EM's Convergence in Gaussian Latent Tree Models

TL;DR

This paper analyzes the optimization landscape of Gaussian latent tree models, proving the global optimality of the EM algorithm's convergence in certain cases and supporting its practical effectiveness.

Contribution

It establishes the global convergence of EM in Gaussian latent tree models and characterizes the landscape of the log-likelihood function, including the uniqueness of the non-trivial stationary point.

Findings

Unique non-trivial stationary point is the global maximum.

EM algorithm converges to this maximum in the single latent variable case.

The landscape analysis supports the practical use of maximum likelihood methods.

Abstract

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.