High-Dimensional Procrustes Matching via Tree Counts

Abstract: Suppose we observe two sets of $n$ Gaussian vectors in $\mathbb{R}<sup>d$, with the promise that, after applying a permutation of $[n]$ and a rotation of $\mathbb{R}<sup>d$, 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 $ρ<sup>2&gt;\sqrtα$, 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 $$ρ² \gtrsim \max{\log n/d,\sqrt{\log n/n}}$$. We also carry out a low-degree advantage calculation, which suggests that the condition $ρ² &gt; \sqrtα$ is necessary for any tree-counting algorithm.

• The paper introduces a polynomial-time algorithm that recovers the hidden permutation exactly using weighted tree-count features in a high-dimensional setting with constant correlation.
• It employs a novel mapping of data points to polynomial feature vectors using wide trees and efficient color-coding to control variance and ensure mean separation.
• The work delineates a computational-statistical gap, demonstrating that while information-theoretic recovery is possible at lower correlations, polynomial-time algorithms are limited by tree-based statistical thresholds.

High-Dimensional Procrustes Matching via Tree Counts

Problem Setting and Motivation

The Procrustes matching problem arises in high-dimensional inference tasks where two datasets, $X$ and $Y$, each representing $n$ points in $\mathbb{R}^d$, are observed. These datasets correspond to correlated, unobserved objects but are obscured by an unknown permutation $\pi$ (relabeling of indices) and an unknown orthogonal transformation $Q$ (latent rotation). Mathematically, the generative model is:

$$X_i \sim \mathcal{N}(0, I_d) \qquad Y_i = \rho Q X_{\pi(i)} + \sqrt{1-\rho^2}Z_i$$

with $\rho \in [0,1]$ and $Z_i \sim \mathcal{N}(0, I_d)$ independent. The objective is to recover the hidden permutation $\pi$ given only $Y$0 and $Y$1.

While efficient algorithms exist in the low-dimensional regime ($Y$2), the high-dimensional regime ($Y$3) presents significant statistical and computational barriers, particularly when $Y$4 is a constant bounded away from $Y$5. Existing approaches require nearly-perfect correlation ($Y$6) for tractable matching when $Y$7 is large.

Tree Counting Framework and Algorithmic Innovation

The principal contribution is a polynomial-time algorithm for exact permutation recovery in the high-dimensional regime under constant correlation—specifically, when $Y$8 and $Y$9 for Otter’s tree-counting constant $n$0.

Key Algorithmic Features

• Polynomial Features via Trees:

Each data point $n$1 is mapped to a vector of polynomial features, constructed as weighted counts of rooted, labeled bipartite trees embedded in $n$2. Each tree encodes a monomial in the entries of $n$3; the template family $n$4 comprises trees whose $n$5-nodes all have degree exactly $n$6 and whose root degree is large compared to tree size—so-called “wide” trees.

• Permutation Recovery via Similarity Matching:

For each pair $n$7, a similarity score is computed as the inner product of these signature vectors over a suite of such trees, aggregated and weighted appropriately. Permutation recovery is then cast as solving for the bijection that maximizes aggregate similarity.

• Efficient Counting by Color Coding:

Direct enumeration of tree counts is super-polynomial, but the algorithm leverages color-coding techniques to approximate such counts efficiently, yielding a total runtime of $n$8 for some constant $n$9.

Statistical Mechanisms

The key technical barrier with high-dimensional Procrustes matching is the low-rank structure and the absence of positive edge-wise correlation after the hidden rotation. However, matching pairs of edges that share $\mathbb{R}^d$0-node endpoints between $\mathbb{R}^d$1 and $\mathbb{R}^d$2 restores correlation in the form of rank-2 moments, enabling the design of informative features.

Weighted counts of wide trees serve two purposes:

1. Variance Control: Tree counts indexed by high root degree and bounded local tree-width mitigate spurious correlations induced by low-rank effects.
2. Mean Separation: The aggregate statistics concentrate so that the true matching achieves significantly higher similarity than any false matching with high probability, provided $\mathbb{R}^d$3.

Main Theoretical Results

Polynomial-Time Exact Recovery

• Algorithmic Guarantee:

There exists a polynomial-time estimator recovering the permutation exactly with high probability whenever $\mathbb{R}^d$4 and $\mathbb{R}^d$5.

• Tree Structure:

The required tree width (root degree $\mathbb{R}^d$6) is $\mathbb{R}^d$7; tree size is $\mathbb{R}^d$8.

• Mean-Variance Separation:

For true matches, expected similarity is asymptotically $\mathbb{R}^d$9; for false matches, expectation is zero, and the variance is controlled to $\pi$0 for non-diagonal pairs.

Information-Theoretic Bound

An improved QAP-based information-theoretic analysis shows that (possibly intractable) estimators can succeed under

$\pi$1

matching known lower bounds for related linear assignment models. As $\pi$2 increases, the required correlation for possible recovery decreases, indicating a significant computational-statistical gap.

Computational-Statistical Gap

It is explicitly shown that, although information-theoretic recovery is feasible at vanishing correlation for sufficiently large $\pi$3, no known polynomial-time algorithm can match this bound. The tree-counting approach is barriered at $\pi$4, and low-degree polynomial testing shows this cannot be improved using only tree-based feature statistics.

Structural and Analytical Techniques

• Combining Weingarten Calculus with Circuit Decomposition:

The analysis of joint moments under random $\pi$5 leverages Weingarten calculus for the orthogonal group, expressing moment calculations as sums over circuit decompositions of associated bipartite graphs. This enables explicit tracking of contributions from tree-based statistics and bounds on their concentration.

• Wide Tree Construction:

The choice of “wide” trees suppresses the excess variance from overlapping substructures. The authors formalize this advantage with a detailed combinatorial and probabilistic analysis, linking the achievable correlation threshold to Otter’s constant—mirroring thresholds in random graph matching and matrix models.

• Low-Degree-Polynomial Limitations:

The theoretical analysis demonstrates, via a low-degree framework, that counting trees with degree-$\pi$6 $\pi$7-nodes indeed achieves optimal separation for moment-based statistics: extending to more general trees does not improve the threshold unless the algorithm leaves the space of tree-count-based methods.

Implications and Future Directions

The demonstrated result represents the first polynomial-time method achieving exact Procrustes permutation recovery in the high-dimensional, constant-correlation regime. Key implications are:

• Advancement in Geometric Matching:

The method bridges the algorithmic limitations in high-dimensional geometric and graph matching, indicating that fine control of feature-aggregation using combinatorial tree structures extends optimal recovery regimes well beyond classical spectral or low-rank approaches.

• Barriers and Open Problems:

The robust computational-statistical gap suggests that tree-counting approaches capture the optimal threshold among low-degree algorithms, but further improvement (toward information-theoretic limits) may require fundamentally different algorithmic paradigms—possibly outside the class of polynomial-time algorithms, or using richer (non-tree) structural features.

• Broader Applications and Generalizations:

This analytical template is generalizable to other matching and alignment problems in high-dimensions, including those with more complex latent transformations, heteroskedastic noise, or manifold structures.

Conclusion

This work rigorously characterizes the phase transition and algorithmic feasibility of high-dimensional Procrustes matching based on tree-count polynomial features. The connection to Otter's constant is established as a sharp computational threshold within tree-counting algorithms, and the authors robustly delineate the algorithmic and statistical frontiers for this family of high-dimensional inference problems. The analysis leverages fine combinatorial, probabilistic, and group-integral methods, and clarifies the limits of low-degree polynomial approaches for latent alignment in high dimensions. These results will inform both practical design and theoretical understanding for geometric matching and related inference problems in modern machine learning and statistics (2607.08538).