# Theoretical Guarantees for Sparse Principal Component Analysis based on the Elastic Net

arXiv: https://arxiv.org/abs/2212.14194 · HTML: https://mlsanity.com/paper/2212.14194/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 43 references that are in the map.
- First posted on arXiv 29 Dec 2022.

## Builds on (most cited first, 3 of 3)

- [Sparse PCA via Covariance Thresholding](https://mlsanity.com/paper/1311.5179/) (2013)
- [Do semidefinite relaxations solve sparse PCA up to the information limit?](https://mlsanity.com/paper/1306.3690/) (2013)
- [Sparsistency and agnostic inference in sparse PCA](https://mlsanity.com/paper/1401.6978/) (2014)

## Abstract (arXiv)

Sparse principal component analysis (SPCA) is widely used for dimensionality reduction and feature extraction in high-dimensional data analysis. Despite many methodological and theoretical developments in the past two decades, the theoretical guarantees of the popular SPCA algorithm proposed by Zou, Hastie & Tibshirani (2006) are still unknown. This paper aims to address this critical gap. We first revisit the SPCA algorithm of Zou et al. (2006) and present our implementation. We also study a computationally more efficient variant of the SPCA algorithm in Zou et al. (2006) that can be considered as the limiting case of SPCA. We provide the guarantees of convergence to a stationary point for both algorithms and prove that, under a sparse spiked covariance model, both algorithms can recover the principal subspace consistently under mild regularity conditions. We show that their estimation error bounds match the best available bounds of existing works or the minimax rates up to some logarithmic factors. Moreover, we demonstrate the competitive numerical performance of both algorithms in numerical studies.