Plugin Estimation of Smooth Optimal Transport Maps
This work addresses the challenge of efficient and statistically sound estimation of optimal transport maps, which is crucial for applications in machine learning and statistics, though it is incremental in refining existing plugin methods.
The paper tackles the problem of estimating optimal transport maps between two distributions by analyzing plugin estimators, showing they achieve minimax optimal rates under Lipschitz assumptions and faster rates with higher regularity. It also provides new bounds for quadratic Wasserstein distance estimators and derives central limit theorems for statistical inference.
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.