Statistics Papers

Document Type

Journal Article

Date of this Version

4-2015

Publication Source

Probability Theory and Related Fields

Volume

161

Issue

3

Start Page

781

Last Page

815

DOI

10.1007/s00440-014-0562-z

Abstract

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.

Copyright/Permission Statement

The final publication is available at Springer via http://dx.doi.org/10.1007/s00440-014-0562-z.

Keywords

covariance matrix, group sparsity, low-rank matrix, minimax rate of convergence, sparse principal component analysis, principal subspace, rank detection

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Date Posted: 27 November 2017

This document has been peer reviewed.