An Alternative to EM for Gaussian Mixture Models: Batch and Stochastic Riemannian Optimization

TL;DR

This paper introduces Riemannian optimization techniques as an alternative to EM for Gaussian Mixture Model parameter estimation, demonstrating superior performance through novel algorithms and convergence analysis.

Contribution

It develops new Riemannian gradient algorithms for GMM estimation, improving over EM, with a first non-asymptotic convergence analysis for Riemannian stochastic gradient methods.

Findings

Riemannian formulation initially underperforms compared to EM

Refined Riemannian approach significantly outperforms EM in experiments

First global convergence analysis for Riemannian stochastic gradient methods

Abstract

We consider maximum likelihood estimation for Gaussian Mixture Models (Gmms). This task is almost invariably solved (in theory and practice) via the Expectation Maximization (EM) algorithm. EM owes its success to various factors, of which is its ability to fulfill positive definiteness constraints in closed form is of key importance. We propose an alternative to EM by appealing to the rich Riemannian geometry of positive definite matrices, using which we cast Gmm parameter estimation as a Riemannian optimization problem. Surprisingly, such an out-of-the-box Riemannian formulation completely fails and proves much inferior to EM. This motivates us to take a closer look at the problem geometry, and derive a better formulation that is much more amenable to Riemannian optimization. We then develop (Riemannian) batch and stochastic gradient algorithms that outperform EM, often substantially. We provide a non-asymptotic convergence analysis for our stochastic method, which is also the first (to our knowledge) such global analysis for Riemannian stochastic gradient. Numerous empirical results are included to demonstrate the effectiveness of our methods.