SPECTRAL ALIGNMENT OF CORRELATED GAUSSIAN MATRICES
ADVANCES IN APPLIED PROBABILITY(2022)
摘要
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 two corresponding leading eigenvectors v(1) and v'(1). The algorithm returns the permutation pi circumflex accent such that the rank of coordinate (pi) over cap (i) in v(1) and that of coordinate i in vi (up to the sign of v'(i)) are the same. We consider a model of weighted graphs where the adjacency matrix A belongs to the Gaussian orthogonal ensemble of size N x N, and B is a noisy version of A where all nodes have been relabeled according to some planted permutation pi; that is, B= 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 N7/6+epsilon -> 0 for some epsilon > 0, EIG1 recovers all but a vanishing part of the underlying permutation pi, whereas if sigma N7/6-epsilon -> infinity, 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.
更多查看译文
关键词
Graph alignment, random matrices, spectral theory
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络