The Annals of Statistics

Do semidefinite relaxations solve sparse PCA up to the information limit?

Robert Krauthgamer, Boaz Nadler, and Dan Vilenchik

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Estimating the leading principal components of data, assuming they are sparse, is a central task in modern high-dimensional statistics. Many algorithms were developed for this sparse PCA problem, from simple diagonal thresholding to sophisticated semidefinite programming (SDP) methods. A key theoretical question is under what conditions can such algorithms recover the sparse principal components? We study this question for a single-spike model with an $\ell_{0}$-sparse eigenvector, in the asymptotic regime as dimension $p$ and sample size $n$ both tend to infinity. Amini and Wainwright [ Ann. Statist. 37 (2009) 2877–2921] proved that for sparsity levels $k\geq\Omega(n/\log p)$, no algorithm, efficient or not, can reliably recover the sparse eigenvector. In contrast, for $k\leq O(\sqrt{n/\log p})$, diagonal thresholding is consistent. It was further conjectured that an SDP approach may close this gap between computational and information limits. We prove that when $k\geq\Omega(\sqrt{n})$, the proposed SDP approach, at least in its standard usage, cannot recover the sparse spike. In fact, we conjecture that in the single-spike model, no computationally-efficient algorithm can recover a spike of $\ell_{0}$-sparsity $k\geq\Omega(\sqrt{n})$. Finally, we present empirical results suggesting that up to sparsity levels $k=O(\sqrt{n})$, recovery is possible by a simple covariance thresholding algorithm.

Article information

Ann. Statist., Volume 43, Number 3 (2015), 1300-1322.

Received: September 2014
Revised: January 2015
First available in Project Euclid: 15 May 2015

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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 integrality gap


Krauthgamer, Robert; Nadler, Boaz; Vilenchik, Dan. Do semidefinite relaxations solve sparse PCA up to the information limit?. Ann. Statist. 43 (2015), no. 3, 1300--1322. doi:10.1214/15-AOS1310.

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