Sliced Wasserstein Distance for Learning Gaussian Mixture Models

TL;DR

This paper introduces a novel GMM estimation method using sliced Wasserstein distance, which offers better robustness and fidelity in high-dimensional data compared to traditional EM algorithms.

Contribution

It proposes a new GMM parameter estimation approach based on sliced Wasserstein distance, improving robustness and high-dimensional data modeling.

Findings

More robust to random initializations

Better high-dimensional data distribution estimation

Energy landscape is more well-behaved

Abstract

Gaussian mixture models (GMM) are powerful parametric tools with many applications in machine learning and computer vision. Expectation maximization (EM) is the most popular algorithm for estimating the GMM parameters. However, EM guarantees only convergence to a stationary point of the log-likelihood function, which could be arbitrarily worse than the optimal solution. Inspired by the relationship between the negative log-likelihood function and the Kullback-Leibler (KL) divergence, we propose an alternative formulation for estimating the GMM parameters using the sliced Wasserstein distance, which gives rise to a new algorithm. Specifically, we propose minimizing the sliced-Wasserstein distance between the mixture model and the data distribution with respect to the GMM parameters. In contrast to the KL-divergence, the energy landscape for the sliced-Wasserstein distance is more well-behaved and therefore more suitable for a stochastic gradient descent scheme to obtain the optimal GMM parameters. We show that our formulation results in parameter estimates that are more robust to random initializations and demonstrate that it can estimate high-dimensional data distributions more faithfully than the EM algorithm.