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 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 qconstrained PCA. In particular, our results provide convergence rates for 1-constrained PCA.

• Type: Journal
• Year: 2012
• Where: JMLR
• Authors: Vincent Q. Vu, Jing Lei
• Keywords: Convergence Rates, JMLR 2012, Minimax Estimation, Principal Components Analysis, Programming Languages