The Annals of Statistics

Optimal detection of sparse principal components in high dimension

Quentin Berthet and Philippe Rigollet

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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.

Article information

Ann. Statist., Volume 41, Number 4 (2013), 1780-1815.

First available in Project Euclid: 5 September 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 62F04 90C22: Semidefinite programming

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


Berthet, Quentin; Rigollet, Philippe. Optimal detection of sparse principal components in high dimension. Ann. Statist. 41 (2013), no. 4, 1780--1815. doi:10.1214/13-AOS1127.

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