Spectral Alignment of Correlated Gaussian matrices

TL;DR

This paper analyzes a spectral method for aligning matrices by matching their leading eigenvectors, providing conditions under which it successfully recovers the underlying permutation in a Gaussian matrix model.

Contribution

It offers a rigorous analysis of the spectral alignment method, establishing a sharp phase transition for its success in a Gaussian matrix model.

Findings

Successful recovery when  N^{7/6+\u03b5}  0

Failure to recover more than o(N) matches when  N^{7/6-} o \u221e

Provides insights into the effectiveness of a simple spectral alignment algorithm

Abstract

In this paper we analyze a simple spectral method (EIG1) for the problem of matrix alignment, consisting in aligning their leading eigenvectors: given two matrices $A$ and $B$, we compute $v_1$ and $v'_1$ two corresponding leading eigenvectors. The algorithm returns the permutation $\hat{\pi}$ such that the rank of coordinate $\hat{\pi}(i)$ in $v_1$ and that of coordinate $i$ in $v'_1$ (up to the sign of $v'_1$) are the same. We consider a model of weighted graphs where the adjacency matrix $A$ belongs to the Gaussian Orthogonal Ensemble (GOE) of size $N \times N$, and $B$ is a noisy version of $A$ where all nodes have been relabeled according to some planted permutation $\pi$, namely $B= \Pi^T (A+\sigma H) \Pi $, where $\Pi$ is the permutation matrix associated with $\pi$ and $H$ is an independent copy of $A$. We show the following zero-one law: with high probability, under the condition $\sigma N^{7/6+\epsilon} \to 0$ for some $\epsilon>0$, EIG1 recovers all but a vanishing part of the underlying permutation $\pi$, whereas if $\sigma N^{7/6-\epsilon} \to \infty$, this method cannot recover more than $o(N)$ correct matches. This result gives an understanding of the simplest and fastest spectral method for matrix alignment (or complete weighted graph alignment), and involves proof methods and techniques which could be of independent interest.