Given two probability distributions in R^d, a transport map is a function which maps samples from one distribution into samples from the other. For absolutely continuous measures, Brenier proved a remarkable theorem identifying a unique canonical transport map, which is "monotone" in a suitable sense. We study the question of whether this map can be efficiently estimated from samples. The minimax rates for this problem were recently established by Hutter and Rigollet (2021), but the estimator they propose is computationally infeasible in dimensions greater than three. We propose two new estimators---one minimax optimal, one not---which are significantly more practical to compute and implement. The analysis of these estimators is based on new stability results for the optimal transport problem and its regularized variants. Based on joint work with Manole, Balakrishnan, and Wasserman and with Pooladian.
