Phase Transition in Convex Relaxations for Graph Alignment

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/√1+σ2. 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/log4 n), the solution of either relaxation (X^star) concentrates around the ground-truth permutation matrix (Pi^star), i.e., |X^star-Pi^star|F2 = 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-Pi^star|F2 transitions from o(n) for σ = tildeo(n-1/2) to Omega(n) for σ = tildeOmega(n-1/2). In doing so, our analysis significantly tightens prior results and extends them beyond doubly stochastic relaxations.