# Subspace Recovery in Winsorized PCA: Insights into Accuracy and Robustness

arXiv: https://arxiv.org/abs/2502.16391 · HTML: https://mlsanity.com/paper/2502.16391/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 47 references that are in the map.
- First posted on arXiv 23 Feb 2025.

## Builds on (most cited first, 14 of 14)

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- [CoinPress: Practical Private Mean and Covariance Estimation](https://mlsanity.com/paper/2006.06618/) (2020)
- [Randomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition](https://mlsanity.com/paper/1504.05477/) (2015)
- [Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics](https://mlsanity.com/paper/1605.00353/) (2016)
- [Heteroskedastic PCA: Algorithm, Optimality, and Applications](https://mlsanity.com/paper/1810.08316/) (2018)
- [Differentially Private Algorithms for Learning Mixtures of Separated Gaussians](https://mlsanity.com/paper/1909.03951/) (2019)
- [LazySVD: Even Faster SVD Decomposition Yet Without Agonizing Pain](https://mlsanity.com/paper/1607.03463/) (2016)
- [Subspace Recovery from Heterogeneous Data with Non-isotropic Noise](https://mlsanity.com/paper/2210.13497/) (2022)
- [Optimal Differentially Private PCA and Estimation for Spiked Covariance Matrices](https://mlsanity.com/paper/2401.03820/) (2024)
- [Robust SVD Made Easy: A fast and reliable algorithm for large-scale data analysis](https://mlsanity.com/paper/2402.09754/) (2024)
- [Generalized Spherical Principal Component Analysis](https://mlsanity.com/paper/2303.05836/) (2023)
- [A generalized spatial sign covariance matrix](https://mlsanity.com/paper/1805.01417/) (2018)

## Abstract (arXiv)

In this paper, we explore the theoretical properties of subspace recovery using Winsorized Principal Component Analysis (WPCA), utilizing a common data transformation technique that caps extreme values to mitigate the impact of outliers. Despite the widespread use of winsorization in various tasks of multivariate analysis, its theoretical properties, particularly for subspace recovery, have received limited attention. We provide a detailed analysis of the accuracy of WPCA, showing that increasing the number of samples while decreasing the proportion of outliers guarantees the consistency of the sample subspaces from WPCA with respect to the true population subspace. Furthermore, we establish perturbation bounds that ensure the WPCA subspace obtained from contaminated data remains close to the subspace recovered from pure data. Additionally, we extend the classical notion of breakdown points to subspace-valued statistics and derive lower bounds for the breakdown points of WPCA. Our analysis demonstrates that WPCA exhibits strong robustness to outliers while maintaining consistency under mild assumptions. A toy example is provided to numerically illustrate the behavior of the upper bounds for perturbation bounds and breakdown points, emphasizing winsorization's utility in subspace recovery.