# Solving general elliptical mixture models through an approximate Wasserstein manifold

## TL;DR

This paper introduces a novel stable and robust method for estimating elliptical mixture models using an approximate Wasserstein distance, outperforming traditional approaches in stability and convergence.

## Contribution

It proposes an efficient optimization framework on a statistical manifold with an approximate Wasserstein distance for elliptical mixture models, improving stability and convergence.

## Key findings

- Outperforms existing methods in stability and accuracy
- Provides a unifying account of computable elliptical mixture models
- Demonstrates superior performance through experiments

## Abstract

We address the estimation problem for general finite mixture models, with a particular focus on the elliptical mixture models (EMMs). Compared to the widely adopted Kullback-Leibler divergence, we show that the Wasserstein distance provides a more desirable optimisation space. We thus provide a stable solution to the EMMs that is both robust to initialisations and reaches a superior optimum by adaptively optimising along a manifold of an approximate Wasserstein distance. To this end, we first provide a unifying account of computable and identifiable EMMs, which serves as a basis to rigorously address the underpinning optimisation problem. Due to a probability constraint, solving this problem is extremely cumbersome and unstable, especially under the Wasserstein distance. To relieve this issue, we introduce an efficient optimisation method on a statistical manifold defined under an approximate Wasserstein distance, which allows for explicit metrics and computable operations, thus significantly stabilising and improving the EMM estimation. We further propose an adaptive method to accelerate the convergence. Experimental results demonstrate the excellent performance of the proposed EMM solver.