A Relaxed Wasserstein Distance Formulation for Mixtures of Radially Contoured Distributions

TL;DR

The paper addresses the problem of comparing mixtures of radially contoured distributions when components originate from different families and marginal consistency is not guaranteed. It introduces a relaxed Wasserstein distance that remains identifiable in such mixtures and does not require marginal consistency. The approach is evaluated on color transfer tasks, where it shows more stable error and yields more desirable color distributions than a Wasserstein-type distance designed for Gaussian mixtures. This work broadens the applicability of Wasserstein-based distances to heterogeneous mixture components, enabling robust image processing applications.

Abstract

Recently, a Wasserstein-type distance for Gaussian mixture models has been proposed. However, that framework can only be generalized to identifiable mixtures of general elliptically contoured distributions whose components come from the same family and satisfy marginal consistency. In this paper, we propose a simple relaxed Wasserstein distance for identifiable mixtures of radially contoured distributions whose components can come from different families. We show some properties of this distance and that its definition does not require marginal consistency. We apply this distance in color transfer tasks and compare its performance with the Wasserstein-type distance for Gaussian mixture models in an experiment. The error of our method is more stable and the color distribution of our output image is more desirable.

Figures (2)

• Figure 1: Example figure using external image files.
• Figure 2: Example PGFPLOTS figure.

Theorems & Definitions (5)

• Theorem 6.1: Mean Value Theorem
• Corollary 6.2
• Proof 1
• Claim 6.3
• Proof 2: Proof of main theorem