Entrywise Recovery Guarantees for Sparse PCA via Sparsistent Algorithms

TL;DR

This paper establishes entrywise error bounds for Sparse PCA under high-dimensional subgaussian models, providing a more detailed understanding of estimation accuracy for sparsistent algorithms.

Contribution

It introduces entrywise $ ext{ell}_{2, ext{infty}}$ bounds for Sparse PCA, improving upon existing spectral norm results and applicable to any sparsistent algorithm.

Findings

Provides entrywise $ ext{ell}_{2, ext{infty}}$ bounds for Sparse PCA

Results hold for any support-recovering sparsistent algorithms

Uses entrywise subspace perturbation techniques

Abstract

Sparse Principal Component Analysis (PCA) is a prevalent tool across a plethora of subfields of applied statistics. While several results have characterized the recovery error of the principal eigenvectors, these are typically in spectral or Frobenius norms. In this paper, we provide entrywise $\ell_{2,\infty}$ bounds for Sparse PCA under a general high-dimensional subgaussian design. In particular, our results hold for any algorithm that selects the correct support with high probability, those that are sparsistent. Our bound improves upon known results by providing a finer characterization of the estimation error, and our proof uses techniques recently developed for entrywise subspace perturbation theory.