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. 

Since the moments of a mixture of Gaussians are polynomial expressions in the means, variances and mixture weights, one can address this question from the perspective of algebraic geometry. With the help of tools from polyhedral geometry, we answer this fundamental question for several classes of Gaussian mixture models. Furthermore, these results allow us to present an algorithm that performs parameter recovery and density estimation, applicable even in the high dimensional case.

Based on joint work with Julia Lindberg and Jose Rodriguez (University of Wisconsin-Madison).