In this paper, a regularization of Wasserstein barycenters for random measures supported on @math is introduced via convex penalization. The existence and uniqueness of such barycenters is first proved for a large class of penalization functions. The Bregman divergence associated to the penalization term is then considered to obtain a stability result on penalized barycenters. This allows the comparison of data made of @math absolutely continuous probability measures, within the more realistic setting where one only has access to a dataset of random variables sampled from unknown distributions. The convergence of the penalized empirical barycenter of a set of @math iid random probability measures towards its population counterpart is finally analyzed. This approach is shown to be appropriate for the statistical analysis of either discrete or absolutely continuous random measures. It also allows to construct, from a set of discrete measures, consistent estimators of population Wasserstein barycenters that are absolutely continuous.
A detailed characterization of empirical Wasserstein barycenters in terms of existence, uniqueness and regularity for probability measures with support in @math is given in the seminal paper @cite_26 . The relation of such barycenters with the solution of the multi-marginal problem is also studied in both @cite_26 and @cite_31 . The notion of Wasserstein barycenter has been first generalized in @cite_16 , by establishing existence, uniqueness and consistency for random probability measures supported on a locally compact geodesic space. The more general case of probability measures supported on Riemannian manifolds has then been studied in @cite_24 . Subsequently, trimmed barycenters in the Wasserstein space @math have been introduced in @cite_20 for the purpose of combining informations from different experimental units in a parallelized or distributed estimation setting. The framework of optimal transport has been recently adapted for nonnegative measures supported on a compact of @math with different masses, independently by @cite_12 and @cite_33 . However, in all these papers, incorporating regularization into the computation of Wasserstein barycenters has not been considered, which is of interest when the data are irregular probability measures.
@cite_26 @cite_33 @cite_24 @cite_31 @cite_16 @cite_12 @cite_20
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Abstract In this paper we introduce a new transportation distance between non-negative measures inside a domain Ω . This distance enjoys many nice properties, for instance it makes the space of non-negative measures inside Ω a geodesic space without any convexity assumption on the domain. Moreover we will show that the gradient flow of the entropy functional ∫ Ω [ ρ log ( ρ ) − ρ ] d x with respect to this distance coincides with the heat equation, subject to the Dirichlet boundary condition equal to 1. We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures Ω∈P(P(M))Ω∈P(P(M)) on Wasserstein space, extending on one hand, results in the Euclidean case (for barycenters between finitely many measures) of Agueh and Carlier [1] to the Riemannian setting, and on the other hand, results in the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschlager [12] for barycenters between two measures to the multi-marginal setting. Our work also extends these results to the case where Ω is not finitely supported. As applications, we prove versions of Jensen's inequality on Wasserstein space and a generalized Brunn–Minkowski inequality for a random measurable set on a Riemannian manifold. Abstract We formulate and study an optimal transportation problem with infinitely many marginals; this is a natural extension of the multi-marginal problem studied by Gangbo and Świȩch (1998) [15] . We prove results on the existence, uniqueness and characterization of the optimizer, which are natural extensions of the results in Gangbo and Świȩch (1998) [15] . The proof relies on a relationship between this problem and the problem of finding barycenters in the Wasserstein space, a connection first observed for finitely many marginals by Agueh and Carlier (2011) [1] . In this paper, based on the Frechet mean, we define a notion of barycenter corresponding to a usual notion of statistical mean. We prove the existence of Wasserstein barycenters of random distributions defined on a geodesic space (E, d). We also prove the consistency of this barycenter in a general setting, that includes taking barycenters of empirical versions of the distributions or of a growing set of distributions. This paper defines a new transport metric over the space of nonnegative measures. This metric interpolates between the quadratic Wasserstein and the Fisher–Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher–Rao metric and is averaged with the transportation term. This gives rise to a convex variational problem defining the new metric. Our first contribution is a proof of the existence of geodesics (i.e., solutions to this variational problem). We then show that (generalized) optimal transport and Hellinger metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Dirac measures are made of translating mixtures of Dirac masses. Lastly, we propose a numerical scheme making use of first-order proximal splitting methods and we show an application of this new distance to image interpolation.