A Smoothed Dual Approach for Variational Wasserstein Problems

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 Carlier et al. (2014) 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

Graph Signal Representation with Wasserstein Barycenters

In many applications signals reside on the vertices of weighted graphs. Thus, there is the need to learn low dimensional representations for graph signals that will allow for data analysis and interpretation. Existing unsupervised dimensionality reduction methods for graph signals have focused on dictionary learning. In these works the graph is taken into consideration by imposing a structure or a parametrization on the dictionary and the signals are represented as linear combinations of the atoms in the dictionary. However, the assumption that graph signals can be represented using linear combinations of atoms is not always appropriate. In this paper we propose a novel representation framework based on non-linear and geometry-aware combinations of graph signals by leveraging the mathematical theory of Optimal Transport. We represent graph signals as Wasserstein barycenters and demonstrate through our experiments the potential of our proposed framework for low-dimensional graph signal representation

Absolute continuity of Wasserstein barycenters on manifolds with a lower Ricci curvature bound

Given a complete Riemannian manifold $M$ with a lower Ricci curvature bound, we consider barycenters in the Wasserstein space $\mathcal{W}_2(M)$ of probability measures on $M$. We refer to them as Wasserstein barycenters, which by definition are probability measures on $M$. The goal of this article is to present a novel approach to proving their absolute continuity. We introduce a new class of displacement functionals exploiting the Hessian equality for Wasserstein barycenters. To provide suitable instances of such functionals, we revisit Souslin space theory, Dunford-Pettis theorem and the de la Vall\'ee Poussin criterion for uniform integrability. Our method shows that if a probability measure $\mathbb{P}$ on $\mathcal{W}_2(M)$ gives mass to absolutely continuous measures on $M$, then its unique barycenter is also absolutely continuous. This generalizes the previous results on compact manifolds by Kim and Pass arXiv:1412.7726 [math.AP]