Computing Optimal Transport Maps and Wasserstein Barycenters Using Conditional Normalizing Flows

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.

Wasserstein barycenters provide a natural way of defining the average of a set of probability distributions and are widely used in applications such as shape interpolation, style translation, and fairness. However, computing them remains difficult, particularly in high-dimensional spaces. In this paper, we introduce a new method based on conditional normalizing flows that directly leverages the primal formulation of the problem, unlike prior approaches that rely on the dual formulation. Our approach delivers accurate results in high dimensions, compares favorably with existing state-of-the-art methods, and can efficiently compute barycenters for hundreds of input distributions, which was computationally infeasible with previous methods.