  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 the $l_2$ loss, 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.
