Optimal transport map estimation in general function spaces

Stochastics Seminar
Thursday, March 14, 2024 - 3:30pm for 1 hour (actually 50 minutes)
Skiles 006
Jonathan Niles-Weed – New York University
Cheng Mao

We present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces. Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure P satisfy a Poincare inequality and that the optimal map be the gradient of a smooth convex function that lies in a space whose metric entropy can be controlled. As a special case, we recover known estimation rates for Holder transport maps, but also obtain nearly sharp results in many settings not covered by prior work. For example, we provide the first statistical rates of estimation when P is the normal distribution, between log-smooth and strongly log-concave distributions, and when the transport map is given by an infinite-width shallow neural network. (joint with Vincent Divol and Aram-Alexandre Pooladian.)