A natural way to represent complex probability measures is to couple them with a known reference distribution via some transport map. In this talk, we study nonparametric density estimators which arise from measure transport. Given a smooth reference distribution and random samples from the unknown density, the estimators are given as the pushforward of a transport map minimizing the empirical Kullback-Leibler risk, or a penalized version thereof. For triangular Knothe-Rosenblatt transports on the d-dimensional unit cube, it is shown that both the non-penalized and penalized versions achieve minimax optimal convergence rates over H\"older classes of densities. The same is proved for certain sieved wavelet estimators.
Our results follow from a concentration inequality for general penalized measure-transport estimators, which is proved using techniques from M-estimation. The minimax rates for triangular maps are then derived using certain natural anisotropic regularity classes which correspond to common isotropic smoothness classes of target densities. This is joint work with Y. Marzouk (MIT).