Central Limit Theorems for Smooth Optimal Transport Maps

Abstract: One of the central objects in the theory of optimal transport is the Brenier map: the unique monotone transformation which pushes forward an absolutely continuous probability law onto any other given law. Recent work has identified a class of plugin estimators of Brenier maps which achieve the minimax L^2 risk, and are simple to compute. In this talk, we show that such estimators obey pointwise central limit theorems. This provides a first step toward the question of performing statistical inference for smooth Brenier maps in general dimension. We further show that these results have implications for the problem of estimating the 2-Wasserstein distance. In particular, they allow us to develop the higher-order semiparametric efficiency theory for the Wasserstein distance, and as a consequence, we derive an efficient estimator of the Wasserstein distance under nearly optimal smoothness conditions.

This talk is based on joint work with Sivaraman Balakrishnan, Jonathan Niles-Weed, and Larry Wasserman.

Bio: Tudor Manole is a Norbert Wiener postdoctoral associate in the Statistics and Data Science Center at the Massachusetts Institute of Technology (MIT). He earned his PhD in Statistics at Carnegie Mellon University, where he was advised by Sivaraman Balakrishnan and Larry Wasserman. He is broadly interested in nonparametric statistics and statistical machine learning. Some specific research interests include statistical optimal transport, latent variable models, minimax hypothesis testing, and their applications to the physical sciences.