Entrywise Recovery Guarantees for Sparse PCA via Sparsistent Algorithms
This work addresses a specific theoretical gap in sparse PCA for applied statistics, offering incremental improvements in error characterization.
The paper tackled the problem of providing entrywise recovery guarantees for Sparse PCA under high-dimensional subgaussian designs, resulting in improved bounds that offer a finer characterization of estimation error compared to existing spectral or Frobenius norm results.
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.