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

翻译：在本文中, 我们分析一个简单的光谱方法( EIG1), 解决矩阵校正问题, 包括调合它们的首席导师: 给两个基质 $A$和$B$, 我们计算$v_ 1美元和$v_ 1$ 2对应的首席导师。 算法返回了 $\ hat\ pi} (一) 的调和 $v_ 1美元, 协调美元1美元, 协调美元 1美元 美元 的调和美元 。 我们考虑一个加权图表模型, 给两个基质 基质 $1 美元 和$1 美元 美元 。 算法的调和 美元 美元, 全部调和 美元 美元 。 基质的调和 美元 美元