Gaussian graphical regressions have emerged as a powerful approach for regressing the precision matrix of a Gaussian graphical model on covariates, which, unlike traditional Gaussian graphical models, can help determine how graphs are modulated by high dimensional subject-level covariates, and recover both the population-level and subject-level graphs. To fit the model, a multi-task learning approach {achieves} %has been shown to result in lower error rates compared to node-wise regressions. However, due to the high complexity and dimensionality of the Gaussian graphical regression problem, the important task of statistical inference remains unexplored. We propose a class of debiased estimators based on multi-task learners for statistical inference in Gaussian graphical regressions. We show that debiasing can be performed quickly and separately for the multi-task learners. In a key debiasing step {that estimates} %involving the estimation of the inverse covariance matrix, we propose a novel {projection technique} %diagonalization approach that dramatically reduces computational costs {in optimization} to scale only with the sample size $n$. We show that our debiased estimators enjoy a fast convergence rate and asymptotically follow a normal distribution, enabling valid statistical inference such as constructing confidence intervals and performing hypothesis testing. Simulation studies confirm the practical utility of the proposed approach, and we further apply it to analyze gene co-expression graph data from a brain cancer study, revealing meaningful biological relationships.

翻译：高斯图回归已成为一种强大的方法，用于将高斯图模型的精度矩阵回归到协变量上。与传统高斯图模型不同，该方法有助于确定图结构如何受高维个体层面协变量的调节，并同时恢复群体层面和个体层面的图结构。在模型拟合方面，多任务学习方法相比节点回归方法已被证明能实现更低的错误率。然而，由于高斯图回归问题的高度复杂性和维数，统计推断这一重要任务仍未得到充分探索。我们提出了一类基于多任务学习器的去偏估计量，用于高斯图回归中的统计推断。我们证明，对于多任务学习器，去偏操作可以快速且独立地进行。在一个关键的涉及估计逆协方差矩阵的去偏步骤中，我们提出了一种新颖的投影技术，该技术显著降低了计算成本，使其优化复杂度仅与样本量$n$成比例。我们证明了所提出的去偏估计量具有快速收敛速率，并且渐近服从正态分布，从而能够进行有效的统计推断，如构建置信区间和执行假设检验。仿真研究证实了该方法的实际效用，我们进一步将其应用于分析脑癌研究中的基因共表达图数据，揭示了有意义的生物学关系。