On minimax density estimation via measure transport

We study the convergence properties, in Hellinger and related distances, of nonparametric density estimators based on measure transport. These estimators represent the measure of interest as the pushforward of a chosen reference distribution under a transport map, where the map is chosen via a maximum likelihood objective (equivalently, minimizing an empirical Kullback-Leibler loss) or a penalized version thereof. We establish concentration inequalities for a general class of penalized measure transport estimators, by combining techniques from M-estimation with analytical properties of the transport-based density representation. We then demonstrate the implications of our theory for the case of triangular Knothe-Rosenblatt (KR) transports on the d-dimensional unit cube, and show that both penalized and unpenalized versions of such estimators achieve minimax optimal convergence rates over H"older classes of densities. Specifically, we establish optimal rates for unpenalized nonparametric maximum likelihood estimation over bounded H"older-type balls, and then for certain Sobolev-penalized estimators and sieved wavelet estimators.