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 paper, we present a new sparse multivariate 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 non-zero 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 $\ell_2$ 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.

翻译：尽管协方差矩阵已在多个科学领域得到广泛研究，但在允许大协方差矩阵随高维个体层面协变量变化的条件协方差估计方面，相关进展相对有限。本文提出一种新的稀疏多变量回归框架，将协方差矩阵建模为个体层面协变量的函数。在共表达数量性状基因座研究中，该方法可用于确定基因共表达是否以及如何随遗传变异变化。为应对高维响应变量与协变量，我们规定了一种组合稀疏结构，该结构促使具有非零效应的协变量及受这些协变量调控的边同时保持稀疏性。我们采用块坐标下降算法进行参数估计，并研究了估计参数的$\ell_2$收敛速率。此外，我们提出了一种计算高效的去偏推断程序用于不确定性量化。通过数值实验及脑癌患者基因共表达网络研究的实际应用，验证了所提方法的有效性。