Estimating Gaussian mixtures using sparse polynomial moment systems

The method of moments is a statistical technique for density estimation that solves a system of moment equations to estimate the parameters of an unknown distribution. A fundamental question critical to understanding identifiability asks how many moment equations are needed to get finitely many solutions and how many solutions there are. We answer this question for classes of Gaussian mixture models using the tools of polyhedral geometry. Using these results, we present a homotopy method to perform parameter recovery, and therefore density estimation, for high dimensional Gaussian mixture models. The number of paths tracked in our method scales linearly in the dimension.