Gaussian Graphical Models: An Algebraic and Geometric Perspective

TL;DR

This paper provides an accessible overview of Gaussian graphical models, emphasizing their algebraic and geometric properties, and reviews recent advances in maximum likelihood estimation and related algorithms.

Contribution

It offers a pedagogic introduction and synthesizes recent research on the algebraic, geometric, and optimization aspects of Gaussian graphical models.

Findings

Highlights algebraic and geometric properties of models

Discusses conditions for MLE existence

Connects properties to convex optimization algorithms

Abstract

Gaussian graphical models are used throughout the natural sciences, social sciences, and economics to model the statistical relationships between variables of interest in the form of a graph. We here provide a pedagogic introduction to Gaussian graphical models and review recent results on maximum likelihood estimation for such models. Throughout, we highlight the rich algebraic and geometric properties of Gaussian graphical models and explain how these properties relate to convex optimization and ultimately result in insights on the existence of the maximum likelihood estimator (MLE) and algorithms for computing the MLE.