# EM's Convergence in Gaussian Latent Tree Models

arXiv: https://arxiv.org/abs/2211.11904 · HTML: https://mlsanity.com/paper/2211.11904/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 39 references that are in the map.
- First posted on arXiv 21 Nov 2022.

## Builds on (most cited first, 15 of 15)

- [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)
- [Singularity, Misspecification, and the Convergence Rate of EM](https://mlsanity.com/paper/1810.00828/) (2018)
- [Global analysis of Expectation Maximization for mixtures of two Gaussians](https://mlsanity.com/paper/1608.07630/) (2016)
- [EM Converges for a Mixture of Many Linear Regressions](https://mlsanity.com/paper/1905.12106/) (2019)
- [Global Convergence of EM Algorithm for Mixtures of Two Component Linear Regression](https://mlsanity.com/paper/1810.05752/) (2018)
- [On the Minimax Optimality of the EM Algorithm for Learning Two-Component Mixed Linear Regression](https://mlsanity.com/paper/2006.02601/) (2020)
- [Estimating the Coefficients of a Mixture of Two Linear Regressions by Expectation Maximization](https://mlsanity.com/paper/1704.08231/) (2017)
- [Benefits of over-parameterization with EM](https://mlsanity.com/paper/1810.11344/) (2018)
- [On the Analysis of EM for truncated mixtures of two Gaussians](https://mlsanity.com/paper/1902.06958/) (2019)
- [A Survey on Latent Tree Models and Applications](https://mlsanity.com/paper/1402.0577/) (2014)
- [On the Behavior of the Expectation-Maximization Algorithm for Mixture Models](https://mlsanity.com/paper/1809.08705/) (2018)
- [Maximum Likelihood Estimation for Brownian Motion Tree Models Based on One Sample](https://mlsanity.com/paper/2112.00816/) (2021)
- [Gaussian Tree Constraints Applied to Acoustic Linguistic Functional Data](https://mlsanity.com/paper/1410.0813/) (2014)
- [Efficient recycled algorithms for quantitative trait models on phylogenies](https://mlsanity.com/paper/1512.00890/) (2015)

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

- [Sharp analysis of EM for learning mixtures of pairwise differences](https://mlsanity.com/paper/2302.10066/) (2023)
- [The DeepCAR Method: Forecasting Time-Series Data That Have Change Points](https://mlsanity.com/paper/2302.11241/) (2023)

## Abstract (arXiv)

We study the optimization landscape of the log-likelihood function and the convergence of the Expectation-Maximization (EM) algorithm in latent Gaussian tree models, i.e. tree-structured Gaussian graphical models whose leaf nodes are observable and non-leaf nodes are unobservable. We show that the unique non-trivial stationary point of the population log-likelihood is its global maximum, and establish that the expectation-maximization algorithm is guaranteed to converge to it in the single latent variable case. Our results for the landscape of the log-likelihood function in general latent tree models provide support for the extensive practical use of maximum likelihood based-methods in this setting. Our results for the EM algorithm extend an emerging line of work on obtaining global convergence guarantees for this celebrated algorithm. We show our results for the non-trivial stationary points of the log-likelihood by arguing that a certain system of polynomial equations obtained from the EM updates has a unique non-trivial solution. The global convergence of the EM algorithm follows by arguing that all trivial fixed points are higher-order saddle points.