# Fast convergence of the Expectation Maximization algorithm under a logarithmic Sobolev inequality

arXiv: https://arxiv.org/abs/2407.17949 · HTML: https://mlsanity.com/paper/2407.17949/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 66 references that are in the map.
- First posted on arXiv 25 Jul 2024.

## Builds on (most cited first, 21 of 21)

- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [Singularity, Misspecification, and the Convergence Rate of EM](https://mlsanity.com/paper/1810.00828/) (2018)
- [Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM](https://mlsanity.com/paper/2206.08873/) (2022)
- [Global analysis of Expectation Maximization for mixtures of two Gaussians](https://mlsanity.com/paper/1608.07630/) (2016)
- [Particle algorithms for maximum likelihood training of latent variable models](https://mlsanity.com/paper/2204.12965/) (2022)
- [Homeomorphic-Invariance of EM: Non-Asymptotic Convergence in KL Divergence for Exponential Families via Mirror Descent](https://mlsanity.com/paper/2011.01170/) (2020)
- [Interacting Particle Langevin Algorithm for Maximum Marginal Likelihood Estimation](https://mlsanity.com/paper/2303.13429/) (2023)
- [Error bounds for particle gradient descent, and extensions of the log-Sobolev and Talagrand inequalities](https://mlsanity.com/paper/2403.02004/) (2024)
- [On the Global Convergence of (Fast) Incremental Expectation Maximization Methods](https://mlsanity.com/paper/1910.12521/) (2019)
- [Momentum Particle Maximum Likelihood](https://mlsanity.com/paper/2312.07335/) (2023)
- [Simultaneous Clustering and Estimation of Heterogeneous Graphical Models](https://mlsanity.com/paper/1611.09391/) (2016)
- [Proximal Interacting Particle Langevin Algorithms](https://mlsanity.com/paper/2406.14292/) (2024)
- [Kinetic Interacting Particle Langevin Monte Carlo](https://mlsanity.com/paper/2407.05790/) (2024)
- [High Dimensional Expectation-Maximization Algorithm: Statistical Optimization and Asymptotic Normality](https://mlsanity.com/paper/1412.8729/) (2014)
- [Comparison of Markov chains via weak Poincar'e inequalities with application to pseudo-marginal MCMC](https://mlsanity.com/paper/2112.05605/) (2021)
- [On the rate of convergence of alternating minimization for non-smooth non-strongly convex optimization in Banach spaces](https://mlsanity.com/paper/1911.00404/) (2019)

## Built on by (most cited first, 1 of 1)

- [Wasserstein Contraction of Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2605.30253/) (2026)

## Abstract (arXiv)

We present a new framework for analysing the Expectation Maximization (EM) algorithm. Drawing on recent advances in the theory of gradient flows over Euclidean-Wasserstein spaces, we extend techniques from alternating minimization in Euclidean spaces to the EM algorithm, via its representation as coordinate-wise minimization of the free energy. In so doing, we obtain finite sample error bounds and exponential convergence of the EM algorithm under a natural generalisation of the log-Sobolev inequality. We further show that this framework naturally extends to several variants of EM, offering a unified approach for studying such algorithms.