Spectral Graph Matching and Regularized Quadratic Relaxations I: The Gaussian Model

TL;DR

This paper introduces GRAMPA, a spectral graph matching method that uses pairwise eigenvector alignments and regularized quadratic relaxations, achieving near-optimal recovery in Gaussian models and outperforming prior spectral approaches.

Contribution

The paper proposes GRAMPA, a novel spectral algorithm for graph matching that leverages pairwise eigenvector comparisons and regularized quadratic programming, improving recovery guarantees and efficiency.

Findings

GRAMPA exactly recovers vertex correspondence when noise level is O(1/log n)

Outperforms existing spectral methods in accuracy and computational efficiency

Demonstrated effectiveness on synthetic and real datasets

Abstract

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 $1-\sigma^2$, we show that GRAMPA exactly recovers the correct vertex correspondence with high probability when $\sigma = O(\frac{1}{\log n})$. This matches the state of the art of polynomial-time algorithms, and significantly improves over existing spectral methods which require $\sigma$ 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 Erd\H{o}s-R\'{e}nyi graphs, are deferred to the companion paper.