Minimax Sparse Principal Subspace Estimation in High Dimensions

We study sparse principal components analysis in high dimensions, where p (the number of variables) can be much larger than n (the number of observations), and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix. We prove optimal, non-asymptotic lower and upper bounds on the minimax subspace estimation error under two different, but related notions of ℓq subspace sparsity for 0 ≤ q ≤ 1. Our upper bounds apply to general classes of covariance matrices, and they show that ℓq constrained estimates can achieve optimal minimax rates without restrictive spiked covariance conditions.