Graph Matching: Relax at Your Own Risk

Lyzinski, Vince; Fishkind, Donniell E.; Fiori, Marcelo; Vogelstein, Joshua T.; Priebe, Carey E.; Sapiro, Guillermo

doi:10.1109/tpami.2015.2424894

Cited by 117 publications

(126 citation statements)

References 49 publications

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“…S1; see ref. 27 for a formal proof of failure of convex relaxation on a particular class of random graphs. However, in what follows, we show that by providing additional information in the form of corresponding seeds or vertex attributes disambiguating the symmetry, equivalence of the relaxation to the exact GM problem still holds.…”

Section: Matching Of Symmetric Graphsmentioning

confidence: 99%

On convex relaxation of graph isomorphism

Aflalo

Bronstein

Kimmel

2015

Proc. Natl. Acad. Sci. U.S.A.

106

123

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We consider the problem of exact and inexact matching of weighted undirected graphs, in which a bijective correspondence is sought to minimize a quadratic weight disagreement. This computationally challenging problem is often relaxed as a convex quadratic program, in which the space of permutations is replaced by the space of doubly stochastic matrices. However, the applicability of such a relaxation is poorly understood. We define a broad class of friendly graphs characterized by an easily verifiable spectral property. We prove that for friendly graphs, the convex relaxation is guaranteed to find the exact isomorphism or certify its inexistence. This result is further extended to approximately isomorphic graphs, for which we develop an explicit bound on the amount of weight disagreement under which the relaxation is guaranteed to find the globally optimal approximate isomorphism. We also show that in many cases, the graph matching problem can be further harmlessly relaxed to a convex quadratic program with only n separable linear equality constraints, which is substantially more efficient than the standard relaxation involving 2n equality and n 2 inequality constraints. Finally, we show that our results are still valid for unfriendly graphs if additional information in the form of seeds or attributes is allowed, with the latter satisfying an easy to verify spectral characteristic.graph isomorphism | graph matching | permutation | convex relaxation G raphs are a natural abstraction in a variety of problems and are particularly useful for modeling structures, frequently arising in different domains of science and engineering. In many applications, graphs have to be compared or brought into correspondence. The term "graph isomorphism" or the less precise term "graph matching" (used mainly in the applied community) refers to a class of computational problems consisting of finding an optimal correspondence between the vertices of two graphs that minimizes adjacency disagreement. The uses of graph models in general and graph matching in particular are too numerous to allow a comprehensive review within the scope of this paper. In what follows, we will just list a few prominent ones, referring the reader to a (partial) review of applications with a particular emphasis on the domain of pattern recognition (1). In computer vision and pattern recognition, graph matching is used for stereo vision and 3D reconstruction (2), object detection and recognition (3, 4)-in particular, optical character recognition (5)-and image and video indexing and retrieval (6). In biometric applications, graph-based techniques have been widely used for identification tasks implemented by means of elastic graph matching. These include, among others, face recognition and pose estimation (7) and fingerprint recognition (8). In biomedical applications, graphs have been used to model vascular structures and, more recently, to represent connections between neurons (9). In data mining, graphs are used to model networks, including the Web and social ...

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Section: Matching Of Symmetric Graphsmentioning

confidence: 99%

On convex relaxation of graph isomorphism

Aflalo

Bronstein

Kimmel

2015

Proc. Natl. Acad. Sci. U.S.A.

106

123

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“…There are a number of extensions and open questions that arose during the course of this work. Natural theoretic extensions include lifting Theorems 1 and 2 to non-edge independent models (note that certain localized dependencies amongst edges can easily be handled in the McDiarmind proof framework, while globally dependent errors provide a more significant challenge); formulating the analogues of Theorems 1 and 2 in the weighted, attributed graph settings; and considering the theoretic properties of various continuous relaxations of the multiplex GM problem akin to [1,25,6].…”

Section: Discussionmentioning

confidence: 99%

Multiplex graph matching matched filters

Pantazis

Sussman

Park

et al. 2019

2019 IEEE International Conference on Big Data (Big Data)

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We consider the problem of detecting a noisy induced multiplex template network in a larger multiplex background network. Our approach, which extends the framework of [36] to the multiplex setting, leverages a multiplex analogue of the classical graph matching problem to use the template as a matched filter for efficiently searching the background for candidate template matches. The effectiveness of our approach is demonstrated both theoretically and empirically, with particular attention paid to the potential benefits of considering multiple channels.for a definition of multiplex isomorphism), accounting for the reality that relatively large, complex subgraph templates may only errorfully occur in the larger background network. These errors may be due to missing edges/vertices in the template or background, and arise in a variety of real data settings [34]. The subgraph isomorphism problem-given a template A, determine if an isomorphic copy of A exists in a larger network B and find the isomorphic copy (or copies) if it exists-has been the subject of voluminous research in the monoplex (i.e., single layer) setting, with approaches based on efficient tree search [38], color coding [3, 2], graph homomorphisms [18], rule-based/filter-based matchings [10,29], among others; for a survey of the literature circa 2012, see [24]. In contrast, the problem of multilayer homomorphic/isomorphic subgraph detection is still in its relative infancy, with comparatively fewer existing methods in the literature; see, for example, [41,37,29]. Notation: The following notation will be used throughout. For an integer n > 0, we will define [n] := {1, 2, . . . , n}, J n to be the n × n hollow matrix with all off-diagonal entries identically set to 1, 0 n to be the n × n matrix with all entries identically set to 0,

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“…Furthermore, recent work [30,24] on solving general QAP problems suggests that convex relaxations do not always outperform indefinite relaxations. To this direction, Lim and Wright [22] present a new framework for approximating general QAP problems formulated in terms of sorting networks, and use a continuation procedure [26,23], where they start solving a convex relaxation of the problem and then gradually convert it to a concave one, yielding a local optimum to the original discrete problem.…”

Section: Relation To Existing Methodsmentioning

confidence: 99%

“…For small-scale problems, exact solutions such as branch-and-bound can be applied [5]. Besides generic approximation algorithms, such as simulated annealing, there has been recent interest in relaxation-based approaches for solving QAPs in the context of graph matching [30,24]. This is particularly alluring for the 2-SUM case as it corresponds to the minimization of a convex objective function, and recent works [8,21] have shown how the relaxed 2-SUM problem can be exactly solved using interior-point methods, in either a matrix or vector-based formulation.…”

Section: Introductionmentioning

confidence: 99%

A Graduated Non-Convexity Relaxation for Large Scale Seriation

Evangelopoulos¹,

Brockmeier²,

Mu³

et al. 2017

Proceedings of the 2017 SIAM International Conference on Data Mining

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In this work we propose a highly scalable algorithm for solving the combinatorial data analysis problem of seriation. Seriation is a technique for optimizing a permutation of data instances, with respect to some proximity measure such that nearby instances in the linear arrangement are more similar. One consistent objective function for seriation is the 2-SUM minimization problem, which uses the 2-norm between instance locations to penalize non-zero similarity values, and can be written as a quadratic function of the permutation vector. Recently, two convex relaxations of the 2-SUM problem have been proposed, which can be solved as constrained quadratic programs using interior point methods; however, the interior point solvers become expensive when the problem size increases. In this paper we present a graduated non-convexity method for vector-based relaxations of the 2-SUM that yields better approximate solutions and scales to very large problem sizes. We conduct a number of experiments on real and synthetic datasets. The experimental results demonstrate that our proposed algorithm outperforms other approaches that solve the 2-SUM, and is the only competitive approach that can scale to large problem sizes.

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Graph Matching: Relax at Your Own Risk

Cited by 117 publications

References 49 publications

On convex relaxation of graph isomorphism

On convex relaxation of graph isomorphism

Multiplex graph matching matched filters

A Graduated Non-Convexity Relaxation for Large Scale Seriation

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