Title: Convergence of the Expectation-Maximization Algorithm Through Discrete-Time Lyapunov Stability Theory,
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.