Gaussian graphical models with toric vanishing ideals

TL;DR

This paper characterizes Gaussian graphical models with toric vanishing ideals, identifying the specific graphs for which the ideal is generated in degree 1 and 2, thus advancing the algebraic understanding of these models.

Contribution

It resolves two conjectures by characterizing graphs with toric vanishing ideals in Gaussian graphical models, specifically those formed by 1-clique sums of complete graphs.

Findings

Identifies graphs with vanishing ideals generated in degree 1 and 2.

Characterizes models with toric ideals as 1-clique sums of complete graphs.

Resolves two conjectures by Sturmfels and Uhler.

Abstract

Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. They are widely used throughout natural sciences, computational biology and many other fields. Computing the vanishing ideal of the model gives us an implicit description of the model. In this paper, we resolve two conjectures of Sturmfels and Uhler from \cite{BS n CU}. In particular, we characterize those graphs for which the vanishing ideal of the Gaussian graphical model is generated in degree $1$ and $2$. These turn out to be the Gaussian graphical models whose ideals are toric ideals, and the resulting graphs are the $1$-clique sums of complete graphs.