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.

翻译：我们分析了一系列用于估计两个分布之间最优传输映射的自然估计量，并证明它们达到了极小极大最优性。我们采用插件方法：我们的估计量仅仅是基于观测数据导出的测度之间的最优耦合，并经过适当扩展以定义在 $\mathbb{R}^d$ 上的函数。当基础映射被假设为Lipschitz连续时，我们证明计算经验测度之间的最优耦合，并使用线性平滑器进行扩展，即可得到一个极小极大最优估计量。当基础映射具有更高正则性时，我们证明在适当的非参数密度估计之间进行最优耦合能够获得更快的收敛速率。我们的工作还为二次Wasserstein距离对应插件估计量的风险提供了新的界，并通过光滑且强凸Brenier势函数的稳定性论证，展示了该问题与估计最优传输映射之间的联系。作为我们结果的一个应用，我们推导了平方Wasserstein距离插件估计量的中心极限定理，当基础分布具有足够光滑的密度时，这些估计量以其总体对应值为中心。与已知的经验估计量中心极限定理不同，此结果易于用于二次Wasserstein距离的统计推断。