Minimax Rates of Estimation for Sparse PCA in High Dimensions

Details Minimax Rates of Estimation for Sparse PCA in High Dimensions Minimax Rates of Estimation for Sparse PCA in High Dimensions

2012-02-03

arxiv.org/abs/1202.0786v2

Statistics

Machine Learning

To appear in Proceedings of the 15th International Conference on Artificial Intelligence and Statistics (AISTATS) 2012, La Pal

Scientific paper

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 $\ell_q$ ball for $q \in [0,1]$. Our bounds 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 $\ell_q$-constrained PCA. In particular, our results provide convergence rates for $\ell_1$-constrained PCA.

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