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• W3100498669 abstract "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 $kgeqOmega(n/log p)$, no algorithm, efficient or not, can reliably recover the sparse eigenvector. In contrast, for $kleq O(sqrt{n/log p})$, 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 $kgeqOmega(sqrt{n})$, 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 $kgeqOmega(sqrt{n})$. Finally, we present empirical results suggesting that up to sparsity levels $k=O(sqrt{n})$, recovery is possible by a simple covariance thresholding algorithm." @default.
• W3100498669 created "2020-11-23" @default.
• W3100498669 creator A5034942163 @default.
• W3100498669 creator A5037378898 @default.
• W3100498669 creator A5067064621 @default.
• W3100498669 date "2015-06-01" @default.
• W3100498669 modified "2023-09-30" @default.
• W3100498669 title "Do semidefinite relaxations solve sparse PCA up to the information limit?" @default.
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• W3100498669 doi "https://doi.org/10.1214/15-aos1310" @default.
• W3100498669 hasPublicationYear "2015" @default.
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