Strong recovery of geometric planted matchings

We study the problem of efficiently recovering the matching between an unlabelled collection of n points in ℝd 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 σ2. 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 σ2 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 nδ + o(1) errors for an explicit δ ∈ (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.