Simultaneous Grouping Pursuit and Feature Selection Over an Undirected Graph

Abstract

In high-dimensional regression, grouping pursuit and feature selection have their own merits while complementing each other in battling the curse of dimensionality. To seek a parsimonious model, we perform simultaneous grouping pursuit and feature selection over an arbitrary undirected graph with each node corresponding to one predictor. When the corresponding nodes are reachable from each other over the graph, regression coefficients can be grouped, whose absolute values are the same or close. This is motivated from gene network analysis, where genes tend to work in groups according to their biological functionalities. Through a nonconvex penalty, we develop a computational strategy and analyze the proposed method. Theoretical analysis indicates that the proposed method reconstructs the oracle estimator, that is, the unbiased least-square estimator given the true grouping, leading to consistent reconstruction of grouping structures and informative features, as well as to optimal parameter estimation. Simulation studies suggest that the method combines the benefit of grouping pursuit with that of feature selection, and compares favorably against its competitors in selection accuracy and predictive performance. An application to eQTL data is used to illustrate the methodology, where a network is incorporated into analysis through an undirected graph.

KEY WORDS:

• Network analysis
• Nonconvex minimization
• Prediction
• Structured data

Acknowledgments

The research was supported in part by NSF grants DMS-0906616 and DMS-1207771, NIH grants 1R01GM081535-01 and grant HL65462. The authors thank the editor, the associate editor, and anonymous referees for helpful comments and suggestions.