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 (24)

• Sparse principal component analysis and iterative thresholding
• Optimal detection of sparse principal components in high dimension
• Truncated Power Method for Sparse Eigenvalue Problems
• High-dimensional analysis of semidefinite relaxations for sparse principal components
• Achieving Optimal Misclassification Proportion in Stochastic Block Model
• Statistical analysis of latent generalized correlation matrix estimation in transelliptical distribution
• Sum-of-Squares Lower Bounds for Sparse PCA
• Nonconvex Statistical Optimization: Minimax-Optimal Sparse PCA in Polynomial Time
• Sparsistency and agnostic inference in sparse PCA
• Distributed Estimation of Principal Eigenspaces
• Large Covariance Estimation through Elliptical Factor Models
• Sampled forms of functional PCA in reproducing kernel Hilbert spaces
• Surprising Asymptotic Conical Structure in Critical Sample Eigen-Directions
• Rate-optimal posterior contraction for sparse PCA
• Sparse PCA with Oracle Property
• Subexponential-Time Algorithms for Sparse PCA
• Tests in adaptive regression via the Kac-Rice formula
• Estimation of functionals of sparse covariance matrices
• Influential Feature PCA for high dimensional clustering
• High Dimensional Principal Component Scores and Data Visualization
• High Dimensional Semiparametric Latent Graphical Model for Mixed Data
• Sparse Principal Component Analysis for High Dimensional Vector Autoregressive Models
• Sparse principal component analysis and its $l_1$-relaxation
• High Dimensional Semiparametric Scale-Invariant Principal Component Analysis