# Improved Convergence Guarantees for Learning Gaussian Mixture Models by EM and Gradient EM

arXiv: https://arxiv.org/abs/2101.00575 · HTML: https://mlsanity.com/paper/2101.00575/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 24 references that are in the map.
- First posted on arXiv 3 Jan 2021.

## Builds on (most cited first, 6 of 6)

- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Ten Steps of EM Suffice for Mixtures of Two Gaussians](https://mlsanity.com/paper/1609.00368/) (2016)
- [Local Maxima in the Likelihood of Gaussian Mixture Models: Structural Results and Algorithmic Consequences](https://mlsanity.com/paper/1609.00978/) (2016)
- [Global analysis of Expectation Maximization for mixtures of two Gaussians](https://mlsanity.com/paper/1608.07630/) (2016)
- [On Learning Mixtures of Well-Separated Gaussians](https://mlsanity.com/paper/1710.11592/) (2017)
- [Tight bounds for learning a mixture of two gaussians](https://mlsanity.com/paper/1404.4997/) (2014)

## Built on by (most cited first, 11 of 11)

- [Learning Mixtures of Gaussians Using the DDPM Objective](https://mlsanity.com/paper/2307.01178/) (2023)
- [Active clustering with bandit feedback](https://mlsanity.com/paper/2406.11485/) (2024)
- [Toward Global Convergence of Gradient EM for Over-Parameterized Gaussian Mixture Models](https://mlsanity.com/paper/2407.00490/) (2024)
- [Transformers as Unsupervised Learning Algorithms: A study on Gaussian Mixtures](https://mlsanity.com/paper/2505.11918/) (2025)
- [Differentiable Outlier Detection Enable Robust Deep Multimodal Analysis](https://mlsanity.com/paper/2302.05608/) (2023)
- [Unsupervised spectral-band feature identification for optimal process discrimination](https://mlsanity.com/paper/2212.03800/) (2022)
- [Scalable Clustering: Large Scale Unsupervised Learning of Gaussian Mixture Models with Outliers](https://mlsanity.com/paper/2302.14599/) (2023)
- [Learning Overspecified Gaussian Mixtures Exponentially Fast with the EM Algorithm](https://mlsanity.com/paper/2506.11850/) (2025)
- [Overspecified Mixture Discriminant Analysis: Exponential Convergence, Statistical Guarantees, and Remote Sensing Applications](https://mlsanity.com/paper/2510.27056/) (2025)
- [Low-degree Lower bounds for clustering in moderate dimension](https://mlsanity.com/paper/2602.23023/) (2026)
- [Local linear convergence of gradient methods for overparameterized Gaussian mixtures](https://mlsanity.com/paper/2605.30936/) (2026)

## Abstract (arXiv)

We consider the problem of estimating the parameters a Gaussian Mixture Model with K components of known weights, all with an identity covariance matrix. We make two contributions. First, at the population level, we present a sharper analysis of the local convergence of EM and gradient EM, compared to previous works. Assuming a separation of Omega(√log K), we prove convergence of both methods to the global optima from an initialization region larger than those of previous works. Specifically, the initial guess of each component can be as far as (almost) half its distance to the nearest Gaussian. This is essentially the largest possible contraction region. Our second contribution are improved sample size requirements for accurate estimation by EM and gradient EM. In previous works, the required number of samples had a quadratic dependence on the maximal separation between the K components, and the resulting error estimate increased linearly with this maximal separation. In this manuscript we show that both quantities depend only logarithmically on the maximal separation.