Minimax bounds for sparse PCA with noisy high-dimensional data
Aharon Birnbaum, Iain M. Johnstone, Boaz Nadler, Debashis Paul
The Annals of Statistics · 2013 · Volume 41 · Issue 3
Abstract
We study the problem of estimating the leading eigenvectors of a high-dimensional population covariance matrix based on independent Gaussian observations. We establish a lower bound on the minimax risk of estimators under the $l_{2}$ loss, in the joint limit as dimension and sample size increase to infinity, under various models of sparsity for the population eigenvectors. The lower bound on the risk points to the existence of different regimes of sparsity of the eigenvectors. We also propose a new method for estimating the eigenvectors by a two-stage coordinate selection scheme.
Keywords
62G20, 62H25, High-dimensional data, minimax risk, Principal Component Analysis, Sparsity, spiked covariance model
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