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

TL;DR

This paper extends Wasserstein-type metrics and barycenters to mixtures beyond Gaussian models, including location-scatter and symmetry-invariant measures, with applications in electronic structure calculations.

Contribution

It generalizes Wasserstein metrics to mixtures of various atom types and proves the geodesic space structure for these mixtures, including symmetry-invariant measures.

Findings

Mixtures form geodesic spaces under the new metric.

Optimal transport plans exhibit sparsity and symmetry properties.

Applicable to electronic structure calculations.

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 [A Wasserstein-Type Distance in the Space of Gaussian Mixture Models. SIAM J. Imaging Sci. 13, 936-970 (2020)] 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 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.