Phase Transition in Convex Relaxations for Graph Alignment

• URL: http://arxiv.org/abs/2606.15581v1
• Date: Sun, 14 Jun 2026 03:47:45 GMT
• Title: Phase Transition in Convex Relaxations for Graph Alignment
• Abstract summary: We study the graph alignment problem for correlated Gaussian Orthogonal Ensemble (GOE) matrices.<n>Our results precisely characterize that $|Xstar-star|_F2$ transitions from $o(n)$ for $= tildeo(n-1/2)$ to $(n)$ for $= tilde(n-1/2)$.
• Score: 6.515592049126884
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the graph alignment problem for correlated Gaussian Orthogonal Ensemble (GOE) matrices, where the goal is to recover a hidden vertex permutation given two correlated symmetric Gaussian matrices $(A, B)$ with correlation $1/\sqrt{1+σ^2}$. While the maximum likelihood estimator is information-theoretically optimal, its computation, which reduces to a quadratic assignment problem, is intractable. Motivated by this, we analyze convex relaxations based on minimizing $\|AX - XB\|_F$ over the set of doubly stochastic matrices and the unit hypercube. We show that when the correlation parameter satisfies $σ= o(n^{-1/2}/\log^4 n)$, the solution of either relaxation $(X^\star)$ concentrates around the ground-truth permutation matrix $(Π^\star)$, i.e., $\|X^\star-Π^\star\|_F^2 = o(n)$, implying recovery of all but a vanishing fraction of vertices after simple post-processing. Combined with existing lower bounds, our results precisely characterize that $\|X^\star-Π^\star\|_F^2$ transitions from $o(n)$ for $σ= \tilde{o}(n^{-1/2})$ to $Ω(n)$ for $σ= \tildeΩ(n^{-1/2})$. In doing so, our analysis significantly tightens prior results and extends them beyond doubly stochastic relaxations.

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