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We derive an asymptotic expansion for the log likelihood of Gaussian mixturemodels (GMMs) with equal covariance matrices in the low signal-to-noise regime.The expansion reveals an intimate connection between two types of algorithmsfor parameter estimation: the method of moments and likelihood optimizingalgorithms such as Expectation-Maximization (EM). We show that likelihoodoptimization in the low SNR regime reduces to a sequence of least squaresoptimization problems that match the moments of the estimate to the groundtruth moments one by one. This connection is a stepping stone toward theanalysis of EM and maximum likelihood estimation in a wide range of models. Amotivating application for the study of low SNR mixture models is cryo-electronmicroscopy data, which can be modeled as a GMM with algebraic constraintsimposed on the mixture centers. We discuss the application of our expansion toalgebraically constrained GMMs, among other example models of interest.

Likelihood Maximization and Moment Matching in Low SNR Gaussian Mixture Models