  In this paper we introduce a Wasserstein-type distance on the set of Gaussian
mixture models. This distance is defined by restricting the set of possible
coupling measures in the optimal transport problem to Gaussian mixture models.
We derive a very simple discrete formulation for this distance, which makes it
suitable for high dimensional problems. We also study the corresponding
multi-marginal and barycenter formulations. We show some properties of this
Wasserstein-type distance, and we illustrate its practical use with some
examples in image processing.
