High Dimensional Gaussian Graphical Regression Models with Covariates

TL;DR

This paper introduces a Gaussian graphical regression model linking graph structures to covariates, enabling the analysis of how external factors influence network structures in high-dimensional Gaussian graphical models.

Contribution

It proposes a novel regression framework for the mean and precision matrix, incorporating sparsity and group sparsity, with theoretical guarantees and practical validation.

Findings

Method accurately recovers network structures influenced by covariates.

Theoretical results establish variable selection consistency and convergence rates.

Application demonstrates effectiveness in co-expression QTL studies with brain cancer data.

Abstract

Though Gaussian graphical models have been widely used in many scientific fields, relatively limited progress has been made to link graph structures to external covariates. We propose a Gaussian graphical regression model, which regresses both the mean and the precision matrix of a Gaussian graphical model on covariates. In the context of co-expression quantitative trait locus (QTL) studies, our method can determine how genetic variants and clinical conditions modulate the subject-level network structures, and recover both the population-level and subject-level gene networks. Our framework encourages sparsity of covariate effects on both the mean and the precision matrix. In particular for the precision matrix, we stipulate simultaneous sparsity, i.e., group sparsity and element-wise sparsity, on effective covariates and their effects on network edges, respectively. We establish variable selection consistency first under the case with known mean parameters and then a more challenging case with unknown means depending on external covariates, and establish in both cases the $\ell_2$ convergence rates and the selection consistency of the estimated precision parameters. The utility and efficacy of our proposed method is demonstrated through simulation studies and an application to a co-expression QTL study with brain cancer patients.