High-Dimensional Covariate-Dependent Gaussian Graphical Models

Abstract

In this dissertation, we propose a covariate-dependent Gaussian graphical model (cdexGGM) for capturing network structure that varies with covariates through a novel parameterization. Utilizing a likelihood framework, our methodology jointly estimates all edge and vertex parameters. We further develop statistical inference procedures to test the dynamic nature of the underlying network. Concerning large-scale networks, we perform composite likelihood estimation with an $\ell_1$ penalty to discover sparse covariate-dependent graph structures. We establish the estimation error bound in $\ell_2$ norm and validate the sign consistency in the high-dimensional context. We apply our method to an influenza vaccine data set to model the gene network that evolves with time. We also investigate a Down syndrome data set to model the protein network, which varies with several covariates under a factorial experimental design. These applications demonstrate the applicability and effectiveness of the proposed model.

Moreover, to further address the limitations of GGMs in capturing heterogeneous networks with known structural constraints, we introduce a covariate-dependent colored Gaussian graphical model (CD-CGGM). This model incorporates covariate effects and structured sparsity (through colorings) to model dynamic conditional dependencies. We perform model estimation using penalized composite likelihood, employing coordinate descent and Broyden’s method for optimization under different scenarios. We provide theoretical results ensuring both parameter and sign consistency of the proposed estimator. The method is applied to the same influenza vaccine dataset, where it effectively models the time-evolving gene regulatory network under symmetry constraints, thereby demonstrating its empirical performance and interpretability.