A Smoothed Dual Approach for Variational Wasserstein Problems

TLDR

Variational problems that involve Wasserstein distances have recently been proposed to summarize and learn from probability measures, yet they remain computationally challenging because the distances themselves are difficult to compute. The study shows that entropic smoothing of the dual formulation of Wasserstein variational problems yields smooth, differentiable, convex optimization problems that are simpler to implement and more numerically stable. The authors apply the smoothed dual approach to compute Wasserstein barycenters and gradient flows of spatial regularization functionals. These applications demonstrate the versatility and effectiveness of the method. The paper cites Carlier, Oberman, and Oudet (ESAIM Math.

Abstract

Variational problems that involve Wasserstein distances have been recently proposed to summarize and learn from probability measures. Despite being conceptually simple, such problems are computationally challenging because they involve minimizing over quantities (Wasserstein distances) that are themselves hard to compute. We show that the dual formulation of Wasserstein variational problems introduced recently by G. Carlier, A. Oberman, and E. Oudet [ESAIM Math. Model. Numer. Anal., 6 (2015), pp. 1621--1642] can be regularized using an entropic smoothing, which leads to smooth, differentiable, convex optimization problems that are simpler to implement and numerically more stable. We illustrate the versatility of this approach by applying it to the computation of Wasserstein barycenters and gradient flows of spacial regularization functionals. (A correction is attached.)