Analysis of Generalized Expectation-Maximization for Gaussian Mixture Models: A Control Systems Perspective

The Expectation-Maximization (EM) algorithm is one of the most popular methods used to solve the problem of parametric distribution-based clustering in unsupervised learning. In this paper, we propose to analyze a subclass of generalized EM (GEM) algorithms in the context of Gaussian mixture models, where the maximization step in the EM is replaced by an increasing step. We show that this subclass of GEM algorithms can be understood as a linear time-invariant (LTI) system with a feedback nonlinearity. Therefore, we explore some of its convergence properties by leveraging tools from robust control theory. Lastly, we explain how the proposed GEM can be designed, and present a pedagogical example to understand the advantages of the proposed approach.