Augmented sparse principal component analysis for high dimensional data

Abstract

We study the problem of estimating the leading eigenvectors of a high-dimensional population covariance matrix based on independent Gaussian observations. We establish lower bounds on the rates of convergence of the estimators of the leading eigenvectors under $l^q$-sparsity constraints when an $l^2$ loss function is used. We also propose an estimator of the leading eigenvectors based on a coordinate selection scheme combined with PCA and show that the proposed estimator achieves the optimal rate of convergence under a sparsity regime. Moreover, we establish that under certain scenarios, the usual PCA achieves the minimax convergence rate.

This manuscript was written in 2007, and a version has been available on the first author's website, but it is posted to arXiv now in its 2007 form. Revisions incorporating later work will be posted separately

References in corpus (2)

• Prediction in functional linear regression
• Sparse Principal Components Analysis

Cited by in corpus (4)

• High-dimensional analysis of semidefinite relaxations for sparse principal components
• Nonconvex Statistical Optimization: Minimax-Optimal Sparse PCA in Polynomial Time
• Surprising Asymptotic Conical Structure in Critical Sample Eigen-Directions
• High Dimensional Principal Component Scores and Data Visualization