title: Optimal Estimation and Rank Detection for Sparse Spiked Covariance Matrices.
abstract: This paper considers a sparse spiked covariancematrix 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 [2] where the special case of rank one is considered.
authors.affiliation: Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104.
authors.forename: Tony
authors.initials: T
authors.lastname: Cai
authors.affiliation: Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104.
authors.forename: Zongming
authors.initials: Z
authors.lastname: Ma
authors.affiliation: Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104.
authors.forename: Yihong
authors.initials: Y
authors.lastname: Wu
keywords: Covariance matrix; Group sparsity; Low-rank matrix; Minimax rate of convergence; Principal subspace; Rank detection; Sparse principal component analysis