A Smoothed Dual Approach for Variational Wasserstein Problems

Variational problems that involve Wasserstein distances have been recently\nproposed to summarize and learn from probability measures. Despite being\nconceptually simple, such problems are computationally challenging because they\ninvolve minimizing over quantities (Wasserstein distances) that are themselves\nhard to compute. We show that the dual formulation of Wasserstein variational\nproblems introduced recently by Carlier et al. (2014) can be regularized using\nan entropic smoothing, which leads to smooth, differentiable, convex\noptimization problems that are simpler to implement and numerically more\nstable. We illustrate the versatility of this approach by applying it to the\ncomputation of Wasserstein barycenters and gradient flows of spacial\nregularization functionals.\n