In this paper, we focus on the matching recovery problem between a pair of correlated Gaussian Wigner matrices with a latent vertex correspondence. We are particularly interested in a robust version of this problem such that our observation is a perturbed input $(A+E,B+F)$ where $(A,B)$ is a pair of correlated Gaussian Wigner matrices and $E,F$ are adversarially chosen matrices supported on an unknown $εn * εn$ principal minor of $A,B$, respectively. We propose an approximate message passing (AMP) type iterative algorithm that succeeds in polynomial time as long as the correlation $ρ$ between $(A,B)$ is a non-vanishing constant and $ε= o\big( \tfrac{1}{(\log n)^{20}} \big)$. A key distinction from standard AMP is the introduction of a time-dependent matrix multiplication step within the iteration, which simultaneously enlarges the feature dimension and cancels the correlation during the iteration. The main methodological inputs for our result are the iterative random graph matching algorithm proposed in \cite{DL22+, DL23+} and the spectral preprocessing procedure proposed in \cite{IS24+}. To the best of our knowledge, our algorithm is the first efficient random graph matching type algorithm that is robust under any adversarial perturbations of $n^{1-o(1)}$ size.

翻译：本文研究了具有潜在顶点对应关系的一对相关高斯维格纳矩阵之间的匹配恢复问题。我们特别关注该问题的鲁棒版本：观测数据为受干扰的输入$(A+E,B+F)$，其中$(A,B)$是一对相关的高斯维格纳矩阵，$E,F$分别是由对抗性选择的、仅支持在$(A,B)$的未知$εn × εn$主子矩阵上的矩阵。我们提出了一种近似消息传递型迭代算法，当$(A,B)$之间的相关系数$ρ$为非零常数且$ε= o\big( \tfrac{1}{(\log n)^{20}} \big)$时，该算法能够在多项式时间内成功恢复匹配。与标准AMP的关键区别在于，我们在迭代中引入了一个随时间变化的矩阵乘法步骤，该步骤同时扩大了特征维度并在迭代过程中抵消了相关性。本文结果的主要方法论输入包括文献\cite{DL22+, DL23+}提出的迭代随机图匹配算法和文献\cite{IS24+}提出的谱预处理过程。据我们所知，我们的算法是首个能够在任何$n^{1-o(1)}$规模的对抗性干扰下保持鲁棒性的高效随机图匹配型算法。