Large-Scale Optimal Transport and Mapping Estimation

TL;DR

This paper introduces a scalable two-step method for learning optimal transport maps using regularized OT and neural networks, with theoretical stability guarantees and applications in domain adaptation and generative modeling.

Contribution

It proposes a novel stochastic dual approach for regularized OT and a neural network-based estimation of the Monge map, improving scalability and generalization.

Findings

The method scales better than recent approaches with large sample sizes.

Theoretical stability results show convergence to true OT plan and Monge map.

Successful applications demonstrated in domain adaptation and generative modeling.

Abstract

This paper presents a novel two-step approach for the fundamental problem of learning an optimal map from one distribution to another. First, we learn an optimal transport (OT) plan, which can be thought as a one-to-many map between the two distributions. To that end, we propose a stochastic dual approach of regularized OT, and show empirically that it scales better than a recent related approach when the amount of samples is very large. Second, we estimate a \textit{Monge map} as a deep neural network learned by approximating the barycentric projection of the previously-obtained OT plan. This parameterization allows generalization of the mapping outside the support of the input measure. We prove two theoretical stability results of regularized OT which show that our estimations converge to the OT plan and Monge map between the underlying continuous measures. We showcase our proposed approach on two applications: domain adaptation and generative modeling.