Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections Nabarun Deb Promit Ghosal Bodhisattva Sen Columbia University MIT Columbia University

In practice, these maps need to be estimated from data sampled according to µ and ν. Plugin 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 discretediscrete and semi-discrete estimators of optimal transport maps.