Adjusting for high-dimensional covariates in sparse precision matrix estimation by ℓ1ℓ1-penalization

Motivated by the analysis of genetical genomic data, we consider the problem of estimating high-dimensional sparse precision matrix adjusting for possibly a large number of covariates, where the covariates can affect the mean value of the random vector. We develop a two-stage estimation procedure to first identify the relevant covariates that affect the means by a joint ℓ1ℓ1 penalization. The estimated regression coefficients are then used to estimate the mean values in a multivariate sub-Gaussian model in order to estimate the sparse precision matrix through a ℓ1ℓ1-penalized log-determinant Bregman divergence. Under the multivariate normal assumption, the precision matrix has the interpretation of a conditional Gaussian graphical model. We show that under some regularity conditions, the estimates of the regression coefficients are consistent in element-wise ℓ∞ℓ∞ norm, Frobenius norm and also spectral norm even when p≫np≫n and q≫nq≫n. We also show that with probability converging to one, the estimate of the precision matrix correctly specifies the zero pattern of the true precision matrix. We illustrate our theoretical results via simulations and demonstrate that the method can lead to improved estimate of the precision matrix. We apply the method to an analysis of a yeast genetical genomic data.