Plugin Estimation of Smooth Optimal Transport Maps

Abstract

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 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 a central limit theorem for a density plugin estimator of the squared Wasserstein distance, which is centered at its 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 Wasserstein distances.

Permuted and Unlinked Monotone Regression in $\mathbb{R}^d$: an approach based on mixture modeling and optimal transport

“…Combining this with ( 41), (42), and the above choice of R then yields the assertion. Note that in the first inequality (43), we have used that B = max 1≤j≤p θ j 2 ≤ max 1≤i≤n y i 2 =: Q since ϕ(z) = ϕ( z 2 ) is decreasing in z 2 . Accordingly, we have where P denotes the Euclidean projection, which is a non-expansive operator for convex sets.…”

“…The approach taken in this paper and the techniques used for its analysis bear various connections to recent developments in the literature on optimal transport, e.g., on the estimation of (smooth) optimal transport maps [40,41,42,43,44]. Key steps in our proofs are based on adaptations of parts of the analysis in [42,43,44].…”

Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality

“…All these papers also fail to answer the question that how to determine the network size corresponding to the sampled data number to achieve a desired statistical convergence rate. [32,49] consider the similar problem for the optimal transport problem, i.e. Monge-ampere equation.…”

“…Monge-ampere equation. Nevertheless, the variational problem we considered is different from [32,49] and leads to technical difference. The most related works to ours are two concurrent papers [14,34,35].…”

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“…In particular, the estimator of Pooladian and Niles-Weed (2021) is based on entropic regularization and is computationally friendly, but only works for low regularity of the maps, and is not minimax optimal. Likewise, Manole et al (2021) provide estimators for transportation maps that achieve minimax statistical rates both in smooth and non-smooth regimes. In the smooth regime, which is the setting that we consider in the present paper, Manole et al propose a map estimator for smooth distributions that are supported on the torus.…”