  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 $\mathbb{R}^d$ 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 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. 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.
