Minimax Rates of Estimation for Sparse PCA in High Dimensions

TL;DR

This paper establishes optimal bounds on the estimation error for sparse PCA in high-dimensional settings, providing sharp theoretical guarantees for $ ext{l}_q$-constrained PCA across various distributions.

Contribution

It derives non-asymptotic minimax bounds for sparse PCA's eigenvector estimation, extending to $ ext{l}_q$ constraints and analyzing $ ext{l}_q$-constrained PCA performance.

Findings

Sharp minimax bounds for eigenvector estimation in sparse PCA.

Convergence rates for $ ext{l}_1$-constrained PCA.

Bounds valid for a wide class of distributions.

Abstract

We study sparse principal components analysis in the high-dimensional setting, where $p$ (the number of variables) can be much larger than $n$ (the number of observations). We prove optimal, non-asymptotic lower and upper bounds on the minimax estimation error for the leading eigenvector when it belongs to an $\ell_q$ ball for $q \in [0,1]$. Our bounds are sharp in $p$ and $n$ for all $q \in [0, 1]$ over a wide class of distributions. The upper bound is obtained by analyzing the performance of $\ell_q$-constrained PCA. In particular, our results provide convergence rates for $\ell_1$-constrained PCA.