Graphical-model based high dimensional generalized linear models

Abstract

We consider the problem of both prediction and model selection in high dimensional generalized linear models. Predictive performance can be improved by leveraging structure information among predictors. In this paper, a graphic model-based doubly sparse regularized estimator is discussed under the high dimensional generalized linear models, that utilizes the graph structure among the predictors. The graphic information among predictors is incorporated node-by-node using a decomposed representation and the sparsity is encouraged both within and between the decomposed components. We propose an efficient iterative proximal algorithm to solve the optimization problem. Statistical convergence rates and selection consistency for the doubly sparse regularized estimator are established in the ultra-high dimensional setting. Specifically, we allow the dimensionality grows exponentially with the sample size. We compare the estimator with existing methods through numerical analysis on both simulation study and a microbiome data analysis.