# Stochastic and Private Nonconvex Outlier-Robust PCA

arXiv: https://arxiv.org/abs/2203.09276 · HTML: https://mlsanity.com/paper/2203.09276/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 40 references that are in the map.
- First posted on arXiv 17 Mar 2022.

## Builds on (most cited first, 12 of 12)

- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Private Stochastic Convex Optimization with Optimal Rates](https://mlsanity.com/paper/1908.09970/) (2019)
- [Differentially Private Empirical Risk Minimization Revisited: Faster and More General](https://mlsanity.com/paper/1802.05251/) (2018)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [The Noisy Power Method: A Meta Algorithm with Applications](https://mlsanity.com/paper/1311.2495/) (2013)
- [An Overview of Robust Subspace Recovery](https://mlsanity.com/paper/1803.01013/) (2018)
- [Private Empirical Risk Minimization Beyond the Worst Case: The Effect of the Constraint Set Geometry](https://mlsanity.com/paper/1411.5417/) (2014)
- [Fast, Robust and Non-convex Subspace Recovery](https://mlsanity.com/paper/1406.6145/) (2014)
- [A Well-Tempered Landscape for Non-convex Robust Subspace Recovery](https://mlsanity.com/paper/1706.03896/) (2017)
- [Wishart Mechanism for Differentially Private Principal Components Analysis](https://mlsanity.com/paper/1511.05680/) (2015)
- [Robust Subspace Recovery with Adversarial Outliers](https://mlsanity.com/paper/1904.03275/) (2019)

## Built on by (most cited first, 4 of 4)

- [Differentially private Riemannian optimization](https://mlsanity.com/paper/2205.09494/) (2022)
- [A Subspace-Constrained Tyler's Estimator and its Applications to Structure from Motion](https://mlsanity.com/paper/2404.11590/) (2024)
- [A Riemannian ADMM](https://mlsanity.com/paper/2211.02163/) (2022)
- [Enhancing Differentially Private Mechanisms via Empirical Bayes](https://mlsanity.com/paper/2606.21107/) (2026)

## Abstract (arXiv)

We develop theoretically guaranteed stochastic methods for outlier-robust PCA. Outlier-robust PCA seeks an underlying low-dimensional linear subspace from a dataset that is corrupted with outliers. We are able to show that our methods, which involve stochastic geodesic gradient descent over the Grassmannian manifold, converge and recover an underlying subspace in various regimes through the development of a novel convergence analysis. The main application of this method is an effective differentially private algorithm for outlier-robust PCA that uses a Gaussian noise mechanism within the stochastic gradient method. Our results emphasize the advantages of the nonconvex methods over another convex approach to solving this problem in the differentially private setting. Experiments on synthetic and stylized data verify these results.