$K-$means with learned metrics

We study the Fréchet {\it k-}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 {\it k-}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 {\it k-}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 $k=1$. These include {\it k-}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.}

Finite sample bounds for barycenter estimation in geodesic spaces

We study the problem of estimating the barycenter of a distribution given i.i.d. data in a geodesic space. Assuming an upper curvature bound in Alexandrov's sense and a support condition ensuring the strong geodesic convexity of the barycenter problem, we establish finite-sample error bounds in expectation and with high probability. Our results generalize Hoeffding- and Bernstein-type concentration inequalities from Euclidean to geodesic spaces. Building on these concentration inequalities, we derive statistical guarantees for two efficient algorithms for the computation of barycenters.