Geometric planted matchings in high dimensions: The power of multiple views

2026-07-09

AI Summary

This study addresses the problem of recovering true correspondences from multiple noisy and independently permuted point clouds in high-dimensional space. While single-view observation exhibits an information-theoretic impossibility threshold—where exact matching becomes infeasible when the signal strength parameter $b < 2$—this work demonstrates for the first time that incorporating multiple views circumvents this fundamental limitation. Leveraging a high-dimensional Gaussian model and tools from random matrix theory, the authors devise a polynomial-time algorithm that, given $K$ views, achieves near-perfect recovery with only $o(n)$ mismatches whenever $b > K/(K-1)$. Notably, with three views, the method enables efficient and accurate matching in the regime $3/2 < b < 2$, which is provably impossible under a single view.

Abstract

We study the problem of recovering the correspondence between a collection of $n$ points in $\mathbb{R}^d$ and a noisy, permuted version of those points. In the high-dimensional regime $d=ω(\log n)$, under a Gaussian model with noise variance $σ^2=d/(b\log n)$, prior work identifies $b=2$ as the threshold for almost exact recovery. We prove that this threshold is all-or-nothing: for every fixed $b<2$, no estimator recovers a positive fraction of the matching, and even estimating the matched point cloud in Euclidean distance is asymptotically no better than ignoring the correspondence. On the other hand, we consider a multi-view generalization of the problem where $K$ noisy, independently permuted copies of the same latent point cloud are observed. Here we show that a simple polynomial-time procedure recovers all relative matchings up to $o(n)$ errors whenever $b>K/(K-1)$. Thus multiple views can break the impossibility barrier $b=2$ for the original matching problem: in particular, for $3/2 < b < 2$, the two-view model has no nontrivial recovery, but a third view makes all latent correspondences efficiently recoverable.

Problem

• geometric planted matching
• high-dimensional statistics
• permutation recovery
• multi-view learning
• correspondence problem

Innovation

• geometric planted matching
• high-dimensional statistics
• multi-view learning
• permutation recovery
• all-or-nothing phase transition