# Convex Estimation of Gaussian Graphical Regression Models with Covariates

arXiv: https://arxiv.org/abs/2410.06326 · HTML: https://mlsanity.com/paper/2410.06326/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 73 references that are in the map.
- First posted on arXiv 8 Oct 2024.

## Builds on (most cited first, 14 of 14)

- [Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression](https://mlsanity.com/paper/1804.02605/) (2018)
- [A highly efficient semismooth Newton augmented Lagrangian method for solving Lasso problems](https://mlsanity.com/paper/1607.05428/) (2016)
- [An efficient Hessian based algorithm for solving large-scale sparse group Lasso problems](https://mlsanity.com/paper/1712.05910/) (2017)
- [A Study of Error Variance Estimation in Lasso Regression](https://mlsanity.com/paper/1311.5274/) (2013)
- [Estimating the error variance in a high-dimensional linear model](https://mlsanity.com/paper/1712.02412/) (2017)
- [sparsegl: An R Package for Estimating Sparse Group Lasso](https://mlsanity.com/paper/2208.02942/) (2022)
- [Asymptotically Normal and Efficient Estimation of Covariate-Adjusted Gaussian Graphical Model](https://mlsanity.com/paper/1309.5923/) (2013)
- [On the prediction loss of the lasso in the partially labeled setting](https://mlsanity.com/paper/1606.06179/) (2016)
- [High Dimensional Gaussian Graphical Regression Models with Covariates](https://mlsanity.com/paper/2011.05245/) (2020)
- [Bayesian Covariate-Dependent Graph Learning with a Dual Group Spike-and-Slab Prior](https://mlsanity.com/paper/2409.17404/) (2024)
- [Bayesian Edge Regression in Undirected Graphical Models to Characterize Interpatient Heterogeneity in Cancer](https://mlsanity.com/paper/2101.09587/) (2021)
- [Covariate-Assisted Bayesian Graph Learning for Heterogeneous Data](https://mlsanity.com/paper/2308.07806/) (2023)
- [Joint estimation of sparse multivariate regression and conditional graphical models](https://mlsanity.com/paper/1306.4410/) (2013)
- [Multi-task Learning for Gaussian Graphical Regressions with High Dimensional Covariates](https://mlsanity.com/paper/2205.10672/) (2022)

## Abstract (arXiv)

Gaussian graphical models (GGMs) are widely used to recover the conditional independence structure among random variables. Recent work has sought to incorporate auxiliary covariates to improve estimation, particularly in applications such as co-expression quantitative trait locus (eQTL) studies, where both gene expression levels and their conditional dependence structure may be influenced by genetic variants. Existing approaches to covariate-adjusted GGMs either restrict covariate effects to the mean structure or lead to nonconvex formulations when jointly estimating the mean and precision matrix. In this paper, we propose a convex framework that simultaneously estimates the covariate-adjusted mean and precision matrix via a natural parametrization of the multivariate Gaussian likelihood. The resulting formulation enables joint convex optimization and yields improved theoretical guarantees under high-dimensional scaling, where the sparsity and dimension of covariates grow with the sample size. We support our theoretical findings with numerical simulations and demonstrate the practical utility of the proposed method through a reanalysis of an eQTL study of glioblastoma multiforme and an analysis of diet on the human gut microbiome.