Learning Probability Measures with respect to Optimal Transport Metrics

We study the problem of estimating, in the sense of optimal transport
metrics, a measure which is assumed supported on a manifold embedded 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 convergence rate of the empirical law of
large numbers, which, unlike existing bounds, are applicable to a wide class of
measures.