Convergence of the Expectation-Maximization Algorithm Through Discrete-Time Lyapunov Stability Theory
This provides a theoretical foundation for convergence analysis in statistics and machine learning, but it is incremental as it applies existing stability theory to a known algorithm.
The paper tackled the problem of analyzing the convergence of the Expectation-Maximization (EM) algorithm by reinterpreting it as a nonlinear dynamical system, and established its convergence using discrete-time Lyapunov stability theory to prove asymptotic stability.
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.