  The method of moments is a classical 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. In addition,
we show that a generic Gaussian $k$-mixture model is identifiable from its
first $3k+2$ moments. Using these results, we present a homotopy algorithm that
performs parameter recovery for high dimensional Gaussian mixture models where
the number of paths tracked scales linearly in the dimension.
