Department of Statistics Unitmark
Dietrich College of Humanities and Social Sciences

Minimax Rates of Estimation for Sparse PCA in High Dimensions

Publication Date

February, 2012

Publication Type

Tech Report

Author(s)

Vincent Q. Vu, Jing Lei

Abstract

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 ℓq 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 ℓq-constrained PCA. In particular, our results provide convergence rates for ℓ1-constrained PCA.