Statistics Papers

Document Type

Journal Article

Date of this Version

2013

Publication Source

The Annals of Statistics

Volume

41

Issue

6

Start Page

3074

Last Page

3110

DOI

10.1214/13-AOS1178

Abstract

Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. This paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting. Under mild technical conditions, we first establish the optimal rates of convergence for estimating the principal subspace which are sharp with respect to all the parameters, thus providing a complete characterization of the difficulty of the estimation problem in term of the convergence rate. The lower bound is obtained by calculating the local metric entropy and an application of Fano’s lemma. The rate optimal estimator is constructed using aggregation, which, however, might not be computationally feasible.

We then introduce an adaptive procedure for estimating the principal subspace which is fully data driven and can be computed efficiently. It is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces. A key idea in our construction is a reduction scheme which reduces the sparse PCA problem to a high-dimensional multivariate regression problem. This method is potentially also useful for other related problems.

Keywords

adaptive estimation, aggregation, covariance matrix, eigenvector, group sparsity, low-rank matrix, minimax lower bound, optimal rate of convergence, principal component analysis, thresholding

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Date Posted: 27 November 2017

This document has been peer reviewed.