The Umeyama algorithm for matching correlated Gaussian geometric models
  in the low-dimensional regime
Shuyang Gong Zhangsong Li
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.