Minimax Rates of Estimation for Sparse PCA in High Dimensions
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 minimax lower and upper bounds on the estimation error for the first leading eigenvector when it belongs to an \ell_q ball for q ∈[0,1]. Our bounds are tight 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 \ell_q-constrained PCA. In particular, our results provide convergence rates for \ell_1-constrained PCA. Our methods and arguments are also extendable to multi-dimensional subspace estimation.
Cite
Text
Vu and Lei. "Minimax Rates of Estimation for Sparse PCA in High Dimensions." Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, 2012.Markdown
[Vu and Lei. "Minimax Rates of Estimation for Sparse PCA in High Dimensions." Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, 2012.](https://mlanthology.org/aistats/2012/vu2012aistats-minimax/)BibTeX
@inproceedings{vu2012aistats-minimax,
title = {{Minimax Rates of Estimation for Sparse PCA in High Dimensions}},
author = {Vu, Vincent and Lei, Jing},
booktitle = {Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics},
year = {2012},
pages = {1278-1286},
volume = {22},
url = {https://mlanthology.org/aistats/2012/vu2012aistats-minimax/}
}