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

## TL;DR

This paper introduces a new Wasserstein-type distance specifically designed for Gaussian mixture models, enabling efficient computation and practical applications in high-dimensional image processing tasks.

## Contribution

It proposes a novel Wasserstein-type distance restricted to Gaussian mixtures, with a simple discrete formulation suitable for high-dimensional problems.

## Key findings

- Simple discrete formulation for the distance
- Properties of the Wasserstein-type distance analyzed
- Practical applications demonstrated in image processing

## Abstract

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.