  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.
