Learning Probability Measures with respect to Optimal Transport Metrics

Abstract

We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embed-ded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal quantization, and learning theory, we derive new probabilistic bounds for the performance of a classic algorithm in unsupervised learning (k-means), when used to produce a probability measure derived from the data. In the course of the analysis, we arrive at new lower bounds, as well as probabilistic upper bounds on the conver-gence rate of the empirical law of large numbers, which, unlike existing bounds, are applicable to a wide class of measures. 1