Matrix Manifold Optimization for Gaussian Mixtures

We take a new look at parameter estimation for Gaussian Mixture Model (GMMs). Specifically, we advance Riemannian manifold optimization (on the manifold of positive definite matrices) as a potential replacement for Expectation Maximization (EM), which has been the de facto standard for decades. An out-of-the-box invocation of Riemannian optimization, however, fails spectacularly: it obtains the same solution as EM, but vastly slower. Building on intuition from geometric convexity, we propose a simple reformulation that has remarkable consequences: it makes Riemannian optimization not only match EM (a nontrivial result on its own, given the poor record nonlinear programming has had against EM), but also outperform it in many settings. To bring our ideas to fruition, we develop a well-tuned Riemannian LBFGS method that proves superior to known competing methods (e.g., Riemannian conjugate gradient). We hope that our results encourage a wider consideration of manifold optimization in machine learning and statistics.