Spectral Alignment of Correlated Gaussian matrices

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.