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

## TL;DR

This paper analyzes a generalized EM algorithm for Gaussian mixture models from a control systems perspective, revealing its convergence properties and design advantages using robust control theory tools.

## Contribution

It introduces a control-theoretic analysis of a generalized EM algorithm, providing new insights into its convergence and design for Gaussian mixture models.

## Key findings

- GEM can be modeled as an LTI system with feedback nonlinearity.
- Convergence properties are analyzed using robust control theory.
- The approach offers a pedagogical example demonstrating advantages.

## Abstract

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 generalized EM (GEM) algorithm in the context of Gaussian mixture models, where the maximization step in the EM is replaced by an increasing step. We show that this GEM algorithm 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.