A Wasserstein-type distance in the space of Gaussian Mixture Models

Julie Delon (1) , Agnès Desolneux (2)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 CMLA - Centre de Mathématiques et de Leurs Applications

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.

Mots clés: optimal transport; Wasserstein distance; Gaussian mixture model; multi-marginal optimal trans- port; barycenter; image processing applications; 65K05; 90C05; 62-07; 68Q25; 68U10; 68U05; 68R10; multi-marginal optimal trans- 10 port; image processing applications 11 AMS subject classifications 65K10; 68R10 12

Domaines: Optimisation et contrôle [math.OC]

Julie Delon, Agnès Desolneux. A Wasserstein-type distance in the space of Gaussian Mixture Models. 2019. ⟨hal-02178204v2⟩