## High-dimensional analysis of semidefinite relaxations for sparse principal components.(English)Zbl 1173.62049

Summary: Principal components 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, where 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 where 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, with transitions from success to failure as a function of the rescaled sample size $$\theta _{dia}(n, p, k)=n/[k^{2}\log (p - k)]$$; and (b) a more sophisticated semidefinite programming (SDP) relaxation, which succeeds once the rescaled sample size $$\theta _{sdp}(n, p, k)=n/[k\log (p - k)]$$ 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 $$\theta _{sdp}(n, p, k)$$ is below a threshold. Our results thus highlight an interesting trade-off between computational and statistical efficiency in high-dimensional inference.

### MSC:

 62H25 Factor analysis and principal components; correspondence analysis 90C90 Applications of mathematical programming 62F12 Asymptotic properties of parametric estimators 65C60 Computational problems in statistics (MSC2010) 60F10 Large deviations

DSPCA
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### References:

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