MINIMAX ESTIMATION OF SMOOTH OPTIMAL TRANSPORT MAPS

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

Automated summary

The Brenier's theorem establishes the existence of an optimal transport map T between two probability distributions under certain regularity conditions. This work aims to establish minimax estimation rates for such a transport map from data sampled from the distributions under smoothness assumptions on T. By developing an estimator based on empirical versions of the semidual optimal transport problem and providing numerical experiments supporting the theoretical findings, the study demonstrates the practical benefits of smoothness regularization in achieving near minimax optimality for transport maps in general dimension.