Augmented sparse principal component analysis for high dimensional data

TL;DR

This paper investigates the estimation of leading eigenvectors in high-dimensional covariance matrices, establishing convergence bounds, proposing a sparsity-aware estimator, and comparing its performance to traditional PCA.

Contribution

It introduces an augmented sparse PCA method with optimal convergence rates under sparsity constraints and compares it to standard PCA.

Findings

Proposed estimator achieves optimal convergence rate under sparsity.

Lower bounds on convergence rates established for eigenvector estimators.

Standard PCA can attain minimax rates in certain scenarios.

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.