A Wasserstein-type metric for generic mixture models, including location-scatter and group invariant measures

¹ Université Marie et Louis Pasteur, CNRS, LmB (UMR 6623), F-25000 Besançon, France
² CERMICS, École des Ponts & Inria Paris, France
³ CEA Saclay, DEN/DM2S/STMF/LMSF, France

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 14 November 2025
Accepted: 7 January 2026

Abstract

In this article, we study Wasserstein-type metrics and corresponding barycenters for mixtures of a chosen subset of probability measures called atoms hereafter. In particular, this works extends what was proposed by Delon and Desolneux [SIAM J. Imaging Sci. 13 (2020) 936-970] for mixtures of Gaussian measures to other mixtures. We first prove in a general setting that for a set of atoms equipped with a metric that defines a geodesic space, the set of mixtures based on this set of atoms is a also geodesic space for the defined modified Wasserstein metric. We then focus on two particular cases of sets of atoms: (i) the set of location-scatter atoms and (ii) the set of measures that are invariant with respect to some symmetry group. Both cases are particularly relevant for various applications among which electronic structure calculations. Along the way, we also prove some sparsity and symmetry properties of optimal transport plans between measures that are invariant under some well-chosen symmetries.