  This paper considers sparse spiked covariance matrix models 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 [1] where the special case of
rank one is considered.
