Sample complexity of optimal transport barycenters with discrete support

TL;DR

This paper establishes statistical generalization bounds for empirical optimal transport barycenters with discrete support, applicable across various divergences and settings, advancing understanding of approximation quality in computational optimal transport.

Contribution

It provides the first $O( oot{N}{n})$ bounds for empirical sparse optimal transport barycenters, covering multiple divergences and practical applications.

Findings

Bounds apply to Wasserstein, Sinkhorn, and Sliced-Wasserstein divergences.

Results include applications to K-means and constrained barycenter problems.

Analysis demonstrates how sample size influences approximation accuracy.

Abstract

Computational implementation of optimal transport barycenters for a set of target probability measures requires a form of approximation, a widespread solution being empirical approximation of measures. We provide an $O(\sqrt{N/n})$ statistical generalization bounds for the empirical sparse optimal transport barycenters problem, where $N$ is the maximum cardinality of the barycenter (sparse support) and $n$ is the sample size of the target measures empirical approximation. Our analysis includes various optimal transport divergences including Wasserstein, Sinkhorn and Sliced-Wasserstein. We discuss the application of our result to specific settings including K-means, constrained K-means, free and fixed support Wasserstein barycenters.