Open Access
August 2013 Optimal detection of sparse principal components in high dimension
Quentin Berthet, Philippe Rigollet
Ann. Statist. 41(4): 1780-1815 (August 2013). DOI: 10.1214/13-AOS1127


We perform a finite sample analysis of the detection levels for sparse principal components of a high-dimensional covariance matrix. Our minimax optimal test is based on a sparse eigenvalue statistic. Alas, computing this test is known to be NP-complete in general, and we describe a computationally efficient alternative test using convex relaxations. Our relaxation is also proved to detect sparse principal components at near optimal detection levels, and it performs well on simulated datasets. Moreover, using polynomial time reductions from theoretical computer science, we bring significant evidence that our results cannot be improved, thus revealing an inherent trade off between statistical and computational performance.


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Quentin Berthet. Philippe Rigollet. "Optimal detection of sparse principal components in high dimension." Ann. Statist. 41 (4) 1780 - 1815, August 2013.


Published: August 2013
First available in Project Euclid: 5 September 2013

zbMATH: 1277.62155
MathSciNet: MR3127849
Digital Object Identifier: 10.1214/13-AOS1127

Primary: 62H25
Secondary: 62F04 , 90C22

Keywords: High-dimensional detection , minimax lower bounds , planted clique , semidefinite relaxation , sparse principal component analysis , spiked covariance model

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 4 • August 2013
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