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

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-σ2, we show that GRAMPA exactly recovers the correct vertex correspondence with high probability when σ = O(frac1log 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 ErdHos-R'enyi graphs, are deferred to the companion paper.