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

Optimal transport maps between two probability distributions \mu and u on \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 u . 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.