Convergence of the Expectation-Maximization Algorithm Through Discrete-Time Lyapunov Stability Theory

TL;DR

This paper models the EM algorithm as a nonlinear dynamical system and uses Lyapunov stability theory to analyze its convergence, providing a novel theoretical framework for understanding its behavior.

Contribution

It introduces a dynamical systems perspective to analyze EM algorithm convergence using Lyapunov stability theory, a novel approach in this context.

Findings

EM algorithm limit points are equilibria in the dynamical system model.

Convergence of EM is established as asymptotic stability via Lyapunov methods.

Provides a new theoretical foundation for analyzing iterative algorithms' stability.

Abstract

In this paper, we propose a dynamical systems perspective of the Expectation-Maximization (EM) algorithm. More precisely, we can analyze the EM algorithm as a nonlinear state-space dynamical system. The EM algorithm is widely adopted for data clustering and density estimation in statistics, control systems, and machine learning. This algorithm belongs to a large class of iterative algorithms known as proximal point methods. In particular, we re-interpret limit points of the EM algorithm and other local maximizers of the likelihood function it seeks to optimize as equilibria in its dynamical system representation. Furthermore, we propose to assess its convergence as asymptotic stability in the sense of Lyapunov. As a consequence, we proceed by leveraging recent results regarding discrete-time Lyapunov stability theory in order to establish asymptotic stability (and thus, convergence) in the dynamical system representation of the EM algorithm.