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

arXiv: https://arxiv.org/abs/1810.02022 · HTML: https://mlsanity.com/paper/1810.02022/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Oct 2018.

## Built on by (most cited first, 2 of 2)

- [Bridging Associative Memory and Probabilistic Modeling](https://mlsanity.com/paper/2402.10202/) (2024)
- [Geometric path augmentation for inference of sparsely observed stochastic nonlinear systems](https://mlsanity.com/paper/2301.08102/) (2023)

## Abstract (arXiv)

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.