Smoothed estimation of Wasserstein barycenters

TL;DR

This paper introduces a smoothness-aware method for estimating Wasserstein barycenters, leveraging Sobolev geometry to improve statistical convergence rates in high-dimensional settings.

Contribution

It develops a novel approach combining density estimation with Sobolev geometry, achieving better nonparametric convergence rates for Wasserstein barycenters.

Findings

Non-asymptotic results show improved convergence rates with smoothness.

The method mitigates the curse of dimensionality in barycenter estimation.

Theoretical analysis demonstrates the benefits of Sobolev structure in estimation accuracy.

Abstract

This paper studies the statistical estimation of exact Wasserstein barycenters. Existing non-asymptotic results for empirical barycenters exhibit a severe curse of dimensionality. Motivated by the semi-dual formulation of the barycenter problem and its associated Sobolev optimization geometry, we develop a smoothness-aware approach that combines density estimation with Sobolev geometric structure to estimate the population barycenter. We establish nonparametric convergence rates for estimating both the barycenter functional and its minimizer, demonstrating how smoothness can substantially improve statistical performance.