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.

翻译：我们研究了相关高斯正交系综（GOE）矩阵的图对齐问题，其目标是从两个相关对称高斯矩阵$(A, B)$（相关系数为$1/\sqrt{1+σ^2}$）中恢复隐藏的顶点置换。虽然最大似然估计在信息论意义上是最优的，但其计算（归结为二次分配问题）是难以处理的。受此启发，我们分析了基于在双随机矩阵集合和单位超立方体上最小化$\|AX - XB\|_F$的凸松弛。我们证明，当相关系数满足$σ= o(n^{-1/2}/\log^4 n)$时，任一松弛的解$(X^\star)$都集中在地面真值置换矩阵$(Π^\star)$附近，即$\|X^\star-Π^\star\|_F^2 = o(n)$，这意味着经简单后处理后，除消失比例的顶点外，其余顶点均可恢复。结合现有下界，我们的结果精确刻画了$\|X^\star-Π^\star\|_F^2$从$σ= \tilde{o}(n^{-1/2})$时的$o(n)$到$σ= \tildeΩ(n^{-1/2})$时的$Ω(n)$的相变。在此过程中，我们的分析显著紧缩了先前结果，并将其推广至双随机松弛之外。