Minimax Sparse Principal Subspace Estimation in High Dimensions Vincent Q. Vu Jing Lei 10.1184/R1/6586742.v1 https://kilthub.cmu.edu/articles/journal_contribution/Minimax_Sparse_Principal_Subspace_Estimation_in_High_Dimensions/6586742 <p>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 ℓ<sub>q</sub> subspace sparsity for 0 ≤ q ≤ 1. Our upper bounds apply to general classes of covariance matrices, and they show that ℓ<sub>q</sub> constrained estimates can achieve optimal minimax rates without restrictive spiked covariance conditions.</p> 2005-12-01 00:00:00 principal components analysis subspace estimation sparsity high-dimensional statistics minimax bounds random matrices empirical process