The Annals of Statistics

Minimax bounds for sparse PCA with noisy high-dimensional data

Aharon Birnbaum, Iain M. Johnstone, Boaz Nadler, and Debashis Paul

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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.

Article information

Ann. Statist., Volume 41, Number 3 (2013), 1055-1084.

First available in Project Euclid: 13 June 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties
Secondary: 62H25: Factor analysis and principal components; correspondence analysis

Minimax risk high-dimensional data principal component analysis sparsity spiked covariance model


Birnbaum, Aharon; Johnstone, Iain M.; Nadler, Boaz; Paul, Debashis. Minimax bounds for sparse PCA with noisy high-dimensional data. Ann. Statist. 41 (2013), no. 3, 1055--1084. doi:10.1214/12-AOS1014.

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