Sparse PCA: Optimal rates and adaptive estimation

TL;DR

This paper establishes the optimal convergence rates for estimating sparse principal subspaces in high dimensions, introduces a computationally feasible adaptive estimator, and reduces sparse PCA to a multivariate regression problem for broader applicability.

Contribution

It provides the first sharp minimax rates for sparse PCA, constructs an adaptive estimator that attains these rates, and introduces a reduction scheme to multivariate regression.

Findings

Optimal convergence rates for sparse PCA estimation.

An adaptive, computationally efficient estimator matching these rates.

Reduction of sparse PCA to multivariate regression for broader use.

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.