Strong recovery of geometric planted matchings

TL;DR

This paper analyzes the conditions under which the maximum likelihood estimator can perfectly or strongly recover a planted matching between points in Euclidean space, extending previous graph-based results to a geometric setting.

Contribution

It establishes thresholds for perfect and strong recovery of planted matchings in a geometric setting with Gaussian perturbations, using novel combinatorial analysis techniques.

Findings

Thresholds for perfect recovery of the matching.

Thresholds for strong recovery with o(n) errors.

Characterization of error rates between thresholds.

Abstract

We study the problem of efficiently recovering the matching between an unlabelled collection of $n$ points in $\mathbb{R}^d$ and a small random perturbation of those points. We consider a model where the initial points are i.i.d. standard Gaussian vectors, perturbed by adding i.i.d. Gaussian vectors with variance $\sigma^2$. In this setting, the maximum likelihood estimator (MLE) can be found in polynomial time as the solution of a linear assignment problem. We establish thresholds on $\sigma^2$ for the MLE to perfectly recover the planted matching (making no errors) and to strongly recover the planted matching (making $o(n)$ errors) both for $d$ constant and $d = d(n)$ growing arbitrarily. Between these two thresholds, we show that the MLE makes $n^{\delta + o(1)}$ errors for an explicit $\delta \in (0, 1)$. These results extend to the geometric setting a recent line of work on recovering matchings planted in random graphs with independently-weighted edges. Our proof techniques rely on careful analysis of the combinatorial structure of partial matchings in large, weakly dependent random graphs using the first and second moment methods.