The Annals of Statistics

Statistical and computational trade-offs in estimation of sparse principal components

Tengyao Wang, Quentin Berthet, and Richard J. Samworth

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In recent years, sparse principal component analysis has emerged as an extremely popular dimension reduction technique for high-dimensional data. The theoretical challenge, in the simplest case, is to estimate the leading eigenvector of a population covariance matrix under the assumption that this eigenvector is sparse. An impressive range of estimators have been proposed; some of these are fast to compute, while others are known to achieve the minimax optimal rate over certain Gaussian or sub-Gaussian classes. In this paper, we show that, under a widely-believed assumption from computational complexity theory, there is a fundamental trade-off between statistical and computational performance in this problem. More precisely, working with new, larger classes satisfying a restricted covariance concentration condition, we show that there is an effective sample size regime in which no randomised polynomial time algorithm can achieve the minimax optimal rate. We also study the theoretical performance of a (polynomial time) variant of the well-known semidefinite relaxation estimator, revealing a subtle interplay between statistical and computational efficiency.

Article information

Ann. Statist., Volume 44, Number 5 (2016), 1896-1930.

Received: May 2015
Revised: July 2015
First available in Project Euclid: 12 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H25: Factor analysis and principal components; correspondence analysis 68Q17: Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) [See also 68Q15]

Computational lower bounds planted clique problem polynomial time algorithm sparse principal component analysis


Wang, Tengyao; Berthet, Quentin; Samworth, Richard J. Statistical and computational trade-offs in estimation of sparse principal components. Ann. Statist. 44 (2016), no. 5, 1896--1930. doi:10.1214/15-AOS1369.

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Supplemental materials

  • Supplementary material to “Statistical and computational trade-offs in estimation of sparse principal components”. Ancillary results and a brief introduction to computational complexity theory.