Statistical and computational trade-offs in estimation of sparse principal components
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.
Additional Information
© 2016 Institute of Mathematical Statistics. Received May 2015; revised July 2015. Supported by a Benefactors' scholarship from St. John's College, Cambridge. Supported by Air Force Office of Scientific Research (AFOSR) Grant FA9550-14-1-0098 at the Center for the Mathematics of Information at the California Institute of Technology. Supported by Engineering and Physical Sciences Research Council Early Career Fellowship EP/J017213/1 and Leverhulme Trust Grant PLP-2014-353.Attached Files
Published - euclid.aos.1473685263.pdf
Supplemental Material - euclid.aos.1473685263_si.pdf
Files
Name | Size | Download all |
---|---|---|
md5:d0a2a705ad03fac785946e30151e2bb4
|
426.3 kB | Preview Download |
md5:94a791ad2c2a8b7123335f5463928f38
|
373.1 kB | Preview Download |
Additional details
- Eprint ID
- 71319
- Resolver ID
- CaltechAUTHORS:20161020-132624352
- St. John's College
- FA9550-14-1-0098
- Air Force Office of Scientific Research (AFOSR)
- EP/J017213/1
- Engineering and Physical Sciences Research Council (EPSRC)
- PLP-2014-353
- Leverhulme Trust
- Created
-
2016-10-20Created from EPrint's datestamp field
- Updated
-
2023-06-01Created from EPrint's last_modified field