# Sparse PCA from Sparse Linear Regression

arXiv: https://arxiv.org/abs/1811.10106 · HTML: https://mlsanity.com/paper/1811.10106/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Nov 2018.

## Built on by (most cited first, 6 of 6)

- [Canonical normalizing flows for manifold learning](https://mlsanity.com/paper/2310.12743/) (2023)
- [Oja's Algorithm for Streaming Sparse PCA](https://mlsanity.com/paper/2402.07240/) (2024)
- [Optimal Sketching Bounds for Sparse Linear Regression](https://mlsanity.com/paper/2304.02261/) (2023)
- [Computational-Statistical Gaps for Improper Learning in Sparse Linear Regression](https://mlsanity.com/paper/2402.14103/) (2024)
- [Lasso with Latents: Efficient Estimation, Covariate Rescaling, and Computational-Statistical Gaps](https://mlsanity.com/paper/2402.15409/) (2024)
- [Combinatorial Sparse PCA Beyond the Spiked Identity Model](https://mlsanity.com/paper/2603.02607/) (2026)

## Abstract (arXiv)

Sparse Principal Component Analysis (SPCA) and Sparse Linear Regression (SLR) have a wide range of applications and have attracted a tremendous amount of attention in the last two decades as canonical examples of statistical problems in high dimension. A variety of algorithms have been proposed for both SPCA and SLR, but an explicit connection between the two had not been made. We show how to efficiently transform a black-box solver for SLR into an algorithm for SPCA: assuming the SLR solver satisfies prediction error guarantees achieved by existing efficient algorithms such as those based on the Lasso, the SPCA algorithm derived from it achieves near state of the art guarantees for testing and for support recovery for the single spiked covariance model as obtained by the current best polynomialtime algorithms. Our reduction not only highlights the inherent similarity between the two problems, but also, from a practical standpoint, allows one to obtain a collection of algorithms for SPCA directly from known algorithms for SLR. We provide experimental results on simulated data comparing our proposed framework to other algorithms for SPCA.