Motivated by the problem of matching two correlated random geometric graphs, we study the problem of matching two Gaussian geometric models correlated through a latent node permutation. Specifically, given an unknown permutation $\pi^*$ on $\{1,\ldots,n\}$ and given $n$ i.i.d. pairs of correlated Gaussian vectors $\{X_{\pi^*(i)},Y_i\}$ in $\mathbb{R}^d$ with noise parameter $\sigma$, we consider two types of (correlated) weighted complete graphs with edge weights given by $A_{i,j}=\langle X_i,X_j \rangle$, $B_{i,j}=\langle Y_i,Y_j \rangle$. The goal is to recover the hidden vertex correspondence $\pi^*$ based on the observed matrices $A$ and $B$. For the low-dimensional regime where $d=O(\log n)$, Wang, Wu, Xu, and Yolou [WWXY22+] established the information thresholds for exact and almost exact recovery in matching correlated Gaussian geometric models. They also conducted numerical experiments for the classical Umeyama algorithm. In our work, we prove that this algorithm achieves exact recovery of $\pi^*$ when the noise parameter $\sigma=o(d^{-3}n^{-2/d})$, and almost exact recovery when $\sigma=o(d^{-3}n^{-1/d})$. Our results approach the information thresholds up to a $\operatorname{poly}(d)$ factor in the low-dimensional regime.

翻译：受匹配两个相关随机几何图问题的启发，我们研究了通过潜在节点排列相关的高斯几何模型的匹配问题。具体而言，给定$\{1,\ldots,n\}$上的未知排列$\pi^*$，以及$\mathbb{R}^d$中$n$个独立同分布的相关高斯向量对$\{X_{\pi^*(i)},Y_i\}$（含噪声参数$\sigma$），我们考虑两类（相关）加权完全图，其边权重分别为$A_{i,j}=\langle X_i,X_j \rangle$和$B_{i,j}=\langle Y_i,Y_j \rangle$。目标是基于观测矩阵$A$和$B$恢复隐藏的顶点对应关系$\pi^*$。对于$d=O(\log n)$的低维体制，Wang、Wu、Xu和Yolou [WWXY22+] 建立了匹配相关高斯几何模型中精确恢复和几乎精确恢复的信息阈值。他们还对经典Umeyama算法进行了数值实验。在我们的工作中，我们证明当噪声参数$\sigma=o(d^{-3}n^{-2/d})$时，该算法能够实现$\pi^*$的精确恢复；当$\sigma=o(d^{-3}n^{-1/d})$时，实现几乎精确恢复。我们的结果在低维体制下逼近信息阈值，仅相差一个$\operatorname{poly}(d)$因子。