MINIMAX SPARSE PRINCIPAL SUBSPACE ESTIMATION IN HIGH DIMENSIONS

• 成果类型: Article
• 署名作者: Vu, Vincent Q.; Lei, Jing
• 署名单位: University System of Ohio; Ohio State University; Carnegie Mellon University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1151
• 发表日期: 2013
• 页码: 2905-2947
• 关键词: component analysis; PCA; Consistency; regression

摘要

We study sparse principal components analysis in high dimensions, where p (the number of variables) can be much larger than n (the number of observations), and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix. We introduce two complementary notions of eq subspace sparsity: row sparsity and column sparsity. We prove nonasymptotic lower and upper bounds on the minimax subspace estimation error for 0 <= q <= I. The bounds are optimal for row sparse subspaces and nearly optimal for column sparse subspaces, they apply to general classes of covariance matrices, and they show that l(q) constrained estimates can achieve optimal minimax rates without restrictive spiked covariance conditions. Interestingly, the form of the rates matches known results for sparse regression when the effective noise variance is defined appropriately. Our proof employs a novel variational sine theorem that may be useful in other regularized spectral estimation problems.