Likelihood Maximization and Moment Matching in Low SNR Gaussian Mixture Models |
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Authors: | Anya Katsevich Afonso S Bandeira |
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Institution: | 1. Courant Institute, 251 Mercer St, New York, NY, 10012 USA;2. HG G 23.1, ETH Zurich, Ramistrasse 101, 8092 Zurich, Switzerland |
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Abstract: | 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 towards 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. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. |
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