starting matrix X0 = Hp and the intermediate ma-
+trices Xn generated during the course of RFOE. Our numerical
+studies on smaller size representative Hp arising in DFT show
+that εtr ≈ O(10−4) gives an acceptable error of O(10−7) with
+respect to fully double precision (FP64) computation of the
+density matrix.
+C.
+Implementation details
+The multinode parallel implementation of RFOE codes was
+done in C++ employing Message Passing Interface (MPI)
+library.
+Software for Linear Algebra Targeting Exascale
+(SLATE) library17 was used for storing the parallel matrices
+encountered during the course of RFOE. SLATE stores the
+matrix in a 2-D block-cyclic manner on both CPUs and GPUs
+within a node. The tile size is the most basic parameter that
+can affect the SLATE routines’ performance. Numerical ex-
+periments were conducted by varying the tile size to decide
+the optimal tile size.
+Some of the key aspects of the implementation are high-
+lighted below:
+1.
+Trace Calculations
+Traces of matrix squares are required during the course of
+RFOE iterations and are computed by evaluating the square of
+the Frobenius norm of the given symmetric matrix (Tr(A2) =
+||A||2
+F). To this end, Frobenius norm function available in the
+SLATE library was used. Further, the computations of matrix
+traces which was required only in the beginning and end of
+RFOE involved a traversal through the diagonal elements of
+the global matrix and the use of an MPI collective function.
+2.
+Matrix-matrix multiplication
+The computationally dominant step in all the RFOE algo-
+rithms implemented is the matrix-matrix multiplication step.
+We used the SLATE library functions to perform this step
+in parallel across multinode CPUs and GPUs. The perfor-
+mance of the Communication-optimal Matrix Multiplication
+(COSMA)18 and cuBLASMg (made available from CUDA
+Math Library Early Access Program19) library was also ex-
+plored to compute parallel matrix-matrix multiplications on
+CPUs and GPUs.
+Our studies indicates that COSMA was
+slower in terms of computational times compared to the
+SLATE library. And the cuBLASMg library is restricted to
+multi-GPUs within a single node.
+D.
+Metrics for accuracy benchmarking of RFOE
+For accuracy benchmarking of the RFOE methods imple-
+mented, we computed two errors: (a) Relative error between
+the exact and approximated density matrix (f(H)) using the
+Frobenius norm, i.e ε1 = (||D − Dref ||F)/||Dref ||F, (b) Rela-
+tive error between the trace of actual and the approximated
+f(H)H, ε2 = (tr(DH) − tr(Dref H))/tr(Dref H).
+Dref was
+computed by explicit diagonalization using ELPA library13.
+IV.
+RESULTS AND DISCUSSION
+In order to assess the accuracy and performance of the pro-
+posed methods, we employ synthetic matrices representative
+of the subspace projected Hamiltonians (Hp) arising in DFT
+calculations.
+To this end, the matrix Hp is constructed in
+such a way that the spectral width is smaller and remains con-
+stant with increase in matrix size. We choose H p
+ij = H p
+ji =
+e−d∗|i−j| ∗ sin(i + 1), and the matrix sizes used were 8192 ×
+8192, 16384 × 16384, 32768 × 32768, and 65536 × 65536.
+The multinode CPU study was done on PARAM Pravega hav-
+ing Intel Xeon Cascade Lake 8268 CPU (2.9 GHz) with 48
+cores (96 threads) on each node, while multinode GPU study
+was done on a local lab cluster having 16x (8 on each node)
+NVIDIA Tesla V100 GPUs with 32 GB of memory.
+
+4
+FIG. 1: General implementation details flowchart for all the
+RFOE codes
+The performance metrics used for comparisons are:
+• Node-hrs ⇒ Execution time (in hours) × the number
+of nodes taken in the best scaling regime. It gives a
+measure of computational efficiency on CPUs.
+• GPU-hrs ⇒ Execution time (in hours) × the number
+of GPUs taken in the best scaling regime. It gives a
+measure of computational efficiency on GPUs.
+• Minimum walltime ⇒ Least possible time for the job
+execution using as many resources as possible. It is a
+measure of scaling efficiency of the implementation.
+• Energy consumption ⇒ Upper bound of the energy re-
+quired by the job in kWh to run it on the supercomputer.
+Indicative of the rupee cost required for the calculations
+on the supercomputer. For the energy consumption cal-
+culation we used the Thermal Design Power (TDP) rat-
+ings for both CPUs and GPUs.
+1.
+Multinode CPU comparisons
+Figure 2 shows that, all our RFOE implementations for
+zero-temperature case are better than ELPA in terms of node-
+hrs, which indicates that all of our implementations are com-
+putationally efficient compared to ELPA. For instance, the A-
+Z-RFOE results in a speedup of around 2x in comparison to
+ELPA for the 65536 size matrix. In the minimum walltime
+regime, we find that ELPA is slightly faster than the RFOE im-
+plementation for the matrix sizes considered. Figure 3 shows
+(a)
+(b)
+FIG. 2: (a) Node-hrs vs. matrix size plot, and (b) Min.
+walltime vs. matrix size plot (Note: c stands for CPUs, n
+stands for nodes) for different implementations of RFOEs for
+zero-temperature case on multinode CPUs.
+that, both our T-RFOE and mixed-precision T-RFOE imple-
+mentations for finite-temperature case are better than ELPA in
+terms of node-hrs (4.2x speedup of mixed-precision T-RFOE
+implementation over ELPA for the 65536 size matrix), which
+indicates that both of our implementations are computation-
+ally efficient compared to ELPA. And, even in the minimum
+walltime, mixed precision T-RFOE was found to be slightly
+better than ELPA. The number of CPUs on which we got min-
+imum walltime for different matrix sizes is also shown on the
+minimum walltime plots.
+2.
+Multinode GPU comparisons
+Figure 4 shows that, all our implementations for zero-
+temperature case are better than ELPA up to 16384 size ma-
+trix, and beyond this size, the mixed-precision Z-RFOE and
+A-Z-RFOE are better than ELPA for GPU-hrs timings, which
+indicates that both of our implementations are computation-
+ally efficient compared to ELPA. Our A-Z-RFOE implemen-
+tation gave 1.5x speedup over ELPA for the 32768 size ma-
+trix. Due to the memory issue of ELPA, it does not work
+on multinode GPUs up to 16 GPUs for the 65536 size ma-
+trix, which indicates that the RFOE implementations use the
+memory efficiently compared to ELPA. And for minimum
+walltime, ELPA shows a better behaviour, suggesting that
+the RFOE implementations are not scaling well on multinode
+GPUs. Figure 5 shows that, our T-RFOE and mixed-precision
+T-RFOE implementations for finite-temperature case are bet-
+
+Initializations
+Nocc : Occupancy number
+So= Hp : Hamiltonian
+(Defined on Distributed System)
+Xo = WoSo+bol (on Parallel architectures)
+Initial scaling using spectral bound estimates of H
+Ns = Tr[Xo] (Using MPl_Allreduce)
+While
+(TrEr < Tolerance)
+Final D Matrix
+Xn = WXn-12 + bXn-1
+Nx = IIXn-1ll-=2
+FP32/FP64 GEMM for Xn-12
+( IIXn-1]l==[Tr[Xn-13]1/2)
+Xn and Xn-1 stored in 2-D
+block-cyclic manner on
+CPUs/GPUs
+Compute w = f(Ns, Nx, Nocc)
+and b = g(Ns, Nx, Nocc)
+where f() and g(.) depends on
+Update Ns using Nx and Ns
+the expansion scheme (Z-RFOE
+TrEr = abs(Ns - Nocc)
+A-Z-RFOE, T-RFOE)5
+ELPA
+Double-PrecisionZ-RFOE
+4
+Mixed-Precision Z-RFOE
+node-hrs
+Double-Precision A-Z-RFOE
+m
+1
+0
+8192
+16384
+32768
+65536
+Matrix Size350
+ELPA
+(secs)
+300
+Double-Precision Z-RFOE
+250
+Mixed-PrecisionZ-RFOE
+(6144c128n)
+Double-Precision A-Z-RFOE
+walltime
+200
+150
+100
+Min.
+(6144c 128n)
+(6144c128n)
+（12288c256n)
+50
+(3072c64n)
+←(12288c256n)
+0
+（3072c64n）
+(12288c256n)
+8192
+16384
+32768
+65536
+Matrix Size5
+(a)
+(b)
+FIG. 3: (a) Node-hrs vs. matrix size plot, and (b) Min.
+walltime vs. matrix size plot (Note: c stands for CPUs, n
+stands for nodes) for different implementations of RFOEs for
+finite-temperature case on multinode CPUs.
+ter than ELPA for GPU-hrs timings (2.5x speedup of mixed-
+precision T-RFOE implementation over ELPA for the 65536
+size matrix), indicating that both implementations are com-
+putationally efficient compared to ELPA. And, for minimum
+walltime, our mixed-precision T-RFOE implementation is al-
+most similar to or better than ELPA. The number of GPUs on
+which we got minimum walltime for different matrix sizes is
+shown on the minimum walltime plot.
+3.
+Energy consumption comparisons
+Figure 6 shows that, in both the regimes of node-hrs/GPU-
+hrs and minimum walltime, we are better than ELPA in terms
+of energy consumption for zero-temperature case. Figure 7
+shows that, in both the regimes of node-hrs/GPU-hrs and min-
+imum walltime case, we are better than ELPA in terms of en-
+ergy consumption for finite-temperature case. This indicates
+that the rupee cost required for our calculations on the super-
+computer will be less than ELPA for both zero-temperature
+case and finite-temperature case of approximating the density
+matrix.
+4.
+Accuracy benchmarking
+The errors ε1 and ε2 (defined earlier) were of the O(10−10)
+and O(10−09) for double-precision implementation of Z-
+RFOE and A-Z-RFOE. And, were of the O(10−07) and
+O(10−09) for mixed-precision implementation of Z-RFOE.
+(a)
+(b)
+FIG. 4: (a) GPU-hrs vs. matrix size plot, and (b) Min.
+walltime vs. matrix size plot for different implementations of
+RFOEs for zero-temperature case on multinode GPUs.
+(a)
+(b)
+FIG. 5: (a) GPU-hrs vs. matrix size plot, and (b) Min.
+walltime vs. matrix size plot for different implementations of
+RFOEs for finite-temperature case on multinode GPUs.
+
+5
+ELPA
+Double-PrecisionT-RFOE
+4
+Mixed-Precision T-RFOE
+node-hrs
+m
+N
+1
+0
+8192
+16384
+32768
+65536
+Matrix Size160
+ELPA
+(12288c256n)
+(secs)
+140
+Double-Precision T-RFOE
+Mixed-PrecisionT-RFOE
+120
+(6144c128n)
+Min. walltime
+100
+80
+60
+(6144c128n)
+40
+(6144c128n)
+20
+(3072c 64n)
+(12288c256n)
+0
+(3072c64n)
+(12288c256n）
+8192
+16384
+32768
+65536
+Matrix Size1.75
+ELPA
+1.50
+Double-Precision Z-RFOE
+1.25
+Mixed-PrecisionZ-RFOE
+GPU-hrs
+Double-Precision A-Z-RFOE
+1.00
+0.75
+0.50
+0.25
+0.00
+8192
+16384
+32768
+65536
+Matrix size700
+ELPA
+(SDos)
+Double-Precision Z-RFOE
+600
+Mixed-Precision Z-RFOE
+(8 GPUs)
+Min. walltime 
+500
+Double-Precision A-Z-RFOE
+400
+300
+200
+(8 GPUs)
+100
+(4 GPUs)
+(6 GPUs)
+(16 GPUS)
+0
+（6GPUS)
+(16GPUs)
+8192
+16384
+32768
+65536
+Matrix size0.8
+ELPA
+0.7
+Double-Precision T-RFOE
+0.6
+Mixed-PrecisionT-RFOE
+GPU-hrs
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+8192
+16384
+32768
+65536
+Matrix Size350
+ELPA
+(secs)
+300
+Double-PrecisionT-RFOE
+Mixed-Precision T-RFOE
+250
+(8 GPUs)
+Min. walltime
+200
+150
+(8 GPUs)
+100
+(6 GPUs)
+50
+(4 GPUS)
+(16GPUS)
+0
+(6GPUs)
+(16-GPUs)
+8192
+16384
+32768
+65536
+Matrix size6
+(a)
+(b)
+FIG. 6: Energy consumption (kWh) vs. matrix size plot in
+terms of (a) Node-hrs/GPU-hrs, and (b) Min. walltime for the
+best implementation of RFOE for zero-temperature case.
+(a)
+(b)
+FIG. 7: Energy consumption (kWh) vs. matrix size plot in
+terms of (a) Node-hrs/GPU-hrs, and (b) Min. walltime for the
+best implementation of RFOE for finite-temperature case.
+For T-RFOE, the error ε1 was of the O(10−03) and ε2 was
+of O(10−06). The density matrix approximated by T-RFOE
+has an higher error compared to the Fermi-Dirac based den-
+sity matrix. However, in DFT, the computation of material
+properties relies on differences in energies and hence, the T-
+RFOE approach can be viewed as an alternative approach to
+smearing the zero-temperature density matrix, which can be
+practically helpful in approximating finite-temperature den-
+sity matrix.
+V.
+CONCLUSIONS
+RFOE schemes, as expected, had a lesser computational
+prefactor which made them computationally efficient com-
+pared to ELPA in the node-hrs/GPU-hrs regime. In the case of
+minimum walltimings, ELPA timings were better as it scaled
+better than our RFOE implementations. Energy efficiency-
+wise, the RFOE implementations were better on both multin-
+ode CPUs and GPUs, which is directly proportional to the cost
+required for the computations. In terms of memory utiliza-
+tion, multinode GPU implementations of RFOE were better
+than ELPA. From all the observations we can conclude that,
+these techniques can be used whenever we have fewer com-
+putational resources and have cost constraints.
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+necke, H.-J. Bungartz, and H. Lederer, “The elpa library: Scalable parallel
+
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+Energy
+1.0
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+0.0
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+65536
+Matrix Size3.5
+CPU ELPA
+3.0
+CPUDouble-PrecisionA-Z-REOE
+(kWh)
+GPU ELPA
+2.5
+GPUDouble-Precision A-Z-RFOE
+2.0
+Energy
+1.5
+1.0
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+16384
+32768
+65536
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+CPUELPA
+CPUMixed-PrecisionT-RFOE
+(yM)
+GPU ELPA
+1.5
+GPUMixed-Precision T-RFOE
+Energy
+1.0
+0.5
+0.0
+8192
+16384
+32768
+65536
+Matrix Size3.5
+CPU ELPA
+3.0
+CPUMixed-PrecisionT-RFOE
+(kWh)
+GPU ELPA
+2.5
+GPUMixed-PrecisionT-RFOE
+2.0
+Energy
+1.5
+1.0
+0.5
+0.0
+8192
+16384
+32768
+65536
+Matrix Size7
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+

5NE1T4oBgHgl3EQfBAIU/content/tmp_files/2301.02845v1.pdf.txt

@@ -0,0 +1,1784 @@
+A one-dimensional model for axisymmetric deformations of an
+inflated hyperelastic tube of finite wall thickness
+Xiang Yua,˚, Yibin Fub
+aSchool of Computer Science and Technology, Dongguan University of Technology, Dongguan, China
+bSchool of Computing and Mathematics, Keele University, Staffordshire ST5 5BG, UK
+Abstract
+We derive a one-dimensional (1d) model for the analysis of bulging or necking in an inflated hyper-
+elastic tube of finite wall thickness from the three-dimensional finite elasticity theory by applying
+the dimension reduction methodology proposed by Audoly and Hutchinson (J. Mech. Phys. Solids,
+97, 2016). The 1d model makes it much easier to characterize fully nonlinear axisymmetric defor-
+mations of a thick-walled tube using simple numerical schemes such as the finite difference method.
+The new model recovers the diffuse interface model for analyzing bulging in a membrane tube and
+the 1d model for investigating necking in a stretched solid cylinder as two limiting cases. It is con-
+sistent with, but significantly refines, the exact linear and weakly nonlinear bifurcation analyses.
+Comparisons with finite element simulations show that for the bulging problem, the 1d model is
+capable of describing the entire bulging process accurately, from initiation, growth, to propagation.
+The 1d model provides a stepping stone from which similar 1d models can be derived and used to
+study other effects such as anisotropy and electric loading, and other phenomena such as rupture.
+Keywords:
+localized bulging; necking; reduced models; tubes; stability; nonlinear elasticity
+1. Introduction
+Hyperelastic tubes are commonly found in various applications ranging from soft robotics (Ma
+et al., 2015; Lu et al., 2015, 2020) to energy harvesting (Lu & Suo, 2012; Bucchi & Hearn, 2013;
+Smith, 2016). They are also used to model human arteries in order to understand pathological
+conditions such as aneurysms (Fu et al., 2012; Alhayani et al., 2014; Demirkoparan & Merodio,
+2017; Varatharajan & DasGupta, 2017; Hejazi et al., 2021). Inflation of a hyperelastic tube is one
+of the few boundary value problems in nonlinear elasticity that have closed-form solutions, and it
+provides the simplest setup to explain bifurcation, localization, loss of convexity, and “two-phase”
+deformations. Thus, understanding this problem is not only important for applications, but may
+also shed light on other more complicated stability and bifurcation problems.
+˚Corresponding author
+Email address: [email protected] (Xiang Yu)
+Preprint submitted to Journal of the Mechanics and Physics of Solids
+January 10, 2023
+arXiv:2301.02845v1  [cond-mat.soft]  7 Jan 2023
+
+Simple inflation experiments with commercially available latex rubber tubes show that localized
+bulging is the dominant deformation form. For almost all realistic constitutive models for rubber,
+the pressure versus volume curve has an N shape under the condition of fixed resultant axial force
+(Green & Adkins, 1960). This led Yin (1977) and Chater & Hutchinson (1984) to analyze the final
+observable configuration as that corresponding to a “two-phase” deformation.
+The subsequent
+experimental studies carried out by Kyriakides & Chang (1990, 1991), Pamplona et al. (2006) and
+Goncalves et al. (2008) have provided a clear picture on how a localized bulge initiates, grows and
+then propagates under fixed axial force or fixed-ends conditions.
+When the membrane assumption is made, the governing equations for tube inflation can be
+viewed as a finite-dimensional spatial dynamical system that has two conservation laws/integrals
+(Pipkin, 1968). This realization enabled Fu et al. (2008) to demonstrate explicitly how a localized
+solution initiates as a zero-wave-number mode from the uniform deformation and how it evolves
+into a “two-phase” state. The stability of bulging solutions and their sensitivity to imperfections
+have been studied under the same framework (Pearce & Fu, 2010; Fu & Il’ichev, 2015; Fu & Xie,
+2010). Fresh analytical insight into the case of fixed ends has also been obtained. It is shown
+that the bifurcation condition for this case corresponds to the axial force reaching a maximum at
+a fixed pressure (Fu & Il’ichev, 2015); in other words, as pressure is increased, the critical pressure
+is the value of pressure at which the axial force reaches a maximum when viewed as a function of
+the axial stretch. Also, in contrast with the case of fixed axial force where the measured pressure
+approaches a constant value (the propagation pressure), the measured pressure in the case of fixed
+ends has an N shape where the right branch approaches a master curve that is independent of the
+pre-axial-stretch or the tube length (Guo et al., 2022).
+In some practical applications, however, the tube wall may be of moderate or even large thickness
+and the membrane model no longer applies. For example, in the context of aneurysm formation,
+a human artery can be as thick as a quarter of its outer radius (M¨uller et al., 2008), and fiber-
+reinforcement also seems to reduce the range of validity of the membrane assumption (Wang &
+Fu, 2018). Thus, recent studies have begun to consider hyperelastic tubes of finite wall thickness.
+Fu et al. (2016) showed that the associated bifurcation condition for localized bulging corresponds
+to the vanishing of the Jacobian determinant of the internal pressure and resultant axial force as
+functions of the azimuthal stretch and the axial stretch; see also Yu & Fu (2022) for an alternative
+derivation.
+This provides a framework under which additional effects such as rotation (Wang
+et al., 2017), double-fiber-reinforcement (Wang & Fu, 2018), bi-laying (Liu et al., 2019; Ye et al.,
+2019), torsion (Althobaiti, 2022), and surface tension (Emery & Fu, 2021a,b,c; Emery, 2023) can
+be assessed in a systematic manner. Ye et al. (2020) conducted a weakly non-linear analysis and
+derived the bulging solution explicitly. The analytic predictions were corroborated by numerical
+simulations and experiments (Wang et al., 2019).
+For tubes of finite wall thickness, the equations that govern their axisymmetric deformations
+are coupled nonlinear partial differential equations. Although analytic solutions can be obtained in
+2
+
+the near-critical regime using asymptotic methods (Ye et al., 2020), the complexity of the governing
+equations forbids any further analytic attempts to understand the bulging evolution further away
+from the bifurcation point. The post-bifurcation behavior in the fully nonlinear regime has so far
+only been investigated by resorting to Abaqus simulations (Wang et al., 2019; Lin et al., 2020).
+This is not satisfactory since the insight provided by full-scale simulations tends to be limited and
+there are situations where repeated calculations of the bulging profile are required (e.g. in the
+assessment of the rupture potential (Hejazi et al., 2021)).
+A recent series of studies by Audoly and coworkers has opened the possibility that a 1d reduced
+model can be derived to describe the fully nonlinear evolution of bulging or necking. In the first
+of this series, Audoly & Hutchinson (2016), the authors derived a 1d model for tensile necking
+localization in a 3d prismatic solid of arbitrary cross-section. The key idea of their derivation is a
+dimension reduction assuming slow variation in the axial direction that respects self-consistency.
+In terms of the language of perturbation analysis, the leading-order solution is almost correct and
+higher-order terms are only added to restore self-consistency. The method was later applied by
+Lestrigant and Audoly to obtain a diffuse interface model for the characterization of propagating
+bulges in membrane tubes (Lestringant & Audoly, 2018) and a 1d model for predicting surface
+tension-driven necking in soft elastic cylinders (Lestringant & Audoly, 2020b). It has also been
+used recently to derive a 1d model for elastic ribbons (Audoly & Neukirch, 2021) and for tape springs
+(Kumar et al., 2022). The systematic reduction method for deriving 1d strain-gradient models for
+nonlinear slender structures was further generalized by Lestringant & Audoly (2020a). It is worth
+pointing out that although the 1d models are built on the assumption that localized solutions
+vary slowly in the longitudinal direction, it is surprisingly accurate, even in the region where the
+localization is well developed. This is illustrated by the numeric examples in the aforementioned
+work and in the comparative studies by Wang & Fu (2021) and Fu et al. (2021).
+This work aims to extend the diffuse interface model of Lestringant & Audoly (2018) for mem-
+brane tubes to tubes of finite wall thickness, in a similar spirit as the previous work Fu et al. (2016)
+and Ye et al. (2020) that extend the bifurcation condition and the weakly nonlinear analysis from
+membrane tubes to thick-walled tubes. In contrast with the case under the membrane assumption
+where the original governing equations are already one-dimensional, the governing equations for the
+current case are two-dimensional, and the uniformly inflated deformation is no longer homogeneous
+since the solution depends on the radial variable. It will be shown that a 1d reduced model can
+still be derived and simplified to the form
+E1dras “
+ż L
+´L
+´
+Gpa, λpaqq ` 1
+2Dpaqa1pZq2¯
+dZ ` Cpaqa1pZq|L
+´L,
+(1.1)
+where L is the initial half length, Z is the axial coordinate, apZq is the azimuthal stretch on the inner
+surface (a constant multiple of the deformed inner radius ) and the expressions for Gpa, λpaqq, Dpaq
+and Cpaq are given in (3.10), (4.21) and (4.22), respectively. The first term G in (1.1) corresponds
+to the energy of the uniform deformation, which determines the amplitudes of the two phases in
+3
+
+the bulge propagation stage; the second term accounts for the contribution of the strain gradient
+to the total energy, which describes how these two phases are connected.
+The Euler-Lagrange
+equation associated with the energy functional (1.1) is a second-order nonlinear ode for apZq,
+which is a drastic simplification from the original nonlinear partial differential equations. This 1d
+model is validated by comparison with finite element simulations, showing excellent agreement with
+numerical results even for the propagation stage.
+The outline of this paper is as follows. In Sect. 2, we formulate the 3d axisymmetric finite-
+strain model for a tube of finite wall thickness under inflation and axial stretching. In Sect. 3, we
+summarize solutions corresponding to uniform inflation of the tube, making preparation for the
+subsequent dimension reduction. In Sect. 4, we carry out the dimension reduction and derive the
+above-mentioned 1d strain-gradient model. The connection of the 1d model with prior work is
+given in Sect. 5. In Sect. 6, we validate the 1d model by making comparisons with finite element
+simulations. Finally, concluding remarks are given in Sect. 7.
+2. Three-dimensional finite-strain model
+We consider a circular cylindrical tube that has a length 2L, inner radius A and outer radius B
+in its reference configuration; see Fig. 1(a). The ratio of the outer radius to the length ε “ B{2L is
+assumed to be small, thus ε ! 1. The tube deforms axisymmetrically under the combined action of
+an internal pressure P and a resultant axial force N, as shown in Fig. 1(b). In terms of cylindrical
+coordinates, the current position vector of a representative point is given by
+x “ zpZ, Rqez ` rpZ, Rqer,
+(2.1)
+where pR, Θ, Zq and pr, θ, zq are the coordinates of a representative point before and after defor-
+mation, and per, eθ, ezq are the standard basis vectors associated with both pR, Θ, Zq and pr, θ, zq.
+The deformation gradient related to (2.1) is given by
+F “ r
+Reθ b eθ ` zZez b ez ` zRez b er ` rZer b ez ` rRer b er,
+(2.2)
+where zZ :“ Bz{BZ, zR :“ Bz{BR, etc.
+We assume that the tube is made of an incompressible isotropic hyperelastic material, associated
+with the strain energy function Wpλ1, λ2, λ3q, where λ1, λ2, λ3 denote the three principal stretches.
+Throughout this paper, we identify the indices 1, 2, 3 such that in uniform inflation they coincide
+with the θ-, z- and r-directions, respectively.
+The total potential energy of the tube is composed of the elastic energy and the load potential,
+which reads
+E “
+ż L
+´L
+´ ż B
+A
+`
+wpλ1, λ2q ´ N˚zZ
+˘
+2πR dR ´ Pπr2zZ
+ˇˇ
+R“A
+¯
+dZ,
+(2.3)
+4
+
+m
+B
+G
+(a)
+m2
+G2
+H2
+(b)
+Figure 1: A hyperelastic cylindrical tube of finite thickness in (a) reference (undeformed) configuration and (b)
+current configuration.
+where wpλ1, λ2q “ Wpλ1, λ2, λ´1
+1 λ´1
+2 q is the reduced strain energy function and N˚ “ N{pπpB2 ´
+A2qq is the resultant axial force per unit cross-sectional area. The elastic model governed by the
+energy functional (2.3) will be used as a starting point for the subsequent dimension reduction.
+The governing equations for the two unknown functions rpZ, Rq and zpZ, Rq can be derived by
+setting the first variation of E to zero, but these equations are not required in the approach that
+we follow.
+3. Uniform inflation
+We now summarise the solution that corresponds to uniform inflation and extension of the tube.
+This solution will be referred to as the uniform solution and is indicated by a superposed bar. For
+a more detailed derivation, see Haughton & Ogden (1979).
+First, incompressibility implies that a uniform solution must necessarily be of the form
+¯z “ λZ,
+¯r “
+a
+a2A2 ` λ´1pR2 ´ A2q,
+(3.1)
+where λ and a denote the constant axial stretch and azimuthal stretch on the inner surface, respec-
+tively. The three principal stretches are simply
+¯λ1 “ ¯r
+R,
+¯λ2 “ λ,
+¯λ3 “ d¯r
+dR “ ¯λ´1
+1 ¯λ´1
+2 ,
+(3.2)
+and the azimuthal stretch on the outer surface, denoted by b, is given by
+b “ ¯λ1|R“B “
+a
+a2A2 ` λ´1pB2 ´ A2q
+B
+.
+(3.3)
+The three associated principal Cauchy stresses ¯σ11, ¯σ22 and ¯σ33 satisfy the relations
+¯σ11 ´ ¯σ33 “ ¯λ1w1,
+¯σ22 ´ ¯σ33 “ λw2,
+(3.4)
+5
+
+where w1 “ Bwp¯λ1, ¯λ2q{B¯λ1 and w2 “ Bwp¯λ1, ¯λ2q{B¯λ2.
+The only equilibrium equation that is not satisfied automatically is
+d¯σ33
+d¯r
+“ ¯σ11 ´ ¯σ33
+¯r
+“
+¯λ1w1
+¯r
+.
+(3.5)
+On integrating this equation from R “ A to R “ B and making use of the boundary conditions
+that ¯σ33|R“A “ ´P and ¯σ33|R“B “ 0, we obtain
+P “ Qpa, λq :“
+ż a
+b
+w1p¯λ1, λq
+¯λ2
+1λ ´ 1 d¯λ1,
+(3.6)
+where the second equation defines the function Qpa, λq and we have made use of the identity
+d¯r
+¯r “ ´
+d¯λ1
+¯λ1p¯λ2
+1λ ´ 1q,
+(3.7)
+which can be deduced from (3.1)2.
+The overall equilibrium in the axial direction implies
+Mpa, λq ´ 1
+2a2P ´
+N
+2πA2 “ 0,
+(3.8)
+where Mpa, λq is given by
+Mpa, λq :“ 1
+A2
+ż B
+A
+λ´1¯σ22R dR “
+ż a
+b
+p¯λ2
+1 ´ a2qw1p¯λ1, λq ` 2¯λ1λpa2λ ´ 1qw2p¯λ1, λq
+2p¯λ2
+1λ ´ 1q2
+d¯λ1.
+(3.9)
+In view of (2.3), the total potential energy of the uniform deformation (3.1) per unit reference
+length, after scaling by 2π, is
+Gpa, λq “
+ż B
+A
+wp¯λ1, λq R dR ´ 1
+2PA2a2λ ´ N
+2πλ.
+(3.10)
+The equilibrium equations (3.6) and (3.8) can also be obtained from BG{Ba “ 0 and BG{Bλ “ 0,
+respectively. Once the loads P and N are specified, the deformation parameters a and λ can be
+found by solving the equilibrium equations (3.6) and (3.8).
+On differentiating the left-hand side of (3.8) with respect to λ, we find that it takes the form
+Hw22pa, λq{A ` OpH2q, where H “ B ´ A is the thickness of the tube. We assume that the strong
+ellipticity condition is satisfied pointwise which guarantees that w22pa, λq is positive (Knowles &
+Sternberg, 1976). This, combined with the implicit function theorem, implies that (3.8) can be
+inverted to express λ in terms of a uniquely at least when H is small. We assume that this remains
+true for arbitrary H. This enables us to view (3.8) as an implicit equation for λ “ λpaq. We remark
+that λ is also dependent on P, but this dependence is not indicated explicitly for notational brevity.
+Thus, by definition, λpaq is the solution to the implicit equation
+Mpa, λpaqq ´ 1
+2a2P ´
+N
+2πA2 “ 0.
+(3.11)
+6
+
+Since λ has been viewed as a function of a, all quantities (except ¯z which also depends on Z) related
+to the uniform solution are functions of a and R. For instance, ¯r denotes the function
+¯rpa, Rq “
+a
+a2A2 ` λpaq´1pR2 ´ A2q.
+(3.12)
+We denote ¯σ33 by ´qpa, Rq so that
+qpa, Rq :“ ´¯σ33 “
+ż ¯λ1
+b
+w1p˜λ1, λq
+˜λ2
+1λ ´ 1
+d˜λ1.
+(3.13)
+We also define another function mpa, Rq through
+mpa, Rq :“ 1
+R2
+ż B
+R
+λ´1¯σ22RdR “
+ż ¯λ1
+b
+p˜λ2
+1 ´ ¯λ2
+1qw1p˜λ1, λq ` 2˜λ1λp¯λ2
+1λ ´ 1qw2p˜λ1, λq
+2p˜λ2
+1λ ´ 1q2
+d˜λ1,
+(3.14)
+and record the connections
+qpa, Aq “ Qpa, λpaqq,
+mpa, Aq “ Mpa, λpaqq.
+(3.15)
+The 1d reduced model to be derived in the next section will be expressed in terms of the two
+functions qpa, Rq and mpa, Rq. The integrals in these two functions can be evaluated explicitly for
+some commonly used strain energy functions, including the Gent material model that will be used
+in our illustrative examples.
+4. Derivation of the one-dimensional model
+In this section, we apply the dimension reduction methodology proposed by Audoly & Hutchin-
+son (2016) to derive a one-dimensional model from the full three-dimensional theory formulated in
+Sect. 2.
+4.1. Optimal correction
+We start our dimension reduction by assuming that all dependent variables related to the
+axisymmetric configuration vary slowly in the axial direction. More precisely, it is assumed that
+all variables depend on Z through the “far distance” variable
+S “ εZ.
+(4.1)
+Recall that ε is the ratio of the outer radius to the length, which is assumed to be small.
+In
+particular, we now allow a and λ to depend on S and write a “ apSq, λ “ λpapSqq. Our aim is to
+derive a reduced model, an ordinary differential equation, that is satisfied by apSq. We recall that
+apSq is the deformed inner radius divided by a constant (i.e. A).
+A naive approach would be to use a “ apSq and λ “ λpapSqq to compute the two principal
+stretches and then derive the equation satisfied by a “ apSq by minimizing the energy functional
+7
+
+(2.3). This would yield an equation for apSq that is not self-consistent. The correct way is to allow
+for higher-order correction terms by looking for an asymptotic solution of the form
+zpZ, Rq “ 1
+ε
+ż S
+0
+λpapTqq dT ` εv˚pS, Rq ` Opε3q,
+rpZ, Rq “ ¯rpapSq, Rq ` ε2u˚pS, Rq ` Opε4q.
+(4.2)
+We note that the correction terms in zpZ, Rq and rpZ, Rq are of order ε and ε2, respectively.
+This is because the Op1q-term in zpZ, Rq and the Opεq-term in rpZ, Rq correspond to a uniform
+perturbation and can thus be absorbed into the leading terms.
+On substituting (4.2) into (2.2) and truncating at order ε2, we obtain the deformation gradient
+F “
+¨
+˚
+˝
+¯r{R ` ε2u˚{R
+0
+0
+0
+λpapSqq ` ε2v˚
+S
+εv˚
+R
+0
+ε¯raa1pSq
+¯rR ` ε2u˚
+R
+˛
+‹‚,
+(4.3)
+where the subscripts represent partial differentiation with respect to the indicated variables (in
+particular ¯ra “ B¯r{Ba). Consequently, the three principal stretches are given by
+λ1 “ ¯λ1 ` ε2 u˚
+R ,
+λ2 “ ¯λ2 ` ε2´
+v˚
+S ` λp¯r2
+aa1pSq2 ` v˚2
+R q ` 2¯λ3¯raa1pSqv˚
+R
+2pλ2 ´ ¯λ3q
+¯
+,
+(4.4)
+where ¯λ1 and ¯λ2 are given by (3.2) but with a and λ replaced by apSq and λpapSqq, respectively.
+Substituting (4.4) into (2.3) and expanding to second order, we see that E can be written, in
+terms of the un-scaled variables, as
+E “ 2π
+´ ż L
+´L
+GpapZq, λpapZqqq dZ ` E2
+¯
+` OpLε3q,
+(4.5)
+where E2 represents the term of order ε2 and is given by
+E2 “
+ż L
+´L
+´ ż B
+A
+´
+pw2 ´ N˚qvZ ` w2
+λp¯r2
+aa12 ` v2
+Rq ` 2¯λ3¯raa1vR
+2pλ2 ´ ¯λ2
+3q
+¯
+R dR
+`
+ż B
+A
+w1u dR ´ 1
+2PA2a2vZ|R“A ´ PAaλu|R“A
+¯
+dZ.
+(4.6)
+In the above expression, vpZ, Rq “ εv˚pS, Rq and upZ, Rq “ ε2u˚pS, Rq denote the unscaled dis-
+placements, and here and hereafter we write apZq for apSq and so a1 now denotes a1pZq. It is seen
+that the only reason for introducing S above is to identify all terms of order ε2 that should be kept
+in (4.6). With this task accomplished, the scaled variable S will no longer appear in the subsequent
+analysis. Also, w1 “ w1p¯λ1, λq, w2 “ w2p¯λ1, λq in which λ is a function of a and ¯λ1 is a function of
+a and R.
+8
+
+Our formulation in terms of the reduced strain energy function requires the solution (4.2) to
+satisfy the incompressibility condition automatically. This can be achieved by eliminating u in
+(4.6) with the use of detpF q “ 1 which takes the form
+λp¯ruqR ` ¯rp¯rRvZ ´ ¯raa1vRq “ 0.
+(4.7)
+To this end, we make use of the equilibrium equation (3.5) and write
+ż B
+A
+w1u dR “ λ
+ż B
+A
+¯σ33,R¯ru dR “ λ¯σ33¯ru|B
+A ´ λ
+ż B
+A
+¯σ33p¯ruqR dR
+“ λPAau|R“A ´
+ż B
+A
+q¯rp¯rRvZ ´ ¯raa1vRq dR,
+(4.8)
+where we have replaced ¯σ33 by ´qpa, Rq (cf. (3.13)) and have used (4.7) to eliminate p¯ruqR.
+On eliminating u in (4.6) with the use of (4.8), we can recast E2 in the form
+E2 “
+ż L
+´L
+´ ż B
+A
+`
+pλ´1¯σ22 ´ N˚qvZ ` 1
+2ζp¯r2
+aa12 ` v2
+Rq ` ξ¯raa1vR
+˘
+R dR
+´ 1
+2PA2a2vZ|R“A
+¯
+dZ,
+(4.9)
+where ζ “ λw2{pλ2 ´ ¯λ2
+3q, ξ “ q¯λ1 ` ¯λ3ζ{λ, and we have made use of the connection λw2 ´ q “ ¯σ22
+that follows from (3.4)2 with ¯σ33 “ ´q. Then upon using integration by parts, we obtain
+E2 “
+ż L
+´L
+´ ż B
+A
+KpR, v, vRq dR ` PA2aa1v|R“A
+¯
+dZ
+`
+´ ż B
+A
+pλ´1¯σ22 ´ N˚qvR dR ´ 1
+2PA2a2v|R“A
+¯ˇˇˇ
+Z“L
+Z“´L,
+(4.10)
+where KpR, v, vRq is given by
+KpR, v, vRq “ ´pλ´1¯σ22qaa1Rv ` 1
+2Rζp¯r2
+aa12 ` v2
+Rq ` Rξ¯raa1vR.
+(4.11)
+In the last expression, pλ´1¯σ22qa denotes the partial derivative of λ´1¯σ22 with respect to a with R
+fixed. To find the remaining correction field v “ vpZ, Rq, we assume that the leading-order stretch
+apZq is prescribed and seek the correction v such that the total potential energy is stationary
+(Audoly & Hutchinson, 2016). As a result, the optimal v satisfies the Euler-Lagrange equation and
+the boundary conditions
+BK
+Bv ´ d
+dR
+´ BK
+BvR
+¯
+“ 0,
+A ď R ď B,
+(4.12)
+BK
+BvR
+“ PA2aa1,
+R “ A,
+(4.13)
+BK
+BvR
+“ 0,
+R “ B.
+(4.14)
+9
+
+Solution of the above boundary value problem requires satisfaction of the solvability condition
+ż B
+A
+BK
+Bv dR “ ´PA2aa1,
+that is
+ż B
+A
+pλ´1¯σ22qaa1R dR “ PA2aa1.
+This is automatically satisfied in view of the definition (3.9) for Mpa, λq and the equilibrium con-
+dition (3.8).
+Written out explicitly, Eqs. (4.12) and (4.14) take the form
+d
+dRpRζvRq “ ´
+´
+pλ´1¯σ22qaR ` d
+dRpRξ¯raq
+¯
+a1,
+A ď R ď B,
+(4.15)
+RζvR “ ´Rξ¯raa1,
+R “ B.
+(4.16)
+Integrating (4.15) subject to the boundary condition (4.16) leads to
+vR “ cpa, Rqa1pZq,
+(4.17)
+where cpa, Rq is defined by
+cpa, Rq “ ´ ¯ra
+¯λ1λ2 ` 1
+Rζ
+´
+R2 B
+Bampa, Rq ´ ¯r¯raqpa, Rq
+¯
+.
+(4.18)
+Once vR is found, the optimal correction v can be obtained by integrating (4.17) from B to R,
+which yields
+v “ ´
+ˆż B
+R
+cpa, Tq dT
+˙
+a1pZq,
+(4.19)
+where we have neglected the function arising from integration since it can be absorbed into λpapZqq.
+4.2. One-dimensional energy functional
+Substituting the correction function v found in (4.19) back into (4.10), after some simplification
+(which is detailed in Appendix A), we obtain the final expression for the energy functional of the
+1d model
+E1dras “
+ż L
+´L
+´
+Gpa, λpaqq ` 1
+2Dpaqa1pZq2¯
+dZ ` Cpaqa1pZq|L
+´L,
+(4.20)
+where the gradient moduli D and C are given by
+Dpaq “
+ż B
+A
+Rζp¯r2
+a ´ cpa, Rq2q dR,
+(4.21)
+Cpaq “
+ż B
+A
+pλ´1¯σ22 ´ N˚q˜cpa, RqR dR ´ 1
+2PA2a2˜cpa, Aq,
+(4.22)
+10
+
+with ˜cpa, Rq “ ´
+şB
+R cpa, Tq dT.
+The associated equilibrium equation is obtained by extremizing (4.20) with respect to apZq and
+is found to take the form
+A2aλpaqpQpa, λpaqq ´ Pq ´ 1
+2D1paqa1pZq2 ´ Dpaqa2pZq “ 0,
+(4.23)
+where we have used the fact that BG{Bλ “ 0 as it is used to find the implicit relation between λ and
+a (see (3.11)). Since Z does not explicitly appear in the integrand of (4.20) due to the translational
+invariance of the current problem in Z, by the Beltrami identity, the equilibrium equation (4.23)
+admits a first integral of the form
+Gpa, λpaqq ´ 1
+2Dpaqa1pZq2 “ constant.
+(4.24)
+We remark that the variational problem (4.20) is ill-posed due to the presence of the boundary
+terms Cpaqa1pZq|L
+´L.
+This is because the variational structure of the problem is broken when
+higher-order terms are dropped. There are two possible ways to get around this issue (Lestringant
+& Audoly, 2020a). The first one is to simply ignore the boundary terms, i.e., to set Cpaq “ 0. The
+second one is to add an Opε2q-term to apZq so that the boundary terms go away, which is rigorous
+but slightly more complex. It has previously been verified in Lestringant & Audoly (2020a) that
+the simple and rigorous approaches yield curves that can hardly be distinguished visually in any of
+the plots.
+To summarize, the second-order nonlinear ordinary differential equation (4.23) is our approxi-
+mate 1d model that governs the variation of the inner radius (which is A times apZq) in the axial
+direction. Once apZq is determined, the 3d deformation is given by (3.1). We note that the func-
+tion Qpa, λpaqq is explicit for most of the commonly used strain energy functions. The only slight
+complication is that the function Dpaq is given by an integral; see (4.21), but the functions mpa, Rq,
+qpa, Rq, and hence cpa, Rq and the integrand in (4.21) all have explicit expressions for most of the
+commonly used strain energy functions. Thus, only one numerical integration is required. This can
+easily be implemented on a symbolic manipulation platform such as Mathematica (Wolfram, 1991)
+as we shall show later.
+5. Connections with previous work
+We now demonstrate that the 1d model derived in Sect. 4 can recover the 1d model of Lestringant
+& Audoly (2018) for membrane tubes and that of Audoly & Hutchinson (2016) for solid cylinders
+under appropriate limits, and it can also reproduce the same weakly nonlinear bulging solution as
+that based on the exact 3d theory (Ye et al., 2020).
+11
+
+5.1. Membrane limit
+We first consider the reduction of the 1d model in the membrane limit when the tube thickness
+H approaches zero. The general axisymmetric deformation is now described by
+r “ rpZq,
+θ “ Θ,
+z “ zpZq,
+(5.1)
+and the three principal stretches are given by
+λ1 “ r
+R,
+λ2 “
+a
+r12 ` z12,
+λ3 “ 1{pλ1λ2q,
+(5.2)
+where R denotes the constant radius of the mid-surface. The total energy (2.3) reduces to
+E “ 2π
+ż L
+´L
+´
+w ´ 1
+2P ˚λ2
+1z1 ´ N˚z1¯
+dZ,
+(5.3)
+where P ˚ denotes the pressure scaled by H{R. Setting the first variation δE to zero then gives the
+governing equations
+w1 ´ R
+´w2
+λ2
+r1¯1
+´ P ˚λ1z1 “ 0,
+(5.4)
+w2
+λ2
+z1 ´ 1
+2P ˚λ2
+1 “ N˚.
+(5.5)
+Under the assumption that |r1| ! 1, we have
+λ2 “ z1 ` r12
+2z1 ` ¨ ¨ ¨ .
+(5.6)
+As an algebraic equation for z1, Eq. (5.5) has an asymptotic solution of the form
+z1 “ gpλ1q ` k1pλ1qr12 ` ¨ ¨ ¨ ,
+(5.7)
+where the leading-order term gpλ1q obviously satisfies the algebraic equation
+w2pλ1, gpλ1qq ´ 1
+2P ˚λ2
+1 ´ N˚ “ 0,
+(5.8)
+and the function k1pλ1q can easily be found but is not required. Eq. (5.8) determines gpλ1q uniquely
+under the assumption w22 ą 0.
+With the use of (5.6) and (5.7), we may expand the integrand in (5.3) to order r12 and obtain
+E “ 2π
+ż L
+´L
+´
+wpλ1, gpλ1qq ´ 1
+2P ˚λ2
+1gpλ1q ´ N˚gpλ1q ` 1
+2
+w2pλ1, gpλ1qq
+gpλ1q
+r12¯
+dZ.
+(5.9)
+This is the reduced model derived by Lestringant & Audoly (2018).
+We now show that our general 1d model (4.20) can recover this 1d model under the limit H Ñ 0.
+To this end, we first note that the uniformly deformed configuration is now described by
+¯r “ aR,
+¯z “ λZ.
+(5.10)
+12
+
+In particular, we have ¯ra “ R. Since qpa, Rq and mpa, Rq involve integrals from R to B, they go to
+zero as H Ñ 0. Consequently, the cpa, Rq defined in (4.18) takes the simple form
+cpa, Rq “ ´ R
+aλ2 .
+(5.11)
+Taking the limit H Ñ 0 in ζ “ λw2{pλ2 ´ ¯λ2
+3q yields
+ζ “ a2λ3w2
+a2λ4 ´ 1.
+(5.12)
+Substituting (5.11) and (5.12) into (4.20), we obtain
+lim
+HÑ0
+E1dras
+RH
+“
+ż L
+´L
+´
+wpa, λpaqq ´ 1
+2P ˚a2λpaq ´ N˚λpaq ` 1
+2R2 w2pa, λpaqq
+λpaq
+a1pZq2¯
+dZ.
+(5.13)
+Note that the modulus Cpaq vanishes in the membrane limit because of the equilibrium in the axial
+direction. The integrand on the right-hand side of (5.13) is the same as that on the right-hand side
+of (5.9) if we identify λ1, gpλ1q and r1 with apZq, λpaq, and Ra1pZq, respectively.
+5.2. Solid cylinder limit
+Next we consider the other extreme limit corresponding to A Ñ 0 and P Ñ 0. The uniform
+solution takes the form
+¯z “ λZ,
+¯r “ aR
+(5.14)
+with a “ λ´1{2. The three principal stretches are
+¯λ1 “ ¯λ3 “ λ´1{2,
+¯λ2 “ λ.
+(5.15)
+In particular, we have
+w1p¯λ1, ¯λ2q “ 0,
+w2p¯λ1, ¯λ2q “ ˆw1pλq,
+(5.16)
+where ˆwpλq “ Wpλ´1{2, λ, λ´1{2q. It follows from (5.16)1 that qpa, Rq “ 0. Note that the deforma-
+tion (5.14) is homogeneous, so (3.14) implies that
+mpa, Rq “ A2pB2 ´ R2q
+R2pB2 ´ A2qmpa, Aq “ A2pB2 ´ R2q
+R2pB2 ´ A2qMpa, λpaqq.
+Differentiating this expression with respect to a and noting (3.11), we obtain Bmpa, Rq{Ba “ 0.
+Thus cpa, Rq reduces to
+cpa, Rq “ ´ R
+λ3{2 .
+(5.17)
+The elastic modulus ζ is easily calculated as
+ζ “ λ2 ˆw1pλq
+λ3 ´ 1 .
+(5.18)
+13
+
+Substituting (5.17) and (5.18) into (4.20), we obtain
+2πE1drλs “
+ż L
+´L
+´
+πB2 ˆwpλq ` πB4
+16
+ˆw1pλq
+λ4
+λ1pZq2 ´ Nλ
+¯
+dZ,
+(5.19)
+where we have made use of the relation a1pZq “ λ1pZq{p2λ3{2q. This recovers the 1d model of
+Audoly & Hutchinson (2016) specialized to an incompressible circular cylinder.
+5.3. Comparison with exact weakly nonlinear analysis
+Finally, we carry out a weakly nonlinear near-critical analysis using our 1d model and compare
+the resulting amplitude equation with that obtained by Ye et al. (2020) from the exact 3d theory.
+We focus on localized solutions in an infinitely long tube of finite wall thickness.
+Denote by a8 the limit of apZq as Z Ñ 8 and λ8 “ λpa8q. It follows from (3.6) and (3.11)
+that
+P “ Qpa8, λ8q,
+N “ 2πA2Fpa8, λ8q,
+(5.20)
+where Fpa8, λ8q is defined by
+Fpa8, λ8q “ Mpa8, λ8q ´ 1
+2a2
+8Qpa8, λ8q.
+(5.21)
+We look for a localized solution that bifurcates from the uniform solution by writing
+apZq “ a8 ` ypZq,
+(5.22)
+where ypZq is a small perturbation. Substituting (5.22) into the 1d equilibrium equation (4.23)
+and expanding in terms of ypZq to quadratic order with the use of (5.20), we obtain
+Dpa8qy2pZq “ ωpa8, λ8qypZq ` γpa8, λ8qypZq2,
+(5.23)
+where the two coefficient functions ωpa, λq and γpa, λq are given by
+ωpa, λq “ A2
+2aλ
+a2Qλ ` 2Fλ
+Ωpa, λq,
+(5.24)
+γpa, λq “ A2 aλpa2Qa ` 2Faq
+Fapa2Qλ ` 2Fλq2 Γpa, λq ` A2ψpa, λqΩpa, λq.
+(5.25)
+In the above expressions, Qa “ BQpa, λq{Ba, Qλ “ BQpa, λq{Bλ, etc. and Ωpa, λq and Γpa, λq are
+defined by
+Ωpa, λq “ BQ
+Ba
+BF
+Bλ ´ BQ
+Bλ
+BF
+Ba ,
+(5.26)
+Γpa, λq “ BΩ
+Ba
+BF
+Bλ ´ BΩ
+Bλ
+BF
+Ba ,
+(5.27)
+and ψpa, λq is not written out as it is not required in the weakly nonlinear analysis.
+14
+
+The solution to the linearized equation of (5.23) changes character when the sign of ωpa8, λ8q
+changes. Thus a bifurcation occurs when ωpa8, λ8q “ 0, or equivalently,
+Ωpa8, λ8q “ 0.
+(5.28)
+Note that Qpa8, λ8q and Fpa8, λ8q represent respectively the functional dependence of P and
+N on a8 and λ8. Thus the above bifurcation condition is simply the vanishing of the Jacobian
+determinant of P and N as functions of a8 and λ8. This is consistent with previous work Fu et al.
+(2016) and Yu & Fu (2022).
+We consider two typical loading scenarios: either the resultant axial force N or the axial stretch
+at infinity λ8 is fixed. The latter case is used to approximate the case of fixed axial length, which
+can be realized more easily experimentally or in Abaqus simulations.
+Let us first assume that the resultant axial force N “ Nc is fixed, where Nc is the prescribed
+axial force. Denote by pacr, λcrq the root of the system of equations
+ωpa8, λ8q “ 0,
+Fpa8, λ8q “ Nc,
+(5.29)
+at which the bifurcation occurs according to the previous discussion. In the vicinity of the bifurca-
+tion point, the amplitude equation (5.23) reduces to
+Dpacrqy2pZq “ ω1pacr, λcrqpa8 ´ acrqypZq ` γpacr, λcrqypZq2,
+(5.30)
+where the prime on ω denotes d{da8 “ B{Ba8 ` pB{Bλ8qpdλ8{da8q. The above equation admits
+a localized solution of the form
+ypZq “ ´3ω1pacr, λcrq
+2γpacr, λcrq pa8 ´ acrq sech2 ´1
+2
+d
+ω1pacr, λcrq
+Dpacrq
+pa8 ´ acrqZ
+¯
+.
+(5.31)
+On the other hand, the weakly nonlinear amplitude equation derived from the 3d theory (Ye
+et al., 2020) takes the form
+c2
+1pZq “ λ2
+crk1pa8 ´ acrqc1pZq ` λ2
+crk2c1pZq2,
+(5.32)
+where c1pZq and ypZq are related by
+ypZq “ kc1pZq
+(5.33)
+with k “ ´2λpaq{λ1paq|a“acr, and k1 and k2 are constants available in Ye et al. (2020). One can
+see that (5.30) and (5.32) are identical provided
+k1 “ ω1pacr, λcrq
+λ2crDpacrq ,
+k2 “ kγpacr, λcrq
+λ2crDpacrq .
+(5.34)
+We have verified numerically that this is indeed the case, but the current expressions on the right
+hand sides of (5.34) are more compact and revealing.
+15
+
+The case of fixed λ8 can be handled similarly. Let pacr, λcrq be the solution to the system of
+equations
+ωpa8, λ8q “ 0,
+λ8 “ λc,
+(5.35)
+where λc is a given constant.
+In the vicinity of the bifurcation point, the amplitude equation
+parallel to (5.30) is of the form
+Dpacrqy2pZq “ ω1pacr, λcrqpa8 ´ acrqypZq ` γpacr, λcrqypZq2.
+(5.36)
+where the prime on ω now signifies B{Ba8. Similar to the previous case, one can verify that the
+above amplitude equation is the same as its counterparts in Ye et al. (2020).
+6. Comparison with Abaqus simulations
+In this section, we demonstrate the power of the 1d model by applying it to investigate localized
+bulging in an inflated tube of finite wall thickness in the fully nonlinear regime. Previous studies on
+this problem usually treat the tube as a finite length tube, but the problem can be analyzed more
+easily and very accurately by assuming the tube to be of infinite length. This assumption only fails
+when the tube is very short and when bulging is no longer localized in the axial direction (Wang
+& Fu, 2021). The reason is that bulging solutions decay exponentially towards the two ends. Thus
+in the following analysis, we shall assume that the tube is effectively infinite and focus on solutions
+subject to decaying boundary conditions. This assumption is validated by comparison with Abaqus
+simulations based on tubes of finite lengths. We shall consider the two loading scenarios discussed
+in Subsection 5.3 and compare the predictions of the 1d model with Abaqus simulations, which
+allows us to quantify the accuracy of our 1d model and determine its range of validity.
+In all numerical calculations and Abaqus simulations, we use the Gent material model
+W “ ´µ
+2 Jm ln
+´
+1 ´ λ2
+1 ` λ2
+2 ` λ2
+3 ´ 3
+Jm
+¯
+,
+(6.1)
+where µ is the shear modulus and Jm is a material constant. We take µ “ 1 which is equivalent
+to scaling all stress variables by µ and Jm “ 97.2 which is typical for rubber. The geometry of the
+tube is taken to be H{Rm “ 0.4 and 2L{Rm “ 40, where Rm “ pA ` Bq{2 is the average radius.
+In the Abaqus simulations, to ensure that localized bulging occurs in the middle of the tube, a
+small section with length 0.1L around the middle point of the tube is weakened by taking its shear
+modulus to be 0.9999 times that of the rest of the tube.
+The 1d differential equation (4.20) subject to appropriate end conditions (see (6.7) later) can
+be solved numerically with the aid of the symbolic computation software Mathematica. Although
+the gradient modulus Dpaq involves an integral that cannot be evaluated analytically, this integral
+can be defined numerically in Mathematica with the built-in command ?NumericQ and can be
+manipulated as elementary functions. Numerically solving the 1d equation is significantly faster
+than Abaqus simulations. The 1d equation can typically be solved in a few seconds on a personal
+computer.
+16
+
+6.1. The case of fixed axial force
+We first consider the loading scenario whereby the resultant axial force N is fixed. As mentioned
+earlier, we assume that the tube is infinitely long and focus on the solution that satisfies the decaying
+boundary condition
+lim
+ZÑ8 apZq “ a8.
+(6.2)
+A linear analysis shows that the solution to (4.23) satisfying (6.2) decays exponentially as Z Ñ 8.
+Thus we have limZÑ8 a1pZq “ 0 automatically. We assume that the bulging solution is symmetric
+with respect to Z “ 0 so that a1p0q “ 0. We write λ8 “ λpa8q, a0 “ ap0q and λ0 “ λpap0qq. Since
+pa8, λ8q satisfy the equations (3.6) and (3.8), we have
+Mpa8, λ8q ´ 1
+2a2
+8Qpa8, λ8q ´
+N
+2πA2 “ 0,
+(6.3)
+Qpa8, λ8q ´ P “ 0.
+(6.4)
+From the definition of λ0 and the conservation law (4.24), we see that pa0, λ0q satisfies
+Mpa0, λ0q ´ 1
+2a2
+0Qpa8, λ8q “ Mpa8, λ8q ´ 1
+2a2
+8Qpa8, λ8q,
+(6.5)
+Gpa0, λ0q “ Gpa8, λ8q.
+(6.6)
+Either a8 or P can be taken to be the load parameter. When a8 is specified, one can first obtain
+λ8 from (6.3). The associated P is computed according to (6.4). Then by solving Eqs. (6.6) and
+(6.5), one obtains the “initial” values a0 and λ0. The localized solution can be found by solving
+the initial value problem
+A2aλpaqpQpa, λpaqq ´ Pq ´ 1
+2D1paqa1pZq2 ´ Dpaqa2pZq “ 0,
+(6.7)
+ap0q “ a0,
+a1p0q “ 0.
+(6.8)
+As a first example, fixing the axial force N to be zero, we find from the bifurcation condition (5.29)
+that localized bulging takes place at a8 “ acr “ 1.86 with a critical pressure Pcr “ 0.308. As we
+trace the bifurcation solution away from the bifurcation point, the pressure drops while the bulge
+grows until it has almost reached a maximum amplitude after which the bulge will propagate at a
+constant pressure. From Maxwell’s equal-areal rule, the propagation pressure is PM “ 0.197.
+Fig. 2 shows the dependence of the pressure on ap0q and the bulging amplitude on a8 based on
+Abaqus simulations and use of the 1d model. The bulging solutions given by Abaqus simulations
+and the 1d model at the four states marked in Fig. 2(a) are shown in Fig. 3. It is seen that the 1d
+solution agrees well with Abaqus simulations in the entire post-bifurcation regime. Remarkably,
+the 1d solution remains highly accurate even in the final propagation stage, as shown in Fig. 3(d).
+Note also that the Abaqus simulations and 1d calculations are conducted for 2L “ 40Rm and 8,
+respectively. This verifies our earlier claim that the tube can effectively be viewed to be infinitely
+long.
+17
+
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+● ● ● ●
+●
+●
+●
+● ● ●
+●
+●
+●
+●
+● ●
+●
+●
+●
+1d model
+Abaqus simulation
+1
+2
+3
+4
+5
+6
+7 a(0)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+P
+a
+b
+c
+d
+(a)
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+1d model
+Abaqus simulation
+1.2
+1.3
+1.4
+1.5
+1.6
+1.7
+1.8
+1.9a∞
+0
+1
+2
+3
+4
+5
+6
+a(0) - a∞
+(b)
+Figure 2: Dependence of (a) pressure on ap0q and (b) bulging amplitude on a8 based on Abaqus simulations and the
+1d model for fixed N “ 0.
+Abaqus simulation
+1d model
+0
+2
+4
+6
+8 Z
+1.6
+1.7
+1.8
+1.9
+2.0
+2.1
+2.2
+2.3
+2.4
+a(Z)
+(a)
+Abaqus simulation
+1d model
+0
+2
+4
+6
+8 Z
+1.5
+2.0
+2.5
+3.0
+3.5
+a(Z)
+(b)
+Abaqus simulation
+1d model
+0
+2
+4
+6
+8 Z
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+a(Z)
+(c)
+Abaqus simulation
+1d model
+0
+2
+4
+6
+8 Z
+1
+2
+3
+4
+5
+6
+a(Z)
+(d)
+Figure 3: Bulging solutions given by Abaqus simulations and the 1d model at the four states marked in Fig. 2(a) for
+fixed N “ 0: (a) P “ 0.3, (b) P “ 0.25, (c) P “ 0.22, (d) P “ 0.197.
+18
+
+6.2. The case of fixed ends
+Next, we consider the loading scenario whereby the tube is first stretched to a specified length
+2ℓ and then its two ends are fixed to prevent further axial displacement (whether the radial dis-
+placement is restricted or not at the ends is immaterial since the tube is assumed to be sufficiently
+long). In the previous subsection, we have solved the problem for a specified axial force N or
+equivalently a specified λ8. For the current problem with a given ℓ, we define λc “ ℓ{L and we
+need to find λ8 such that the following condition is satisfied:
+ż L
+0
+λpapZqq dZ “ λcL.
+(6.9)
+This can be achieved by a shooting procedure: for each N, we compute the left-hand side using
+the procedure outlined in the previous subsection and adjust N such that the left-hand side and
+the right-hand side are equal. The procedure may be started by taking λ8 “ λc. However, solving
+the present problem by the shooting procedure requires a lot of adjustments by hand due to the
+fact that the bulging solution may start to grow after decaying for a range of Z values. To find
+solutions for the current case in a more robust way, we use the finite difference method instead.
+To implement the finite difference method, we partition the domain r0, Ls using a uniform mesh
+Z0, Z1, . . . , Zn with mesh size h “ L{n and coordinate of the j-th grid point given by Zj “ jh. We
+use aj to represent the numerical approximation of apZjq. Applying the central difference scheme,
+we discretize the differential equation (6.7) as
+A2ajλpajqpQpaj, λpajqq ´ Pq ´ 1
+2D1pajq
+´aj`1 ´ aj´1
+2h
+¯2
+´ Dpajqaj`1 ´ 2aj ` aj´1
+h2
+“ 0,
+j “ 1, 2, . . . , n ´ 1.
+(6.10)
+The left boundary condition is given by
+a1p0q “ 0.
+(6.11)
+We see from (5.23) that the solution to (6.7) subject to (6.2) has the asymptotic behavior
+apZq „ a8 ` a1e´κZ
+as Z Ñ 8,
+(6.12)
+where a1 is a constant and
+κ “
+d
+ωpa8, λ8q
+Dpa8q
+.
+Because of this, we may replace the decaying condition boundary (6.2) by the “soft” asymptotic
+condition
+a1pLq ` κpapLq ´ a8q “ 0.
+(6.13)
+19
+
+To avoid the loss of accuracy at the two endpoints, we introduce two additional unknowns a´1 and
+an`1. Then the left and right boundary conditions yield
+a1 ´ a´1
+2h
+“ 0,
+(6.14)
+an`1 ´ an´1
+2h
+` κpan ´ a8q “ 0.
+(6.15)
+Solving for a´1 and an`1 from the above equations, and substituting them into the difference
+equations (6.10) at j “ 0 and j “ n, we obtain the discrete boundary conditions with truncation
+errors of order h2:
+A2a0λpajqpQpa0, λpa0qq ´ Pq ´ 2Dpa0qa1 ´ a0
+h2
+“ 0,
+(6.16)
+A2anλpanqpQpan, λpanqq ´ Pq ´ 1
+2D1pajqκ2pan ´ a8q2
+´ 2Dpanqan´1 ´ an ´ hκpan ´ a8q
+h2
+“ 0.
+(6.17)
+Finally, the fixed-length restriction (6.9) gives
+1
+2λpa0q `
+n´1
+ÿ
+j“1
+λpajq ` 1
+2λpanq ´ λcL
+h
+“ 0.
+(6.18)
+We use the pressure P as the loading parameter. When P is given, one can first solve (6.4) to
+express a8 as a function of λ8. Then N can be viewed as a function of λ8 due to (6.3). It follows
+that λpµq and Dpµq also depend on λ8 through their dependence on N. This implicit dependence
+should be considered when solving the above algebraic equations.
+Setting n to be a sufficiently large number, say n “ 100, and solving the system of nonlinear
+algebraic equations consisting of (6.10), (6.16), (6.17) and (6.18) for aj’s and λ8 with a suitable
+initial guess, we obtain the finite-difference solution for the present problem.
+We may use the
+weakly nonlinear solution with fixed λ8 “ λc “ ℓ{L as an initial guess in the near-critical regime
+and continue the solution to the fully nonlinear regime by always using the solution at the previous
+step as the initial guess for the current step.
+When the total length is fixed to be ℓ “ 2L, then initially λ8 “ 2 and localized bulging takes
+place at a8 “ acr “ 1.74 with a critical pressure Pcr “ 0.198 according to (5.35). In Fig. 4, we have
+shown the dependence of the pressure on ap0q and the bulging amplitude on a8 based on Abaqus
+simulations and use of the 1d model. The bulging solutions determined by Abaqus simulations
+and the 1d model at the four states indicated in Fig. 4(a) are presented in Fig. 5. It is observed
+that the agreement between the 1d model and Abaqus simulations is again excellent in the fully
+nonlinear regime.
+Finally, Fig. 6 shows the actual variation of P against ap0q predicted by the 1d model when L
+is varied and the averaged stretch λc is fixed or λc is varied but L is fixed. These results confirm
+the theoretical prediction of Guo et al. (2022) that the right branches of these curves all converge
+20
+
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+● ● ●
+●
+●
+● ●
+●
+●
+● ● ●
+● ● ●
+●
+● ●
+●
+●
+●
+1d model
+Abaqus simulation
+0
+1
+2
+3
+4
+5
+6
+7
+a(0)
+0.00
+0.05
+0.10
+0.15
+0.20
+P
+a
+b
+c
+d
+(a)
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+1d model
+Abaqus simulation
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+1.7
+1.8a∞
+1
+2
+3
+4
+5
+6
+a(0) - a∞
+(b)
+Figure 4: Dependence of (a) pressure on ap0q and (b) bulging amplitude on a8 based on Abaqus simulations and
+using the 1d model for fixed length ℓ{L “ 2.
+Abaqus simulation
+1d model
+0
+2
+4
+6
+8
+10
+12Z
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+2.4
+2.6
+2.8
+a(Z)
+(a)
+Abaqus simulation
+1d model
+0
+2
+4
+6
+8
+10
+12Z
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+a(Z)
+(b)
+Abaqus simulation
+1d model
+0
+2
+4
+6
+8
+10
+12Z
+1
+2
+3
+4
+5
+a(Z)
+(c)
+Abaqus simulation
+1d model
+0
+2
+4
+6
+8
+10
+12Z
+1
+2
+3
+4
+5
+6
+7
+a(Z)
+(d)
+Figure 5: Bulging solutions based on Abaqus simulations and the 1d model the at the four states indicated in Fig.
+4(a) for fixed length ℓ{L “ 2: (a) P “ 0.19, (b) P “ 0.18, (c) P “ 0.173, (d) P “ 0.198.
+to a master curve that is independent of L or λc. These curves all terminate at the point where
+the axial stress near each end of the tube has become compressive enough so that secondary Euler
+buckling or axisymmetric wrinkling becomes possible.
+21
+
+L=15
+L=20
+L=40
+0
+1
+2
+3
+4
+5
+6
+7
+a(0)
+0.12
+0.14
+0.16
+0.18
+0.20
+0.22
+P
+(a)
+λc=1.5
+λc=2
+λc=2.8
+0
+1
+2
+3
+4
+5
+6
+7
+a(0)
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+P
+(b)
+Figure 6: Variation of P against ap0q predicted by the 1d model when (a) the total length is fixed with λc “ 2 and
+L “ 15, 20 and 40, respectively and (b) L is fixed at 20 and λc “ 1.5, 2 and 2.8, respectively.
+7. Conclusion
+We have derived a 1d model for the analysis of axisymmetric deformations of an inflated cylin-
+drical tube of finite wall thickness, and established its range of validity by comparing its predictions
+with those of Abaqus simulations for two typical loading scenarios. The comparison shows that
+the 1d model performs extremely well in both the near-critical and fully nonlinear regimes. The
+dimension reduction started from three-dimensional finite elasticity theory and is performed in
+terms of the energy functional and principal stretches. A key ingredient of the dimension reduc-
+tion is the assumption of slow variation of the leading-order solution in the axial direction without
+any restriction on its amplitude, which results in a 1d model that is simple but is still capable of
+capturing the strain-gradient effect. This is in contrast with the traditional asymptotic analysis
+where the leading order solution is assumed to be a small-amplitude perturbation from the primary
+deformation. It is because of this difference that the 1d model has a much larger range of validity
+than the expansion methods around the bifurcation point. The nonlinearity of the strain is kept
+in the 1d model, reflected by the nonlinear potential Gpa, λpaqq and the nonlinear strain gradient
+modulus Dpaq. Our expression for the strain gradient coefficient Dpaq is quite simple. For the
+Gent material model, Dpaq can be calculated by integrating once. We remark that although the
+derivation presented in this work is variational, the 1d model can also be derived by substituting
+the asymptotic solution (4.2) into the 3d governing equations and solving the resulting equations
+at successive orders.
+The 1d model is amenable to asymptotic and numerical solutions. The bifurcation condition and
+the weakly nonlinear amplitude equation predicted by the model are exact. In fact, the expressions
+(5.24) and (5.25) derived using the 1d model are more compact and more revealing than their
+counterparts in Ye et al. (2020). A major advantage of the 1d model is that the entire evolution
+process of bulging or necking can be determined using the finite difference method which is more
+accessible and much easier to implement than commercial packages such as Abaqus. This advantage
+22
+
+would become even more significant when other fields such as electric loading and residual stresses
+were also present. Such extra fields and new geometries (e.g. axisymmetric necking of a stretched
+plate (Wang et al., 2022) ) will be considered in our future studies.
+A Mathematica code that produces all the results presented in the paper is available on GitHub
+(https://github.com/yfukeele).
+Acknowledgments
+This work was supported by the National Natural Science Foundation of China (Grant No
+12072224) and the Engineering and Physical Sciences Research Council, UK (Grant No EP/W007150/1).
+Appendix A. Simplifying the one-dimensional energy functional
+Substituting (4.19) into (4.11), we can write the integral of KpR, v, vRq as
+ż B
+A
+KpR, v, vRq dR “ pI1 ` I2 ` I3qa12,
+(A.1)
+where
+I1 “
+ż B
+A
+pλ´1¯σ22qaR
+ż B
+R
+cpa, Tq dT dR,
+(A.2)
+I2 “ 1
+2
+ż B
+A
+Rζp¯r2
+a ` cpa, Rq2q dR,
+(A.3)
+I3 “
+ż B
+A
+Rξ¯racpa, Rq dR.
+(A.4)
+By interchanging the order of integration, we can rewrite I1 as
+I1 “
+ż B
+A
+ż B
+R
+pλ´1¯σ22qaRcpa, Tq dT dR
+“
+ż B
+A
+ż T
+A
+pλ´1¯σ22qaRcpa, Tq dR dT
+“
+ż B
+A
+cpa, Tq B
+Ba
+´ ż T
+A
+λ´1¯σ22R dR
+¯
+dT.
+(A.5)
+From (3.14), we have
+ż T
+A
+λ´1¯σ22R dR “ A2mpa, Aq ´ T 2mpa, Tq.
+(A.6)
+Inserting (A.6) into (A.5) and noting (3.15)2 and (3.11), we can simplify I1 as
+I1 “ PA2a
+ż B
+A
+cpa, Rq dR ´
+ż B
+A
+cpa, RqR2 B
+Bampa, Rq dR.
+(A.7)
+23
+
+Noting that ξ “ q¯λ1 ` ¯λ3ζ{λ, the integral I3 can be calculated as
+I3 “
+ż B
+A
+R
+´
+q¯λ1 `
+¯λ3
+λ ζ
+¯
+¯racpa, Rq dR “
+ż B
+A
+´
+¯r¯raq ` Rζ ¯ra
+¯λ1λ2
+¯
+cpa, Rq dR.
+(A.8)
+Adding up the three integrals, we obtain
+ż B
+A
+KpR, v, vRq dR ` PA2aa1v|R“A “pI1 ` I2 ` I3qa12 ´ PA2aa12
+ż B
+A
+cpa, Rq dR
+“a12
+ż B
+A
+´
+´ cpa, RqR2 B
+Bampa, Rq ` 1
+2Rζp¯r2
+a ` cpa, Rq2q
+`
+´
+¯r¯raqpa, Rq ` Rζ ¯ra
+λ1λ2
+¯
+cpa, Rq
+¯
+dR
+“a12
+ż B
+A
+p1
+2Rζp¯r2
+a ` cpa, Rq2q ´ Rζcpa, Rq2q dR
+“1
+2a12
+ż B
+A
+Rζp¯r2
+a ´ cpa, Rq2q dR.
+(A.9)
+This gives the expression of the coefficient Dpaq announced in (4.21). The expression of Cpaq in
+(4.22) follows by a straightforward substitution.
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+27
+

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+RIEMANNIAN GEOMETRY AND MOLECULAR SIMILARITY II:
+KÄHLER QUANTIZATION
+A PREPRINT
+Rachael Pirie
+School of Natural and Environmental Sciences
+Newcastle University
+[email protected]
+Stuart J. Hall
+School of Mathematics, Statistics, and Physics
+Newcastle University
+[email protected]
+Daniel J. Cole
+School of Natural and Environmental Sciences
+Newcastle University
+[email protected]
+Thomas Murphy
+Department of Mathematics
+California State University, Fullerton
+[email protected]
+January 12, 2023
+ABSTRACT
+Shape-similarity between molecules is a tool used by chemists for virtual screening, with the goal of
+reducing the cost and duration of drug discovery campaigns. This paper reports an entirely novel
+shape descriptor as an alternative to the previously described RGMolSA descriptors [1], derived from
+the theory of Riemannian geometry and Kähler quantization (KQMolSA). The treatment of a molecule
+as a series of intersecting spheres allows us to obtain the explicit Riemannian metric which captures
+the geometry of the surface, which can in turn be used to calculate a Hermitian matrix M as a directly
+comparable surface representation. The potential utility of this method is demonstrated using a series
+of PDE5 inhibitors considered to have similar shape. The method shows promise in its capability to
+handle different conformers, and compares well to existing shape similarity methods. The code and
+data used to produce the results are available at: https://github.com/RPirie96/KQMolSA.
+Keywords Riemannian Geometry · Kähler Quantization · Molecular Shape · Ligand-Based Virtual Screening
+1
+Introduction and Summary of Part I
+The concept that shared biological activity exists between similar molecules is used widely in drug discovery [2].
+Molecules with known activity can be used as templates to screen large databases for other potential hits. This is
+more efficient and allows coverage of a greater area of chemical space than is possible with experimental screening
+alone [3]. Estimating similarity between molecules based on their 3D shape has gained popularity due to the
+requirement for protein-drug shape complementarity to enable strong binding. However no fixed notion of shape
+exists. Instead, comparison relies on mathematical approximation of the molecule’s shape based on its volume,
+distribution of atomic distances or surface (most commonly treated as the van der Waals or solvent accessible surface) [4].
+In the accompanying paper [1], the RGMolSA method was presented.
+The descriptor developed there ap-
+proximates the shape of the molecular surface using a simple nine-element vector containing the surface area and an
+approximation to the first eight non-zero eigenvalues of the ordinary Laplacian. The descriptor can be viewed as an
+approximation to the Riemannian metric, the underlying mathematical object that describes the shape of a surface.
+In this paper we present an entirely different method of approximating the Riemannian metric by using ideas from
+the theory of Kähler quantization; we call this method Kähler quantization for Molecular Surface Approximation
+(KQMolSA). The theory was originally developed by mathematicians and string theorists in order to give explicit
+representations of the shapes of 4-dimensional objects (Calabi–Yau manifolds) that appear in physical theories (see [5]
+arXiv:2301.04424v1  [math.DG]  11 Jan 2023
+
+Geometry and Molecular Surfaces
+A PREPRINT
+for the paper that pioneered its use as a numerical technique). In a nutshell, a function called the Kähler potential
+is associated to the metric. We then compute something analogous to a Taylor expansion of this function with the
+coefficients being stored in a Hermitian matrix. While the matrices themselves do depend upon the precise position and
+parameterisation of the molecular surface in three-dimensional space R3, the dependence is easy to calculate. Hence we
+can perform our calculations in the ‘quantized’ space of Hermitian matrices and assign a distance between the shapes of
+two molecular surfaces this way. The final distance is independent of the position of the molecules and the choices
+made in their parameterisations.
+1.1
+Summary of Previous Work
+As in the accompanying paper [1], our approach begins by treating the molecule as a series of intersecting spheres, with
+their radii given by the van der Waals radii of the constituent atoms. The surface is assumed to have a genus of zero, so
+any rings (e.g. benzene) are replaced with a single sphere of radius 2.25 Å to facilitate this. The molecular structure
+is then defined by the number of spheres N (with each ring counted as a single sphere, and excluding any hydrogen
+atoms), the centres ci and radii ri for each sphere and the adjacency matrix T describing intersection of spheres, where
+Tij =
+�
+1
+if spheres i and j intersect
+0
+otherwise (or i = j).
+The surface area A of the molecule is calculated as the area of each sphere minus the “missing parts" where two spheres
+intersect:
+A = 2π
+�
+i
+�
+�2r2
+i −
+�
+�ri
+�
+j
+Tij|ri − λij|
+�
+�
+�
+� .
+(1)
+This value is used to re-scale each of the starting constructs such that the surface area of the molecule is equal to that of
+a unit sphere (or 4π) to address the observation that Riemannian geometry treats two objects which differ only in size
+as having equivalent shape. This re-scaling is accounted for in the final descriptors with some weighting so as not to
+dominate the similarity calculation.
+From the initial data, a map is constructed to ‘unwrap‘ the surface onto the complex plane C in a process
+we refer to as piecewise stereographic projection. This requires an atom to be selected as a starting point from which to
+construct our map, which we refer to as the base sphere. This is taken to be the atom closest to the centre of mass by
+first finding the centroid of the molecule and then taking the atom with the smallest Euclidean distance from this point.
+The Riemannian metric g = Φ∗
+ps(gEuc) induced by the mapping Φps : C → S ⊂ R3 takes the form
+g =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+4r2
+B
+(1+|z|2)2 (dx2 + dy2)
+if z ∈ C
+C1
+(|z−A1|2+B1)2 (dx2 + dy2)
+if z ∈ D(a1, R1),
+C2
+(|z−A2|2+B2)2 (dx2 + dy2)
+if z ∈ D(a2, R2),
+...
+...
+CN−1
+(|z−AN−1|2+BN−1)2 (dx2 + dy2)
+if z ∈ D(aN−1, RN−1),
+(2)
+where rB is the radius of the base sphere and
+C = C\D(a1, R1) ∪ D(a2, R2) ∪ · · · ∪ D(aN−1, RN−1),
+is the complement of the discs D(a1, R1), . . . , D(aN−1, RN−1) which corresponds to the points in the base sphere.
+The RGMolSA descriptor uses the explicit form of the Riemannian metric provided by piecewise stereo-
+graphic projection to approximate the low-lying eigenfunctions of the Laplacian ∆. In [1], we compared the RGMolSA
+descriptor for Sildenafil, Vardenafil and Tadalafil, a series of PDE5 inhibitors that are known to occupy a similar volume
+in the binding pocket of their target protein, and thus have similar shape (Figure 1) [6]. Vardenafil is a classic example
+of a “me-too" drug, where only a few small modifications have been made to the structure of Sildenafil. As these are
+both highly similar chemically, they would be expected to have close to the same shape. Tadalafil on the other hand is
+chemically quite different from the other two, but inspection of the molecules in the pocket of PDE5 reveals they occupy
+a similar binding pose, and thus would also be expected to have similar shape. In this article, for ease of compari-
+son with the previous work [1], we again use these three molecules as the basis for investigating the new shape descriptor.
+2
+
+Geometry and Molecular Surfaces
+A PREPRINT
+(a) Sildenafil
+Pfizer
+First Sold: 1998
+(b) Vardenafil
+Bayer
+First Sold: 2003
+(c) Tadalafil
+Lilly
+First Sold: 2003
+Figure 1: PDE5 inhibitors Sildenafil, Vardenafil and Tadalafil of known shape similarity. Tadalafil (different chemical structure, similar shape) is an example of a scaffold
+hop from the first in class drug Sildenafil, and offers greatly improved performance, while Vardenafil (a "me-too" follow-up drug) only offers minor improvements.
+While RGMolSA was found to give a good description of shape, it has a possible deficiency due to the dependence
+of the results on the choice of base sphere, which in turn determines the trial functions for calculating the integrals
+used to construct the descriptor. The geometry of the surface near the base sphere is well described, but for atoms
+further away a greater number of eigenvalues would be needed for accurate description of the surface. This problem is
+greater for larger molecules and can lead to the introduction of numerical errors when the molecule is large enough.
+We handled such errors by ignoring any contributions from regions with numerical radii less than 10−9; however, this
+forces a somewhat artificial ‘locality’ upon the shape descriptor meaning that it probably only accurately captures the
+shape near to the base sphere.
+In the following section we outline the theory underpinning the KQMolSA descriptors, that again uses the Riemannian
+metric to approximate the geometry of the surface. The resulting descriptors lie in the manifold GL(N, C)/U(N) to
+give a global descriptor of molecular geometry with reduced dependence on the starting position. Figure 2 summarises
+the steps in computing these, using Sildenafil as an example. While the descriptor itself does depend upon the choices
+made and the position of the surface within R3, this is easily accounted for within the space GL(N, C)/U(N). This
+makes computing the ‘distance’ between the shape descriptors particularly straightforward.
+2
+The Mathematics of Kähler Quantization
+2.1
+Overview of the Theory
+We should say immediately that the theory of Kähler quantization is far too advanced to be able to detail in the current
+paper. For readers with sufficient mathematical background, a good account (and the original account of its use as
+a numerical technique) is given in [5]. An exposition, aimed at readers with a general scientific background, of the
+mathematical theory is currently being written by two of the authors [7].
+The theory is concerned with the geometry of complex manifolds (shapes that locally look like Cn); any sur-
+face that sits in R3 is a complex manifold as it locally looks like a copy of the complex numbers C (i.e. n = 1). More
+concretely, we will be concerned with the surfaces that are topologically equivalent to S2; in the language of complex
+manifolds, the sphere is often referred to as the Riemann Sphere and denoted CP1. The restriction on the topology
+of the surface is justified by the fact that chemists do not expect any activity in the centre of rings occurring in most
+3
+
+N
+N
+HN
+N
+N
+NN
+HN
+N
+N
+N
+NH
+N
+N
+N
+010000Geometry and Molecular Surfaces
+A PREPRINT
+Figure 2: Key steps involved in the computation of the KQMolSA surface descriptor for Sildenafil (a PDE5 inhibitor).
+drug-like molecules. The exceptions to this are macrocyclic molecules (those with large rings of more than 12 atoms)
+where genuine activity occurs in the centre of the ring. Such molecules are therefore excluded from comparison by both
+methods proposed.
+The natural class of functions to work with when dealing with complex manifolds are those that are com-
+plex differentiable, often called holomorphic functions. We consider a general complex manifold X; unfortunately,
+if the manifold X is compact, the only holomorphic functions f : X → C are constant. Thus we cannot hope to
+understand X simply by studying the holomorphic functions on X. A generalisation of the notion of a holomorphic
+function is that of a section of a holomorphic line bundle L with base X. For readers familiar with the theory, a function
+is a section of the trivial bundle. A line bundle is positive if there is a Hermitian metric h on L with positive curvature.
+A foundational result of Kodaira [8] says that if the line bundle L is positive then for large enough k the tensor power
+Lk, has a lot of holomorphic sections. In fact, the space of all such sections, denoted H0(Lk), is a complex vector
+space of dimension that has order O(kn) as k → ∞.
+The curvature of a positively curved Hermitian metric h gives rise to an object called a Kähler form, ω,
+which in turn gives rise to a Riemannian metric g (the mathematical object being used in [1] to describe shape). It
+turns out that the set of all positively curved Hermitian metrics on a line bundle L can be identified with the set of all
+real-valued functions ϕ : X → R that satisfy, in some local coordinate z, the ∂ ¯∂-equation
+√
+−1∂ ¯∂ϕ = ω − ω0
+where ω is the Kähler form of the metric and ω0 is a fixed reference Kähler form. We will give more detail on the
+differential operators ∂ and ¯∂ in Section 2.3; in particular, we will explain that in the molecular surface setting, the
+∂ ¯∂-equation is really just the familiar Poisson equation in the plane. The function ϕ is called a Kähler potential for
+ω. The associated potential is not unique but any two differ by a constant; this does not affect the metric which is
+constructed by taking two derivatives of the potential. However, we will see that the addition of a constant to a potential
+will have the affect of scaling the Hermitian matrix we produce as a shape descriptor by a positive real number and we
+will be required to find the ‘optimal’ rescaling in our distance calculation.
+To summarise, what we have for a positive Hermitian line bundle (L, h) → X are:
+• a Kähler form ω and a Kähler potential ϕ : X → R,
+• a complex vector space H0(Lk).
+4
+
+e.g. Sildenafil
+Space Filling Model
+Replace Rings, Base Sphere (Grey)
+- AiilD
+01
+Map to Complex Plane
+Surface Area
+Matrix of Levels
+2r?
+ifz εc
+(1 + [z/2)2
+C1
+2.16 + 0j
+-2.44 + 0.86j
+2.41 - 1.93j 1
+ifz E D(ai,R1)
+F(z) =
+zje-kF(z)V-1dzΛdz
+M 
+-2.44 - 0.86j
+3.01 + 0j
+(lz - A1/2 + B1)2
+-3.48 + 1.22j
+MI
+-3.48 - 1.22j
+:
+2.41 + 1.93j
+4.41 + 0j
+Cn-1
+ifz E D(an-1,Rn-1)
+(Iz - An-1/2 + Bn-1)2
+Riemannian Metric
+Construct Hermitian Matrix
+Hermitian Matrix Shape DescriptorGeometry and Molecular Surfaces
+A PREPRINT
+What Kähler quantization amounts to is relating the geometry described by the Kähler potentials (an infinite dimensional
+space of functions) to the finite dimensional complex vector space H0(Lk). This theme occurs throughout numerical
+analysis and shape description, for example in the theories of Fourier analysis, spherical harmonics, Taylor series, all of
+which produce a finite-dimensional vector space out of some infinite-dimensional set of functions.
+2.2
+Quantization and Tian’s Theorem
+The data (L, h) → X allows for a natural L2-inner product on the vector space of sections H0(Lk). Given sections
+s1, s2 ∈ H0(Lk), we compute
+⟨s1, s2⟩ :=
+�
+X
+hk(s1, s2)ωn
+n! ,
+where hk is the Hermitian metric induced on Lk by h, and ωn/n! is the volume element produced by the Kähler form.
+It is this inner product that is the quantization of the data (L, h) → X. The space of all (Hermitian) inner products on a
+complex N-dimensional vector space can be thought of as GL(N; C)/U(N). This is a negatively curved symmetric
+space and has a natural notion of distance on it; it is this distance that we will use to measure shape similarity (see
+Section 2.5).
+To recover the geometry defined by (L, h) → X from the quantization, we choose a basis {sj} of the vec-
+tor space H0(Lk) which gives rise to the matrix representation of the inner product
+Mij := ⟨si, sj⟩.
+If we let v be the vector of sections
+v = (s1, s2, . . . sN) ,
+then we can define a Kähler potential (recalling that the sections are locally defined holomorphic functions) ˜ϕ by
+˜ϕ := −1
+k log
+�
+v∗M−1v
+�
+.
+Theorem 2.1 (Tian, [9]). Let (X, L, h) be a complex manifold with holomorphic line bundle L and positively curved
+Hermitian metric h with curvature ω. If we produce another Kähler form
+�ω = ω0 +
+√
+−1∂ ¯∂ ˜ϕ,
+then
+∥ω − �ω∥C0 = O(k−2).
+Paraphrasing this theorem, we can say any Kähler form coming from a Kähler potential ϕ can be well approximated by
+the Kähler form coming from the ‘algebraic’ function �ϕ. If we pick local complex coordinates z1, z2, . . . , zn then the
+term v∗M−1v is just a power series in the coordinates. In the case of a molecular surface, we will have something like a
+polynomial. This is the sense in which the function �ϕ is similar to a truncated Taylor series for the original function ϕ.
+The theorem then says that this series really does converge.
+Tian’s Theorem is stated for smooth metrics (those where one can take an arbitrary number of derivatives of
+the Kähler potential ϕ); in practice (see Section 2.3), we will be working with metrics where the potentials are in
+C2(X), that is twice continuously differentiable. The theory of approximating such metrics algebraically has not
+been written down but we will demonstrate that we get a method that does produce meaningful shape comparisons.
+We expect that, suitably adapted to this setting, something like Tian’s Theorem is still true; for example, the case of
+potentials with lower regularity is discussed in [10].
+2.3
+Implementation in Practice
+As mentioned already, in practice we take X = CP1 the Riemann sphere and the line bundle to be the anticanonical
+bundle K∗
+CP1 = O(2). The Kähler form ω, can be explicitly constructed from the Riemannian metric g, and in the
+coordinates furnished by the piecewise stereographic projection map Φps, we can use the form of the metric (2) to get
+ω = F(z)
+√
+−1dz ∧ dz,
+5
+
+Geometry and Molecular Surfaces
+A PREPRINT
+where F : C → R+ is the ‘metric function’ given by
+F(z) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+2r2
+B
+(1+|z|2)2
+if z ∈ C,
+C1
+(|z−A1|2+B1)2
+if z ∈ D(a1, R1),
+C2
+(|z−A2|2+B2)2
+if z ∈ D(a2, R2),
+...
+...
+CN−1
+(|z−AN−1|2+BN−1)2
+if z ∈ D(aN−1, RN−1).
+(3)
+Note we have replaced, in the metric g, the real symmetric 2-tensor dx2 + dy2 with the antisymmetric form
+(√−1/2)dz ∧ d¯z, where dz = dx + √−1dy and d¯z = dx − √−1dy.
+To find the Kähler potential ϕ : C → R, we solve the ‘∂∂-equation’
+ω =
+√
+−1∂∂ϕ.
+If we consider the complex differential operators
+∂
+∂z = 1
+2
+� ∂
+∂x −
+√
+−1 ∂
+∂y
+�
+and
+∂
+∂z = 1
+2
+� ∂
+∂x +
+√
+−1 ∂
+∂y
+�
+,
+then the ∂∂-equation is equivalent to solving the Poisson equation
+∂2ϕ
+∂z∂z = 1
+4∆Eucϕ = F,
+where ∆Euc is the usual 2-dimensional Laplacian. We can solve the Poisson problem explicitly to find ϕ. The solution
+can be thought of as having two parts: a ‘local’ part that is found by simply observing that
+∂2
+∂z∂z
+�C log(|z − A|2 + B)
+B
+�
+=
+C
+(|z − A|2 + B)2 ,
+and a ‘correction term’, named thus as the term is needed to ensure the function is in C2(C). The correction term is a
+linear combination of functions of the form
+log(|αz + β|2),
+where we get one term for each sphere. As each of the correction terms is a harmonic function, that is
+∆ log(|αz + β|2) = 0,
+the addition of the correction terms is still a solution of the Poisson equation. It would appear the correction terms
+are singular at the points z = −β/α; however, these points always lie outside the disc where the function takes this
+particular form. We record the form of the potential as a theorem and refer the reader to the appendix (Section 5) for a
+derivation of the solution.
+Theorem 2.2 (Form of Kähler potential). Let g be of the form Equation (2). In the region associated to the ith sphere,
+the Kähler potential can be written
+ϕ(z) = Ci
+Bi
+log(|z − Ai|2 + Bi) +
+N
+�
+j=1
+Kij log(|αijz + βij|2),
+where K ∈ M N×N(R), and α, β ∈ M N×N(C).
+The matrices K, α, and β in the previous theorem are easily calculated from the geometric data associated to the
+molecule and so it is straightforward to describe the Kähler potential explicitly.
+The space of global sections H0(O(2k)) ∼= C2k+1 can be identified with the span of the functions
+⟨1, z, z2, . . . , z2k⟩.
+Thus the shape descriptor associated to the surface is the (2k + 1) × (2k + 1) Hermitian matrix M where (considering
+indices that run from 0 to 2k)
+Mij =
+��
+C
+zizje−kϕF(z)
+√
+−1dz ∧ dz.
+(4)
+6
+
+Geometry and Molecular Surfaces
+A PREPRINT
+2.4
+Computing the Relevant Integrals
+A naïve numerical calculation of the integrals described by Equation (4) gives rise to two obvious problems:
+firstly, the domain of integration is unbounded (being the whole complex plane C); secondly, the domains and
+values describing the metric and the Kähler potential ϕ could become so small that numerical instabilities start
+to dominate the contribution of the associated atom.
+The second problem has been discussed as a limitation
+in the approximation of the spectrum of the Laplacian [1]. In this paper, we exploit the fact that the automor-
+phism group of CP1 is the group of Möbius transformations, PSL(2, C); we can use elements of this group to
+ensure the coordinates we perform calculations in are always in a numerically controlled region (here we use a unit disc).
+Put more concretely, let m ∈ {1, 2, . . . , N} index the mth sphere making up the molecular surface, then
+there is an element Tm ∈ PSL(2, C) that maps the unit disc
+D = {z ∈ C | |z| < 1},
+onto the region D(am, Rm) from Equation (2). We note that if the mth sphere has level l, then the pre-image of the
+regions corresponding to level (l + 1) spheres which intersect the mth sphere will describe certain discs properly
+contained in D. Hence the contribution of the mth sphere to the matrix described by Equation (4) is given by
+��
+D− ˆ
+D
+(Tm(w))i(Tm(w))je−kϕ(Tm(w))F(Tm(w)) dTm(w) ∧ dTm(w),
+(5)
+where ˆD represents the union of the discs corresponding to the next level spheres intersecting the mth sphere. In practice,
+we account for these higher-level spheres by assigning the value 0 to the volume form F(Tm(w)) dTm(w) ∧ dTm(w)
+whenever w ∈ ˆD (note this produces a jump discontinuity in the volume form). Numerical calculation of integrals
+of the form of Equation (5) is done by splitting into an angular and radial direction and then performing successive
+applications of the trapezium rule; we choose a radial step size corresponding to nr = 15 integration points and an
+angular step size corresponding to taking nθ = 10 points. This seems to achieve a reasonable accuracy; for example,
+one can check the area integral for a given integration scheme. We have also determined that the distance between
+shape descriptors does not seem to be significantly changed by taking smaller step sizes (Section 3.1).
+2.5
+Finding the Distance Between Shape Descriptors
+Given two positive definite Hermitian matrices M1, M2, such as those generated by Equation (4), there are innumerable
+ways of defining a notion of distance between such matrices. With regards to the theory of Kähler quantization, it is
+natural to consider M1, M2 as two Hermitian inner products on the fixed complex vector space H0(O(2k)). This space
+is naturally seen as the manifold GL(2k + 1; C)/U(2k + 1). An inner product is specified by declaring a particular
+basis to be orthonormal; any basis conjugate under the action of U(2k + 1) defines the same inner product. This
+space has a natural distance on it; one characterisation of this distance is that shortest paths (geodesics) are given
+by one-parameter subgroups of GL(2k + 1; C), that is by paths of matrices of the form exp(tA) where A is some
+(2k + 1) × (2k + 1) complex matrix.
+More explicitly, if {v1, v2, . . . , v2k+1} is a basis of H0(O(2k))such that both inner products are represented
+by diagonal matrices
+M1 = Diag
+�
+eλ1, eλ2, . . . , eλ2k+1�
+,
+M2 = Diag (eµ1, eµ2, . . . , eµ2k+1) ,
+then
+d(M1, M2) = k− 3
+2
+�
+�
+�
+�
+2k+1
+�
+i=1
+(λi − µi)2.
+(6)
+The factor of k−3/2 ensures that the distances stabilise as k → ∞ (see Theorem 1.1 in [11]). It will be useful to consider
+the following more compact form for the distance
+d(M1, M2) = k− 3
+2
+�
+�
+�
+�
+2k+1
+�
+i=1
+(log(ηi))2,
+(7)
+where {ηi} are the eigenvalues of the matrix M−1
+1 M2.
+7
+
+Geometry and Molecular Surfaces
+A PREPRINT
+It is a well-known fact that the automorphism group of the Riemann sphere CP1 is the group of Möbius
+transformations PSL(2, C). Roughly speaking, the subgroup PSU(2) ⊂ PSL(2, C) corresponds to rotations of the
+original surface and the remaining maps correspond to reparameterisations that preserve the complex structure. If
+ϖ ∈ PSL(2, C) is an automorphism of the form
+ϖ(z) = αz + β
+γz + δ ,
+then ϖ also acts on the vector space H0(O(2k)). In representation theoretic terms, this action is the representation
+induced on Sym2k(C2) by the standard representation of SL(2, C). If we denote the element of SL(2k+1, C) by ϑ(ϖ)
+(see [12], Lemma 8) and the original shape descriptor computed in the z-coordinate by M, then the shape descriptor
+computed in the ϖ(z)-coordinate will be
+(ϑ(ϖ))∗ M (ϑ(ϖ)) .
+As mentioned in Section 2, the fact that the Kähler potential is only defined up to the addition of a constant means we
+can also scale the Hermitian matrix M by a positive constant. Hence our calculation of distance between two shape
+descriptors M1 and M2 becomes the concrete problem of minimising, over (p, ϑ) ∈ R × SL(2, C),
+ζ(p, ϑ) =
+2k+1
+�
+i=1
+(log(ηi))2,
+where {ηi} are the eigenvalues of the matrix M−1
+1 ep (ϑ(ϖ))∗ M2 (ϑ(ϖ)).
+It is easy to see that the value of p at a critical point of ζ is independent of the element ϑ.
+Elementary
+calculus yields that the value of p is given by
+p = −
+1
+2k + 1
+2k+1
+�
+i=1
+log(˜ηi),
+where {˜ηi} are the eigenvalues of the matrix M−1
+1 M2. As the matrix (ϑ(ϖ)) has unit determinant, the value of p
+does not depend up the SL(2, C) action on the Hermitian matrix M2. We thus reduce the distance calculation to a
+minimisation over the six-dimensional Lie group SL(2, C).
+Note that the distance between the shape descriptors given by Equation (6) is the distance between the molecular shapes
+after they have been re-scaled to have area 4π. Hence the distance between two molecular surfaces S1 and S2 should
+include a component to reflect the difference in area between S1 and S2. As we are interested in producing a similarity
+score rather than a distance between two inputs, we do not take this point up further in the article. Our initial attempts at
+creating a similarity score are detailed in the subsequent section.
+The remaining minimisation over SL(2, C) is done by parameterising a generic matrix by the 6 real variables x1, ...x6
+and taking
+ϖ(x1, x2, . . . , x6) =
+�
+x1 + √−1x2
+x3 + √−1x4
+x5 + √−1x6
+∗
+�
+,
+where ∗ is chosen to ensure det(ϖ) = 1. To perform the minimisation, we use algorithms that do not require the
+input of a gradient vector, such as Nelder–Mead or Powell methods [13]. These are implemented using off-the-shelf
+packages in SciPy [14]. We found that for k = 1 there was very little difference between the results for either
+method; the minimisation algorithm converges to produce a robust distance value. For k = 2 the minimisation
+methods appear to be a little less stable and occasionally did not converge. One way around this was to use the
+element of SL(2, C) found by the k = 1 minimisation as the initial guess for the k = 2 step (otherwise the identity
+matrix was used). We anticipate that one might be able to improve this process; for example, by computing the
+gradient of the function to be minimised explicitly and then using this in an algorithm such as conjugate gradient descent.
+One further consideration in implementing the distance measure between two matrices was in shape descrip-
+tors for k > 2 (and for k = 2 in some cases), where numerical instability exists within the method. Occasionally
+non-positive definite matrices are produced, that cannot be compared using the above approach. As Hermitian matrices
+that differ only by scale can be considered equivalent, such cases have been treated by scaling one matrix by a factor of
+10, 100 or 1000 as needed in order to bring the eigenvalues into the range required for consideration with Python.
+8
+
+Geometry and Molecular Surfaces
+A PREPRINT
+3
+Initial Case Study: Phosphodiesterase 5 (PDE5) Inhibitors
+3.1
+Tuning the Parameters nr and nθ
+To determine the effect of varying the parameters nr and nθ (Section 2.4) on the quality of the shape descriptors
+produced, we considered three sets of parameters: nr = 200 and nθ = 100; nr = 50 and nθ = 25; nr = 15 and
+nθ = 10. The distances produced between the descriptor for each set and the area returned during the computation of
+the relevant integrals (which should be ∼ 12.57 for an accurate descriptor, as constrained by the choice of scaling the
+surface area to 4π) are reported here for Sildenafil (Table 1), Vardenafil (Table 2) and Tadalafil (Table 3).
+Table 1: Computed distances between descriptors of Sildenafil generated using different values of nr and nθ for k = 1. The area reported is that returned by the
+integration step.
+(200, 100), area = 12.59
+(50, 25), area = 12.62
+(15, 10), area = 12.62
+(200, 100)
+-
+0.032
+0.032
+(50, 25)
+0.038
+-
+0.040
+(15, 10)
+0.038
+0.040
+-
+Table 2: Computed distances between descriptors of Vardenafil generated using different values of nr and nθ for k = 1. The area reported is that returned by the
+integration step.
+(200, 100) area = 12.57
+(50, 25), area = 12.58
+(15, 10), area = 12.58
+(200, 100)
+-
+0.005
+0.005
+(50, 25)
+0.005
+-
+0.004
+(15, 10)
+0.005
+0.004
+-
+Table 3: Computed distances between descriptors of Tadalafil generated using different values of nr and nθ for k = 1. The area reported is that returned by the
+integration step.
+(200, 100), area = 14.32
+(50, 25), area = 14.37
+(15, 10), area = 14.37
+(200, 100)
+-
+0.003
+0.003
+(50, 25)
+0.003
+-
+0.001
+(15, 10)
+0.003
+0.001
+-
+As these distances are small in each case, there is no significant loss of quality when the number of points considered is
+reduced. The areas for both Sildenafil and Vardenafil are also close to 12.57, indicating high quality descriptors. The
+area for Tadalafil is overestimated slightly, however this is due to an issue with the replacement of the rings for motifs
+with a 5-membered ring between two other rings rather than the choice of nr and nθ. Similar results were observed for
+the consideration of k = 2. As the quality is unaffected, the minimum parameters of nr = 15 and nθ = 10 were used
+in the final descriptors to increase the speed of calculation.
+3.2
+Constructing a Similarity Score
+In order to facilitate familiar comparison of molecules, we wish to construct a similarity score rather than simply taking
+the distance between two matrices. In chemoinformatics, this score typically takes a value between 0 (no similarity)
+and 1 (identical) [4]. To achieve this we take the inverse distance, and account for size by taking the ratio of two surface
+areas. Equation 8 gives the similarity score between two molecular surfaces S1 and S2,
+score(S1, S2) = x(Amin/Amax) + y
+1
+1 + d(M1, M2),
+(8)
+where Amin is the smaller of the two surface areas, and Amax is the larger, in order to give a score bounded by 0 and 1.
+We therefore need to choose an appropriate set of weights x and y such that x + y = 1, and x < 0.5, to ensure the
+shape is the primary contributor to the score.
+Table 4 gives the resulting similarity scores for pairwise comparison of the PDE5 inhibitors. In all three cases,
+the similarity increases with increasing contribution from the surface area term as expected. The increase for
+Sildenafil-Vardenafil is only small, while for Tadalafil there is a greater effect of including the area. Final weights of
+x = 0.3 and y = 0.7 were selected to balance the contribution of the surface area without it dominating over the shape
+contribution. The PDE5 inhibitors were selected for tuning due to their known similarity, however further refinement of
+9
+
+Geometry and Molecular Surfaces
+A PREPRINT
+Table 4: Similarity scores for the PDE5 inhibitors for surface area weights ranging from 0 to 0.5.
+x
+y
+Sildenafil-Vardenafil
+Sildenafil-Tadalafil
+Vardenafil-Tadalafil
+0
+1
+0.884
+0.286
+0.275
+0.1
+0.9
+0.892
+0.340
+0.328
+0.2
+0.8
+0.900
+0.394
+0.380
+0.3
+0.7
+0.908
+0.449
+0.432
+0.4
+0.6
+0.916
+0.503
+0.485
+0.5
+0.5
+0.924
+0.557
+0.537
+these parameters with a larger set of examples may be required for full scale virtual screening.
+3.3
+Investigating Variation in 3D Conformers
+As discussed in the previous work [1], consideration of the different orientations a molecule can adopt (known as
+conformers) is important when using 3D shape descriptors. Conformers of the same molecule should theoretically have
+scores in the range 0.7 < score < 1, as high self-similarity is expected (scores above 0.7 in chemoinformatics), while
+retaining the ability to distinguish between them.
+As with RGMolSA, two small sets of 10 conformers of the PDE5 inhibitors are used to investigate how KQ-
+MolSA regards different conformers. One set contains 10 random conformers, in which we would expect slightly
+more variance, while the other has 10 low energy conformers, for which higher similarity is expected. Both sets
+were produced using the ETKDG algorithm [15] with energy optimisation using the MMFF94 force field [16], both
+implemented in RDKit [17]. The minimum, maximum and average shape similarity as well as the average RMSD
+(which compares conformers based on their atomic positions) for each set are given in Figure 3. The full set of RMSD
+and shape similarity comparisons are available in the Supporting Data.
+The RMSD and shape similarity for each set are compared in the swarm plots shown in Figure 4. For k = 1, generally
+high similarity was observed, with some scores for the random conformers of Tadalafil falling slightly below 0.7.
+Greater variation is observed for k = 2, where some conformer pairs have scores below 0.6. This reduction in similarity
+is expected for k = 2 as the descriptors represent a more detailed approximation to the original surface than those
+for k = 1 and hence will be more sensitive to differences in the geometry. However, the similarity scores obtained
+were on the whole lower than for RGMolSA, where the similarity between most conformer pairs is greater than 0.8 [1].
+For the random sets, the similarity between conformers showed more variation than for RGMolSA, where clusters of
+similar conformers were observed. While KQMolSA does handle conformers well, RGMolSA appears to do a better
+job of this, due to the insensitivity to surface deformation of the spectrum of the Laplace–Beltrami operator. For virtual
+screening, this consideration of conformers as similar negates the need for a pre-alignment step prior to shape similarity
+calculation, and may allow molecules that can deform to fit in the binding pocket to be identified as potential hits, where
+these would otherwise be classified as the wrong shape by methods that depend on atomic coordinates.
+3.4
+Comparison to Existing Methods
+The PDE5 inhibitor series was also used to investigate how well KQMolSA compares to the previous work, and to
+other open source shape similarity methods. Table 5 provides the shape-similarity scores observed between the PDE5
+inhibitors for KQMolSA (for k = 1 and k = 2), RGMolSA [1], USRCAT [18, 17], Shape-It [19] and MolSG [20]. A
+2D representation, in the form of the 1024-bit Morgan fingerprint using radius 3, is also included. Each descriptor uses
+a similarity score between 0 (different) and 1 (identical).
+Table 5: Comparison of the work presented here (KQMolSA) to the previous work (RGMolSA) [1] and existing atomic-distance [18], atomic-centred [19] and molecular
+surface based [20] descriptors. In all cases the similarity scores given are bound by 0 (no similarity) and 1 (identical).
+KQMolSA
+(k=1)
+KQMolSA
+(k=2)
+RGMolSA
+USRCAT
+Shape-It
+MolSG
+Morgan
+Fingerprint
+Sildenafil-
+Vardenafil
+0.907
+0.652
+0.903
+0.384
+0.388
+0.704
+0.667
+Sildenafil-
+Tadalafil
+0.449
+0.482
+0.809
+0.269
+0.278
+0.746
+0.201
+Vardenafil-
+Tadalafil
+0.432
+0.470
+0.725
+0.291
+0.353
+0.887
+0.209
+10
+
+Geometry and Molecular Surfaces
+A PREPRINT
+(a) k = 1
+(b) k = 2
+Figure 3: Overlay of the most and least shape-similar conformers of Sildenafil, Vardenafil and Tadalafil and the average shape similarity and RMSD for each set for (a)
+k = 1 and (b) k = 2. On average the conformers display a high degree of self-similarity despite the variance in atom-position similarity.
+As discussed in the prequel to this paper, as Sildenafil and Vardenafil are close structural analogues they should display
+both high shape and fingerprint similarity. As Tadalafil is known to occupy a similar volume in PDE5 compared to
+the other inhibitors, we’d expect high shape similarity scores also, but lower 2D similarity. One conformer of each
+molecule is considered for simplicity.
+As for RGMolSA, Sildenafil and Vardenafil are scored as highly similar, with a score of 0.907 (k
+= 1).
+However Tadalafil is not scored as highly, and for KQMolSA would be classed as dissimilar if the typical threshold
+of 0.7 was used. Lower similarity is observed for k = 2, which is expected as discussed previously. The similarity
+score for k = 2 has a small dependence on the order of comparison (A compared to B yields a score which may
+differ at the second decimal place from B compared to A, Table 6). This is due to the distance calculation involving
+a numerical minimisation procedure rather than an exact expression, but this will have no practical implications in
+chemoinformatics applications. Both proposed methods (RGMolSA and KQMolSA) perform well in this simple study,
+with a higher predicted similarity for Sildenafil and Vardenafil than all the other 3D methods, and a more intuitive
+ordering of the relative similarity measures than MolSG. However, a full scale benchmarking study will be required to
+verify their performance.
+11
+
+Geometry and Molecular Surfaces
+A PREPRINT
+(a) RMS Similarity
+(b) Shape Similarity (k = 1)
+(c) Shape Similarity (k = 2)
+Figure 4: Swarm plots of the RMSD (in Å) and shape similarity for our set of conformers highlight the general trend that different conformers are classed as having
+similar shape, despite significant variance in their atomic positions. Conformers with RMSD less than 1 Å are considered similar, while those over 3 Å have significant
+differences.
+Table 6: Similarity scores for the PDE5 inhibitors for k=2 highlighting the dependence on the order of comparison.
+Sildenafil
+Vardenafil
+Tadalafil
+Sildenafil
+-
+0.652
+0.462
+Vardenafil
+0.648
+-
+0.470
+Tadalafil
+0.482
+0.470
+-
+3.5
+Similarity to Potential Decoys
+As for RGMolSA, we also wanted to check how the method handles molecules that should be classed as genuinely
+different from the PDE5 inhibitor molecules. We therefore present a comparison to four other molecules (Figure 5):
+Arginine (supplement) which has a lower molecular weight, but similar general shape (a long chain of spheres);
+Lymecycline (antibiotic), with a higher molecular weight and a four-ring motif potentially giving part of the molecule a
+similar shape to Sildenafil; Diflorasone (topical corticosteroid), which has a similar molecular weight and four rings, but
+has a different therapeutic target/indication and S-octylglutathione (oligopeptide), which again has similar molecular
+weight, but no rings and the potential for similarity due to the branching in the centre of the molecule.
+The results of this comparison are presented in Figure 6. Most of the scores obtained for both k = 1 and k = 2 fall
+significantly below the typical threshold of 0.7 for similarity, and as such these molecules would be classed as genuinely
+different and likely inactive against PDE5. The exception is the comparison between Tadalafil and Diflorasone, where a
+higher score of 0.74 (k = 1) is obtained. Due to the similarity between their structures (both contain a motif of 4 fused
+rings), we would expect to see some similarity between the two. Inspection by eye of both the space filling model and
+surface of the two molecules also suggests they do have genuinely similar shapes (Figure 7). These were also classed as
+potentially similar by RGMolSA (similarity of 0.872).
+4
+Conclusion
+We have outlined the theory underpinning an entirely novel shape descriptor,
+Mij =
+��
+C
+zizje−kϕF(z)
+√
+−1dz ∧ dz,
+(9)
+12
+
+·····
+Sildenafil Random
+Sildenafil Low Energy
+Vardenafil Random
+Vardenafil Low Energy
+：
+Tadalafil Random
+Tadalafil Low Energy
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Root Mean Square Deviation··…··…
+Sildenafil Random
+：8
+Sildenafil Low Energy
+Vardenafil Random
+（
+8
+Vardenafil Low Energy
+Tadalafil Random
+Tadalafil Low Energy
+0.60
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+Shape SimilaritySildenafil Random
+Sildenafil Low Energy
+Vardenafil Random
+Vardenafil Low Energy
+Tadalafil Random
+Tadalafil Low Energy
+0.6
+0.7
+0.8
+0.9
+1.0
+Shape SimilarityGeometry and Molecular Surfaces
+A PREPRINT
+Arginine
+Lymecycline
+Diflorasone
+S-Octylglutathione
+Figure 5: Chemical structures of potential decoy molecules.
+the (2k + 1) × (2k + 1) Hermitian matrix which captures the geometry of the molecular surface. The distance between
+two such matrix representations is then given as
+d(M1, M2) = k− 3
+2
+�
+�
+�
+�
+2k+1
+�
+i=1
+(λi − µi)2.
+(10)
+An overall similarity score of 1 for identical molecules and 0 for no similarity is then obtained as
+score(S1, S2) = 0.3(Amin/Amax) + 0.7
+1
+1 + d(M1, M2).
+(11)
+As with the previously reported work, the capabilities of KQMolSA were investigated using a series of PDE5 inhibitors
+known to have similar shape. The method generally handles conformers well, with similarity scores generally higher
+than 0.7. The scores obtained were higher for k = 1 than k = 2, which is expected due to the greater detail leading to
+more sensitivity to changes in geometry. The insensitivity to deformation of the surface lead to RGMolSA outperforming
+KQMolSA in this area. KQMolSA performs relatively well compared to existing methods, identifying Sildenafil
+and Vardenafil as highly similar, but assigning lower similarity scores to Tadalafil. This small study suggests that
+RGMolSA might still perform better, but a full retrospective benchmarking study is required to confirm this. Compared
+to RGMolSA, KQMolSA does have the advantage of a lower dependence on the choice of base sphere. There may
+therefore be some instances where the use of KQMolSA is more appropriate despite its seemingly poorer performance,
+for example in the consideration of long chain molecules with few rings, where numerical errors are often observed for
+RGMolSA. Comparison to a set of potential decoy molecules yielded low scores for all except comparison of Tadalafil
+to Diflorasone, which were also classed as similar by RGMolSA. Inspection by eye of both the space filling and surface
+models of the molecules suggests that this assignment is reasonable, as they look similar in shape. Identification of such
+similarity evidences the potential for scaffold hopping by these methods.
+Whilst the above tests suggest that the matrix M does give a promising description of molecular shape, the method
+does have some drawbacks, primarily in the calculation of the distance between two descriptors. While the notion of
+the distance between two Hermitian inner products (represented by the matrices M1 and M2) is well understood, the
+calculation of the distance between molecular surfaces requires the distance between a point on an SL(2, C)-orbit to be
+minimised. Despite the use of existing optimised minimisation algorithms, this process is still quite slow, depending
+on the extent of the required minimisation, and further does not guarantee that the global minimum has been found.
+This step typically takes a few seconds per pair, compared to a near-instantaneous calculation for RGMolSA. Further
+refinement of this step would be required for use of the method in screening ultra-large chemical libraries as part of a
+drug discovery pipeline.
+13
+
+NH2
+H2N
+N
+HO
+NH2OH
+N
+H
+H
+N
+N
+OH
+OH
+OH
+HO
+OH
+NH2HO
+OH
+OHOH
+NH
+s
+N
+HO
+H
+NH2Geometry and Molecular Surfaces
+A PREPRINT
+Figure 6: KQMolSA similarity (for k = 1 and k = 2) of four ‘different’ molecules (blue) to the PDE5 inhibitor test series (red). The overlay of the structures was
+computed using Open3DAlign [21]
+Of course, there are many other ways of measuring the distance between two Hermitian matrices.
+One
+might hope that some form of machine learning, trained on an appropriate data set, might discern other useful
+geometries on the space of descriptors.
+The method also contains numerical instability above k = 2 (and for k = 2 in a few instances), producing Hermitian
+matrices that are not positive definite. As Hermitian matrices differing only by a scale factor can be considered
+equivalent, we have handled such cases by scaling one matrix by a factor of 10-1000 to bring the eigenvalues into the
+range of Python’s numerical tolerance.
+Along with addressing these issues, both of the methods proposed could be further improved through the consideration
+of pharmacorphoric features, such as aromatic rings, hydrogen bond donors and acceptors, alongside the shape. As
+these features are important for binding, this may lead to improved predictions compared to the consideration of shape
+alone. As for RGMolSA, there would also be scope to investigate the use the Hermitian matrix descriptors produced by
+KQMolSA as a feature descriptor in machine learning.
+5
+Appendix: Finding the Kähler potential
+Before giving the proof of the form of the Kähler potential, we dispense with a small technical point. From the point of
+view of describing the Kähler form ω via
+√
+−1∂ ¯∂ϕ = ω,
+the Kähler potential ϕ is only locally defined and adding any function H satisfying √−1∂ ¯∂H = 0
+will also define a Kähler potential for ω. In our setting where the underlying complex manifold is CP1 and we are using
+the standard coordinate z, we can add any harmonic function H : C → R to obtain a valid Kähler potential.
+14
+
+Arginina
+Lymecyclina
+Diforaaone
+S-octylglubthiona
+K=1:0.469
+k=1:0.513
+k=1:0.367
+k=1:0.467
+k=2: 0.413
+k=2: 0.411
+k=2: 0.3B9
+k=2: 0.378
+sildenfl
+K=1:0.493
+K=1:0.535
+K=1:0.354
+K=1:0.467
+K=2:0.445
+K=2:0.45
+k=2: 0.377
+K=2:0.378
+Vardanafil
+K=1:0.282
+k=1:0.347
+K=1:0.74
+K=1:0.421
+k=2: 0.315
+K=2: 0.339
+k=2: 0.614
+k=2: 0.4
+dalahlGeometry and Molecular Surfaces
+A PREPRINT
+(a) Tadalafil - Space Filling Model
+(b) Diflorasone - Space Filling Model
+(c) Tadalafil - Surface
+(d) Diflorasone - Surface
+Figure 7: Comparison by eye of both the space filling model and the surface of Tadalafil and Diflorasone highlights their similarity.
+However, in Kähler Quantization, the potential ϕ actually describes a global object, the Hermitian metric h
+on the line bundle L. This means that the functions
+h(zj, zj) = e−kϕ(z)|z|2j,
+are defined over whole sphere CP1. In particular, they extend to functions over the point at infinity. For example the
+round metric has Kähler potential ϕ = −2 log(|z|2 + 1) and so, if we add a harmonic function H we require
+|z|4k
+(1 + |z|2)2k e−kH
+to be bounded. The Liouville Theorem then implies H must be constant.
+Theorem 5.1 (Form of Kähler potential). Let ω be a Kähler metric of the form given by Equation (3). If we denote the
+region corresponding to the ith sphere as Ri ⊂ C, then the Kähler potential potential ϕ, which satisfies √−1∂∂ϕ = ω,
+is of the form
+ϕ(z) = Ci
+Bi
+log(|z − Ai|2 + Bi) +
+N
+�
+j=1
+Kij log(|αijz + βij|2),
+where K ∈ M N×N(R), and α, β ∈ M N×N(C).
+Proof. The proof is by induction on the number of spheres N. For N = 1 the metric ω is the round metric and we can
+take K = 0. Adding a new sphere to the surface changes the metric by adding a new region Rk which is a disc where
+the metric takes the form
+ω(z)|Rk =
+Ck
+(|z − Ak|2 + Bk)2
+√
+−1dz ∧ dz.
+15
+
+Geometry and Molecular Surfaces
+A PREPRINT
+We can map Rk to the unit disc about the origin by a Möbius transformation M in such a way that, in the coordinate of
+the unit disc, the metric is given by
+�ω(w) =
+�
+�
+�
+�
+�
+F(w)√−1dw ∧ dw
+if
+|w| > 1,
+κ
+(|w|2 + ε)2
+√−1dw ∧ dw
+if
+|w| ≤ 1,
+for some function F : C → R and constants κ, ε ∈ R.
+We solve the ¯∂-equation using the Dolbeault method; for a compactly supported1 continuous function H : C → C,
+ψ(w) =
+1
+2π√−1
+��
+C
+H(p)
+p − wdp ∧ dp,
+solves ∂ψ = H(w)dw. We split the integral according to the form of the metric and consider
+ψ(w) =
+1
+2π√−1
+��
+D
+κ
+(|p|2 + ε)2(p − w)dp ∧ dp +
+1
+2π√−1
+��
+C\D
+F(p)
+p − wdp ∧ dp.
+To compute the first integral we use the Cauchy–Pompeiu integral formula and the fact that
+κ
+(|p|2 + ε)2 = ∂
+∂p
+� (κ/ε)p
+(|p|2 + ε)
+�
+,
+to give
+1
+2π√−1
+��
+D
+κ
+(|p|2 + ε)2(p − w)dp ∧ dp =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+� (κ/ε)w
+(|w|2 + ε)
+�
+−
+1
+2π√−1
+�
+∂D
+(κ/ε)p
+(|p|2 + ε)(p − w)dp
+if
+|w| < 1,
+−
+1
+2π√−1
+�
+∂D
+(κ/ε)p
+(|p|2 + ε)(p − w)dp
+if
+|w| > 1.
+The contour integral
+1
+2π√−1
+�
+∂D
+(κ/ε)p
+(|p|2 + B)(p − w)dp,
+can be easily computed using the Cauchy Residue Formula and this yields
+1
+2π√−1
+�
+∂D
+(κ/ε)p
+(|p|2 + ε)(p − w)dp =
+�
+0
+if |w| < 1,
+− (κ/ε)
+(1+ε)w
+if |w| > 1.
+Finally, we arrive at
+1
+2π√−1
+��
+D
+κ
+(|p|2 + ε)2(p − w)dp ∧ dp =
+� �
+(κ/ε)w
+|w|2+ε
+�
+if |w| < 1,
+(κ/ε)
+(1+ε)w
+if |w| > 1.
+To compute the second integral, we again split the domain and consider
+1
+2π√−1
+��
+C\D
+F(p)
+p − wdp ∧ dp =
+1
+2π√−1
+��
+C
+F(p)
+p − wdp ∧ dp −
+1
+2π√−1
+��
+D
+F(p)
+p − wdp ∧ dp.
+The integral
+S(w) =
+1
+2π√−1
+��
+C
+F(p)
+p − wdp ∧ dp,
+is a solution to
+∂S
+∂w = F(w).
+1Our function is not compactly supported but we could cut off at an arbitrary radius to produce such a function.
+16
+
+Geometry and Molecular Surfaces
+A PREPRINT
+In the unit disc D, F has the form
+F(w) =
+˜κ
+(|w|2 + ˜ε)2 ,
+where ˜κ and ˜ε are positive constants. Hence
+ψ(w) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+S(w) +
+� (κ/ε)
+|w|2 + ε −
+(˜κ/˜ε)
+|w|2 + ˜ε
+�
+w
+if
+|w| < 1,
+S(w) +
+�(κ/ε)
+1 + ε − (˜κ/˜ε)
+1 + ˜ε
+�
+w−1
+if
+|w| > 1,
+solves dw ∧ ∂ψ = �ω(w).
+If Q(w) is a Kähler potential for F(w)√−1dw ∧ dw then
+�ϕ(w) =
+�
+�
+�
+Q(w) + (κ/ε) log(|w|2 + ε) − (˜κ/˜ε) log(|w|2 + ˜ε) − K
+if
+|w| < 1,
+Q(w) +
+�(κ/ε)
+1 + ε − (˜κ/˜ε)
+1 + ˜ε
+�
+log(|w|2)
+if
+|w| > 1,
+where
+K = (κ/ε) log(1 + ε) − (˜κ/˜ε) log(1 + ˜ε),
+is a Kähler potential for �ω. Pulling back the function �ϕ via the Möbius transformation
+M(z) = αz + β
+γz + δ
+we see
+ϕk(z) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Q
+�αz + β
+γz + δ
+�
++ (κ/ε) log
+�����
+αz + β
+γz + δ
+����
+2
++ ε
+�
+− K
+if
+z ∈ Rk
+Q
+�αz + β
+γz + δ
+�
++
+�(κ/ε)
+1 + ε − (˜κ/˜ε)
+1 + ˜ε
+�
+log
+�����
+αz + β
+γz + δ
+����
+2�
+if
+z ̸∈ Rk
+is a Kähler potential for the metric which is singular at at the point z = −δ/γ. We can replace the Q-term by the
+appropriate function for the previous ϕ and then add the appropriate multiple of log(|γz + δ|2) to produce a Kähler
+potential of the appropriate form.
+6
+Acknowledgements
+The authors acknowledge support from an EPSRC Doctoral Training Partnership studentship (grant EP/R51309X/1),
+the Alan Turing Institute Enrichment Scheme (R.P.), and a UKRI Future Leaders Fellowship (grant MR/T019654/1)
+(D.J.C.). S.J.H. would like to thank Dr R. L. Hall for his interest and for useful conversations about the project. T.M.
+would like to thank University of California, Irvine for their hospitality whilst some of the work on this paper was
+completed.
+References
+[1] Daniel J. Cole, Stuart J. Hall, and Rachael Pirie. Riemannian geometry and molecular surfaces I: Spectrum of the
+Laplacian, (preprint), 2022.
+[2] Mark A. Johnson and Gerald M. Maggiora. Concepts and Applications of Molecular Similarity. 1990.
+[3] Sumudu P. Leelananda and Steffen Lindert. Computational methods in drug discovery. Beilstein J. Org. Chem.,
+12:2694–2718, 2016.
+[4] Ashutosh Kumar and Kam Y. J. Zhang. Advances in the development of shape similarity methods and their
+application in drug discovery. Front. Chem., 6:1–21, 2018.
+[5] S. K. Donaldson. Some numerical results in complex differential geometry. Pure Appl. Math. Q., 5(2):571–618,
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+17
+
+Geometry and Molecular Surfaces
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+[6] Ann E. Cleves and Ajay N. Jain. Effects of inductive bias on computational evaluations of ligand-based modelling
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+University Press, Cambridge, third edition, 2007. The art of scientific computing.
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+ligand alignment. Journal of Computer-Aided Molecular Design, 25(8):777–783, 2011.
+18
+

7dE0T4oBgHgl3EQffQBI/content/tmp_files/2301.02401v1.pdf.txt

@@ -0,0 +1,1848 @@
+You Truly Understand What I Need
+: Intellectual and Friendly Dialogue Agents grounding
+Knowledge and Persona
+Jungwoo Lim1, Myunghoon Kang1∗, Yuna Hur1∗, Seungwon Jung1∗, Jinsung Kim1∗,
+Yoonna Jang1, Dongyub Lee3, Hyesung Ji2, Donghoon Shin2,
+Seungryong Kim1§ and Heuiseok Lim1§
+1Korea University, 2Dialogue Tech Division, NCSOFT, 3Naver Corporation
+{wjddn803,chaos8527,yj72722,redlion0929,jin62304,seungryong_kim,limhseok}@korea.ac.kr,
+{hyesung84,dhshin}@ncsoft.com, [email protected]
+Abstract
+To build a conversational agent that interacts
+fluently
+with
+humans,
+previous
+studies
+blend knowledge or personal profile into the
+pre-trained language model. However, the
+model that considers knowledge and persona
+at the same time is still limited, leading to
+hallucination and a passive way of using
+personas. We propose an effective dialogue
+agent that grounds external knowledge and
+persona simultaneously. The agent selects
+the proper knowledge and persona to use for
+generating the answers with our candidate
+scoring implemented with a poly-encoder.
+Then, our model generates the utterance with
+lesser hallucination and more engagingness
+utilizing
+retrieval
+augmented
+generation
+with
+knowledge-persona
+enhanced
+query.
+We conduct experiments on the persona-
+knowledge chat and achieve state-of-the-art
+performance in grounding and generation
+tasks on the automatic metrics. Moreover,
+we validate the answers from the models
+regarding
+hallucination
+and
+engagingness
+through human evaluation and qualitative
+results. We show our retriever’s effectiveness
+in extracting relevant documents compared
+to
+the
+other
+previous
+retrievers,
+along
+with the comparison of multiple candidate
+scoring
+methods.
+Code
+is
+available
+at
+https://github.com/dlawjddn803/INFO
+1
+Introduction
+To
+build
+an
+ultimate
+conversational
+agent
+that interacts with humans fluently, previous
+studies provide generative neural network-based
+models (Sordoni et al., 2015; Vinyals and Le,
+2015). Although the answers generated from those
+models are plausible, they lack informativeness
+and engagingness resulting in bland responses
+compared to humans (Li et al., 2016; Gao et al.,
+*
+Equal Contributors
+§ Corresponding author
+Dialogue
+Human: Is it in England?
+Machine: No, it is actually in Scotland where you are going.
+Human: Where in Scotland?
+Human’s Persona
+I will travel through North Ayrshire.
+I am going to Scotland.
+I like history.
+I am interested in architecture.
+I love to garden.
+Ground Truth Knowledge
+Eglinton Castle was a large Gothic castellated mansion in
+Kilwinning, North Ayrshire, Scotland.
+Predicted Answers
+BARTbase
+It is in Scotland, which is a place you love.
+BARTlarge
+It is in Scotland. in Scotland. in Scotland. in
+Ground Truth Response
+It is in North Ayrshire so you could visit when you travel through.
+Table 1: Example of the generated answers from a
+typical generative model, i.e., BART. We can find that
+BARTbase uses different persona sentence which has
+not appeared human’s personal profiles resulting in
+hallucinated answer. Also, BARTlarge generates less
+engaging answers by making use of the knowledge only
+to answer the question. Both generated responses are in
+the situation of hallucination and are less engaging.
+2018). However, for knowledgeable and attractive
+conversation, people usually provide informative
+replies by considering the background of the person
+whom they are talking to. Towards a human-like
+manner of dialogue, Ghazvininejad et al. (2018)
+and Dinan et al. (2018) introduce the knowledge-
+grounded conversation for the knowledgeable
+and informative responses, whereas Zhang et al.
+(2018a) suggest the persona-grounded dialogue for
+the personalized responses to the users.
+To improve the machine’s answer with the
+external knowledge base, one injects the factual
+knowledge into the parameters of the language
+model (Raffel et al., 2020; Roberts et al., 2020).
+Despite the models’ capability of utilizing external
+knowledge implicitly, they produce “hallucinations”
+in the responses (Marcus, 2020). The hallucination
+arXiv:2301.02401v1  [cs.CL]  6 Jan 2023
+
+in the dialogue involves the situation where
+the generated output contradicts the reference
+knowledge. Also, it includes the situation when
+the generated output cannot be confirmed from the
+knowledge source (Ji et al., 2022). To mitigate these
+hallucinated answers, hybrid models employing
+parametric memory with non-parametric (i.e.,
+retrieval-based) memory are introduced to directly
+access external memories, leading the source to be
+inspected and interpreted (Karpukhin et al., 2020;
+Petroni et al., 2020; Lewis et al., 2020b).
+On the other hand, Zhang et al. (2018a) suggest
+persona-chat dialogues with the corresponding
+personal profiles of each interlocutor to avoid
+general and monotonous answers from the machine.
+Though See et al. (2019); Liu et al. (2020) show
+comparable quality in generating personalized
+conversation, the generated utterances merely
+confirm each interlocutor’s persona resulting
+in a passive manner of speaking such as “I
+have four children”. In addition, the incoherent
+topics of the dialogues lead to shallow levels
+of conversation between the interlocutors. To
+elaborate on this chit-chat conversation supported
+by external knowledge, Jang et al. (2022) presents
+a novel persona-knowledge chat with a generative
+model that considers persona information and
+world knowledge altogether. Despite obtaining
+the knowledge and persona when generating the
+answers, the generative models’ responses still
+exhibit both hallucination and lesser engagingness
+as in Table 1.
+In this paper, we propose INFO (Intellectual
+and Friendly dialOg agents) that responds with
+external knowledge and persona simultaneously.
+Owing to the enhanced capturing relevancy
+between the context and each candidate set,
+the knowledge selector and persona selector for
+the grounding task are implemented with the
+poly-encoder. To alleviate hallucinated responses
+from the model, we adopt retrieval-augmented
+generation (RAG) (Lewis et al., 2020b) by
+utilizing non-parametric memory and parametric
+generator in addition to the enhanced input
+query. By injecting predicted sources as input
+to the retrieved-augmented generator, our model
+maintains consistency between grounding and
+generation while training. Therefore, our model
+generates more knowledgeable and engaging
+answers in an active manner with less hallucination.
+We show that INFO achieves the highest
+scores on both grounding and generation tasks in
+empirical experiments. Also, we compare diverse
+candidate scoring modules including bi-encoder,
+cross-encoder, and poly-encoder and demonstrate
+their effect on generation. We additionally conduct
+experiments to show the effectiveness of the
+retriever module compared to sparse and dense
+retrievers. The qualitative results and human
+evaluation are also presented to validate our
+model’s capability to generate human-like answers.
+Our contributions are as follows:
+• We propose the model that grounds persona
+information and external knowledge with
+lesser hallucination and adequate utilization of
+persona in an active manner simultaneously.
+• Our approach suggests that the generated
+responses from the model are interpretable
+regarding what the model refers to while
+generating.
+• We show that INFO achieves the SoTA
+performance in all of the automatic metrics
+and demonstrate its comparable quality with
+human evaluation and qualitative analysis.
+2
+Related Works
+2.1
+Knowledge Grounded Conversation
+To let the neural network models ground external
+knowledge and generate informative answers,
+Ghazvininejad et al. (2018) suggests a data-
+driven neural conversational agent that provides
+knowledgeable
+answers.
+Also,
+Dinan
+et
+al.
+(2018) introduces open-domain dialogue where
+the two speakers are talking with Wikipedia
+knowledge. To inject the external knowledge
+into the pre-trained language model efficiently,
+Raffel et al. (2020); Roberts et al. (2020)
+success in equipping the knowledge into the
+parameters and show comparable performance
+in open-domain question and answering tasks.
+However, the approach is not capable of expand
+or revise their inherent knowledge and provides
+hallucination (Marcus, 2020). To overcome the
+limitations, Lewis et al. (2020b) combines a
+pre-trained parametric model and non-parametric
+memory for the open-domain question and
+answering to reduce hallucination. Since their non-
+parametric memory can be updated without extra
+pre-training, revising knowledge is more efficient.
+Furthermore, it is found that a retrieval-augmented
+
+Figure 1: Overview of our method. U is the input comprises dialogue history and knowledge snippet, and cand
+denotes each candidate from the grounding tasks. The grounding score is obtained through the dot product
+operation with the representation of input context Udial and candidate at. The predicted sources convert into the
+knowledge-persona enhanced query (KPEQ) with dialogue history and KPEQ is fed into the retrieval-augmented
+generator to generate the responses.
+generator also reduces hallucination in knowledge-
+grounded conversation as well (Shuster et al.,
+2021), and a similar approach recently achieves
+outstanding performance in knowledge-grounded
+conversation (Paranjape et al., 2021).
+2.2
+Persona Grounded Conversation
+In order to alleviate bland and general answers
+with consistent personality, Zhang et al. (2018a)
+constructs a persona-chat dataset. In the dataset,
+the two interlocutors chat with the persona
+profile sentences. Along with this dataset, Zhang
+et al. (2018a) introduces the model with a
+profile memory network by considering the
+dialogue history to perform attention over the
+persona. They enlarge the persona-chat dataset
+with Reddit corpus, and pre-trained the model
+with these dataset. After that, they fine-tune pre-
+trained model on the persona-chat (Mazare et al.,
+2018). Also, Liu et al. (2020) trains a receiver
+to reinforce the mutual persona understanding
+between interlocutors, and Wolf et al. (2019) utilize
+pre-trained models (Radford et al., 2019) to build
+personalized dialogue agents.
+2.3
+Encoders for Sentence Scoring
+There exist diverse encoder structures for sentence
+scoring. Bi-encoder scores the relevance between
+sentences by feeding context and candidates into
+separate encoders. An example of bi-encoders
+are memory networks
+(Zhang et al., 2018a),
+transformer memory networks (Dinan et al.,
+2018), LSTM (Lowe et al., 2015). Since bi-
+encoder calculates with cached encoded sentence
+representations, it is relatively fast in computation.
+However, the bi-encoder has a limitation of
+capturing mutual information between context
+and candidates. Cross-encoder, on the other hand,
+scores by aligning context and candidates in
+one sequence. A type of cross-encoders is a
+sequential matching network that is based on
+deep matching networks (Yang et al., 2018) and
+gated self-attention (Zhang et al., 2018b). Although
+using a cross-encoder can achieve rich interaction
+between the sentences within the encoder, the
+problem of slow processing still remains. To
+exploit both benefits of each model, poly-encoder
+adopts attention mechanism into the bi-encoder
+architecture and shows satisfactory performances
+as cross-encoder with fast inference time (Humeau
+et al., 2019). For the enhanced representation of
+grounding knowledge and persona, we employ a
+poly-encoder as a selector for each grounding task.
+3
+Method
+To generate more knowledgeable and engaging
+dialogue, we introduce our conversational model
+that grounds external knowledge and persona
+information as in Figure 1. We first encode the
+input with the pre-trained language model, and
+then choose the proper knowledge and persona
+from the given candidates for each selector. We
+employ poly-encoder
+(Humeau et al., 2019)
+as knowledge selector and persona selector to
+exploit its enhanced capability of capturing
+relevance between candidate set and context (i.e.,
+dialogue history). Then, the predicted persona
+and knowledge are aligned into one sequence
+
+KPEQ
+Retriever(Non-Parametric)
+Poly-encoder
+ Knowledge Selector
+ Document Index
+Uaial
+O-
+ Score
+ Poly-
+U
+Z1-08
+encoder
+Attention
+Z7777
+Z1-03
+Z2-02
+Uaial
+Uaal
+acand
+Knowledge
+Z1-01
+Candidate
+Z2-09
+C
+CM
+Dialogue
+Z2-05
+Z2-07
+Attention
+Attention
+Candidate Aggregator
+Persona
+Candidate
+个
+↑
+h1
+hn
+h2
+a2
+a1
+aT
+Persona Selector
+↑
+↑
+Persona
+Marginalize
+Generator
+Generated
+Context Encoder
+Candidate Encoder
+ Poly-
+Level
+(Parametric)
+Answer
+↑
+T
+↑
+ encoder
+Indicator
+U
+ candi
+cand2
+candTto the dialogue history for consistency between
+grounding and generation. The sequence is defined
+as a knowledge-persona enhanced query (KPEQ),
+then it feeds into the retriever-augmented generator
+(RAG). The generator then extracts the relevant
+paragraphs to refer from the knowledge index to
+reduce hallucination.
+3.1
+Input Construction
+The
+given
+dialogue
+is
+notated
+as
+{(uhm
+1 , umc
+1 ), ...(uhm
+o
+, umc
+o )},
+where
+o
+is
+the
+number of rounds. uhm and umc indicate the
+utterances of human and machines, respectively.
+We first take o-th round dialogue history, except
+for the final machine’s reply umc
+o , for the initial
+input for the model. We define the clue of the
+dialogue as knowledge snippet clk to inform the
+machine of which topic the user is interested in.
+The knowledge snippet is the name of the landmark
+that the user encounters, which is given topic from
+the dialogue. We then align the dialogue history
+and knowledge snippet into the one sequence for
+the model input as U = {uhm
+1 , umc
+1 , ...uhm
+o
+, clk}.
+3.2
+Model Components
+3.2.1
+Poly-Encoder Based Candidate Scoring
+For knowledge and persona grounding tasks, we
+suggest poly-encoder-based candidate scoring to
+leverage the capability of capturing the semantic
+similarities between the context input and the
+candidates. It is employed to select proper sources
+to be used when generating the utterance. When
+the context input U comes in, we compute
+the grounding scores of each candidate utilizing
+the embeddings of context input and encoded
+candidates in the poly-encoder. The grounding
+score is used to select the most suitable source(s) in
+the knowledge selector and persona selector, which
+will be introduced in the following Section 3.2.2
+and 3.2.3.
+In poly-encoder architecture (Humeau et al.,
+2019), candidates are fed into the candidate
+encoder and denoted as {a1, ..., aT } where T is
+the number of candidates in the set. Each candidate
+embedding at is the first output of the candidate
+encoder, which is represented by the transformer
+model. After encoding candidates, the context
+input (i.e., dialogue history) is embedded with
+a separate context encoder. Unlike the candidate
+encoder, the context encoder embeds the dialogue
+into multiple vectors through M context codes
+{c1, ...cM}, which are learned for capturing diverse
+aspects of a given context rather than using one
+embedding. Each context code is used to extract
+U m
+dial by attending over all the previous layer’s
+output as follows.
+U m
+dial =
+�
+j
+wcm
+j hj
+(1)
+Note that the h1, ..., hn is the output of the pre-
+trained language model and n is the number of
+tokens in the input. The weights are computed as
+(wcm
+1 , ..., wcm
+n ) = softmax(cm · h1, ..., cm · hn).
+Then, the final attention proceeds between
+the global features of the input and a given
+candidate. In other words, the final dialogue feature
+Udial is obtained by aggregating each dialogue
+feature U m
+dial, while gaining richer interactions with
+context codes as in Equation 2.
+Udial =
+�
+m
+wmU m
+dial,
+(2)
+where
+w1, ..., wM
+can
+be
+obtained
+from
+softmax(at · U 1
+dial, ..., at · U M
+dial).
+The final predicted candidate is chosen based
+on the highest score that is acquired from the dot
+product operation as (Udial · at).
+3.2.2
+Knowledge Selector (KS)
+We build a knowledge selector for the knowledge
+grounding task, employing poly-encoder-based
+candidate scoring. When the grounding scores are
+produced from the candidate scoring module, the
+label with the highest score is selected as the
+predicted knowledge.
+The knowledge loss LKG for the knowledge
+grounding task is computed with cross-entropy
+loss (Brier et al., 1950) as in Equation 3.
+LKG = −
+�
+j
+klj · log ˆ
+klj,
+(3)
+klj is the ground-truth label from the knowledge
+candidates of the j-th example.
+3.2.3
+Persona Selector (PS)
+We also implement a persona selector for
+the persona grounding task. Since multiple
+personas can be chosen to generate the responses,
+consideration of one or more persona sentences
+are needed. Similar to the knowledge selector,
+we assign the grounding score to each persona
+
+candidate with the candidate scoring module as
+in Equation 1 and 2.
+When the scores of each candidate are computed
+from the candidate scoring module, then the
+persona level indicator classifies which the number
+of the persona should be selected with the [CLS]
+token of the model input U. After predicting the
+level of persona-engagingness, we pick persona
+sentences to be grounded according to the number
+predicted. For example, if the persona level
+indicator predicts 2, then top-2 persona sentences
+are chosen in the persona grounding task. The
+selected persona sentence(s) are marked as 1
+otherwise, 0. We use binary cross-entropy loss for
+persona grounding as in Equation 4.
+LPG =
+−
+�
+j
+plj · log ˆ
+plj + (1 − plj) · log(1 − ˆ
+plj)
+(4)
+Note that plj is the ground-truth label from the
+knowledge candidates of the j-th example.
+3.2.4
+Query-Enhanced Generator
+Following the works of Lewis et al. (2020b),
+we exploit the retrieval augmented generation’s
+capability to reduce hallucination and access the
+memory directly. For a consistent way of training
+while solving grounding and generation tasks,
+we reconstruct the query that feeds into the
+retriever. When the knowledge and persona are
+predicted from each selector, we aggregate them
+with dialogue history into one sequence. Then, the
+final query is denoted as KPEQ = {U; ˆP; ˆK} and
+defined as a knowledge-persona enhanced query. ˆP
+and ˆK are predicted persona and knowledge from
+each candidate set, respectively.
+The retriever rη aims to search top-K latent
+paragraphs with the KPEQ. We utilize a pre-
+trained dense passage retriever (DPR) (Karpukhin
+et
+al.,
+2020)
+trained
+on
+natural
+question
+dataset (Kwiatkowski et al., 2019) which has
+parametric memory and bi-encoder architecture to
+retrieve a latent document embedding following
+Lewis et al. (2020b) :
+rη(z|KPEQ) ∝ exp(d(z)⊤q(KPEQ)),
+(5)
+where d(·) is an embedding from a document
+encoder and q(·) is a representation from query
+encoder, both implemented with BERTbase. z
+denotes the list of document.
+With the relevant paragraphs from the retriever,
+we employ RAG-Token architecture as the
+generator to borrow its strength of predicting
+each target token based on top-K different
+paragraphs. Since RAG-Sequence, which has a
+different architecture to RAG-Token, uses the same
+document from the retriever to predict each token
+as depicted in Equation 6, the result may opt to
+depend on the retrieved document (Lewis et al.,
+2020a). The two different versions of RAGs (Lewis
+et al., 2020b) are as follows:
+SRS(y|x) ≈
+�
+z∈top-k(p(·|x))
+rη(z|x)
+N
+�
+i
+gθ(yi|x, z, y1:i−1) (6)
+SRT(y|x) ≈
+N
+�
+i
+�
+z∈top-k(p(·|x))
+rη(z|x)gθ(yi|x, z, y1:i−1),
+(7)
+where SRS indicates our method with RAG-
+Sequence architecture and SRT denotes ours with
+the RAG-Token model. x is a token of KPEQ and
+yi is a single token from the ground truth responses.
+Also, z is a retrieved paragraph from the retriever
+and N is the maximum sequence length.
+The SRT generator g(·) marginalizes the loss
+from different paragraphs when generating answers.
+In detail, the generator outputs a distribution
+for the next token for each document before
+marginalizing as in Equation 7 where η denotes
+the parameter of the retriever, and θ indicates the
+parameter of the generator. After that, the generator
+repeats the process with the following output
+token. Finally, the SRT aims to generate the next
+token following an auto-regressive manner with a
+standard beam search. In other words, the model
+minimizes the negative marginal log-likelihood for
+each input/output pair (KPEQj, yj). The language
+model loss is formulated as :
+LS = −
+�
+j
+logp(yj|KPEQj)
+(8)
+3.3
+Final Objectives
+We then train the full model in the multi-tasking
+manner. The full objectives of the model is
+indicated as Equation 9.
+L = λKGLKG + λPGLPG + λSLS
+(9)
+
+Models
+Generation
+Grounding (Acc.)
+chrF++
+BLEU
+R-1
+R-2
+R-L
+BERTScore
+Persona
+Knowledge
+GPT2small
+28.73
+11.43
+36.58
+19.44
+32.62
+88.56
+67.44
+69.59
+GPT2medium
+30.12
+12.31
+38.29
+21.17
+34.12
+88.92
+67.44
+72.42
+BARTbase
+29.77
+11.99
+36.24
+19.73
+32.13
+88.35
+67.45
+72.18
+BARTlarge
+30.69
+11.91
+36.57
+19.83
+32.05
+88.10
+67.44
+71.01
+INFO (SRS)
+51.33
+29.36
+53.36
+40.36
+51.16
+92.00
+82.70
+99.24
+INFO (SRT )
+53.29
+31.46
+58.26
+42.35
+53.06
+92.29
+80.87
+99.22
+Table 2: Main results on the official validation set. SRS denotes our method with RAG-Sequence architecture and
+SRT indicates the model with RAG-Token model as generator. The models are evaluated by generation metrics,
+including chrF++, BLEU, ROUGE-1 (R-1), ROUGE-2 (R-2), ROUGE-L (R-L), and BERTScore.
+We control the proportion of each task and we
+set λKG, λPG, and λS as 1:1:5 for the experiments,
+respectively. We find the value of each λ with
+manual search.
+4
+Experiments
+4.1
+Experiment Details
+Dataset
+FoCus (Jang et al., 2022) is the dataset
+for customized dialogue benchmark, where each
+conversation is directly grounded with knowledge
+and persona. The dataset includes knowledge-
+aware dialogue with personal profiles between
+humans and machines. There are 12,484 dialogues
+about 5,152 knowledge sources from Wikipedia
+and 32,855 persona sentences. To validate the
+knowledge grounding capability and customized
+dialogue generation, we evaluate our method
+with the official FoCus validation set for the
+effectiveness of experiments since the result from
+the official test set can be tested only through the
+leaderboard*.
+Experimental Setup
+For each candidate scoring
+module, we implement poly-encoder (Humeau
+et al., 2019) with BERTlarge, and the number of
+context codes is 16. For the dialogue generation, we
+implement our method with Hugging Face (Wolf
+et al., 2020) and use facebook/rag-token-nq as
+the backbone model. We use the same architecture
+of retriever and generator from RAG along
+with the decoding and leverage our knowledge
+index for non-parametric query-document ranking
+with FAISS library (Johnson et al., 2019). The
+knowledge index consists of the paragraphs from
+the given Wikipedia knowledge entitled with the
+name of the given landmark. We set learning rate
+as 6.25e-6 with AdamW (Kingma and Ba, 2014)
+*https://codalab.lisn.upsaclay.fr/competitions/3754
+for the optimization. The batch size is set as 32,
+and the number of dialogue history is 1. The whole
+model was trained for three epochs on RTX A6000
+GPU and took 8 hours per one epoch.
+Baselines
+We implement the baselines from
+previous study (Jang et al., 2022) and we conduct
+experiments with GPT-2 (Radford et al., 2019) and
+BART (Lewis et al., 2020a) as well. For a fair
+comparison, we demonstrate the results on GPT-
+2small, which has 12 layers, and BARTbase, which
+has 6 encoders and 6 decoder layers. Also, GPT-
+2medium contains 24 layers of the decoder, and
+BARTlarge possesses 12 layers for each encoder
+and decoder.
+4.2
+Automatic Evaluation
+We show the main results on the FoCus dataset
+with automatic metrics in grounding and generation
+tasks. The official metrics for the benchmark are
+chrF++ (Popovi´c, 2017), BLEU (Papineni et al.,
+2002), ROUGE-1, ROUGE-2, and ROUGE-L (Lin,
+2004). To consider the semantic similarity score
+for each token between candidate and reference
+sentences using contextual representation, we
+additionally adopt BERTscore (Zhang* et al.,
+2020). For grounding task, we used accuracy for
+both knowledge and persona grounding, and F1
+score for the persona grounding.
+In Table 2, it is found that our method shows
+substantial improvements in all the metrics from
+generation to grounding compared to the baselines.
+Especially, the performances of INFO increase
+over 18% at least regarding the generation metrics
+except for BERTScore. Furthermore, our model
+achieves remarkable success in persona and
+knowledge accuracy. Unlike the performance in
+other generation metrics, SRS demonstrates better
+persona accuracy than SRT . This result might be
+
+Model
+Generation
+Grounding
+chrF++
+BLEU
+R-1
+R-2
+R-L
+BERTScore
+Persona
+(Acc.)
+Persona
+(F1)
+Knowledge
+(Acc.)
+SRT
+Bi-encoder
+51.83
+29.51
+56.35
+40.80
+51.37
+91.86
+88.10
+38.20
+99.18
+Cross-encoder
+49.90
+27.18
+53.57
+38.25
+49.29
+91.52
+87.09
+35.32
+99.49
+Poly-encoder
+53.29
+31.46
+58.26
+42.35
+53.06
+92.29
+80.87
+39.56
+99.22
+Table 3: Performances comparison between the encoding modules for grounding tasks
+attributed to the architecture of the generator, which
+is more applicable to sentence classification tasks
+such as persona grounding. The official test result is
+also demonstrated in Appendix A, but BERTscore
+is missing due to the unreleased ground truth.
+4.3
+Human Evaluation
+We conduct a human evaluation to validate
+the responses from our model through Amazon
+Mturk services†. The assessment criteria are
+fluency, adequacy, provenance, engagingness, and
+hallucination. In specific, provenance is the level of
+utilization of the ground truth knowledge into the
+responses, whereas engagingness means how much
+the answers are persona-related. Also, hallucination
+indicates whether the answer contradicts the
+persona and knowledge or cannot be verified
+from the source content. We randomly chose 50
+dialogues from the official test set, and three
+workers were allocated to evaluate each dialogue
+generated by our model and baselines. We asked
+the workers to rank the answers according to each
+criterion following Cho and May (2020). Rank is
+scaled from 1 to 5, and the lower number is mapped
+to the better quality except for hallucination. The
+agreement between the annotators is calculated
+with Fleiss’ Kappa coefficient and is 0.4185
+indicating fair agreement. The relations between
+the annotators hardly exist since we collect the
+results from the Amazon Mturk workers.
+As in Table 4, INFO surpasses BARTbase,
+BARTlarge, GPT-2small and GPT-2medium in all
+of the criteria. INFO achieves the highest rank in
+adequacy, fluency, and provenance and generates
+a more human-like response than other generative
+models. Also, the workers ranked our model the
+lowest when they were asked to rank the responses
+in the most hallucinated order. Thus, it can be found
+that INFO generates more engaging and fewer
+hallucination utterances with respect to the human.
+The distribution of the rank per each criterion is
+illustrated in Appendix B.
+†https://www.mturk.com/
+Models
+Avg. Rank
+Ad. ↓
+Fl. ↓
+Prov. ↓
+Eng. ↓
+Hall. ↑
+GPT-2small
+3.57
+3.41
+3.58
+3.46
+2.49
+GPT-2medium
+3.11
+3.10
+3.04
+3.25
+3.02
+BARTbase
+3.43
+3.29
+3.47
+3.22
+2.45
+BARTlarge
+3.31
+3.63
+3.29
+3.44
+2.69
+INFO (Ours)
+1.57
+1.57
+1.62
+1.63
+4.35
+Table 4: Human evaluation. The value in the table
+is the average rank of the each model’s response.
+The
+abbreviation
+Ad.
+Fl.
+Prov.
+Eng.
+and
+Hall
+denote adequacy, fluency, provenance, engaginess, and
+hallucination, respectively.
+5
+Results and Analysis
+5.1
+Variants on Candidate Scoring Module
+To validate the poly-encoder as a candidate
+scoring module, we apply diverse candidate scoring
+modules, including the bi-encoder and cross-
+encoder. From the results in Table 3, we can find
+that the poly-encoder outperforms in the generation
+task. In the grounding task, SRT with cross-encoder
+scoring shows improved accuracy on grounding
+persona and knowledge. The result seems to be
+SRT with bi-encoder and cross-encoder are better
+than that with poly-encoder. However, the F1
+score of INFO is higher than the two candidate
+scoring modules implying that low accuracy in
+persona is due to the tendency of active use on the
+persona in poly-encoder while the other two models
+opt to predict not to use persona sentence. The
+results suggest that the high accuracy of persona
+not always guarantees the engagingness in the
+dialogue.
+5.2
+Comparison on other Retrievers
+We show that INFO is effective in retrieving
+knowledge compared to other sparse and dense
+retrievers. We retrieve the knowledge from our
+knowledge index built with Wikipedia paragraphs.
+We utilize TF-IDF (Joachims, 1996), and deep
+passage retrieval (DPR) (Karpukhin et al., 2020).
+In the case of TF-IDF, we set the sum of query
+
+and knowledge tokens less than or equal to
+512, which is the maximum sequence length of
+DPR and INFO. We use bert-base-uncased
+as the tokenizer. For DPR, we extract less than
+40 knowledge using TF-IDF due to memory
+limitations. We first retrieve the five paragraphs
+related to the query that comprises knowledge
+snippet, dialogue history, predicted knowledge
+candidate, and selected persona sentences. In Table
+5, we find that the retriever we used outperforms
+compared to the TF-IDF and DPR in all the
+metrics, including BERTscore. The results imply
+that INFO’s retriever is suitable for extracting
+similar paragraphs rather than other retrievers.
+Model
+chrF++
+BLEU
+R-1
+R-2
+R-L
+BERTScore
+TF-IDF
+19.91
+3.52
+13.91
+9.96
+12.43
+51.54
+DPR
+20.57
+3.86
+12.44
+6.55
+10.20
+47.48
+INFO
+26.36
+7.40
+15.48
+12.18
+14.32
+53.14
+Table 5: Comparison with other retrievers
+5.3
+Effect of Selectors on Generation
+We measure each selector module’s effect on
+the generation task by changing the query which
+feds into the retriever on a validation set. The
+experimental results are shown in Table 6, where
+GTK, GTP represents ground truth knowledge and
+persona. Although the query that comprises the
+ground truth source shows the highest scores, INFO
+demonstrates comparable results on the generation
+task. From the result where the performance
+increase of INFO + GTP is larger than that of
+INFO + GTK about 2.8%p, we can identify that
+our persona selector still has more space to achieve
+its maximum level.
+Query
+chrF++
+BLEU
+R-1
+R-2
+R-L
+BERTScore
+INFO (RT)
+53.29
+31.46
+58.26
+42.35
+53.06
+92.29
++GTK
+53.35
+31.56
+58.31
+42.55
+53.18
+92.29
++GTP
+56.19
+34.39
+61.61
+45.46
+56.01
+92.79
++GTK+GTP
+56.40
+34.60
+61.88
+45.64
+56.16
+92.84
+Table
+6:
+Comparison
+between
+the
+generation
+performances based on the variants of query with
+ground truth knowledge and persona. Note that all the
+performance is evaluated with the official validation
+set.
+5.4
+Qualitative Analysis
+In Table 7, an example from the predicted results
+is illustrated. In the case of BARTlarge, and GPT-
+2medium, the responses only reflect the ground
+Given Landmark
+Finding Nemo Submarine Voyage
+Dialogue
+Human: What area of the park is this ride in?
+Machine: This ride is located in the Tomorrowland area of Disneyland.
+Human: Has this ride always been about Finding Nemo?
+Human’s Persona
+I’ve never been to California.
+My favorite cartoon is Finding Nemo.
+I would like to visit Disneyland.
+My favorite color is yellow.
+I enjoy swimming.
+Ground Truth Knowledge (Grounding)
+Based on the characters and settings of the 2003 Disney · Pixar, Finding
+Nemo, it is a re-theming of the classic Submarine Voyage attraction that
+operated from 1959 to 1998.
+Retrieved Knowledge (Generation)
+The original Submarine Voyage was built in 1959 as part of the then new
+Tomorrowland...
+In 2008, Finding Nemo Submarine Voyage received an award for
+outstanding achievement from the Themed Entertainment Association.
+Finding Nemo Submarine Voyage is ... which opened on June 11, 2007. ...
+it is a re-theming of the classic Submarine Voyage attraction that operated
+from 1959 to 1998.
+On July 15, 2005, two days before the 50th Anniversary of Disneyland,
+the Finding Nemo Submarine Voyage was officially announced at the
+new Turtle Talk...
+The attraction reuses the eight original 1959 Submarine Voyage through
+Liquid Space attraction...
+Predicted Answers
+BARTbase
+Yes, that’s right. You’re a fan of the “Fantasy” film,
+so I.
+BARTlarge
+Yes, the ride is based on the characters and settings
+of the 2003 Disney · Pixar film
+GPT-2small
+No, it was originally a way to show that you love
+Finding Nemo.
+GPT-2medium
+Yes, it has operated from 1959 to 1998.
+INFO (Ours)
+No, this attraction is actually a re-theme of the
+classic submarine voyage attraction that operated
+from 1959 to 1998. The attraction is based on the
+characters and settings of the 2003 Disney Pixar
+film Finding Nemo, which is your favorite cartoon.
+Ground Truth Response
+No, your favorite cartoon is a new addition to this ride. The current
+Finding Nemo ride is a re-theming of the classic “Submarine Voyage”
+attraction that operated here from 1959 to 1998.
+Table 7: Qualitative result. All the predicted results
+in grounding task are from our model, INFO and it
+predicts the correct answers in both tasks. We add other
+baselines’ responses for comparative analysis.
+truth knowledge resulting in less engaged answers
+without any persona-related phrases. Although
+BARTbase seems to employ a persona sentence in
+the form of the phrase “You’re fan of the Fantasy
+film”, its used sentence does not appear in human’s
+personal profiles. This result also indicates that
+the utterance is hard to identify its provenance
+on the knowledge source. Moreover, GPT-2small
+generates the utterance that contradicts the ground
+truth knowledge. From the result, we can find that
+the generated responses from the baselines show
+hallucinations on both persona and knowledge.
+Unlike other baselines, our model blends ground
+truth knowledge and persona sentence into the
+
+response with less hallucination and engagingness.
+In addition, the retrieved knowledge source that
+our model refers to provides interpretability and
+provenance of the responses to the users. More
+examples are also depicted in Appendix C.
+6
+Conclusions
+In this paper, we presented a conversational
+agent that generates responses grounding the
+user’s persona and external knowledge. We
+utilized poly-encoder-based candidate scoring
+for
+each
+grounding
+task.
+We
+additionally
+implement persona level indicator to consider
+multiple persona selections for delicate persona
+grounding. With predicted sources, we construct
+a
+knowledge-persona
+enhanced
+query
+to
+retrieve latent paragraphs, and they are used
+to generate informative and engaging responses by
+marginalizing loss for each token. We show that
+our method achieves the state-of-the-art (SoTA)
+score in both grounding and generation tasks in the
+persona-knowledge conversation dataset. We also
+demonstrate that the responses from INFO show
+less hallucination and more engagingness through
+human evaluation and qualitative analysis. We also
+compare the grounding modules and retrievers to
+show INFO’s effectiveness.
+7
+Limitations
+The proposed model INFO has limitations. Given
+the INFO’s settings, the model cannot deal with
+real-world application, which means the absence
+of ground truth knowledge or persona candidates
+in the grounding task. We also conducted the
+human evaluation to evaluate the capability of
+the proposed model’s mitigating hallucination
+in dialogue generation. However, the number
+of cases is relatively small for evaluating the
+capability of mitigating hallucination. Finally,
+INFO demands high GPU computation resources,
+since it marginalizes loss at the token level.
+We plan to improve the INFO for future work.
+We will train and evaluate the INFO in open-
+domain settings as well as real-world settings for
+the applicable conversational agents. Moreover, we
+will conduct human evaluations with more cases.
+Especially, we will enhance the way of quantitative
+measurement for the model’s hallucinated answers.
+Last but not least, we will improve the generator
+of INFO with more computationally efficient
+components.
+8
+Acknowledgement
+This
+work
+was
+supported
+by
+Institute
+of
+Information
+&
+communications
+Technology
+Planning & Evaluation(IITP) grant funded by the
+Korea government(MSIT) (No. 2020-0-00368,
+A
+Neural-Symbolic
+Model
+for
+Knowledge
+Acquisition and Inference Techniques), This
+research was supported by the MSIT(Ministry
+of
+Science
+and
+ICT),
+Korea,
+under
+the
+ITRC(Information Technology Research Center)
+support
+program(IITP-2022-2018-0-01405)
+supervised by the IITP(Institute for Information
+& Communications Technology Planning &
+Evaluation), This work was supported by Institute
+for Information & communications Technology
+Planning & Evaluation(IITP) grant funded by the
+Korea government(MSIT) (No. 2022-0-00369,
+(Part 4) Development of AI Technology to support
+Expert Decision-making that can Explain the
+Reasons/Grounds for Judgment Results based on
+Expert Knowledge)
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+
+A
+Automatic Evaluation on Official Test Set
+Models
+Generation
+Grounding (Acc.)
+chrF++
+BLEU
+R-1
+R-2
+R-L
+Persona
+Knowledge
+GPT2small
+28.83
+11.60
+36.28
+19.56
+32.42
+67.83
+70.95
+GPT2medium
+30.34
+12.58
+38.35
+21.16
+34.34
+67.64
+72.46
+BARTbase
+29.80
+12.15
+36.26
+19.73
+32.06
+67.66
+72.02
+BARTlarge
+30.63
+11.86
+36.36
+19.42
+31.73
+67.62
+70.53
+INFO (RS)
+52.81
+29.41
+56.37
+40.41
+51.16
+82.74
+98.88
+INFO (RT)
+54.61
+32.33
+58.27
+42.39
+53.09
+80.83
+99.10
+Table 8: Main results on the official test set. RT indicates the model with RAG-Token model as generator. The
+models are evaluated by generation metrics, including chrF++, BLEU, ROUGE-1 (R-1), ROUGE-2 (R-2) and
+ROUGE-L (R-L). The accuracy for persona grounding task and knowledge grounding task are also noted. Since
+BERTscore is not the official generation metric, we cannot evaluate the result on the metric as the ground truth of
+the test is not yet disclosed.
+B
+Human Evaluation Distribution on Each Criteria
+(a) Adequacy
+(b) Fluency
+Figure 2: The distribution of the rank on the adequacy and fluency criteria. Guide A to E indicates INFO, BARTbase,
+BARTlarge, GPT-2small, and GPT-2medium, in the order.
+
+Guide A
+Guide B
+100
+Guide C
+Guide D
+Guide E
+80
+f evaluation
+60
+of
+40
+#
+20 
+0
+1
+2
+3
+4
+5
+RankGuide A
+Guide B
+100
+Guide C
+Guide D
+Guide E
+80
+f evaluation
+60
+JO
+40
+#
+20 
+0
+1
+2
+3
+4
+5
+Rank(a) Provenance
+(b) Engagingness
+Figure 3: The distribution of the rank on the provenance and engagingness criteria. Guide A to E indicates INFO,
+BARTbase, BARTlarge, GPT-2small, and GPT-2medium, in the order.
+Figure 4: The distribution of the rank on the less hallucination criterion. Note that the highest rank (1) means the
+most hallucinated. Guide A to E indicates INFO, BARTbase, BARTlarge, GPT-2small, and GPT-2medium, in the
+order.
+
+Guide A
+Guide B
+100
+Guide C
+Guide D
+Guide E
+80
+f evaluation
+60
+of
+40
+#
+20 
+0
+1
+2
+3
+4
+5
+RankGuide A
+100
+Guide B
+Guide C
+Guide D
+Guide E
+80
+f evaluation
+60
+of
+40
+#
+20 
+0
+1
+2
+3
+4
+5
+RankGuide A
+Guide B
+100
+Guide C
+Guide D
+Guide E
+80
+f evaluation
+60
+of
+40
+#
+20 
+0
+2
+3
+5
+1
+4
+RankC
+Qualitative Results
+Given Landmark
+Nocton Hall
+Dialogue
+Human: I know this place, but I don’t remember the name of this place.
+Human’s Persona
+I have a hall in my house.
+I have worked with military hospital.
+I would like to go to England.
+I have heard about Frederick John Robinson.
+I have respect to American Officers.
+Ground Truth Knowledge (Grounding)
+Today the site is maintained by English Heritage.
+Retrieved Knowledge (Generation)
+“Hogesdon” is first recorded in the Domesday Book, meaning an Anglo-
+Saxon farm (or “fortified enclosure”) belonging to Hoch, or Hocq....
+By 1677, the church was in such poor structural condition that the Dean
+of Chichester convened a consistory court with three parishioners and
+asked them why repairs had not been carried out...
+he south transept had historically served as the mausoleum for the
+Cavendish family, who resided at nearby Latimer...
+Following the diminished influence of the ancient Priory, predecessor of
+the two parishes of St Bartholomew, disputes began to arise over rights
+to tithes and taxes payable by lay residents who claimed allegiance with
+the nearby and anciently associated parish of St Botolph Aldersgate -
+an unintended consequence and legacy of King Kenry VIII’s religious
+reforms.
+a wall-mounted stone pulpit, a stone altar, a series of tall, pointed
+windows high in the walls, an unadorned stone font and a short wooden
+steeple-little more than an extended belfry-extending from the nave
+roof...a wall-mounted stone pulpit, a stone altar, a series of tall, pointed
+windows high in the walls, an unadorned stone font and a short wooden
+steeple-little more than an extended belfry-extending from the nave roof...
+Predicted Answers
+BARTbase
+This is the Nocton Hall, a historic Grade II listed
+building in the village of Nocton,
+BARTlarge
+Nocton Hall is a historic Grade II listed building in
+the village of Nocton, in Lincoln
+GPT-2small
+This is Nocton Hall, a historic Grade II listed
+building in the village of Nocton, in
+GPT-2medium
+This is the Nocton Hall, a historic Grade II listed
+building in the village of Nocton,
+INFO (Ours)
+This will be a nice vision for you. Nocton Hall is
+a historic Grade II listed building in England that
+you would like to visit.
+Ground Truth Response
+You can see Nocton Hall in the village of Nocton, in Lincolnshire of
+England, the country you want to visit.
+Given Landmark
+Maiden Castle, Dorset
+Dialogue
+Human: Wow, this is amazing! What is this?
+Machine: It is Maiden Castle in Dorset. I thought you would like it since
+you are interested in historic forts.
+Human: Who owns the site today?
+Human’s Persona
+I like Britain.
+I have been to Dorset.
+I am interested in historic forts.
+I hope to work for English Heritage.
+I would like to visit an old fort.
+Ground Truth Knowledge (Grounding)
+Today the site is protected as a Scheduled Ancient Monument and is
+maintained by English Heritage.
+Retrieved Knowledge (Generation)
+Portland Castle is an artillery fort constructed by Henry VIII on the Isle
+of Portland, Dorset, between 1539 and 1541...
+this version of events, or even that the hill fort was attacked by the
+Romans...
+Between 1985 and 1986 further excavations under Niall Sharples were
+prompted by the hill fort’s deteriorating condition, partly caused by the
+large number of visitors to the site...
+a Tudor rose and the initials E.R. (Elizabeth Regina), has been preserved
+and can be seen in the inner bailey of the castle mounted on a replica
+carriage...
+Constructed on a territorial boundary in about 600 BC, the first hill fort
+at Maiden Castle was a 6.4-hectare (16-acre) area surrounded by a single
+ditch...
+Predicted Answers
+BARTbase
+The site is maintained by English Heritage, the
+country you are from.
+BARTlarge
+Today the site is owned by English Heritage.....
+GPT-2small
+Today the site is protected as a Scheduled Ancient
+Monument and is maintained by English Heritage.
+GPT-2medium
+Today the site is maintained by English Heritage.
+INFO (Ours)
+Today the site is owned by English Heritage. You
+may wish to research this further since you hope to
+work for English Heritage.
+Ground Truth Response
+It is owned by English Heritage; a company you hope to work for.
+Table 9: Qualitative results. All the predicted results in grounding task are from our model, INFO and it predicts
+the correct answers in both tasks. We add other baselines’ responses for comparative analysis.
+

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+Reinforcement Learning-based Joint Handover and
+Beam Tracking in Millimeter-wave Networks
+Sara Khosravi∗, Hossein S. Ghadikolaei‡, Jens Zander∗, and Marina Petrova ∗†
+∗School of EECS, KTH Royal Institute of the Technology, Stockholm, Sweden,
+† Mobile Communications and Computing, RWTH Aachen University, Germany, ‡ Ericsson Research, Sweden
+Email: {sarakhos, jenz, petrovam} @kth.se, [email protected]
+Abstract—In this paper, we develop an algorithm for joint
+handover and beam tracking in millimeter-wave (mmWave)
+networks. The aim is to provide a reliable connection in terms of
+the achieved throughput along the trajectory of the mobile user
+while preventing frequent handovers. We model the association
+problem as an optimization problem and propose a reinforcement
+learning-based solution. Our approach learns whether and when
+beam tracking and handover should be performed and chooses
+the target base stations. In the case of beam tracking, we
+propose a tracking algorithm based on measuring a small spatial
+neighbourhood of the optimal beams in the previous time slot.
+Simulation results in an outdoor environment show the superior
+performance of our proposed solution in achievable throughput
+and the number of handovers needed in comparison to a multi-
+connectivity baseline and a learning-based handover baseline.
+Index Terms—Millimeter-wave, user association, beam track-
+ing, handover, reinforcement learning.
+I. INTRODUCTION
+Millimeter-wave (mmWave) is a key radio access technol-
+ogy for beyond 5G communication systems, offering ultra-
+high data rates due to a large amount of free spectrum [1].
+However, due to the fewer scattering paths and significant
+penetration loss, mmWave links are vulnerable to static or
+dynamic obstacles. To overcome such severe loss, both base
+station (BS) and user equipment (UE) may need directional
+communication using a large number of antennas, which may
+result in frequent misalignment of beams due to mobility and
+blockage. Hence, finding and maintaining the optimal beam
+directions (beam alignment) is necessary. The lengthy period
+to achieve the beam alignment (hundreds of milliseconds
+to seconds [2]) results in a high cell search time or BS
+discovery time in mmWave systems. As reported in [3], the
+BS discovery time which is the time required to search the
+target BS when the handover command is received by the
+UE is about 200 ms. Moreover, to improve the capacity and
+coverage the density of the BSs is usually high in mmWave
+systems [1]. Hence, conventional handover methods based on
+instantaneous received signal power can cause unnecessarily
+frequent handovers and a ping-pong effect. This leads to a
+severe drop in service reliability. Therefore, fast BS discovery
+(finding target BS in the handover process), and efficient
+handover execution techniques, will be required to use the
+full promise of mmWave cellular networks.
+The spatial mmWave channel can be approximated by a
+few dominant paths, where each path can be defined with
+its angle of departure (AoD), angle of arrival (AoA) and
+gain [4]. Hence, one can only estimate these path parameters
+instead of a large dimensional channel matrix [5], [6]. The
+process of identifying the dominant paths is called beam
+training. However, due to the dynamic environment, frequent
+beam training may cause high overhead1. Temporal correlation
+of spatial mmWave channel can be employed to accelerate
+the beam training process by tracking the variation of the
+dominant path directions [6].
+A. Related Work
+To address the link failure and throughput degradation in
+a dynamic environment, the multi-connectivity technique has
+been vastly analyzed in literature [7], [8]. In this technique, the
+UE keeps its connection to multiple BSs (either at mmWave
+band or sub-6 GHz band). However, power consumption,
+synchronization and the need for frequent tracking are the
+main challenges. In the 3GPP standard (release 16) two
+handover techniques are introduced to improve the link robust-
+ness during mobility: dual active protocol stack (DAPS), and
+conditional handover (CHO) [9]. In the DAPS, the connection
+to the current serving BS is maintained until the connection
+to the target BS is fully established. In the CHO, the UE is
+configured with multiple target BSs. During the handover, the
+UE can select one of the configured BSs as the target BS
+during the RRC reconfiguration message. Although CHO can
+decrease the handover failure probability, it may increase the
+handover latency if the UE asks for multiple handovers during
+a single RRC reconfiguration [7].
+Applying machine learning as the main decision-maker tool
+to make the optimal handover decision and choose the target
+BS has been also studied in the literature [10], [11]. The
+authors in [10] proposed a reinforcement learning (RL) based
+handover policy to reduce the number of handovers while
+keeping the quality of service in heterogeneous networks.
+In [11] an intelligent handover method based on choosing
+the backup solution for each serving link to maximize the
+aggregate rate along a trajectory has been proposed.
+1Overhead depends on the training time compared with the changes in the
+environment.
+arXiv:2301.05305v1  [eess.SY]  12 Jan 2023
+
+In terms of beam tracking, authors in [12] applied the
+correlation of spatial mmWave channel in adjacent locations
+and proposed the beam steering method based on searching
+over a small angular space in the vicinity of the previously
+known valid beams. The authors in [6] applied machine
+learning to the tracking procedure to extract useful information
+from the history of AoD tracking.
+All the aforementioned works only take handover or beam
+tracking issues into account. Additionally, they do not study
+the impact of selecting beam tracking and handover on the
+achieved throughput of the UE along its trajectory and instead
+focus on the achieved rate as the primary performance metric.
+B. Our Contributions
+In this paper, we develop a novel joint handover and beam
+tracking algorithm in a mmWave network under mobility. The
+algorithm aims to associate the UEs to BSs that maximize
+the sum achieved throughput along the trajectory and ensure
+the achieved throughput in each location of the trajectory
+is higher than a pre-defined threshold. The user association
+process is defined as the process of determining whether a user
+is associated with a particular BS before data transmissions
+commence. In the case of handover, the UE is associated with a
+new BS, whereas in the case of beam tracking, the UE remains
+associated with the serving BS from the previous time slot. The
+main contributions of our paper are summarized as below:
+• System Modeling: We model the user association prob-
+lem as a non-convex optimization problem. Unlike the
+existing works in the literature, we consider achieved
+throughput as the main performance metric to measure
+the effect of handover or beam tracking on the UEs’
+quality of service.
+• Learning-based Solution: The objective function in our
+proposed user association problem highly depends on the
+user association mechanism. We utilize the reinforcement
+learning (RL) algorithm to approximate the solution to
+this problem. The aim is to decide whether to run a beam
+tracking algorithm or a handover algorithm.
+• Joint Handover and Beam Tracking Algorithm: In the
+case of a handover decision, the target BS will be
+recognized as the output of the RL algorithm. In the
+case of beam tracking, the search space will be defined
+based on our proposed tracking algorithm by searching
+the directions in the small spatial neighbourhood of the
+previously selected optimal directions.
+• Empirical Evaluation: We apply ray tracing with a
+real building data map as the input. The results show
+the effectiveness of our proposed method in achieving
+throughput along trajectories and decreasing the number
+of handovers.
+The rest of the paper is organized as follows. We introduce
+the system model and problem formulation in Section II. In
+Section III, we propose our method. We present the numerical
+results in Section IV and, conclude our work in Section V.
+Notations: Throughout the paper, vectors and scalars are
+shown by bold lower-case (x) and non-bold (x) letters, respec-
+tively. The conjugate transpose of a vector x is represented by
+xH. We define set [M] := {1, 2, .., M} for any integer M. The
+indicator function 1{·} equals to one if the constraint inside
+{·} is satisfied.
+II. SYSTEM MODEL AND PROBLEM FORMULATION
+In this section, first, we introduce the mmWave channel
+model. Then, we present the user association problem formu-
+lation.
+We consider a downlink communication with |B| mmWave
+BSs, where each is equipped with NBS antennas, communi-
+cating with a single antenna mobile UE. We consider analog
+beamforming with a single RF chain. We assume all BSs
+allocate equal resources to their serving UEs. The channel
+between BS j ∈ B and its serving UE during time slot i is
+[13]:
+hj =
+L
+�
+ℓ=1
+hℓaH(φℓ, θℓ),
+(1)
+where L is the number of available paths. Each path ℓ has
+complex gain hℓ (include path-loss) and horizontal φℓ and
+vertical θℓ, AoD. Due to the notation simplicity, we drop the
+index j and i from the channel parameters. The array response
+vector is a(.) where its exact expression depends on the array
+geometry and possible hardware impairments. The signal-to-
+noise ratio (SNR) in time slot i is
+SNR(i)
+j
+= p|hH
+j fj|2
+σ2
+,
+(2)
+where σ2 is the noise power, p is the transmit power, fj ∈ CNBS
+is the beamforming vector of BS j.
+We define variable x(i)
+j
+∈ {0, 1} for j ∈ B as an association
+indicator in time slot i, where is equal 1 if UE is associated to
+the BS j and 0 otherwise. Hence, the achieved rate per second
+per hertz in time slot i is
+R(i) = x(i)
+jS log2(1 + SNR(i)
+jS ) =
+�
+j∈B
+x(i)
+j log2(1 + SNR(i)
+j ),
+where jS is the index of the serving BS of the UE during time
+slot i. Here, we assume each UE is served by only one BS.
+We define the achievable throughput per hertz of the UE by
+multiplying its rate by the data transmission time as
+Γ(i) = (1 − τ (i)
+b
+τc
+)R(i),
+(3)
+where, τ (i)
+b
+is the beam training duration which may have a
+different value in each time slot i, and τc is the duration of
+the time slot that is a fixed value for all time slots, see Fig. 1.
+A. Beam Training and Beam Tracking
+As depicted in Fig. 1a, when the UE is connected to a
+BS j ∈ B, initial beam training is performed by sending
+pilots over all combination of the beam directions in the
+codebook during τb. Based. on the UE’s feedback of the
+received signal strength (or estimated SNR), the best beam pair
+directions are selected. Then, the BS and the UE would use this
+
+Initial beam training
+Data Transmission
+τb
+τc
+(a)
+Tracking
+Data Transmission
+τb
+(b)
+Fig. 1: τc is the time slot duration. τb is (a) the initial beam
+training duration when the UE is associated with the new
+BS (handover case), (b) the beam tacking duration when the
+serving BS is the same for the consecutive slots.
+direction (φℓ⋆, θℓ⋆) during the data transmission phase. The
+beamforming vector, f is chosen to maximize the achievable
+rate of the UE. Due to the monotonicity of the logarithm
+function, this is equivalent to maximising the SNR term in
+(2). Hence
+f ∗
+j = arg max
+fj∈F
+|hH
+j fj|2
+(4)
+where F is the beamforming codebook that contains all
+the feasible beamforming vectors. The n-th element of the
+codebook F is defined as f(n) = a(φn, θn), where (φn, θn)
+are steering angles and a(.) is the array response vector.
+When the BS continues serving the same UE in a consecu-
+tive time slot, only searching the neighbouring beam directions
+of the main directions can be sufficient to maintain the link
+quality. This process is called beam tracking. As shown in
+Fig. 1b, the duration of τb is much smaller than the initial
+beam training duration.
+B. Problem Formulation
+The UE association depends on the channel quality between
+the BS and the UE. Due to UE mobility or temporary
+blockage, the channel quality changes and consequently the
+UE association. Based on the UEs’ velocity, we determine how
+quickly the channel quality can change and predict the time
+at which the current UE association needs to be updated. We
+define TA seconds as the frequency of updating the association.
+Hence, we need to make the decision every TA whether to run
+the handover execution or beam tracking procedure if SNR is
+lower than the pre-defined SNR threshold (SNRthr). Note that
+we can have an on-demand reactive handover at any time slot
+if the link toward the serving BS fails abruptly. However, with
+a proper choice of TA, the frequency of those reactive events
+could be very small. We define the duration of the trajectory
+as M and consider the discrete time index i to describe the
+association update at each interval.
+The goal is to maximize the aggregate throughput of the UE
+along the trajectory while ensuring the achieved throughput in
+each time slot i is higher than a predefined threshold. To this
+end, we define functions F1 and F2 as
+• F1 is the averaged throughput along the trajectory as
+F1 =
+M
+�
+i=1
+E
+�
+Γ(i)�
+,
+where the expectation is with respect to the randomness
+of channel fading and the blockage, M is the duration of
+the trajectory, and Γ(i) is defined in (3).
+• F2 is the expected number of time slots whose throughput
+is lower than the threshold (Γthr).
+F2 = E
+� M
+�
+i=1
+1
+�
+Γ(i) ≤ Γthr
+��
+=
+M
+�
+i=1
+Pr
+�
+Γ(i) ≤ Γthr
+�
+.
+We formulate the user association at time slot i ∈ [M]
+as an optimization problem which involves finding the x(i)
+j
+corresponding to the association indicator as
+max
+{x(i)
+j
+}i,j
+F1 − λF2
+(5a)
+s.t.
+�
+j∈B
+x(i)
+j
+= 1, ∀, i ∈ [M]
+(5b)
+x(i)
+j
+∈ {0, 1},
+∀j ∈ B, i ∈ [M]
+(5c)
+where λ is a large constant controlling the importance of F2.
+Constraint (5b) guarantees that each UE is served by one BS.
+The optimization problem (5) is nonlinear. Solving this
+optimization problem requires estimating the expectation value
+in F1 and F2 which requires running many realizations.
+Moreover, the impact of choosing the x(i)
+j
+(the target BSs
+in the handover case or choosing beam tracking procedure)
+propagates in time and can affect the UEs’ performance in
+the next time slots. Therefore, we need to consider the long-
+term benefits of selecting association indicators besides their
+immediate effects on the UEs’ performance. Furthermore, In
+order to select the target BSs, we need to model or predict the
+UEs’ performance in the next time slots, which can add more
+complexity to the network due to the mobility of the UE and
+obstacles in mmWave networks. These motivate us to utilize
+the RL to approximate the solution of (5).
+III.
+PROPOSED METHOD
+We transform the problem (5) to an RL problem in which
+the objective function is turned into a reward function, and
+the constraints are transformed into the feasible state and
+action spaces. In the following, first, we start with defining
+the Markov decision process, and then we will describe our
+joint handover and beam tracking algorithm.
+A. Markov Decision Process Formulation
+RL problems are formulated based on the idea of the
+Markov decision process (MDP), which is the agent’s interac-
+tion with different states of the environment to maximize the
+expected long-term reward. The agent is the main decision-
+maker who can sit on the edge cloud. All BSs are connected
+to the agent. Now, we define different elements of an MDP.
+
+1) State Space: The state space describes the environ-
+ment by which the agent is interacting through different
+actions. We define the state at time slot i as s(i)
+=
+(ℓ(i)), j(i)
+S , SNR(i), I(i)) ∈ S, where ℓ(i) is the location index
+of the UE along the trajectory 2, j(i)
+S is the index of the serving
+BS, SNR(i) is the SNR value of the UE with serving BS j(i)
+S
+in time slot i. I(i) ∈ {0, 1} is the beam tracking activation
+indicator. I(i) = 1 means the i-th time slot is the tracking slot
+for the UE.
+2) Action Space: The action space includes all possible
+actions that can be taken by the agent. The action can change
+the state of the environment from the current state to the target
+state. In our problem, a(i) ∈ A = {0, 1, 2, ..., [|B|]} is the
+decision regarding beam tracking (a(i) = 0) or choosing the
+index of new serving BS in the case of handover decision
+(a(i) ∈ [|B|]). In other words, if a(i) ̸= 0 means the handover
+decision is made and the value of a(i) shows the target BS.
+Hence, the action is to specify a serving BS for the UE along
+its trajectory.
+3) Policy: A policy π(.) maps the state of the environment
+to the action of the agent. In our case, π is a function from S
+to A, i.e., π : S → {0, 1, ..., [|B|]}
+4) Rewards: The agent obtains the reward after taking an
+action a(i) when current state is s(i) and moves to next state
+s(i+1). Here we define reward r(s(i), a(i), s(i+1)) as
+r(s(i), a(i), s(i+1)) = Γ(i) − λ1
+�
+Γ(i) ≤ Γthr
+�
+,
+(6)
+where Γ(i) is defined in (3).
+5) State-action value: The function Qπ(s, a) is the long-
+term reward and is defined as the expected summation of
+discounted reward in the future for the action a ∈ A that
+agent takes in state s under policy π. The RL algorithm aims
+to choose the optimal policy π⋆ in each state s that maximizes
+the Qπ(s, a). With discount factor η ∈ [0, 1], we have
+Qπ(s, a) = E
+��
+i
+ηir(s(i), s(i), s(i+1))
+�
+,
+where the expectation is over the transition probabilities. In
+our problem, transition probabilities model the SNR variations
+due to the randomness of the channel fading and blockage.
+We assume mobility information including the UEs’ current
+location and its trajectory is known3. Therefore, the transition
+to the next location is deterministic.
+The optimal policy in state s ∈ S is found by
+π⋆(s) = arg max
+a∈A
+Qπ(s, a).
+(7)
+Due to the continuous and large number of state spaces, we
+apply deep Q-learning (DQL) [14] to solve (7). In DQL, the
+state-action value function is estimated by the deep neural
+network function approximators.
+2Note that, we discretize the location of the UE along the trajectory. Hence,
+every location dimension (x, y) a trajectory with length M is mapped to a
+location index ℓ(i) ∈ [M].
+3Note that the location information can be easily fed back through lower-
+frequency links.
+B. Joint Handover and Beam Tracking Algorithm
+Algorithm 1 describes our proposed joint handover and
+beam tracking algorithm along a trajectory with duration M.
+If the current association cannot offer the required SNR level,
+the decision regarding handover or beam track is made based
+on a(i) as the output of the RL algorithm. In the case of the
+handover decision, the value of a(i) represents the target BS.
+The beam tracking algorithm based on small spatial mea-
+surement in time slot i is shown in Algorithm 2. In slot i, the
+algorithm starts by using the main beam of the same serving
+BS in the previous time slot i − 1. If the SNR value is lower
+than the threshold, then starts a small spatial measurement over
+the AoD direction of the main beam. To quantify the size of the
+spatial neighbourhood, we define ∆φ and ∆θ as the maximum
+absolute horizontal and vertical deviation from the main AoD
+direction. We define δφ and δθ as the measurement resolution
+in horizontal and vertical, respectively. Inspired by [15], the
+spatial neighbourhood N surrounding the main AoD direction
+can be expressed using the horizontal neighbourhood Nφ and
+vertical neighbourhood Nθ as
+Nφ(∆φ, δφ) =
+�
+i.δφ : i ∈
+�
+−
+�∆φ
+δφ
+�
+,
+�∆φ
+δφ
+���
+(8)
+Nθ(∆θ, δθ) =
+�
+j.δθ : j ∈
+�
+−
+�∆θ
+δθ
+�
+,
+�∆θ
+δθ
+���
+(9)
+where ⌊.⌋ is the floor operation. The complete neighbourhood
+is the Cartesian product of the horizontal and vertical neigh-
+bourhoods as
+N(∆φ, ∆θ, δφ, δθ) = Nφ(∆φ, δφ) × Nθ(∆θ, δθ)
+= {(φ, θ) : φ ∈ Nφ(∆φ, δφ), θ ∈ Nθ(∆θ, δθ)}(10)
+The spatial neighborhoods T (i) in time slot i surrounding the
+main AoD directions (φ(i−1)
+ℓ⋆
+, θ(i−1)
+ℓ⋆
+) in previous time slot is
+T (i) = (φ(i−1)
+ℓ⋆
+, θ(i−1)
+ℓ⋆
+, ) + N(∆φ, ∆θ, δφ, δθ).
+(11)
+Now given the main AoD direction, we need to find the
+transmit direction from neighbourhoods T (i) that offers the
+SNR threshold. We represent the sorted direction pairs as
+[T (i)]I, where I is the sorted indices. It means the directions
+in [T (i)]I increase in distance from the main AoD direction.
+Starting from the main AoD direction, the SNR of each trans-
+mit direction in [T (i)]I is measured until a beam pair meets
+the required SNR level. Afterwards, no further measurements
+are required. If no direction meets the threshold, the entire
+(∆φ, ∆θ)-neighbourhood is measured to find the beam pairs
+that offer the SNR threshold.
+Note that in the worse scenario, if the selected target BS
+based on our proposed algorithm cannot offer the required
+SNR level due to very sudden blockage, the conventional
+handover methods based on searching over the candidate BSs
+in UEs vicinity can be applied. However, as shown in the
+numerical results, such extreme case is rare.
+
+Algorithm 1 Joint handover and beam tracking
+Input: Trajectory with duration M
+1: Initialization: for i = 1 set j(1)
+S =1
+2: for i ∈ 1, ..., M do
+3:
+if SNR(i)
+jS < SNRthr then
+4:
+Choose the optimal action a(i) based on current
+s(i).
+5:
+if a(i) ̸= 0 then.
+▷ handover execution
+6:
+Set j(i)
+S
+= a(i) and run the initial beam training
+process and compute the achieved throughput Γ(i) as (3).
+7:
+else
+8:
+Run Algorithm 2 and compute Γ(i).
+9:
+end if
+10:
+end if
+11: end for
+Output: Γ(i)
+Algorithm 2 Beam tracking in time slot i at the BS j
+Input: [T (i)]I, SNRthr, duration of each beam pair testing (β),
+cnt(i) = 0.
+1: for (φ, θ) ∈ [T ]I do
+2:
+Set f (i)
+j
+= a(φ, θ).
+3:
+Measure SNR(i)
+j
+as (2).
+4:
+Set cnt(i) = cnt(i) + 1.
+▷ number of beam pair
+testing
+5:
+if SNR(i)
+j
+>= SNRthr then
+6:
+(φ(i)
+ℓ⋆ , θ(i)
+ℓ⋆ ) = (φBS, θBS)
+7:
+τ (i)
+b
+= β.cnt(i)
+8:
+break;
+9:
+end if
+10: end for
+compute the achieved throughput Γ(i) as (3)
+IV. NUMERICAL RESULTS
+We evaluate the performance of the proposed method in an
+urban environment using the ray tracing tool in the MATLAB
+toolbox. The output of the ray tracing tool is the L available
+paths between a BS and a UE in a specific location. The ray
+tracing maintains the spatial consistency of mmWave channels.
+As depicted in Fig. 2, we extracted the building map of Kista
+in Stockholm city, Sweden and used it as the input data for
+the ray tracing simulation. In our scenario, we assumed the
+building material is brick and the terrain material is concrete.
+We also add some random obstacles in the street with different
+heights (1 m and 3 m) and widths (2 m and 4 m) as the human
+bodies and various vehicles. These temporary obstacles are
+distributed randomly in the street with density 10−2 per m2.
+The material loss and the location of the temporary obstacles
+are chosen randomly in each realization of the channel. The
+BSs are located on the wall of buildings. The location of the
+BSs is chosen randomly while covering the entire trajectory.
+The BSs’ height is 6 m. We consider a pedestrian mobility
+Fig. 2: Simulation area in Kista, Stockholm. The yellow line
+shows the trajectory. Stars show the location of the BSs.
+model with a speed of 1 m/s. We consider the different lengths
+of the trajectories as 100TA, 200TA, 300TA, 400TA, 500TA.
+The main simulation parameters are listed in Table I.
+In the simulation, we consider the SNRthr = 2 dB and the
+throughput threshold Γthr = 1 bit/Hz. The value of τc is 10 ms.
+In the case of handover, we fix the initial beam training dura-
+tion as τb = 1
+3τc. In the case of beam tracking, τb is not fixed
+and equals the size of measuring neighbourhood multiplied
+by the duration of each beam pair testing (β = 10 µs). We
+compare the performance of our proposed method with two
+baselines. To have a fair comparison, we choose two baselines
+in which the target BS for the handover is pre-determined.
+Hence, we do not take into account the discovery time of
+finding the target BS in the baselines. Just like in our method,
+the handover is triggered if SNR < SNRthr.
+As Baseline 1 we consider the multi-connectivity method
+[8]. We implement a scenario where the UE maintains its
+connection with a nearby BS as a backup solution while
+being connected to the serving BS and once it experiences the
+blockage of the serving link, starts connecting to the backup
+solution. As Baseline 2 we select the learning-based handover
+in [11]. The method shows very good performance in maxi-
+mizing the achieved rate along the trajectory. In this baseline,
+the target BS during the handover process is determined by a
+learning algorithm. Although the target BSs are selected based
+on the long-term effect on the achieved rate, still can cause
+frequent handovers and throughput degradation.
+First, we fix the number of BSs to 10 (see Fig. 2). We
+consider 104 different channel realization as the input of the
+RL algorithm. After getting the optimal policy, we test it
+over real-time measurements and report the average of the
+performance over 500 channel realizations. Fig. 3 shows the
+average number of locations with unmet throughput thresholds
+along the trajectory with different lengths and Fig. 4 shows the
+average number of handovers needed. In comparison to the
+other two baselines, our method provides better throughput
+results by selecting to perform either beam tracking or a
+handover. Furthermore, we note that the two baselines have
+a higher number of handovers than our method due to only
+considering the handover solution. Hence, by considering the
+joint handover and beam tracking problem our method pro-
+vides better-achieved throughput while decreasing the number
+of handovers.
+Fig. 5 shows the average aggregate achieved
+
+Table I: Simulation parameters.
+Parameters
+Values in Simulations
+BS transmit power
+10 dBm
+Noise power level
+σ2=-174 dBm/Hz
+Signal bandwidth
+100 MHz
+BS antenna
+8 × 8 uniform planar array [11]
+Time interval duration
+TA = 1s
+Neighborhood size
+(∆φ, ∆θ) = (10◦, 10◦)
+Measurement resolution
+(δφ, δθ) = (5◦, 5◦)
+Discount factor
+η = 0.99
+λ
+100
+100
+200
+300
+400
+500
+0
+200
+400
+Trajectory length (m)
+Number of locations satisfying Γthr
+Our method
+Baseline 1
+Baseline 2
+Fig. 3: The average number of locations with unmet through-
+put threshold for different lengths of the trajectory.
+throughput along the trajectory with length 300 m for different
+numbers of BSs. By increasing the number of BSs the number
+of the locations satisfying the Γthr also increases hence the
+aggregate throughput along the trajectory increases. Even with
+a small number of BSs, our method outperforms baselines
+in aggregate throughput along the trajectory by determining
+whether to use a handover or beam tracking solution.
+We consider 10000 iterations during the training in our
+method and Baseline 2. With the training machine MacBook
+Pro 2020 M1 with a memory of 16 GB, each iteration takes
+about 15 seconds. Note that the absolute value of the training
+time per iteration depends on the running machine.
+V. CONCLUSIONS
+In this work, we proposed and studied a learning-based joint
+handover and beam tracking method in a mobile mmWave
+network. The aim of our algorithm is to maximize the aggre-
+gate throughput of the UE along a trajectory and ensure the
+achieved throughput in each location is higher than the thresh-
+old. Our evaluation results showed that by making an optimal
+decision regarding handover execution or beam tracking, our
+method provides high achievable throughput and reduces the
+number of handovers. Considering different mobility models
+and studying the effect of neighbouring size can be valuable
+future work.
+REFERENCES
+[1] T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N.
+Wong, J. K. Schulz, M. Samimi, and F. Gutierrez Jr, “Millimeter wave
+mobile communications for 5G cellular: It will work!” IEEE Access,
+vol. 1, no. 1, pp. 335–349, May 2013.
+[2] H. Hassanieh, O. Abari, M. Rodriguez, M. Abdelghany, D. Katabi,
+and P. Indyk, “Fast millimeter wave beam alignment,” in Pro. ACM
+SIGCOM, 2018, pp. 432–445.
+100
+200
+300
+400
+500
+0
+2
+4
+6
+8
+10
+Trajectory length (m)
+Number of handovers
+Our method
+Baseline 1
+Baseline 2
+Fig. 4: The average number of handovers for different lengths
+of the trajectory.
+4
+6
+8
+10
+100
+150
+200
+250
+300
+Number of BSs
+Average aggregate Γ(bits/Hz)
+Our method
+Baseline 1
+Baseline 2
+Fig. 5: The average aggregate achieved throughput per Hz
+along the trajectory with length 300 m.
+[3] 3GPP, “Requirements for support of radio resource management,” Stan-
+dard 3GPP TS 38.138, no. TS 36.133, v15.19.0, Sep. 2022.
+[4] R. W. Heath, N. Gonzalez-Prelcic, S. Rangan, W. Roh, and A. M.
+Sayeed, “An overview of signal processing techniques for millimeter
+wave mimo systems,” IEEE J. Sel. Top. Signal Process., vol. 10, no. 3,
+pp. 436–453, Apr. 2016.
+[5] X. Sun, C. Qi, and G. Y. Li, “Beam training and allocation for multiuser
+millimeter wave massive mimo systems,” IEEE Trans. Wirel. Commun.,
+vol. 18, no. 2, pp. 1041–1053, 2019.
+[6] D. Zhang, S. Shen, C. She, M. Xiao, Z. Pang, Y. Li, and L. Wang,
+“Training beam sequence design for mmwave tracking systems with
+and without environmental knowledge,” IEEE Trans. Wirel. Commun.,
+2022.
+[7] M. F. ¨Ozkoc¸, A. Koutsaftis, R. Kumar, P. Liu, and S. S. Panwar, “The
+impact of multi-connectivity and handover constraints on millimeter
+wave and terahertz cellular networks,” IEEE J-SAC., vol. 39, no. 6, pp.
+1833–1853, 2021.
+[8] M. Gapeyenko, V. Petrov, D. Moltchanov, M. R. Akdeniz, S. Andreev,
+N. Himayat, and Y. Koucheryavy, “On the degree of multi-connectivity
+in 5G millimeter-wave cellular urban deployments,” IEEE Trans. Veh.
+Technol., vol. 68, no. 2, pp. 1973–1978, Feb. 2019.
+[9] “Multi-connectivity; overall description,” Standard 3GPP, vol. v16.1.0,
+no. TS 37.340, 2020.
+[10] Y. Sun, G. Feng, S. Qin, Y. Liang, and T. P. Yum, “The smart handoff
+policy for millimeter wave heterogeneous cellular networks,” IEEE Trans
+Mob Comput., vol. 17, no. 6, pp. 1456–1468, Jun. 2018.
+[11] S. Khosravi, H. S. Ghadikolaei, and M. Petrova, “Learning-based
+handover in mobile millimeter-wave networks,” IEEE TCCN, vol. 7,
+no. 2, pp. 663–674, 2021.
+[12] A. Patra, L. Simi´c, and P. M¨ah¨onen, “Smart mm-wave beam steering
+algorithm for fast link re-establishment under node mobility in 60 ghz
+indoor wlans,” in Proceedings of the 13th ACM International Symposium
+on Mobility Management and Wireless Access, 2015, pp. 53–62.
+[13] M. R. Akdeniz, Y. Liu, M. K. Samimi, S. Sun, S. Rangan, T. S.
+Rappaport, and E. Erkip, “Millimeter wave channel modeling and
+cellular capacity evaluation,” IEEE J-SAC, vol. 32, no. 6, pp. 1164–
+1179, Jun. 2014.
+[14] D. Bertsekas, Reinforcement Learning and optimal control.
+Athena
+Scientific, 2019.
+[15] I. P. Roberts, A. Chopra, T. Novlan, S. Vishwanath, and J. G. Andrews,
+“Steer: Beam selection for full-duplex millimeter wave communication
+systems,” IEEE Trans Commun., pp. 1–1, 2022.
+

9dE4T4oBgHgl3EQf3Q3I/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf,len=425
+page_content='Reinforcement Learning-based Joint Handover and  Beam Tracking in Millimeter-wave Networks  Sara Khosravi∗, Hossein S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Ghadikolaei‡, Jens Zander∗, and Marina Petrova ∗†  ∗School of EECS, KTH Royal Institute of the Technology, Stockholm, Sweden,  † Mobile Communications and Computing, RWTH Aachen University, Germany, ‡ Ericsson Research, Sweden  Email: {sarakhos, jenz, petrovam} @kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='se, hossein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='shokri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='ghadikolaei@ericsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='com  Abstract—In this paper, we develop an algorithm for joint  handover and beam tracking in millimeter-wave (mmWave)  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The aim is to provide a reliable connection in terms of  the achieved throughput along the trajectory of the mobile user  while preventing frequent handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We model the association  problem as an optimization problem and propose a reinforcement  learning-based solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Our approach learns whether and when  beam tracking and handover should be performed and chooses  the target base stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In the case of beam tracking, we  propose a tracking algorithm based on measuring a small spatial  neighbourhood of the optimal beams in the previous time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Simulation results in an outdoor environment show the superior  performance of our proposed solution in achievable throughput  and the number of handovers needed in comparison to a multi-  connectivity baseline and a learning-based handover baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Index Terms—Millimeter-wave, user association, beam track-  ing, handover, reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' INTRODUCTION  Millimeter-wave (mmWave) is a key radio access technol-  ogy for beyond 5G communication systems, offering ultra-  high data rates due to a large amount of free spectrum [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  However, due to the fewer scattering paths and significant  penetration loss, mmWave links are vulnerable to static or  dynamic obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' To overcome such severe loss, both base  station (BS) and user equipment (UE) may need directional  communication using a large number of antennas, which may  result in frequent misalignment of beams due to mobility and  blockage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Hence, finding and maintaining the optimal beam  directions (beam alignment) is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The lengthy period  to achieve the beam alignment (hundreds of milliseconds  to seconds [2]) results in a high cell search time or BS  discovery time in mmWave systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' As reported in [3], the  BS discovery time which is the time required to search the  target BS when the handover command is received by the  UE is about 200 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Moreover, to improve the capacity and  coverage the density of the BSs is usually high in mmWave  systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Hence, conventional handover methods based on  instantaneous received signal power can cause unnecessarily  frequent handovers and a ping-pong effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' This leads to a  severe drop in service reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Therefore, fast BS discovery  (finding target BS in the handover process), and efficient  handover execution techniques, will be required to use the  full promise of mmWave cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  The spatial mmWave channel can be approximated by a  few dominant paths, where each path can be defined with  its angle of departure (AoD), angle of arrival (AoA) and  gain [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Hence, one can only estimate these path parameters  instead of a large dimensional channel matrix [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The  process of identifying the dominant paths is called beam  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' However, due to the dynamic environment, frequent  beam training may cause high overhead1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Temporal correlation  of spatial mmWave channel can be employed to accelerate  the beam training process by tracking the variation of the  dominant path directions [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Related Work  To address the link failure and throughput degradation in  a dynamic environment, the multi-connectivity technique has  been vastly analyzed in literature [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In this technique, the  UE keeps its connection to multiple BSs (either at mmWave  band or sub-6 GHz band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' However, power consumption,  synchronization and the need for frequent tracking are the  main challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In the 3GPP standard (release 16) two  handover techniques are introduced to improve the link robust-  ness during mobility: dual active protocol stack (DAPS), and  conditional handover (CHO) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In the DAPS, the connection  to the current serving BS is maintained until the connection  to the target BS is fully established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In the CHO, the UE is  configured with multiple target BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' During the handover, the  UE can select one of the configured BSs as the target BS  during the RRC reconfiguration message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Although CHO can  decrease the handover failure probability, it may increase the  handover latency if the UE asks for multiple handovers during  a single RRC reconfiguration [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Applying machine learning as the main decision-maker tool  to make the optimal handover decision and choose the target  BS has been also studied in the literature [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The  authors in [10] proposed a reinforcement learning (RL) based  handover policy to reduce the number of handovers while  keeping the quality of service in heterogeneous networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  In [11] an intelligent handover method based on choosing  the backup solution for each serving link to maximize the  aggregate rate along a trajectory has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  1Overhead depends on the training time compared with the changes in the  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='05305v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='SY]  12 Jan 2023  In terms of beam tracking, authors in [12] applied the  correlation of spatial mmWave channel in adjacent locations  and proposed the beam steering method based on searching  over a small angular space in the vicinity of the previously  known valid beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The authors in [6] applied machine  learning to the tracking procedure to extract useful information  from the history of AoD tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  All the aforementioned works only take handover or beam  tracking issues into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Additionally, they do not study  the impact of selecting beam tracking and handover on the  achieved throughput of the UE along its trajectory and instead  focus on the achieved rate as the primary performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Our Contributions  In this paper, we develop a novel joint handover and beam  tracking algorithm in a mmWave network under mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The  algorithm aims to associate the UEs to BSs that maximize  the sum achieved throughput along the trajectory and ensure  the achieved throughput in each location of the trajectory  is higher than a pre-defined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The user association  process is defined as the process of determining whether a user  is associated with a particular BS before data transmissions  commence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In the case of handover, the UE is associated with a  new BS, whereas in the case of beam tracking, the UE remains  associated with the serving BS from the previous time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The  main contributions of our paper are summarized as below:  System Modeling: We model the user association prob-  lem as a non-convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Unlike the  existing works in the literature, we consider achieved  throughput as the main performance metric to measure  the effect of handover or beam tracking on the UEs’  quality of service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Learning-based Solution: The objective function in our  proposed user association problem highly depends on the  user association mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We utilize the reinforcement  learning (RL) algorithm to approximate the solution to  this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The aim is to decide whether to run a beam  tracking algorithm or a handover algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Joint Handover and Beam Tracking Algorithm: In the  case of a handover decision, the target BS will be  recognized as the output of the RL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In the  case of beam tracking, the search space will be defined  based on our proposed tracking algorithm by searching  the directions in the small spatial neighbourhood of the  previously selected optimal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Empirical Evaluation: We apply ray tracing with a  real building data map as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The results show  the effectiveness of our proposed method in achieving  throughput along trajectories and decreasing the number  of handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We introduce  the system model and problem formulation in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In  Section III, we propose our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We present the numerical  results in Section IV and, conclude our work in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Notations: Throughout the paper, vectors and scalars are  shown by bold lower-case (x) and non-bold (x) letters, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The conjugate transpose of a vector x is represented by  xH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We define set [M] := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='., M} for any integer M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The  indicator function 1{·} equals to one if the constraint inside  {·} is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' SYSTEM MODEL AND PROBLEM FORMULATION  In this section, first, we introduce the mmWave channel  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Then, we present the user association problem formu-  lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  We consider a downlink communication with |B| mmWave  BSs, where each is equipped with NBS antennas, communi-  cating with a single antenna mobile UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We consider analog  beamforming with a single RF chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We assume all BSs  allocate equal resources to their serving UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The channel  between BS j ∈ B and its serving UE during time slot i is  [13]:  hj =  L  �  ℓ=1  hℓaH(φℓ, θℓ),  (1)  where L is the number of available paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Each path ℓ has  complex gain hℓ (include path-loss) and horizontal φℓ and  vertical θℓ, AoD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Due to the notation simplicity, we drop the  index j and i from the channel parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The array response  vector is a(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=') where its exact expression depends on the array  geometry and possible hardware impairments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The signal-to-  noise ratio (SNR) in time slot i is  SNR(i)  j  = p|hH  j fj|2  σ2  ,  (2)  where σ2 is the noise power, p is the transmit power, fj ∈ CNBS  is the beamforming vector of BS j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  We define variable x(i)  j  ∈ {0, 1} for j ∈ B as an association  indicator in time slot i, where is equal 1 if UE is associated to  the BS j and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Hence, the achieved rate per second  per hertz in time slot i is  R(i) = x(i)  jS log2(1 + SNR(i)  jS ) =  �  j∈B  x(i)  j log2(1 + SNR(i)  j ),  where jS is the index of the serving BS of the UE during time  slot i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Here, we assume each UE is served by only one BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  We define the achievable throughput per hertz of the UE by  multiplying its rate by the data transmission time as  Γ(i) = (1 − τ (i)  b  τc  )R(i),  (3)  where, τ (i)  b  is the beam training duration which may have a  different value in each time slot i, and τc is the duration of  the time slot that is a fixed value for all time slots, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Beam Training and Beam Tracking  As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' 1a, when the UE is connected to a  BS j ∈ B, initial beam training is performed by sending  pilots over all combination of the beam directions in the  codebook during τb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' on the UE’s feedback of the  received signal strength (or estimated SNR), the best beam pair  directions are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Then, the BS and the UE would use this  Initial beam training  Data Transmission  τb  τc  (a)  Tracking  Data Transmission  τb  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' 1: τc is the time slot duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' τb is (a) the initial beam  training duration when the UE is associated with the new  BS (handover case), (b) the beam tacking duration when the  serving BS is the same for the consecutive slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  direction (φℓ⋆, θℓ⋆) during the data transmission phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The  beamforming vector, f is chosen to maximize the achievable  rate of the UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Due to the monotonicity of the logarithm  function, this is equivalent to maximising the SNR term in  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Hence  f ∗  j = arg max  fj∈F  |hH  j fj|2  (4)  where F is the beamforming codebook that contains all  the feasible beamforming vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The n-th element of the  codebook F is defined as f(n) = a(φn, θn), where (φn, θn)  are steering angles and a(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=') is the array response vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  When the BS continues serving the same UE in a consecu-  tive time slot, only searching the neighbouring beam directions  of the main directions can be sufficient to maintain the link  quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' This process is called beam tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' As shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' 1b, the duration of τb is much smaller than the initial  beam training duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Problem Formulation  The UE association depends on the channel quality between  the BS and the UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Due to UE mobility or temporary  blockage, the channel quality changes and consequently the  UE association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Based on the UEs’ velocity, we determine how  quickly the channel quality can change and predict the time  at which the current UE association needs to be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We  define TA seconds as the frequency of updating the association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Hence, we need to make the decision every TA whether to run  the handover execution or beam tracking procedure if SNR is  lower than the pre-defined SNR threshold (SNRthr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Note that  we can have an on-demand reactive handover at any time slot  if the link toward the serving BS fails abruptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' However, with  a proper choice of TA, the frequency of those reactive events  could be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We define the duration of the trajectory  as M and consider the discrete time index i to describe the  association update at each interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  The goal is to maximize the aggregate throughput of the UE  along the trajectory while ensuring the achieved throughput in  each time slot i is higher than a predefined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' To this  end, we define functions F1 and F2 as  F1 is the averaged throughput along the trajectory as  F1 =  M  �  i=1  E  �  Γ(i)�  ,  where the expectation is with respect to the randomness  of channel fading and the blockage, M is the duration of  the trajectory, and Γ(i) is defined in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  F2 is the expected number of time slots whose throughput  is lower than the threshold (Γthr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  F2 = E  � M  �  i=1  1  �  Γ(i) ≤ Γthr  ��  =  M  �  i=1  Pr  �  Γ(i) ≤ Γthr  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  We formulate the user association at time slot i ∈ [M]  as an optimization problem which involves finding the x(i)  j  corresponding to the association indicator as  max  {x(i)  j  }i,j  F1 − λF2  (5a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  �  j∈B  x(i)  j  = 1, ∀, i ∈ [M]  (5b)  x(i)  j  ∈ {0, 1},  ∀j ∈ B, i ∈ [M]  (5c)  where λ is a large constant controlling the importance of F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Constraint (5b) guarantees that each UE is served by one BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  The optimization problem (5) is nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Solving this  optimization problem requires estimating the expectation value  in F1 and F2 which requires running many realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Moreover, the impact of choosing the x(i)  j  (the target BSs  in the handover case or choosing beam tracking procedure)  propagates in time and can affect the UEs’ performance in  the next time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Therefore, we need to consider the long-  term benefits of selecting association indicators besides their  immediate effects on the UEs’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Furthermore, In  order to select the target BSs, we need to model or predict the  UEs’ performance in the next time slots, which can add more  complexity to the network due to the mobility of the UE and  obstacles in mmWave networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' These motivate us to utilize  the RL to approximate the solution of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  PROPOSED METHOD  We transform the problem (5) to an RL problem in which  the objective function is turned into a reward function, and  the constraints are transformed into the feasible state and  action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In the following, first, we start with defining  the Markov decision process, and then we will describe our  joint handover and beam tracking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Markov Decision Process Formulation  RL problems are formulated based on the idea of the  Markov decision process (MDP), which is the agent’s interac-  tion with different states of the environment to maximize the  expected long-term reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The agent is the main decision-  maker who can sit on the edge cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' All BSs are connected  to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Now, we define different elements of an MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  1) State Space: The state space describes the environ-  ment by which the agent is interacting through different  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We define the state at time slot i as s(i)  =  (ℓ(i)), j(i)  S , SNR(i), I(i)) ∈ S, where ℓ(i) is the location index  of the UE along the trajectory 2, j(i)  S is the index of the serving  BS, SNR(i) is the SNR value of the UE with serving BS j(i)  S  in time slot i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' I(i) ∈ {0, 1} is the beam tracking activation  indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' I(i) = 1 means the i-th time slot is the tracking slot  for the UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  2) Action Space: The action space includes all possible  actions that can be taken by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The action can change  the state of the environment from the current state to the target  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In our problem, a(i) ∈ A = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=', [|B|]} is the  decision regarding beam tracking (a(i) = 0) or choosing the  index of new serving BS in the case of handover decision  (a(i) ∈ [|B|]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In other words, if a(i) ̸= 0 means the handover  decision is made and the value of a(i) shows the target BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Hence, the action is to specify a serving BS for the UE along  its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  3) Policy: A policy π(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=') maps the state of the environment  to the action of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In our case, π is a function from S  to A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=', π : S → {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=', [|B|]}  4) Rewards: The agent obtains the reward after taking an  action a(i) when current state is s(i) and moves to next state  s(i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Here we define reward r(s(i), a(i), s(i+1)) as  r(s(i), a(i), s(i+1)) = Γ(i) − λ1  �  Γ(i) ≤ Γthr  �  ,  (6)  where Γ(i) is defined in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  5) State-action value: The function Qπ(s, a) is the long-  term reward and is defined as the expected summation of  discounted reward in the future for the action a ∈ A that  agent takes in state s under policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The RL algorithm aims  to choose the optimal policy π⋆ in each state s that maximizes  the Qπ(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' With discount factor η ∈ [0, 1], we have  Qπ(s, a) = E  ��  i  ηir(s(i), s(i), s(i+1))  �  ,  where the expectation is over the transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In  our problem, transition probabilities model the SNR variations  due to the randomness of the channel fading and blockage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  We assume mobility information including the UEs’ current  location and its trajectory is known3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Therefore, the transition  to the next location is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  The optimal policy in state s ∈ S is found by  π⋆(s) = arg max  a∈A  Qπ(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  (7)  Due to the continuous and large number of state spaces, we  apply deep Q-learning (DQL) [14] to solve (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In DQL, the  state-action value function is estimated by the deep neural  network function approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  2Note that, we discretize the location of the UE along the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Hence,  every location dimension (x, y) a trajectory with length M is mapped to a  location index ℓ(i) ∈ [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  3Note that the location information can be easily fed back through lower-  frequency links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Joint Handover and Beam Tracking Algorithm  Algorithm 1 describes our proposed joint handover and  beam tracking algorithm along a trajectory with duration M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  If the current association cannot offer the required SNR level,  the decision regarding handover or beam track is made based  on a(i) as the output of the RL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In the case of the  handover decision, the value of a(i) represents the target BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  The beam tracking algorithm based on small spatial mea-  surement in time slot i is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In slot i, the  algorithm starts by using the main beam of the same serving  BS in the previous time slot i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' If the SNR value is lower  than the threshold, then starts a small spatial measurement over  the AoD direction of the main beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' To quantify the size of the  spatial neighbourhood, we define ∆φ and ∆θ as the maximum  absolute horizontal and vertical deviation from the main AoD  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We define δφ and δθ as the measurement resolution  in horizontal and vertical, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Inspired by [15], the  spatial neighbourhood N surrounding the main AoD direction  can be expressed using the horizontal neighbourhood Nφ and  vertical neighbourhood Nθ as  Nφ(∆φ, δφ) =  �  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='δφ : i ∈  �  −  �∆φ  δφ  �  ,  �∆φ  δφ  ���  (8)  Nθ(∆θ, δθ) =  �  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='δθ : j ∈  �  −  �∆θ  δθ  �  ,  �∆θ  δθ  ���  (9)  where ⌊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='⌋ is the floor operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The complete neighbourhood  is the Cartesian product of the horizontal and vertical neigh-  bourhoods as  N(∆φ, ∆θ, δφ, δθ) = Nφ(∆φ, δφ) × Nθ(∆θ, δθ)  = {(φ, θ) : φ ∈ Nφ(∆φ, δφ), θ ∈ Nθ(∆θ, δθ)}(10)  The spatial neighborhoods T (i) in time slot i surrounding the  main AoD directions (φ(i−1)  ℓ⋆  , θ(i−1)  ℓ⋆  ) in previous time slot is  T (i) = (φ(i−1)  ℓ⋆  , θ(i−1)  ℓ⋆  , ) + N(∆φ, ∆θ, δφ, δθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  (11)  Now given the main AoD direction, we need to find the  transmit direction from neighbourhoods T (i) that offers the  SNR threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We represent the sorted direction pairs as  [T (i)]I, where I is the sorted indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' It means the directions  in [T (i)]I increase in distance from the main AoD direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Starting from the main AoD direction, the SNR of each trans-  mit direction in [T (i)]I is measured until a beam pair meets  the required SNR level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Afterwards, no further measurements  are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' If no direction meets the threshold, the entire  (∆φ, ∆θ)-neighbourhood is measured to find the beam pairs  that offer the SNR threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Note that in the worse scenario, if the selected target BS  based on our proposed algorithm cannot offer the required  SNR level due to very sudden blockage, the conventional  handover methods based on searching over the candidate BSs  in UEs vicinity can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' However, as shown in the  numerical results, such extreme case is rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Algorithm 1 Joint handover and beam tracking  Input: Trajectory with duration M  1: Initialization: for i = 1 set j(1)  S =1  2: for i ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=', M do  3:  if SNR(i)  jS < SNRthr then  4:  Choose the optimal action a(i) based on current  s(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  5:  if a(i) ̸= 0 then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  ▷ handover execution  6:  Set j(i)  S  = a(i) and run the initial beam training  process and compute the achieved throughput Γ(i) as (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  7:  else  8:  Run Algorithm 2 and compute Γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  9:  end if  10:  end if  11: end for  Output: Γ(i)  Algorithm 2 Beam tracking in time slot i at the BS j  Input: [T (i)]I, SNRthr, duration of each beam pair testing (β),  cnt(i) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  1: for (φ, θ) ∈ [T ]I do  2:  Set f (i)  j  = a(φ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  3:  Measure SNR(i)  j  as (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  4:  Set cnt(i) = cnt(i) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  ▷ number of beam pair  testing  5:  if SNR(i)  j  >= SNRthr then  6:  (φ(i)  ℓ⋆ , θ(i)  ℓ⋆ ) = (φBS, θBS)  7:  τ (i)  b  = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='cnt(i)  8:  break;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  9:  end if  10: end for  compute the achieved throughput Γ(i) as (3)  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' NUMERICAL RESULTS  We evaluate the performance of the proposed method in an  urban environment using the ray tracing tool in the MATLAB  toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The output of the ray tracing tool is the L available  paths between a BS and a UE in a specific location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The ray  tracing maintains the spatial consistency of mmWave channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' 2, we extracted the building map of Kista  in Stockholm city, Sweden and used it as the input data for  the ray tracing simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In our scenario, we assumed the  building material is brick and the terrain material is concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  We also add some random obstacles in the street with different  heights (1 m and 3 m) and widths (2 m and 4 m) as the human  bodies and various vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' These temporary obstacles are  distributed randomly in the street with density 10−2 per m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  The material loss and the location of the temporary obstacles  are chosen randomly in each realization of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The  BSs are located on the wall of buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The location of the  BSs is chosen randomly while covering the entire trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  The BSs’ height is 6 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We consider a pedestrian mobility  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' 2: Simulation area in Kista, Stockholm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The yellow line  shows the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Stars show the location of the BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  model with a speed of 1 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We consider the different lengths  of the trajectories as 100TA, 200TA, 300TA, 400TA, 500TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  The main simulation parameters are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  In the simulation, we consider the SNRthr = 2 dB and the  throughput threshold Γthr = 1 bit/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The value of τc is 10 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  In the case of handover, we fix the initial beam training dura-  tion as τb = 1  3τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In the case of beam tracking, τb is not fixed  and equals the size of measuring neighbourhood multiplied  by the duration of each beam pair testing (β = 10 µs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We  compare the performance of our proposed method with two  baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' To have a fair comparison, we choose two baselines  in which the target BS for the handover is pre-determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Hence, we do not take into account the discovery time of  finding the target BS in the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Just like in our method,  the handover is triggered if SNR < SNRthr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  As Baseline 1 we consider the multi-connectivity method  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We implement a scenario where the UE maintains its  connection with a nearby BS as a backup solution while  being connected to the serving BS and once it experiences the  blockage of the serving link, starts connecting to the backup  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' As Baseline 2 we select the learning-based handover  in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The method shows very good performance in maxi-  mizing the achieved rate along the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In this baseline,  the target BS during the handover process is determined by a  learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Although the target BSs are selected based  on the long-term effect on the achieved rate, still can cause  frequent handovers and throughput degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  First, we fix the number of BSs to 10 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' We  consider 104 different channel realization as the input of the  RL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' After getting the optimal policy, we test it  over real-time measurements and report the average of the  performance over 500 channel realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' 3 shows the  average number of locations with unmet throughput thresholds  along the trajectory with different lengths and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' 4 shows the  average number of handovers needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' In comparison to the  other two baselines, our method provides better throughput  results by selecting to perform either beam tracking or a  handover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Furthermore, we note that the two baselines have  a higher number of handovers than our method due to only  considering the handover solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Hence, by considering the  joint handover and beam tracking problem our method pro-  vides better-achieved throughput while decreasing the number  of handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' 5 shows the average aggregate achieved  Table I: Simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  Parameters  Values in Simulations  BS transmit power  10 dBm  Noise power level  σ2=-174 dBm/Hz  Signal bandwidth  100 MHz  BS antenna  8 × 8 uniform planar array [11]  Time interval duration  TA = 1s  Neighborhood size  (∆φ, ∆θ) = (10◦, 10◦)  Measurement resolution  (δφ, δθ) = (5◦, 5◦)  Discount factor  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='99  λ  100  100  200  300  400  500  0  200  400  Trajectory length (m)  Number of locations satisfying Γthr  Our method  Baseline 1  Baseline 2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' 3: The average number of locations with unmet through-  put threshold for different lengths of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  throughput along the trajectory with length 300 m for different  numbers of BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' By increasing the number of BSs the number  of the locations satisfying the Γthr also increases hence the  aggregate throughput along the trajectory increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Even with  a small number of BSs, our method outperforms baselines  in aggregate throughput along the trajectory by determining  whether to use a handover or beam tracking solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  We consider 10000 iterations during the training in our  method and Baseline 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' With the training machine MacBook  Pro 2020 M1 with a memory of 16 GB, each iteration takes  about 15 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Note that the absolute value of the training  time per iteration depends on the running machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' CONCLUSIONS  In this work, we proposed and studied a learning-based joint  handover and beam tracking method in a mobile mmWave  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' The aim of our algorithm is to maximize the aggre-  gate throughput of the UE along a trajectory and ensure the  achieved throughput in each location is higher than the thresh-  old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Our evaluation results showed that by making an optimal  decision regarding handover execution or beam tracking, our  method provides high achievable throughput and reduces the  number of handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
+page_content=' Considering different mobility models  and studying the effect of neighbouring size can be valuable  future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'}
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+Learning to compile smartly for program size reduction
+Youwei Liang∗ 1 Kevin Stone∗ 2 Ali Shameli 2 Chris Cummins 2 Mostafa Elhoushi 2 Jiadong Guo 2
+Benoit Steiner 2 Pengtao Xie 1 Hugh Leather 2 Yuandong Tian 2
+Abstract
+Compiler optimization passes are an important
+tool for improving program efficiency and reduc-
+ing program size, but manually selecting opti-
+mization passes can be time-consuming and error-
+prone. While human experts have identified a few
+fixed sequences of optimization passes (e.g., the
+Clang -Oz passes) that perform well for a wide
+variety of programs, these sequences are not con-
+ditioned on specific programs. In this paper, we
+propose a novel approach that learns a policy to
+select passes for program size reduction, allow-
+ing for customization and adaptation to specific
+programs. Our approach uses a search mecha-
+nism that helps identify useful pass sequences
+and a GNN with customized attention that se-
+lects the optimal sequence to use. Crucially it
+is able to generalize to new, unseen programs,
+making it more flexible and general than previ-
+ous approaches. We evaluate our approach on a
+range of programs and show that it leads to size
+reduction compared to traditional optimization
+techniques. Our results demonstrate the potential
+of a single policy that is able to optimize many
+programs.
+1. Introduction
+Finding the right compiler optimization ordering for a given
+program in order to execute them more efficiently with a
+smaller amount of resources (e.g., memory, CPU and stor-
+age) is an important yet challenging problem. Traditionally,
+to tune configurations, human effort and expert knowledge
+are needed, which is a time-consuming and error-prone
+process that often yields sub-par results.
+In recent years, machine learning-guided compiler optimiza-
+tion has emerged as an interesting field to replace this labori-
+ous process (Wang & O’Boyle, 2018). Along this line, many
+*Equal contribution
+1University of California, San Diego
+2Meta AI. Correspondence to:
+Kevin Stone
+<kevinlee-
+[email protected]>.
+works show promising results using various machine learn-
+ing (ML) techniques and optimizers (e.g., reinforcement
+learning (Haj-Ali et al., 2020a), language modelling (Cum-
+mins et al., 2017), evolutionary algorithms (Kulkarni &
+Cavazos, 2012), etc). They focus on automatic decision-
+making of specific components in compilation, e.g., op-
+timizing computational graph of neural network for ML
+compilers (Zhou et al., 2020), optimizing the loop structure
+of neural network computations (Steiner et al., 2021), deter-
+mination of a function to be inlining (MLGO) (Trofin et al.,
+2021), etc. There are only a few works targeting generic
+optimization of compilation (Haj-Ali et al., 2020b; Almakki
+et al., 2022).
+In this work, we take a different path and explore the pos-
+sibility of learning policies for more a general aspect of
+compiler optimization. More specifically, we treat compila-
+tion optimization as a general sequential decision process,
+in which each step is to find an additional compiler pass
+so that when combined with existing passes, can improve
+specific metrics (e.g., the binary size / running speed of the
+codebase). Our goal is to learn a policy, named pass policy,
+to help quickly find the right compiler passes to be used,
+given the current program and existing passes. We aim to
+compare with existing handtuned policies in the compiler,
+e.g., -Oz for code size reduction and “-O3” for running
+speed optimization. Such policies have been tuned by the
+domain experts for decades and used extensively in all the
+computer systems, but are invariant to the program being
+compiled.
+As the first contribution, we propose a novel evaluation
+protocol for compiler pass optimization, named zero-shot
+generalizability (ZSG), to evaluate learned policies applied
+to unseen programs. Here “zero-shot” means that each
+unseen program has never been seen before in the training
+process, and we allow a fixed number of optimization passes
+(say 45). The policy is allowed to adjust the passes based
+on previous outcomes (This includes backtracking if an
+exploratory step turns out to be sub-optimal.) Compared
+to previous works (e.g., Autophase (Haj-Ali et al., 2020b),
+CompilerGym (Cummins et al., 2022), MLGO (Trofin et al.,
+2021), MLGoPerf (Ashouri et al., 2022)) that run adaptive
+search algorithms to optimize a set of programs for many
+hours, we argue that ZSG is a realistic setting for practical
+arXiv:2301.05104v1  [cs.PL]  9 Jan 2023
+
+Learning to compile smartly for program size reduction
+deployment: for most programs except the critical ones,
+there is a limited time budget to optimize them, and often
+there is not enough computational resources to fine-tune
+learned policies for each unseen task separately (e.g., fine-
+tune setting in GO (Zhou et al., 2020)). While similar metric
+has been used in previous works (e.g., MLGO and GO), we
+are the first to test it in more general ML-guided compiler
+optimizations.
+With the ZSG metric, we then evaluate multiple existing
+techniques applied to compiler optimization, with LLVM
+compiler and on large-scale datasets provided by Compiler-
+Gym (Cummins et al., 2022). We use code-size reduction
+as the metric since it is a deterministic quality and relatively
+easy/cheap to compute, and we leave program run-time
+optimization for future work.
+Given the metric, we discover a universal core subset of
+LLVM pass sequences, of size 50, that leads to very strong
+performance across multiple sets of programs from diverse
+domains, ranging from Linux Kernel to BLAS library. The
+total number of steps in the 50 pass sequences is 625 and
+the average length of the sequences is 12.5. Specifically,
+for one unseen program, there exists one of the 50 pass
+sequences that leads to an average code size reduction of
+5.8% compared to the default -Oz setting, across 10 diverse
+codebases of over one million programs. In other words,
+after running trying the 625 steps on an unseen program, the
+smallest code size during the compilation is 5.8% smaller
+than that of -Oz. Considering the huge search space of com-
+piler flags (10104), this is a very surprising finding. We find
+this coreset in a training set and it can generalize reasonably
+well to the evaluation set including datasets not including in
+the training set.
+While it can be time-consuming to find such a compiler
+flag configuration with an exhaustive enumeration of the
+core subset, we find that the optimal pass can be directly
+predicted with high accuracy via GNN with a customized
+attention layer to control information flow with the Pro-
+GraML (Cummins et al., 2021) graph as the input. The
+prediction top-1 and top-5 accuracy is 75% and 95%. There-
+fore, we can run a few sequences selected by the model on
+an unseen program to obtain a good code size reduction.
+This enables us to find a good flag configuration that leads
+to 4% improvement on average, with just 45 compilation
+passes (this is roughly 3 sequences since the average length
+of the sequences in the coreset is 12.5).
+We have compared our approach with extensive baselines,
+including RL-based methods such as PPO and Q-learning
+and black-box optimizers such as evolutionary algorithm.
+It turns out that ML approaches often suffer from severe
+overfitting to the training set and does not perform well
+for new program categories, even combined with heuristic
+search, and optimizers may get lost in the exponential action
+space due to lack of domain knowledge. In comparison, our
+approach is simple, effective and generalizable to unseen
+programs.
+2. Related Work
+Graph structured data are present in numerous applications
+and it has been shown that taking advantage of this data
+can help us train very effective machine learning models.
+(Brauckmann et al., 2020) use abstract syntax trees and con-
+trol flow graphs for learning compiler optimization goals.
+They show that using such graphs allows them to outper-
+form state-of-the-art in the task of heterogeneous OpenCL
+mapping. (Guo et al., 2020) uses a transformer based model
+with a graph guided masked attention that incorporates the
+data flow graph into the training. They achieve state of the
+art performance in four tasks including code search, clone
+detection, code translation, and code refinement.
+As a contender to graph neural networks, (Mialon et al.,
+2021) uses transformers to process graphs. They show that
+if we effectively encode positional and local sub-structures
+of graphs and feed them to the transformer, then the trans-
+former can outperform the classical GNN models. They
+test their model on classification and regression tasks and
+achieve state of the art performance.
+In (Srinivas et al.,
+2020), they used an unsupervised model to learn embed-
+dings of high dimensional pixel data using contrastive learn-
+ing. They then use this embedding for downstream rein-
+forcement learning tasks.
+3. Methodology
+3.1. Action space
+The CompilerGym framework provides a convenient in-
+terface for the compiler pass ordering problem. The de-
+fault environment allows choosing one of 124 discrete ac-
+tions at each step corresponding to running a sequence of
+specific compiler pass. Given that our trajectories have a
+length of 45 steps, this means we have 12445 ∼ 1.6 × 1094
+possible action trajectories to explore. To find an opti-
+mal action sequence for a program, we can apply some
+existing reinforcement learning methods including Q learn-
+ing like DQN (Mnih et al., 2015) and policy gradient like
+PPO (Schulman et al., 2017).
+Action Sequences However for this problem it turns out
+that certain action sequences are good at optimizing many
+different programs (where “good” is defined as better than
+the compiler default -Oz). We found that constraining the
+action space to a learned set of action sequences enables
+state of the performance and also significantly reduces the
+challenge of exploration. This allows us to cast the prob-
+lem as one of supervised learning over this set of action
+
+Learning to compile smartly for program size reduction
+Figure 1. An example of a small action tree created by keeping
+track of the common prefixes of the action sequences.
+sequences. We use the following algorithm to find a good
+set of action sequences.
+• Random search We seed the list of candidate action
+trajectories (action sequences) by running a random
+policy on a subset of the training programs (N). For
+each program we run M episodes and keep track of
+the best action sequence for each program.
+• Canonicalize During the search process we find that
+trajectories often revisit the same state. Whenever this
+happens we truncate all previous actions. On average
+this reduces the trajectory length by a factor of 1/5.
+• All-to-all We then test all N best action sequences
+against all N programs. This gives us a N×N matrix
+where each value is the cumulative reward (return). We
+then normalize this matrix by the maximum return for
+each program. A value of 1 represents that an action
+sequences was optimal for a program (optimal in the
+sense of this limited set of action sequences).
+• Greedy assignment Many action sequences are opti-
+mal for more than one program. We take advantage
+of this to reduce the number of action sequences by
+greedily picking action sequences that are optimal for
+the the largest number of programs. This results in a
+much smaller list of action sequences. We can visual-
+ize this by creating a prefix tree that shows common
+prefixes as single node. See Figure 1 for truncated tree
+for illustration.
+It is interesting to note that some actions are common across
+the beginning of multiple action sequences. These popular
+actions such as 27 (-reg2mem) as seen in Figure 1 are
+pivotal for many programs. For a complete list of all LLVM
+passes refer to Table 5.
+3.2. Offline behavior cloning methods
+Normalized Value Prediction After discovering the “good”
+action sequences (i.e., the coreset), we can turn the problem
+of the sequential decision-making on compiler passes into a
+problem of supervised classification. The target is to train
+a model to predict the best action sequence conditioned on
+the program, where the label of the program is the index
+of the action sequence that results in the highest code size
+reduction. However, one important observation we have
+is that there are typically multiple action sequences in the
+coreset that all result in the highest code size reduction.
+Therefore, instead of using the single-class classification
+method with cross entropy loss, we leverage the fact we have
+access to the values for all action sequences. We predict
+the softmax normalized value of each action sequence with
+a cross entropy loss detailed below. We call this approach
+behavior cloning (BC) over the coreset.
+For a program o, we roll out all the predefined action se-
+quences on it, obtaining a return ro
+i for the i-th sequence
+(i.e., the highest cumulative reward observed during the
+rollout of the action sequence), which forms a value vec-
+tor ro = [ro
+1, . . . , ro
+n]. Then, the normalized values of the
+action sequences are defined by
+vo = Softmax(ro/T)
+(1)
+where T is temperature.
+For an initial observation so
+0 of a program, our model out-
+puts a probability distribution, p = f(so
+0), over the action
+sequences. The target of the training is to make p close to
+the normalized values of the action sequences. We use the
+cross entropy loss to supervise the model
+L(po, vo) = −
+n
+�
+i
+po
+i log vo
+i
+(2)
+3.3. Program Representation
+We are considering LLVM optimization passes on that op-
+erate on the Intermediate Representation (IR) of a program.
+The IR contains very rich structures, while its size can be
+quite large for large programs. It also contains informa-
+tion (e.g., strings and constants) irrelevant to the task we
+are considering. We found that working with compressed
+representations made this problem tractable and run in a
+reasonable amount of time.
+ProGraML We leverage a graph based representation that
+encodes semantic information of the program covering three
+layers: control flow, data flow, and data types. This rep-
+resentation has the advantage that it is not a fixed size - it
+does oversimplify large programs - and yet it is still a more
+compact format than the original IR format.
+3.4. Network Architecture
+One of the ways to model our policy function is to use a
+graph neural network (GNN). To achieve this goal, we use
+
+53,122,31,36,111,10,97
+10
+64,31,10,52,111,116,36,40,48,54,30,53,114,29,120,10
+36,103,24,53,97,53,38,69,97,57,10,29
+39,64,55,53,38,122,31,111,64,10,39,21,105,36
+27
+104,55,57,26,103,10,29,31,36,120,102,53
+root 
+29
+55,39,61,27,41,36,25,103,10
+30,48,29,120,103,96,47,29,78,21,122,41,36,10
+72,55,103,36,122,59,30,65,53,10
+103,102,30,36,61,29,41,71,10,61,41,52Learning to compile smartly for program size reduction
+1
+2
+5
+3
+𝑋!"
+#
+𝑋!$
+#
+𝑋!%
+#
+𝑋"&
+#
+𝑋"!
+𝑋%!
+𝑋$!
+𝑋&"
+4
+Figure 2. Graph attention. Circles denote nodes and solid arrows
+denote edges. Squares are the calculated features, and dash arrows
+represent feature aggregation. The orange/green squares denote the
+features to be aggregated in the target/source nodes of the edges.
+the ProGraML graph structure proposed in (Cummins et al.,
+2021) in which individual statements are connected to other
+statements through relational dependencies. Compared to
+the approach of treating the program scripts as text and
+encoding the programs with a language model, using the
+ProGraML graph representations and encoding them with
+GNNs has several advantages. Firstly, the long-range de-
+pendencies of instructions are automatically captured by
+the edges in the graphs, whereas the NLP approaches need
+to use an LSTM or Transformer to capture the dependen-
+cies. An LSTM could lose early memory when the program
+is long, and a Transformer could cause out of memory is-
+sues in such cases. Secoundly, changing the names of the
+variables/constants/functions/classes in a program will not
+affect its ProGraML representation but will change the rep-
+resentations in texts. Our goal is to use the structure and
+relational dependencies of this graph to learn an embed-
+ding which allows us to learn a better policy. We experi-
+mented with several different architectures such as Gated
+Graph Convolutions (Li et al., 2015), Graph Attention Net-
+works (Veliˇckovi´c et al., 2017), as well as our own custom
+variation described bellow.
+Edges types ProGraML supports multiple directed edge
+types representing control flow, data flow, function call, and
+data type definition. The other edge features include the
+integer position of an edge among all the edges pointing
+to the same node, a boolean feature indicating whether the
+Notation
+Meaning
+E
+The set of edges in the graph
+(i, j)
+Edge from node i pointing to node j
+X(t)
+i
+Repr. of node i at layer t
+E(t)
+i→j
+Repr. of the edge (i, j) at layer t
+X(t)
+ij
+Repr. for node i associated with edge (i, j)
+X′(t)
+ij
+Repr. for node i associated with edge (j, i)
+a(t)
+ij
+Raw attention associated with repr. X(t)
+ij
+a′(t)
+ij
+Raw attention associated with repr. X′(t)
+ij
+α(t)
+ij
+Normalized attention associated with a(t)
+ij
+α′(t)
+ij
+Normalized attention associated with a′(t)
+ij
+Ti
+Target neighbors of node i: {j|(i, j) ∈ E}
+Si
+Source neighbors of node i: {j|(j, i) ∈ E}
+Table 1. The notations in GNN (“Repr.” means representation)
+two nodes connected by the edge are in the same LLVM
+basic block, and the distance of the two nodes if they are
+in the same basic block. Equipped with the rich features of
+the edges in ProGraML graphs, we propose a dynamic edge
+encoding approach to capture the edge representations.
+Dynamic edge representation Most existing GNNs that
+exploit the edge feature basically use a static edge feature,
+which means the same edge feature is repeatedly used for
+all layers. It turns out that it is important to use a dynamic
+edge representation during graph encoding, where the edge
+representation gets updated in each GNN layer. The initial
+edge representations are the concatenations of the embed-
+ding of edge types, edge positions, and other edge features
+discussed in the last paragraph.
+Attention with edge features It was also helpful to modify
+the default attention mechanism to support these custom
+edge types. We propose to incorporate the edge features
+by encoding a triplet containing the features of two nodes
+and the feature of the edge that connects them. For clar-
+ity, we show a table containing the notations used in the
+GNN in Table 1. Then, the feature update process can be
+mathematically defined by the following equations, where
+Mi, i = 1, . . . , 5 is an encoding MLP.
+X′(t+1)
+ij
+= M1(X(t)
+i , E(t)
+i→j, X(t)
+j ),
+(3)
+a′(t+1)
+ij
+= M2(X(t)
+i , E(t)
+i→j, X(t)
+j ),
+(4)
+X(t+1)
+ji
+= M3(X(t)
+i , E(t)
+i→j, X(t)
+j ),
+(5)
+a(t+1)
+ji
+= M4(X(t)
+i , E(t)
+i→j, X(t)
+j ),
+(6)
+E(t+1)
+i→j
+= M5(X(t)
+i , E(t)
+i→j, X(t)
+j ),
+(7)
+In words, the 3-tuple, (X(t)
+i , E(t)
+i→j, X(t)
+j ), associated with
+
+Learning to compile smartly for program size reduction
+edge (i, j), is encoded by MLPs to output 5 features, includ-
+ing X′(t+1)
+ij
+and a′(t+1)
+ij
+(a representation and attention to
+be aggregated in node i), and X(t+1)
+ji
+and a(t+1)
+ji
+(a repre-
+sentation and attention to be aggregated in node j), and the
+updated edge representation E(t+1)
+i→j . Note that the features
+to be aggregated to a target node are marked with the ′, and
+those to a source node are without the ′. After the feature
+encoding, we perform an attention-weighted neighborhood
+aggregation for each node, which can be mathematically
+described by the following equations.
+�
+{α(t+1)
+ij
+}j∈Ti ∪ {α′(t+1)
+ij
+}j∈Si
+�
+= Softmax
+�
+{a(t+1)
+ij
+}j∈Ti ∪ {a′(t+1)
+ij
+}j∈Si
+�
+(8)
+X(t+1)
+i
+=
+�
+j∈Ti
+α(t+1)
+ij
+X(t+1)
+ij
++
+�
+j∈Si
+α′(t+1)
+ij
+X′(t+1)
+ij
+(9)
+3.5. Dataset preparation
+Overfitting issues could happen if training is performed on
+a small subset of programs, or the set of programs is not
+diverse enough. To mitigate this we found it helpful to
+create an aggregate dataset that uses many different public
+datasets as curated by CompilerGym. CompilerGym gives
+us access to 14 different datasets constructed using two
+different methods.
+• Curated These are small collections of hand-picked
+programs. They are curated to be distinct from one
+another, so splitting curated suites can be challenging.
+Typically programs are larger as they may comprise
+multiple source files combined into a single program.
+These are commonly used for evaluating compiler op-
+timization improvements.
+• Uncurated These are comprised of individual source
+files scraped from open source repositories such as
+Linux, Tensorflow, or synthetically generated pro-
+grams, normally targeted for compiler testing (not op-
+timization). They may not be as ”representative” of
+human written test programs.
+For our aggregate dataset we decided to hold-out the entirety
+of the four curated datasets for use as an out-of-domain test
+set. This is important because they represent the types of
+programs we expect to see in the wild. We also split the
+uncurated datasets into train, validaton, and test programs.
+3.6. Evaluation
+For all our metrics and rewards we leverage the IR instruc-
+tion count as value we are trying to minimize. We also
+report metrics on each CompilerGym dataset as well as
+Type
+Dataset
+Splits
+Uncurated
+anghabench-v1
+train,val,test
+blas-v0
+train,val,test
+github-v0
+train,val,test
+linux-v0
+train,val,test
+opencv-v0
+train,val,test
+poj104-v1
+train,val,test
+tensorflow-v0
+train,val,test
+clgen-v0
+train,val,test
+csmith-v0
+train,val,test
+llvm-stress-v0
+train,val,test
+Curated
+cbench-v1
+test
+chstone-v0
+test
+mibench-v1
+test
+npb-v0
+test
+Table 2. CompilerGym dataset types and training splits.
+the mean over datasets to get a single number to compare
+overall results.
+• The mean percent improved over -Oz (MeanOverOz)
+defined as following:
+MeanOverOz :=
+1
+|P|
+�
+p
+IOz
+p
+− Iπθ
+p
+IOz
+p
+,
+(10)
+where p is a specific program from the set of programs
+P in the dataset. IOz
+p
+is the number of IR instructions
+in program after running the default compiler pass -Oz.
+Iπθ
+p
+is the number of IR instruction in the program after
+applying the policy under consideration. We can think
+of this as a simple average of the percent improvement
+over -Oz.
+• We
+also
+look
+compare
+the
+geometric
+mean
+(GMeanOverOz) of final size relative to -Oz.
+This metric is less sensitive to outliers and is used
+by (Cummins et al., 2022).
+GMeanOverOz :=
+��
+p
+IOz
+p
+Iπθ
+p
+�
+1
+|P|
+(11)
+4. Experiments
+4.1. Baselines
+• Oracle-All We consider a brute-force search over the
+action tree in order to find the best action sequence for
+a given program. This gives us an upper-bound of the
+downstream policy network. In this case the action tree
+has a total of 625 nodes.
+
+Learning to compile smartly for program size reduction
+• Oracle-Top-45 We also consider how well we would
+do if the oracle is only allowed to use the most popular
+action sequences but limited to 45 steps. We use 45
+steps because this is maximum allowed for our all other
+baselines and our proposed method.
+• Autophase-RL We start with the strong baseline of us-
+ing Autophase features. Autophase is a mapping from
+a program to fixed size feature vector of 54 dimensions.
+It contains integer counts of various program proper-
+ties such as number of instructions, maximum loop
+depth, etc. This is used in combined with a 2-layer
+MLP model and trained with the PPO algorithm. This
+is the approach presented in (Haj-Ali et al., 2020b).
+• Autophase-BC We also consider the performance of
+using the action sequences combined with a MLP
+model using Autophase features to isolate the contribu-
+tion of the GNN from the action sequences search.
+• GNN-RL We compare with using ProGraML features
+and our GNN model, but trained with the PPO algo-
+rithm. This helps motivate the reason for performing
+the search for action sequences in the first phase.
+4.2. Results
+In Table 3 we present the results of our experiments com-
+paring our proposed model GNN-BC as compared to the
+various baselines. The test programs were completely held
+out during both data-driven learning phases (action sequence
+search and model training).
+The results show that our model achieves strong perfor-
+mance over the prior method (Autophase-RL) proposed in
+(Haj-Ali et al., 2020b). Additionally we can see that both
+the GNN model and the action sequences were needed to
+achieve our final performance. See Figure 3 for a visualiza-
+tion of the improvement in program size over the 45 steps
+on some of the programs from the holdout set.
+The Oracle-All shows strong performance but requires a
+large number of interactions with the compiler. But, this
+shows that the action sequence search generalizes to new
+unseen programs. This is somewhat unsurprising given that
+the compilers built-in hand tuned pass list (-Oz) works
+reasonably well for most programs.
+The performance of Oracle-Top-45 by itself is weak show-
+ing that in order to achieve good results in a reasonable
+number of passes (45) we need to leverage a general pol-
+icy and search to select the most likely candidate action
+sequences to evaluate.
+Both RL baselines using the original action space of single
+passes Autophase-RL and GNN-RL performed poorly on
+the generalization task. We hypothesis that this is partly
+Method
+Test MeanOverOz
+Test GMeanOverOz
+Compiler (-Oz)
+0%
+0
+Autophase-RL
+-16.3%
+0.960
+Autophase-BC
+4.2%
+1.056
+GNN-RL
+-9.6%
+1.005
+GNN-BC
+4.7%
+1.062
+Oracle-Top-45
+-7.5%
+0.992
+Oracle-All
+5.8%
+1.075
+Table 3. Evaluation results on held out test set averaged over all
+datasets.
+due to the challenging exploration problem of the large
+search space (12445). This is also a very hard setting for RL
+because each program has its own cumulative reward upper
+bound (which is unknown even for the training set). This
+makes approximating the value function very difficult for
+the baseline RL methods.
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+
+Learning to compile smartly for program size reduction
+Figure 3. Program optimization example over many steps comparing the Autophase-RL (blue) approach with our GNN-BC (orange)
+approach. The dashed line represents the compiler default -Oz performance and higher is better.
+
+benchmark://cbench-v1/diikstra
+benchmark://cbench-v1/stringsearch
+1.5
+1.0
+0.5
+0.0
+benchmark://cbench-v1/blowifish
+benchmark://cbench-v1/stringsearch2
+1.5
+1.0
+0.5
+0.0
+benchmark://cbench-vl/gsort
+benchmark://cbench-v1/bitcount
+1.5
+1.0
+0.5
+0.0
+benchmark://cbench-y1/rindae!
+benchmark://cbench-v1/sha
+1.5
+1.0
+0.5
+0.0
+i
+0
+5
+10
+15
+20
+25
+30
+35
+40
+0
+5
+10
+15
+20
+25
+30
+35
+40Learning to compile smartly for program size reduction
+Dataset
+Oracle-All
+Oracle-Top-45
+Autophase-RL
+Autophase-BC
+GNN-RL
+GNN-BC
+anghabench-v1
+0.7%/1.011
+-1.0%/0.996
+-15.9%/0.974
+-0.1%/1.002
+-2.5%/0.988
+-0.0%/1.003
+blas-v0
+2.6%/1.028
+-0.4%/0.997
+-1.7%/0.984
+1.2%/1.013
+-1.2%/0.989
+2.4%/1.026
+cbench-v1
+3.5%/1.041
+-2.4%/0.984
+-10.1%/0.925
+1.5%/1.021
+2.4%/1.030
+2.2%/1.028
+chstone-v0
+9.3%/1.106
+1.2%/1.016
+1.3%/1.018
+7.0%/1.079
+6.4%/1.071
+8.8%/1.101
+clgen-v0
+5.4%/1.060
+3.1%/1.034
+-0.5%/0.998
+4.5%/1.050
+2.2%/1.024
+5.0%/1.056
+csmith-v0
+21.2%/1.320
+-96.3%/0.851
+-116.0%/0.954
+21.1%/1.318
+-125.4%/0.994
+21.1%/1.320
+github-v0
+1.0%/1.011
+0.2%/1.002
+0.1%/1.001
+0.9%/1.009
+0.1%/1.002
+0.9%/1.010
+linux-v0
+0.6%/1.007
+-0.4%/0.998
+-0.5%/0.997
+0.6%/1.006
+-0.9%/0.996
+0.6%/1.007
+llvm-stress-v0
+6.3%/1.087
+-18.9%/0.885
+-67.0%/0.731
+1.6%/1.040
+-22.0%/0.872
+2.1%/1.045
+mibench-v1
+1.7%/1.020
+0.0%/1.003
+-2.8%/0.976
+-0.4%/0.999
+0.6%/1.008
+-0.1%/1.003
+npb-v0
+9.8%/1.159
+5.7%/1.085
+0.9%/1.035
+6.0%/1.088
+4.8%/1.074
+5.5%/1.085
+opencv-v0
+5.2%/1.061
+1.0%/1.013
+0.5%/1.007
+4.5%/1.054
+0.7%/1.009
+4.8%/1.057
+poj104-v1
+7.8%/1.105
+3.9%/1.055
+-17.5%/0.876
+5.7%/1.075
+0.1%/1.014
+6.3%/1.082
+tensorflow-v0
+6.1%/1.077
+-0.2%/0.998
+0.2%/1.004
+5.1%/1.063
+0.8%/1.011
+5.9%/1.075
+Average
+5.8%/1.075
+-7.5%/0.992
+-16.3%/0.960
+4.2%/1.056
+-9.6%/1.005
+4.7%/1.062
+Table 4. Evaluation results on held out test set averaged over all datasets.
+
+Learning to compile smartly for program size reduction
+Index
+Flag
+Index
+Flag
+Index
+Flag
+0
+-add-discriminators
+42
+-globalsplit
+84
+-lower-expect
+1
+-adce
+43
+-guard-widening
+85
+-lower-guard-intrinsic
+2
+-aggressive-instcombine
+44
+-hotcoldsplit
+86
+-lowerinvoke
+3
+-alignment-from-assumptions
+45
+-ipconstprop
+87
+-lower-matrix-intrinsics
+4
+-always-inline
+46
+-ipsccp
+88
+-lowerswitch
+5
+-argpromotion
+47
+-indvars
+89
+-lower-widenable-condition
+6
+-attributor
+48
+-irce
+90
+-memcpyopt
+7
+-barrier
+49
+-infer-address-spaces
+91
+-mergefunc
+8
+-bdce
+50
+-inferattrs
+92
+-mergeicmps
+9
+-break-crit-edges
+51
+-inject-tli-mappings
+93
+-mldst-motion
+10
+-simplifycfg
+52
+-instsimplify
+94
+-sancov
+11
+-callsite-splitting
+53
+-instcombine
+95
+-name-anon-globals
+12
+-called-value-propagation
+54
+-instnamer
+96
+-nary-reassociate
+13
+-canonicalize-aliases
+55
+-jump-threading
+97
+-newgvn
+14
+-consthoist
+56
+-lcssa
+98
+-pgo-memop-opt
+15
+-constmerge
+57
+-licm
+99
+-partial-inliner
+16
+-constprop
+58
+-libcalls-shrinkwrap
+100
+-partially-inline-libcalls
+17
+-coro-cleanup
+59
+-load-store-vectorizer
+101
+-post-inline-ee-instrument
+18
+-coro-early
+60
+-loop-data-prefetch
+102
+-functionattrs
+19
+-coro-elide
+61
+-loop-deletion
+103
+-mem2reg
+20
+-coro-split
+62
+-loop-distribute
+104
+-prune-eh
+21
+-correlated-propagation
+63
+-loop-fusion
+105
+-reassociate
+22
+-cross-dso-cfi
+64
+-loop-guard-widening
+106
+-redundant-dbg-inst-elim
+23
+-deadargelim
+65
+-loop-idiom
+107
+-rpo-functionattrs
+24
+-dce
+66
+-loop-instsimplify
+108
+-rewrite-statepoints-for-gc
+25
+-die
+67
+-loop-interchange
+109
+-sccp
+26
+-dse
+68
+-loop-load-elim
+110
+-slp-vectorizer
+27
+-reg2mem
+69
+-loop-predication
+111
+-sroa
+28
+-div-rem-pairs
+70
+-loop-reroll
+112
+-scalarizer
+29
+-early-cse-memssa
+71
+-loop-rotate
+113
+-separate-const-offset-from-gep
+30
+-early-cse
+72
+-loop-simplifycfg
+114
+-simple-loop-unswitch
+31
+-elim-avail-extern
+73
+-loop-simplify
+115
+-sink
+32
+-ee-instrument
+74
+-loop-sink
+116
+-speculative-execution
+33
+-flattencfg
+75
+-loop-reduce
+117
+-slsr
+34
+-float2int
+76
+-loop-unroll-and-jam
+118
+-strip-dead-prototypes
+35
+-forceattrs
+77
+-loop-unroll
+119
+-strip-debug-declare
+36
+-inline
+78
+-loop-unswitch
+120
+-strip-nondebug
+37
+-insert-gcov-profiling
+79
+-loop-vectorize
+121
+-strip
+38
+-gvn-hoist
+80
+-loop-versioning-licm
+122
+-tailcallelim
+39
+-gvn
+81
+-loop-versioning
+123
+-mergereturn
+40
+-globaldce
+82
+-loweratomic
+41
+-globalopt
+83
+-lower-constant-intrinsics
+Table 5. A list of LLVM compiler pass indices and their corresponding command line flag.
+

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