High-Dimensional Procrustes Matching via Tree Counts

Abstract

Suppose we observe two sets of $n$ Gaussian vectors in ${\mathbb{R}}^d$, with the promise that, after applying a permutation of $[n]$ and a rotation of ${\mathbb{R}}^d$, the two sets are $\ddot{I}$-correlated. The Procrustes matching problem asks us to recover the unknown permutation of $[n]$ that aligns the two sets. The problem is well-studied in the low-dimensional regime $d=O(\log n)$, but the high-dimensional regime $d\gg \log n$ has remained largely uncharted: prior matching guarantees require nearly perfect correlation $\ddot{I}=1-o(1)$, even for information-theoretic recovery. Our main result is a polynomial-time algorithm for exact recovery at constant correlation. The algorithm works by computing and comparing weighted counts of a specially chosen family of ``wide'' trees. So long as $d\ge \mathrm{polylog}(n)$, the algorithm succeeds with high probability for any $\ddot{I}{}^2>\sqrt{\hat{I}}\pm$, where $\hat{I}\pm \approx 0.338$ is Otter's tree-counting constant. We complement this algorithmic result with an improved information-theoretic guarantee, showing that exact recovery is possible when $\ddot{I}{}^2\gtrsim \max \{\log n/d,\sqrt{\log n/n}\}$. We also carry out a low-degree advantage calculation, which suggests that the condition $\ddot{I}{}^2>\sqrt{\hat{I}}\pm$ is necessary for any tree-counting algorithm.