Minimax Rates of Estimation for Sparse PCA in High Dimensions

We study sparse principal components analysis in the high-dimensional setting, where p (the number of variables) can be much larger than n (the number of observations). We prove optimal minimax lower and upper bounds on the estimation error for the first leading eigenvector when it belongs to an \ell_q ball for q ∈[0,1]. Our bounds are tight in p and n for all q ∈[0, 1] over a wide class of distributions. The upper bound is obtained by analyzing the performance of \ell_q-constrained PCA. In particular, our results provide convergence rates for \ell_1-constrained PCA. Our methods and arguments are also extendable to multi-dimensional subspace estimation.