title: MINIMAX BOUNDS FOR SPARSE PCA WITH NOISY HIGH-DIMENSIONAL DATA.
abstract: 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 thel2loss, 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.
authors.affiliation: School of Computer Science and Engineering Hebrew University of Jerusalem The Edmond J. Safra Campus Jerusalem, 91904 Israel aharob01@cs.huji.ac.il.
authors.forename: Aharon
authors.initials: A
authors.lastname: Birnbaum
authors.affiliation: Department of Statistics Stanford University Stanford, California 94305 USA imj@stanford.edu.
authors.forename: Iain M
authors.initials: IM
authors.lastname: Johnstone
authors.affiliation: Department of Computer Science and Applied Mathematics Weizmann Institute of Science P.O. Box 26, Rehovot, 76100 Israel boaz.nadler@weizmann.ac.il.
authors.forename: Boaz
authors.initials: B
authors.lastname: Nadler
authors.affiliation: Department of Statistics University of California Davis, California 95616 USA debashis@wald.ucdavis.edu.
authors.forename: Debashis
authors.initials: D
authors.lastname: Paul
keywords: Minimax risk; high-dimensional data; principal component analysis; sparsity; spiked covariance model