High-Dimensional Procrustes Matching via Tree Counts
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 $\rho$-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 $\rho=1-o(1)$, 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 $\rho^2>\sqrt{\alpha}$, where $\alpha\approx 0.338$ is Otter's tree-counting constant. We complement this algorithmic result with an improved information-theoretic guarantee, showing that exact recovery is possible when $\rho^2 \gtrsim \max\{\log n/d,\sqrt{\log n/n}\}$. We also carry out a low-degree advantage calculation, which suggests that the condition $\rho^2>\sqrt{\alpha}$ is necessary for any tree-counting algorithm.