We consider the problem of estimating the optimal transport map between a (fixed) source distribution P and an unknown target distribution Q, based on samples from Q. The estimation of such optimal transport maps has become increasingly relevant in modern statistical applications, such as generative modeling. At present, estimation rates are only known in a few settings (e.g. when P and Q have densities bounded above and below and when the transport map lies in a Holder class), which are often not reflected in practice. On the contrary, we present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces, based on the optimization of the empirical dual transport problem.
Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure P satisfies a Poincaré inequality and that the optimal map is 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 bounded densities and Holder transport maps, but also obtain nearly sharp results in many settings not covered by prior work. For example, we are able to give statistical rates of estimation when P is the normal distribution and the transport map is given by an infinite-width shallow neural network.
Joint work with Aram-Alexandre Pooladian and Jonathan Niles-Weed