Georgia Institute of Technology College of Sciences

This presentation is based on the papers by D. Paul and I. Johnstone (2007) and V.Q. Vu and J. Lei (2012). Here is the abstract of the second paper. We study the 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-aymptotics lower bounds and upper bounds on the minimax estimation error for the leading eigenvector when it belongs to an $l_q$ ball for $q\in [0,1]$. Our bound are sharp in $p$ and $n$ for all $q\in[0,1]$ 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 $l_1$-constrained PCA.