$K-$means with leraned metrics

ArXi:2603.14601v1 Announce Type: cross We study the Fr\'echet {\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 Gromo-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.