Wang, T;
Berthet, Q;
Samworth, RJ;
(2016)
Statistical and computational trade-offs in estimation of sparse principal components.
Annals of Statistics
, 44
(5)
pp. 1896-1930.
10.1214/15-AOS1369.
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Abstract
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.
Type: | Article |
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Title: | Statistical and computational trade-offs in estimation of sparse principal components |
Open access status: | An open access version is available from UCL Discovery |
DOI: | 10.1214/15-AOS1369 |
Publisher version: | http://dx.doi.org/10.1214/15-AOS1369 |
Language: | English |
Additional information: | This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions. |
Keywords: | Science & Technology, Physical Sciences, Statistics & Probability, Mathematics, Computational lower bounds, planted clique problem, polynomial time algorithm, sparse principal component analysis, HIGH-DIMENSIONAL DATA, ADAPTIVE ESTIMATION, POWER METHOD, PCA, EIGENVALUE, MINIMIZATION, RELAXATIONS, CONSISTENCY, OPERATORS, CLIQUES |
UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science |
URI: | https://discovery.ucl.ac.uk/id/eprint/10055407 |
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