Generalized Wasserstein barycenters between probability measures living on different subspaces

In this paper, we introduce a generalization of the Wasserstein barycenter, to a case where the initial probability measures live on different subspaces of Rd. We study the existence and uniqueness of this barycenter, we show how it is related to a larger multimarginal optimal transport problem, and we propose a dual formulation. Finally, we explain how to compute numerically this generalized barycenter on discrete distributions, and we propose an explicit solution for Gaussian distributions.