  We study the problem of estimating the leading eigenvectors of a
high-dimensional population covariance matrix based on independent Gaussian
observations. We establish lower bounds on the rates of convergence of the
estimators of the leading eigenvectors under $l^q$-sparsity constraints when an
$l^2$ loss function is used. We also propose an estimator of the leading
eigenvectors based on a coordinate selection scheme combined with PCA and show
that the proposed estimator achieves the optimal rate of convergence under a
sparsity regime. Moreover, we establish that under certain scenarios, the usual
PCA achieves the minimax convergence rate.
