GAN Estimation of Lipschitz Optimal Transport Maps

TL;DR

This paper presents a novel neural network-based estimator for optimal transport maps that is both statistically consistent and practically feasible, with convergence guarantees and promising numerical results.

Contribution

It introduces the first consistent neural network estimator for optimal transport maps with theoretical convergence and practical performance insights.

Findings

Estimator converges uniformly to the true transport map with increasing data.

Numerical experiments show the learned map performs well in practice.

Provides a feasible method with statistical guarantees for optimal transport applications.

Abstract

This paper introduces the first statistically consistent estimator of the optimal transport map between two probability distributions, based on neural networks. Building on theoretical and practical advances in the field of Lipschitz neural networks, we define a Lipschitz-constrained generative adversarial network penalized by the quadratic transportation cost. Then, we demonstrate that, under regularity assumptions, the obtained generator converges uniformly to the optimal transport map as the sample size increases to infinity. Furthermore, we show through a number of numerical experiments that the learnt mapping has promising performances. In contrast to previous work tackling either statistical guarantees or practicality, we provide an expressive and feasible estimator which paves way for optimal transport applications where the asymptotic behaviour must be certified.