Phase Transition in Convex Relaxations for Graph Alignment

2026-06-13

AI Summary

This work addresses the problem of graph alignment under two correlated Gaussian Orthogonal Ensemble (GOE) matrices, aiming to recover a hidden vertex permutation. The authors propose a convex relaxation approach that minimizes the Frobenius norm $\|AX - XB\|_F$ subject to doubly stochastic and unit hypercube constraints, providing an efficient approximation to the NP-hard maximum likelihood estimator. Their theoretical analysis precisely characterizes a phase transition in estimation error at noise level $\sigma = \widetilde{O}(n^{-1/2})$, substantially improving upon existing bounds and extending to a broader class of convex relaxations. When $\sigma = o(n^{-1/2}/\log^4 n)$, the squared Frobenius distance between the relaxed solution and the true permutation is $o(n)$, enabling near-perfect recovery after post-processing, with the phase transition threshold matching the information-theoretic lower bound.

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.

Problem

• graph alignment
• phase transition
• convex relaxation
• Gaussian Orthogonal Ensemble
• vertex permutation

Innovation

• graph alignment
• convex relaxation
• phase transition
• Gaussian Orthogonal Ensemble
• quadratic assignment problem