# Approximating Sparse PCA from Incomplete Data

arXiv: https://arxiv.org/abs/1503.03903 · HTML: https://mlsanity.com/paper/1503.03903/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Mar 2015.

## Built on by (most cited first, 2 of 2)

- [Support Recovery in Sparse PCA with Incomplete Data](https://mlsanity.com/paper/2205.15215/) (2022)
- [Support Recovery in Sparse PCA with Non-Random Missing Data](https://mlsanity.com/paper/2302.01535/) (2023)

## Abstract (arXiv)

We study how well one can recover sparse principal components of a data matrix using a sketch formed from a few of its elements. We show that for a wide class of optimization problems, if the sketch is close (in the spectral norm) to the original data matrix, then one can recover a near optimal solution to the optimization problem by using the sketch. In particular, we use this approach to obtain sparse principal components and show that for mathm data points in mathn dimensions, mathO(ε-2tilde kmaxm,n) elements gives an mathε-additive approximation to the sparse PCA problem (mathtilde k is the stable rank of the data matrix). We demonstrate our algorithms extensively on image, text, biological and financial data. The results show that not only are we able to recover the sparse PCAs from the incomplete data, but by using our sparse sketch, the running time drops by a factor of five or more.