Generalized Thresholding of Large Covariance Matrices

• 成果类型: Article
• 署名作者: Rothman, Adam J.; Levina, Elizaveta; Zhu, Ji
• 署名单位: University of Michigan System; University of Michigan
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1198/jasa.2009.0101
• 发表日期: 2009
• 页码: 177-186
• 关键词: nonconcave penalized likelihood; principal components; wavelet shrinkage; oracle properties; selection; Lasso; regularization

摘要

We propose a new class of generalized thresholding operators that combine thresholding with shrinkage, and Study generalized thresholding of the sample covariance matrix in high dimensions. Generalized thresholding of the covariance matrix has good theoretical properties and carries almost no computational burden. We obtain in explicit convergence rate in the operator norm that shows the tradeoff between the sparsity of the true model, dimension, and the sample size, and shows that generalized thresholding is consistent over a large class of models as long as the dimension p and the sample size it satisfy log p/n -> 0. In addition, we show that generalized thresholding has the sparsistency property, meaning it estimates true zeros a, zeros with probability tending to 1, and, under an additional mild condition, is sign consistent for nonzero elements. We show that generalized thresholding covers, as special cases, hard and soft thresholding, smoothly clipped absolute deviation, and adaptive lasso, and compare different types of generalized thresholding in a simulation study and in an example of gene clustering from a microarray experiment with tumor tissues.