Learning Probability Measures with respect to Optimal Transport Metrics

TL;DR

This paper investigates estimating probability measures on manifolds using optimal transport metrics, linking it to quantization and learning theory, and providing new bounds for k-means performance and convergence rates.

Contribution

It establishes a novel connection between optimal transport, quantization, and learning theory, deriving new probabilistic bounds and convergence rates for measure estimation.

Findings

New probabilistic bounds for k-means in measure learning

Lower bounds on convergence rates of empirical measures

Bounds applicable to a wide class of measures

Abstract

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.