PCA is one of the most commonly used statistical procedures with a wide range of
applications. We consider both minimax and adaptive estimation of the principal
subspace in the high dimensional setting. Under mild technical conditions, we
establish the optimal rates of convergence for estimating the principal subspace
which are sharp with respect to all the parameters. The lower bound is obtained
by calculating the local metric entropy and an application of Fano's Lemma. The
rate optimal estimator is constructed using aggregation, which, however, might
not be computationally feasible. We then introduce an adaptive procedure for
estimating the principal subspace. A key idea is a reduction scheme which
reduces the sparse PCA problem to a high-dimensional multivariate regression
problem.