High-Dimensional Covariance Regression with Application to Co-Expression QTL Detection

While covariance matrices have been widely studied in many scientific fields, relatively limited progress has been made on estimating conditional covariances that permits a large covariance matrix to vary with high-dimensional subject-level covariates. In this article, we present a new sparse covariance regression framework that models the covariance matrix as a function of subject-level covariates. In the context of co-expression quantitative trait locus (QTL) studies, our method can be used to determine if and how gene co-expressions vary with genetic variations. To accommodate high-dimensional responses and covariates, we stipulate a combined sparsity structure that encourages covariates with nonzero effects and edges that are modulated by these covariates to be simultaneously sparse. We approach parameter estimation with a blockwise coordinate descent algorithm, and investigate the l1 and l2 convergence rate of the estimated parameters. In addition, we propose a computationally efficient debiased inference procedure for uncertainty quantification. The efficacy of the proposed method is demonstrated through numerical experiments and an application to a gene co-expression network study with brain cancer patients. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.