The Umeyama algorithm for matching correlated Gaussian geometric models in the low-dimensional regime

TL;DR

This paper analyzes the Umeyama algorithm's ability to recover hidden permutations in correlated Gaussian geometric models, showing it achieves exact recovery under certain noise conditions in low-dimensional settings.

Contribution

We prove that the Umeyama algorithm attains exact and almost exact recovery thresholds close to the information-theoretic limits in low-dimensional Gaussian geometric graph matching.

Findings

Exact recovery when noise $\sigma = o(d^{-3}n^{-2/d})$

Almost exact recovery when noise $\sigma = o(d^{-3}n^{-1/d})$

Results approach information thresholds up to a polynomial factor in $d$

Abstract

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.