Fast Computation of Wasserstein Barycenters

TL;DR

This paper introduces efficient algorithms for computing Wasserstein barycenters by smoothing the optimal transport problem with entropic regularization, enabling faster gradient computations for large-scale applications.

Contribution

It proposes novel algorithms leveraging entropic regularization to compute Wasserstein barycenters more efficiently than traditional methods.

Findings

Algorithms enable faster computation of Wasserstein barycenters.

Application to image visualization and constrained clustering.

Significant reduction in computational cost.

Abstract

We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter, is the measure that minimizes the sum of its Wasserstein distances to each element in that set. We propose two original algorithms to compute Wasserstein barycenters that build upon the subgradient method. A direct implementation of these algorithms is, however, too costly because it would require the repeated resolution of large primal and dual optimal transport problems to compute subgradients. Extending the work of Cuturi (2013), we propose to smooth the Wasserstein distance used in the definition of Wasserstein barycenters with an entropic regularizer and recover in doing so a strictly convex objective whose gradients can be computed for a considerably cheaper computational cost using matrix scaling algorithms. We use these algorithms to visualize a large family of images and to solve a constrained clustering problem.