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.

我们研究了用于两个分布间优化传输图的自然估计器，并证明它们是极小极大最优的。我们采用插件法，并展示了最优耦合与经验度量之间的计算，以及使用线性平滑器的扩展，已经给出了一个极小极大最优估计器。当底层地图具有更高的正则性时，我们展示了合适的非参数密度估计之间的最优耦合，能够提供更快的速率。我们的工作为插件估计器提供了新的风险上限，并展示了如何使用平稳性论据和光滑且强凸的 Brenier 势函数来估计优化传输图的问题。作为我们结果的一个应用，当底层分布具有足够光滑的密度时，我们推导了插件估计器的平方 Wasserstein 距离的中心极限定理，与经验估计器已知的中心极限定理不同，此结果更容易用于统计推断的二次 Wasserstein 距离。