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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Published in:Foundations of computational mathematics Vol. 23; no. 5; pp. 1511 - 1565
Main Authors: Fan, Zhou, Mao, Cheng, Wu, Yihong, Xu, Jiaming
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2023
Springer Nature B.V
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-022-09570-y