The Annals of Statistics

High-dimensional analysis of semidefinite relaxations for sparse principal components

Arash A. Amini and Martin J. Wainwright

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Principal component analysis (PCA) is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix. However, PCA is well known to behave poorly in the “large p, small n” setting, in which the problem dimension p is comparable to or larger than the sample size n. This paper studies PCA in this high-dimensional regime, but under the additional assumption that the maximal eigenvector is sparse, say, with at most k nonzero components. We consider a spiked covariance model in which a base matrix is perturbed by adding a k-sparse maximal eigenvector, and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector, as follows: (a) a simple diagonal thresholding method, which transitions from success to failure as a function of the rescaled sample size θdia(n, p, k)=n/[k2log(pk)]; and (b) a more sophisticated semidefinite programming (SDP) relaxation, which succeeds once the rescaled sample size θsdp(n, p, k)=n/[klog(pk)] is larger than a critical threshold. In addition, we prove that no method, including the best method which has exponential-time complexity, can succeed in recovering the support if the order parameter θsdp(n, p, k) is below a threshold. Our results thus highlight an interesting trade-off between computational and statistical efficiency in high-dimensional inference.

Article information

Ann. Statist., Volume 37, Number 5B (2009), 2877-2921.

First available in Project Euclid: 17 July 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 62F12: Asymptotic properties of estimators

Principal component analysis spectral analysis spiked covariance ensembles sparsity high-dimensional statistics convex relaxation semidefinite programming Wishart ensembles random matrices


Amini, Arash A.; Wainwright, Martin J. High-dimensional analysis of semidefinite relaxations for sparse principal components. Ann. Statist. 37 (2009), no. 5B, 2877--2921. doi:10.1214/08-AOS664.

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