# Solving Large-Scale Sparse PCA to Certifiable (Near) Optimality

arXiv: https://arxiv.org/abs/2005.05195 · HTML: https://mlsanity.com/paper/2005.05195/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 May 2020.

## Built on by (most cited first, 11 of 11)

- [A new perspective on low-rank optimization](https://mlsanity.com/paper/2105.05947/) (2021)
- [Manifold constrained steepest descent for smooth and closed-set optimization](https://mlsanity.com/paper/2601.21487/) (2026)
- [Variable Selection for Kernel Two-Sample Tests](https://mlsanity.com/paper/2302.07415/) (2023)
- [Sparse PCA With Multiple Components](https://mlsanity.com/paper/2209.14790/) (2022)
- [Efficient Sparse PCA via Block-Diagonalization](https://mlsanity.com/paper/2410.14092/) (2024)
- [Coordinate Descent Methods for Fractional Minimization](https://mlsanity.com/paper/2201.12691/) (2022)
- [Principal Component Hierarchy for Sparse Quadratic Programs](https://mlsanity.com/paper/2105.12022/) (2021)
- [Prescriptive PCA: Dimensionality Reduction for Two-stage Stochastic Optimization](https://mlsanity.com/paper/2306.02223/) (2023)
- [Isometry pursuit](https://mlsanity.com/paper/2411.18502/) (2024)
- [A Randomized Algorithm for Sparse PCA based on the Basic SDP Relaxation](https://mlsanity.com/paper/2507.09148/) (2025)
- [Video-based detection of cessation of breathing in pre-term infants using machine learning](https://mlsanity.com/paper/2607.05230/) (2026)

## Abstract (arXiv)

Sparse principal component analysis (PCA) is a popular dimensionality reduction technique for obtaining principal components which are linear combinations of a small subset of the original features. Existing approaches cannot supply certifiably optimal principal components with more than p=100s of variables. By reformulating sparse PCA as a convex mixed-integer semidefinite optimization problem, we design a cutting-plane method which solves the problem to certifiable optimality at the scale of selecting k=5 covariates from p=300 variables, and provides small bound gaps at a larger scale. We also propose a convex relaxation and greedy rounding scheme that provides bound gaps of 1-2% in practice within minutes for p=100s or hours for p=1,000s and is therefore a viable alternative to the exact method at scale. Using real-world financial and medical datasets, we illustrate our approach's ability to derive interpretable principal components tractably at scale.