Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections

TL;DR

This paper analyzes the convergence rates of plug-in estimators for optimal transport maps using barycentric projections, highlighting how smoothness assumptions can improve estimation speed and reduce dimensionality issues.

Contribution

It introduces a new stability estimate for barycentric projections applicable under minimal smoothness, enabling comprehensive analysis of various plug-in estimators and their convergence rates.

Findings

Wavelet and kernel estimators accelerate convergence under smoothness assumptions.

Faster convergence rates for Wasserstein distance estimators under smoothness.

Application to Wasserstein barycenters and independence testing thresholds.

Abstract

Optimal transport maps between two probability distributions $\mu$ and $\nu$ on $\mathbb{R}^d$ have found extensive applications in both machine learning and statistics. In practice, these maps need to be estimated from data sampled according to $\mu$ and $\nu$. Plug-in estimators are perhaps most popular in estimating transport maps in the field of computational optimal transport. In this paper, we provide a comprehensive analysis of the rates of convergences for general plug-in estimators defined via barycentric projections. Our main contribution is a new stability estimate for barycentric projections which proceeds under minimal smoothness assumptions and can be used to analyze general plug-in estimators. We illustrate the usefulness of this stability estimate by first providing rates of convergence for the natural discrete-discrete and semi-discrete estimators of optimal transport maps. We then use the same stability estimate to show that, under additional smoothness assumptions of Besov type or Sobolev type, wavelet based or kernel smoothed plug-in estimators respectively speed up the rates of convergence and significantly mitigate the curse of dimensionality suffered by the natural discrete-discrete/semi-discrete estimators. As a by-product of our analysis, we also obtain faster rates of convergence for plug-in estimators of $W_2(\mu,\nu)$, the Wasserstein distance between $\mu$ and $\nu$, under the aforementioned smoothness assumptions, thereby complementing recent results in Chizat et al. (2020). Finally, we illustrate the applicability of our results in obtaining rates of convergence for Wasserstein barycenters between two probability distributions and obtaining asymptotic detection thresholds for some recent optimal-transport based tests of independence.