Minimax rates of estimation for smooth optimal transport maps
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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 rates 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 semi-dual optimal transport problem, restricted to truncated wavelet expansions. This estimator is shown to achieve near minimax optimality using new stability arguments for the semi-dual and a complementary minimax lower bound. These are the first minimax estimation rates for transport maps in general dimension.
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