$K-$means with learned metrics
Abstract
We study the Fr\'echet means of a metric measure space when both the measure and the distance are unknown and have to be estimated. We prove a general result that states that the means are continuous with respect to the measured Gromov-Hausdorff topology. In this situation, we also prove a stability result for the Voronoi clusters they determine. We do not assume uniqueness of the set of means, but when it is unique, the results are stronger. This framework provides a unified approach to proving consistency for a wide range of metric learning procedures. As concrete applications, we obtain new consistency results for several important estimators that were previously unestablished, even when . These include means based on: (i) Isomap and Fermat geodesic distances on manifolds, (ii) difussion distances, (iii) Wasserstein distances computed with respect to learned ground metrics. Finally, we consider applications beyond the statistical inference paradigm like (iv) first passage percolation and (v) discrete approximations of length spaces.
Cite
@article{arxiv.2603.14601,
title = {$K-$means with learned metrics},
author = {Pablo Groisman and Matthieu Jonckheere and Jordan Serres and Mariela Sued},
journal= {arXiv preprint arXiv:2603.14601},
year = {2026}
}