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, non-asymptotic lower and upper bounds on the minimax estimation error for the leading eigenvector when it belongs to an ℓ_q ball for q ∈ [0, 1]. Our bounds are sharp 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 ℓ_q constrained PCA. In particular, our results provide convergence rates for ℓ₁-constrained PCA.