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.

中文翻译：

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低信噪比高斯混合模型中的似然最大化和矩匹配

我们推导出在低信噪比状态下具有相等协方差矩阵的高斯混合模型 (GMM) 的对数似然性的渐近展开。扩展揭示了用于参数估计的两种算法之间的密切联系：矩量法和似然优化算法，例如期望最大化 (EM)。我们表明，低 SNR 状态下的似然优化可简化为一系列最小二乘优化问题，这些问题将估计矩与地面真值矩逐一匹配。这种联系是在各种模型中分析 EM 和最大似然估计的垫脚石。研究低 SNR 混合模型的一个激励应用是低温电子显微镜数据，可以将其建模为对混合中心施加代数约束的 GMM。我们讨论了我们的扩展在代数约束 GMM 中的应用，以及其他感兴趣的示例模型。© 2022 作者。Communications on Pure and Applied Mathematics由 Wiley Periodicals LLC 出版。