Positive Semidefinite Rank-Based Correlation Matrix Estimation With Application to Semiparametric Graph Estimation

Abstract

Many statistical methods gain robustness and flexibility by sacrificing convenient computational structures. In this article, we illustrate this fundamental tradeoff by studying a semiparametric graph estimation problem in high dimensions. We explain how novel computational techniques help to solve this type of problem. In particular, we propose a nonparanormal neighborhood pursuit algorithm to estimate high-dimensional semiparametric graphical models with theoretical guarantees. Moreover, we provide an alternative view to analyze the tradeoff between computational efficiency and statistical error under a smoothing optimization framework. Though this article focuses on the problem of graph estimation, the proposed methodology is widely applicable to other problems with similar structures. We also report thorough experimental results on text, stock, and genomic datasets.

Key Words:

• Graphical model
• High-dimensional statistics
• Model selection/Variable selection
• Multivariate analysis
• Numerical optimization

ACKNOWLEDGMENTS

Tuo Zhao and Han Liu are supported by NSF Grant III-1116730, while Kathryn Roeder was supported by National Institute of Mental Health Grant MH057881 (PI Bernie Devlin).