Spectral alignment of correlated Gaussian random matrices
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In this paper we analyze a simple method (EIG1) for the problem of matrix alignment, consisting in aligning their leading eigenvectors: given A and B, we compute v_1 and v'_1 two leading eigenvectors of A and B. The algorithm returns a permutation Π̂ such that the rank of the coordinate Π̂(i) in v_1 is the rank of the coordinate i in v'_1 (up to the sign of v'_1). We consider a model where A belongs to the Gaussian Orthogonal Ensemble (GOE), and B= Π^T (A+σ H) Π, where Π is a permutation matrix and H is an independent copy of A. We show the following 0-1 law: under the condition σ N^7/6+ϵ→ 0, the EIG1 method recovers all but a vanishing part of the underlying permutation Π. When σ N^7/6-ϵ→∞, this algorithm 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.
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