Spectral Graph Matching and Regularized Quadratic Relaxations I: The Gaussian Model
This work addresses the computationally intractable quadratic assignment problem for graph matching, offering improved accuracy and efficiency for applications in network analysis and data alignment, though it is incremental with a novel method for a known bottleneck.
The authors tackled the graph matching problem by proposing GRAMPA, a spectral method that constructs a similarity matrix from all eigenvector pairs and uses a Cauchy kernel on eigenvalue separations, achieving exact recovery with high probability for Gaussian Wigner models when σ = O(1/log n), matching state-of-the-art polynomial-time algorithms and outperforming existing spectral methods.
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 $σ= O(\frac{1}{\log n})$. 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 the companion paper.