Strong recovery of geometric planted matchings

Dmitriy Kunisky, Jonathan Niles-Weed

Research output: Chapter in Book/Report/Conference proceedingConference contribution


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 covariance 2Id. 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 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+o(1) errors for an explicit 2 (0; 1). These results extend a recent line of work on recovering matchings planted in random graphs with independently-weighted edges to the geometric setting. 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.

Original languageEnglish (US)
Title of host publicationACM-SIAM Symposium on Discrete Algorithms, SODA 2022
PublisherAssociation for Computing Machinery
Number of pages43
ISBN (Electronic)9781611977073
StatePublished - 2022
Event33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 - Alexander, United States
Duration: Jan 9 2022Jan 12 2022

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022
Country/TerritoryUnited States

ASJC Scopus subject areas

  • Software
  • General Mathematics


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