  We derive an asymptotic expansion for the log likelihood of Gaussian mixture
models (GMMs) with equal covariance matrices in the low signal-to-noise regime.
The expansion reveals an intimate connection between two types of algorithms
for parameter estimation: the method of moments and likelihood optimizing
algorithms such as Expectation-Maximization (EM). We show that likelihood
optimization in the low SNR regime reduces to a sequence of least squares
optimization problems that match the moments of the estimate to the ground
truth moments one by one. This connection is a stepping stone toward the
analysis of EM and maximum likelihood estimation in a wide range of models. A
motivating application for the study of low SNR mixture models is cryo-electron
microscopy data, which can be modeled as a GMM with algebraic constraints
imposed on the mixture centers. We discuss the application of our expansion to
algebraically constrained GMMs, among other example models of interest.
