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The Wasserstein distance is a distance between two probability distributions and has recently gained increasing popularity in statistics and machine learning, owing to its attractive properties. One important approach to extending this…
Numerous infinite dimensional dynamical systems arising in different fields have been shown to exhibit a gradient flow structure in the Wasserstein space. We construct Two Point Flux Approximation Finite Volume schemes discretizing such…
The theory of Wasserstein gradient flows in the space of probability measures has made an enormous progress over the last twenty years. It constitutes a unified and powerful framework in the study of dissipative partial differential…
This paper is devoted to solving a class of second order Hamilton-Jacobi-Bellman (HJB) equations in the Wasserstein space, associated with mean field control problems involving common noise. The well-posedness of viscosity solutions to the…
The dynamical formulation of the optimal transport can be extended through various choices of the underlying geometry (kinetic energy), and the regularization of density paths (potential energy). These combinations yield different…
We propose a new method for smoothly interpolating probability measures using the geometry of optimal transport. To that end, we reduce this problem to the classical Euclidean setting, allowing us to directly leverage the extensive toolbox…
We propose to compute Wasserstein barycenters (WBs) by solving for Monge maps with variational principle. We discuss the metric properties of WBs and explore their connections, especially the connections of Monge WBs, to K-means clustering…
Statistical solutions have recently been introduced as a an alternative solution framework for hyperbolic systems of conservation laws. In this work we derive a novel a posteriori error estimate in the Wasserstein distance between…
Optimal transport is a notoriously difficult problem to solve numerically, with current approaches often remaining intractable for very large scale applications such as those encountered in machine learning. Wasserstein barycenters -- the…
This paper deals with the estimation of a probability measure on the real line from data observed with an additive noise. We are interested in rates of convergence for the Wasserstein metric of order $p\geq 1$. The distribution of the…
We introduce Wasserstein consensus alternating direction method of multipliers (ADMM) and its entropic-regularized version: Sinkhorn consensus ADMM, to solve measure-valued optimization problems with convex additive objectives. Several…
This paper presents a unified approach based on Wasserstein distance to derive concentration bounds for empirical estimates for two broad classes of risk measures defined in the paper. The classes of risk measures introduced include as…
This paper presents a computational framework for the concise encoding of an ensemble of persistence diagrams, in the form of weighted Wasserstein barycenters [100], [102] of a dictionary of atom diagrams. We introduce a multi-scale…
As interest in graph data has grown in recent years, the computation of various geometric tools has become essential. In some area such as mesh processing, they often rely on the computation of geodesics and shortest paths in discretized…
Since the early nineties, it has been observed that the Schroedinger bridge problem can be formulated as a stochastic control problem with atypical boundary constraints. This in turn has a fluid dynamic counterpart where the flow of…
Wasserstein barycenters correspond to optimal solutions of transportation problems for several marginals, and as such have a wide range of applications ranging from economics to statistics and computer science. When the marginal probability…
This paper studies a distributionally robust chance constrained program (DRCCP) with Wasserstein ambiguity set, where the uncertain constraints should be satisfied with a probability at least a given threshold for all the probability…
Wasserstein distances are widely used in modern data analysis but pose significant computational and statistical challenges in high dimensions. The sliced Wasserstein distance alleviates these challenges by leveraging one-dimensional…
In this paper, a regularization of Wasserstein barycenters for random measures supported on $\mathbb{R}^{d}$ is introduced via convex penalization. The existence and uniqueness of such barycenters is first proved for a large class of…
The sliced Wasserstein barycenter (SWB) is a widely acknowledged method for efficiently generalizing the averaging operation within probability measure spaces. However, achieving marginal fairness SWB, ensuring approximately equal distances…