  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.
