DeepAI
Log In Sign Up

Plugin Estimation of Smooth Optimal Transport Maps

07/26/2021
by   Tudor Manole, et al.
2

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

05/14/2019

Minimax rates of estimation for smooth optimal transport maps

Brenier's theorem is a cornerstone of optimal transport that guarantees ...
05/26/2019

Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport

The problem of estimating Wasserstein distances in high-dimensional spac...
06/24/2021

Sharp Convergence Rates for Empirical Optimal Transport with Smooth Costs

We revisit the question of characterizing the convergence rate of plug-i...
06/15/2020

Faster Wasserstein Distance Estimation with the Sinkhorn Divergence

The squared Wasserstein distance is a natural quantity to compare probab...
07/05/2021

Fast and Scalable Optimal Transport for Brain Tractograms

We present a new multiscale algorithm for solving regularized Optimal Tr...
06/22/2020

On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification

Optimal transport (OT) distances are increasingly used as loss functions...
09/21/2022

Quantitative Stability of Barycenters in the Wasserstein Space

Wasserstein barycenters define averages of probability measures in a geo...