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

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 ℓ0-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≥Omega(n/log p), no algorithm, efficient or not, can reliably recover the sparse eigenvector. In contrast, for k≤ O(√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≥Omega(√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 ℓ0-sparsity k≥Omega(√n). Finally, we present empirical results suggesting that up to sparsity levels k=O(√n), recovery is possible by a simple covariance thresholding algorithm.