On generating random Gaussian graphical models

TL;DR

This paper explores methods for generating random Gaussian graphical models, focusing on creating positive definite matrices that adhere to specific graph structures, including chordal and non-chordal graphs, for better validation of structure learning algorithms.

Contribution

It introduces novel techniques for sampling positive definite matrices consistent with undirected graph constraints, especially for non-chordal graphs using a partial orthogonalization approach.

Findings

Uniform sampling from correlation matrices is possible for chordal graphs.

Partial orthogonalization enables sampling for general undirected graphs.

The methods improve validation of structure learning algorithms.

Abstract

Structure learning methods for covariance and concentration graphs are often validated on synthetic models, usually obtained by randomly generating: (i) an undirected graph, and (ii) a compatible symmetric positive definite (SPD) matrix. In order to ensure positive definiteness in (ii), a dominant diagonal is usually imposed. In this work we investigate different methods to generate random symmetric positive definite matrices with undirected graphical constraints. We show that if the graph is chordal it is possible to sample uniformly from the set of correlation matrices compatible with the graph, while for general undirected graphs we rely on a partial orthogonalization method.