Spectral Graph Matching and Regularized Quadratic Relaxations I Algorithm and Gaussian Analysis

Graph matching aims at finding the vertex correspondence between two unlabeled graphs that maximizes the total edge weight correlation. This amounts to solving a computationally intractable quadratic assignment problem. In this paper, we propose a new spectral method, graph matching by pairwise eige...

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Vydáno v:	Foundations of computational mathematics Ročník 23; číslo 5; s. 1511 - 1565
Hlavní autoři:	Fan, Zhou, Mao, Cheng, Wu, Yihong, Xu, Jiaming
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.10.2023 Springer Nature B.V
Témata:	Algorithms Apexes Applications of Mathematics Assignment problem Computer Science Correlation coefficients Economics Eigenvalues Eigenvectors Graph matching Graph theory Graphs Linear and Multilinear Algebras Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Operations research Polynomials Quadratic programming Similarity Spectral methods Statistical analysis Random matrix theory Convex relaxations Primary 90C25 Quadratic assignment problem Quadratic programming Secondary 68Q87 Spectral methods Graph matching
ISSN:	1615-3375, 1615-3383
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Shrnutí:	Graph matching aims at finding the vertex correspondence between two unlabeled graphs that maximizes the total edge weight correlation. This amounts to solving a computationally intractable quadratic assignment problem. In this paper, we propose a new spectral method, graph matching by pairwise eigen-alignments (GRAMPA). Departing from prior spectral approaches that only compare top eigenvectors, or eigenvectors of the same order, GRAMPA first constructs a similarity matrix as a weighted sum of outer products between all pairs of eigenvectors of the two graphs, with weights given by a Cauchy kernel applied to the separation of the corresponding eigenvalues, then outputs a matching by a simple rounding procedure. The similarity matrix can also be interpreted as the solution to a regularized quadratic programming relaxation of the quadratic assignment problem. For the Gaussian Wigner model in which two complete graphs on n vertices have Gaussian edge weights with correlation coefficient 1 - σ 2 , we show that GRAMPA exactly recovers the correct vertex correspondence with high probability when σ = O ( 1 log n ) . This matches the state of the art of polynomial-time algorithms and significantly improves over existing spectral methods which require σ to be polynomially small in n . The superiority of GRAMPA is also demonstrated on a variety of synthetic and real datasets, in terms of both statistical accuracy and computational efficiency. Universality results, including similar guarantees for dense and sparse Erdős–Rényi graphs, are deferred to a companion paper.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	1615-3375 1615-3383
DOI:	10.1007/s10208-022-09570-y