MINIMAX BOUNDS FOR SPARSE PCA WITH NOISY HIGH-DIMENSIONAL DATA

• 成果类型: Article
• 署名作者: Birnbaum, Aharon; Johnstone, Iain M.; Nadler, Boaz; Paul, Debashis
• 署名单位: Hebrew University of Jerusalem; Stanford University; Weizmann Institute of Science; University of California System; University of California Davis
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS1014
• 发表日期: 2013
• 页码: 1055-1084
• 关键词: principal-components-analysis; Consistency; rates

摘要

We study the problem of estimating the leading eigenvectors of a high-dimensional population covariance matrix based on independent Gaussian observations. We establish a lower bound on the minimax risk of estimators under the l(2) loss, in the joint limit as dimension and sample size increase to infinity, under various models of sparsity for the population eigenvectors. The lower bound on the risk points to the existence of different regimes of sparsity of the eigenvectors. We also propose a new method for estimating the eigenvectors by a two-stage coordinate selection scheme.