Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces

TL;DR

This paper derives upper bounds for the expected Wasserstein distance between probability measures and their empirical counterparts in unbounded functional spaces, extending previous finite-dimensional results and covering Gaussian processes with optimal rates.

Contribution

It generalizes Wasserstein distance bounds to unbounded functional spaces, including infinite-dimensional Gaussian processes, with rate-optimal dependence on dimension.

Findings

Provides upper bounds for Wasserstein distance in unbounded spaces

Achieves rate-optimal bounds for Gaussian processes in Hilbert spaces

Yields exponential tail bounds under Bernstein and log Sobolev conditions

Abstract

We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence on the dimensionality. Our method also covers the important case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper bounds for functional data distributions whose coordinates decay geometrically or polynomially. Moreover, our bounds of the expected value can be combined with mean-concentration results to yield improved exponential tail probability bounds for the Wasserstein error of empirical measures under Bernstein-type or log Sobolev-type conditions.