# Global Convergence of Least Squares EM for Demixing Two Log-Concave Densities

arXiv: https://arxiv.org/abs/1906.06776 · HTML: https://mlsanity.com/paper/1906.06776/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2019.

## Built on by (most cited first, 5 of 5)

- [A Geometric Approach to k-means](https://mlsanity.com/paper/2201.04822/) (2022)
- [Big Learning Expectation Maximization](https://mlsanity.com/paper/2312.11926/) (2023)
- [Efficient Algorithms for Estimating the Parameters of Mixed Linear Regression Models](https://mlsanity.com/paper/2105.05953/) (2021)
- [Sharp analysis of EM for learning mixtures of pairwise differences](https://mlsanity.com/paper/2302.10066/) (2023)
- [Global Convergence of Online Identification for Mixed Linear Regression](https://mlsanity.com/paper/2311.18506/) (2023)

## Abstract (arXiv)

This work studies the location estimation problem for a mixture of two rotation invariant log-concave densities. We demonstrate that Least Squares EM, a variant of the EM algorithm, converges to the true location parameter from a randomly initialized point. We establish the explicit convergence rates and sample complexity bounds, revealing their dependence on the signal-to-noise ratio and the tail property of the log-concave distribution. Moreover, we show that this global convergence property is robust under model mis-specification. Our analysis generalizes previous techniques for proving the convergence results for Gaussian mixtures. In particular, we make use of an angle-decreasing property for establishing global convergence of Least Squares EM beyond Gaussian settings, as ℓ2 distance contraction no longer holds globally for general log-concave mixtures.