High-Dimensional Procrustes Matching via Tree Counts

2026-07-09

AI Summary

This work addresses the problem of exactly recovering an unknown permutation that aligns two sets of Gaussian vectors in high dimensions (where \( d \gg \log n \)), given that the vectors are related by this permutation, an unknown rotation, and a fixed correlation coefficient \( \rho \). The authors propose a polynomial-time algorithm based on weighted “wide-tree” counting, which succeeds with high probability when \( d \geq \mathrm{polylog}(n) \) and \( \rho^2 > \sqrt{\alpha} \) with \( \alpha \approx 0.338 \). This is the first efficient method achieving exact recovery under constant correlation, significantly relaxing the previously required condition that \( \rho \) be close to 1. The tightness and necessity of this condition are corroborated through information-theoretic lower bounds—namely \( \rho^2 \gtrsim \max\{\log n / d, \sqrt{\log n / n}\} \)—and low-degree polynomial analysis.

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

Problem

• Procrustes matching
• high-dimensional
• permutation recovery
• Gaussian vectors
• correlated data

Innovation

• Procrustes matching
• high-dimensional statistics
• tree counts
• polynomial-time algorithm
• information-theoretic limits