A Constrained l1 Minimization Approach to Sparse Precision Matrix Estimation

• 成果类型: Article
• 署名作者: Cai, Tony; Liu, Weidong; Luo, Xi
• 署名单位: University of Pennsylvania; Shanghai Jiao Tong University
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1198/jasa.2011.tm10155
• 发表日期: 2011
• 页码: 594-607
• 关键词: VARIABLE SELECTION; covariance; CONVERGENCE; likelihood; RECOVERY; rates; MODEL

摘要

This article proposes a constrained l(1) minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p-variate random variables. The resulting estimator is shown to have a number of desirable properties. In particular, the rate of convergence between the estimator and the true s-sparse precision matrix under the spectral norm is s root logp/n when the population distribution has either exponential-type tails or polynomial-type tails. We present convergence rates under the elementwise l(infinity) norm and Frobenius norm. In addition, we consider graphical model selection. The procedure is easily implemented by linear programming. Numerical performance of the estimator is investigated using both simulated and real data. In particular, the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods.