Computing Optimal Transport Maps and Wasserstein Barycenters Using Conditional Normalizing Flows

TL;DR

This paper introduces a new method using conditional normalizing flows to efficiently compute optimal transport maps and Wasserstein barycenters in high-dimensional spaces, enabling scalable and accurate solutions.

Contribution

The paper presents a novel approach leveraging conditional normalizing flows for direct primal optimization of transport and barycenters, improving scalability and accuracy.

Findings

Accurately computes transport maps in high dimensions

Enables barycenter computation for hundreds of distributions

Outperforms previous methods in numerical experiments

Abstract

We present a novel method for efficiently computing optimal transport maps and Wasserstein barycenters in high-dimensional spaces. Our approach uses conditional normalizing flows to approximate the input distributions as invertible pushforward transformations from a common latent space. This makes it possible to directly solve the primal problem using gradient-based minimization of the transport cost, unlike previous methods that rely on dual formulations and complex adversarial optimization. We show how this approach can be extended to compute Wasserstein barycenters by solving a conditional variance minimization problem. A key advantage of our conditional architecture is that it enables the computation of barycenters for hundreds of input distributions, which was computationally infeasible with previous methods. Our numerical experiments illustrate that our approach yields accurate results across various high-dimensional tasks and compares favorably with previous state-of-the-art methods.