  We analyze a number of natural estimators for the optimal transport map
between two distributions and show that they are minimax optimal. We adopt the
plugin approach: our estimators are simply optimal couplings between measures
derived from our observations, appropriately extended so that they define
functions on $\mathbb{R}^d$. When the underlying map is assumed to be
Lipschitz, we show that computing the optimal coupling between the empirical
measures, and extending it using linear smoothers, already gives a minimax
optimal estimator. When the underlying map enjoys higher regularity, we show
that the optimal coupling between appropriate nonparametric density estimates
yields faster rates. Our work also provides new bounds on the risk of
corresponding plugin estimators for the quadratic Wasserstein distance, and we
show how this problem relates to that of estimating optimal transport maps
using stability arguments for smooth and strongly convex Brenier potentials. As
an application of our results, we derive central limit theorems for plugin
estimators of the squared Wasserstein distance, which are centered at their
population counterpart when the underlying distributions have sufficiently
smooth densities. In contrast to known central limit theorems for empirical
estimators, this result easily lends itself to statistical inference for the
quadratic Wasserstein distance.
