# Global analysis of Expectation Maximization for mixtures of two Gaussians

arXiv: https://arxiv.org/abs/1608.07630 · HTML: https://mlsanity.com/paper/1608.07630/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Aug 2016.

## Built on by (most cited first, 30 of 31)

- [Learning Mixtures of Gaussians Using the DDPM Objective](https://mlsanity.com/paper/2307.01178/) (2023)
- [Clustering Mixtures with Almost Optimal Separation in Polynomial Time](https://mlsanity.com/paper/2112.00706/) (2021)
- [A theoretical perspective on mode collapse in variational inference](https://mlsanity.com/paper/2410.13300/) (2024)
- [Breaking the gridlock in Mixture-of-Experts: Consistent and Efficient Algorithms](https://mlsanity.com/paper/1802.07417/) (2018)
- [Improved Convergence Guarantees for Learning Gaussian Mixture Models by EM and Gradient EM](https://mlsanity.com/paper/2101.00575/) (2021)
- [Provable Hierarchical Imitation Learning via EM](https://mlsanity.com/paper/2010.03133/) (2020)
- [Towards the Theory of Unsupervised Federated Learning: Non-asymptotic Analysis of Federated EM Algorithms](https://mlsanity.com/paper/2310.15330/) (2023)
- [The EM Algorithm is Adaptively-Optimal for Unbalanced Symmetric Gaussian Mixtures](https://mlsanity.com/paper/2103.15653/) (2021)
- [Optimal Estimation and Computational Limit of Low-rank Gaussian Mixtures](https://mlsanity.com/paper/2201.09040/) (2022)
- [Toward Global Convergence of Gradient EM for Over-Parameterized Gaussian Mixture Models](https://mlsanity.com/paper/2407.00490/) (2024)
- [A sampling-based approach for efficient clustering in large datasets](https://mlsanity.com/paper/2112.14793/) (2021)
- [Convergence Dynamics of Over-Parameterized Score Matching for a Single Gaussian](https://mlsanity.com/paper/2511.22069/) (2025)
- [High-Dimensional Differentially-Private EM Algorithm: Methods and Near-Optimal Statistical Guarantees](https://mlsanity.com/paper/2104.00245/) (2021)
- [A Geometric Approach to k-means](https://mlsanity.com/paper/2201.04822/) (2022)
- [Beyond EM Algorithm on Over-specified Two-Component Location-Scale Gaussian Mixtures](https://mlsanity.com/paper/2205.11078/) (2022)
- [On the Semi-supervised Expectation Maximization](https://mlsanity.com/paper/2211.00537/) (2022)
- [EM's Convergence in Gaussian Latent Tree Models](https://mlsanity.com/paper/2211.11904/) (2022)
- [Sharp-SSL: Selective high-dimensional axis-aligned random projections for semi-supervised learning](https://mlsanity.com/paper/2304.09154/) (2023)
- [Learning Mixtures of Gaussians with Censored Data](https://mlsanity.com/paper/2305.04127/) (2023)
- [Big Learning Expectation Maximization](https://mlsanity.com/paper/2312.11926/) (2023)
- [An Exponentially Increasing Step-size for Parameter Estimation in Statistical Models](https://mlsanity.com/paper/2205.07999/) (2022)
- [Sharp analysis of EM for learning mixtures of pairwise differences](https://mlsanity.com/paper/2302.10066/) (2023)
- [EM for Mixture of Linear Regression with Clustered Data](https://mlsanity.com/paper/2308.11518/) (2023)
- [Joint Problems in Learning Multiple Dynamical Systems](https://mlsanity.com/paper/2311.02181/) (2023)
- [Fast convergence of the Expectation Maximization algorithm under a logarithmic Sobolev inequality](https://mlsanity.com/paper/2407.17949/) (2024)
- [Partial Distribution Matching via Partial Wasserstein Adversarial Networks](https://mlsanity.com/paper/2409.10499/) (2024)
- [In-context Learning for Mixture of Linear Regressions: Existence, Generalization and Training Dynamics](https://mlsanity.com/paper/2410.14183/) (2024)
- [Mean Estimation in High-Dimensional Binary Markov Gaussian Mixture Models](https://mlsanity.com/paper/2206.02455/) (2022)
- [Differentiable Expectation-Maximisation and Applications to Gaussian Mixture Model Optimal Transport](https://mlsanity.com/paper/2509.02109/) (2025)
- [Individual-heterogeneous sub-Gaussian Mixture Models](https://mlsanity.com/paper/2604.05337/) (2026)

## Abstract (arXiv)

Expectation Maximization (EM) is among the most popular algorithms for estimating parameters of statistical models. However, EM, which is an iterative algorithm based on the maximum likelihood principle, is generally only guaranteed to find stationary points of the likelihood objective, and these points may be far from any maximizer. This article addresses this disconnect between the statistical principles behind EM and its algorithmic properties. Specifically, it provides a global analysis of EM for specific models in which the observations comprise an i.i.d. sample from a mixture of two Gaussians. This is achieved by (i) studying the sequence of parameters from idealized execution of EM in the infinite sample limit, and fully characterizing the limit points of the sequence in terms of the initial parameters; and then (ii) based on this convergence analysis, establishing statistical consistency (or lack thereof) for the actual sequence of parameters produced by EM.