Statistical inference on high-dimensional covariate-dependent Gaussian graphical regressions.

In many genomic studies, gene co-expression graphs are influenced by subject-level covariates like single nucleotide polymorphisms. Traditional Gaussian graphical models ignore these covariates and estimate only population-level networks, potentially masking important heterogeneity. Covariate-dependent Gaussian graphical regressions address this limitation by regressing the precision matrix on covariates, thereby modeling how graph structures vary with high-dimensional subject-specific covariates. To fit the model, we adopt a multi-task learning approach that achieves lower error rates than node-wise regressions. Yet, the important problem of statistical inference in this setting remains largely unexplored. We propose a class of debiased estimators based on multi-task learners, which can be computed quickly and separately. In a key step, we introduce a novel projection technique for estimating the inverse covariance matrix, reducing optimization costs to scale with the sample size n. Our debiased estimators achieve fast convergence and asymptotic normality, enabling valid inference. Simulations demonstrate the utility of the method, and an application to a brain cancer gene-expression dataset reveals meaningful biological relationships.