Minimax estimation of smooth optimal transport maps

Brenier’s theorem is a cornerstone of optimal transport that guarantees the existence an map T between two probability distributions P and Q over Rd under certain regularity conditions. The main goal this work to establish minimax estimation rates for such from data sampled additional smoothness assumptions on T. To achieve goal, we develop estimator based minimization empirical version semidual problem, restricted truncated wavelet expansions. This shown near optimality using new stability arguments complementary lower bound. Furthermore, provide numerical experiments synthetic supporting our theoretical findings highlighting practical benefits regularization. These are first maps in general dimension.