Spectral Alignment of Correlated Gaussian matrices

TL;DR

The paper analyzes a simple spectral alignment method (EIG1) for correlational Gaussian matrices modeled by $A$ and $B=\Pi^T(A+\sigma H)\Pi$, where $A$ and $H$ are GOE matrices and $\Pi$ encodes a planted permutation. By deriving a first-order Gaussian perturbation expansion of the leading eigenvector $v'_1$ around $v_1$ and introducing a toy model $\mathcal{J}(N,s)$ that captures rank-preservation under correlated Gaussian noise, the authors establish a sharp zero–one law: if $\sigma = o(N^{-7/6-\epsilon})$, EIG1 recovers all but a vanishing fraction of the permutation, while if $\sigma = \omega(N^{-7/6+\epsilon})$, recovery is $o(N)$. The results illuminate the fundamental limits of the simplest spectral alignment and connect random-matrix diffusion phenomena to exact recovery thresholds. This yields a rigorous, scalable baseline for weighted graph alignment in high-dimensional random settings and may inform the design of more robust spectral methods.

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 $\hatπ$ such that the rank of coordinate $\hatπ(i)$ 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 $π$, namely $B= Π^T (A+σH) Π$, where $Π$ is the permutation matrix associated with $π$ and $H$ is an independent copy of $A$. We show the following zero-one law: with high probability, under the condition $σN^{7/6+ε} \to 0$ for some $ε>0$, EIG1 recovers all but a vanishing part of the underlying permutation $π$, whereas if $σN^{7/6-ε} \to \infty$, 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.

Figures (4)

• Figure 1: Estimated overlap $\mathcal{L}(\hat{\Pi},\Pi)$ reached by EIG1 in model \ref{['GOEmodel']}, for varying $N$ and $\sigma$. With $95\%$ confidence intervals.
• Figure 2: Estimated $p(N,s)$ in the toy model $\mathcal{J}(N,s)$. With $95\%$ confidence intervals.
• Figure 3: Areas corresponding to $\mathcal{N}^{+} (x,y)$ and $\mathcal{N}^{-} (x,y)$.
• Figure 4: Orthogonal projection of $\widetilde{v}_1$ on $\mathcal{P} := \mathrm{span}(v'_1, v_1)$.

Theorems & Definitions (42)

• Remark 1.1
• Theorem 1: Zero-one law for EIG1 method
• Remark 2.1
• Proposition 2.1
• Remark 2.2
• Proposition 2.2: Zero-one law for $p(N,s)$
• Proposition 3.1
• Remark 3.1
• Lemma 3.1
• Remark 3.2