Optimal estimation and rank detection for sparse spiked covariance matrices

• 成果类型: Article
• 署名作者: Cai, Tony; Ma, Zongming; Wu, Yihong
• 署名单位: University of Pennsylvania; University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0562-z
• 发表日期: 2015
• 页码: 781-815
• 关键词: principal-components; Optimal Rates; high dimension; CONVERGENCE; PCA; asymptotics; Consistency; number

摘要

This paper considers a sparse spiked covariance matrix model in the high-dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection. The optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established, which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariance matrices. We also establish the minimax rate under the spectral norm for estimating the principal subspace, the primary object of interest in principal component analysis. In addition, the optimal rate for the rank detection boundary is obtained. This result also resolves the gap in a recent paper by Berthet and Rigollet (Ann Stat 41(4):1780-1815, 2013) where the special case of rank one is considered.