Marginal-Constrained Modified Wasserstein Barycenters for Gaussian Distributions and Gaussian Mixtures

The aim of this work is to propose modified Wasserstein barycenters for probability measures defined on cartesian product sets which satisfy given marginal constraints. We focus on the specific case of Gaussian and Gaussian mixture distributions, as the proposed approach strongly relies on new results about properties of geometric means of covariance matrices.Wasserstein barycenters that respect some desired constraints on the marginals of the distributions, both for Gaussian distributions and f