Subspace Recovery in Winsorized PCA: Insights into Accuracy and Robustness

2025-02-23

AI Summary

This paper investigates the theoretical properties of Winsorized Principal Component Analysis (WPCA) for subspace recovery in high-dimensional data contaminated with outliers. Addressing the lack of rigorous characterization of subspace consistency and robustness in existing work, we establish the first asymptotic consistency theory for WPCA. We introduce a generalized breakdown point for subspace estimators and derive a tight lower bound, thereby revealing the inherent trade-off between estimation accuracy and robustness induced by Winsorization. By integrating data truncation, random matrix perturbation analysis, and projection distance metrics, we prove that the WPCA-estimated subspace converges almost surely to the true subspace as the sample size grows and the outlier proportion vanishes, while achieving an optimal-order perturbation bound. Both theoretical analysis and numerical experiments confirm that WPCA simultaneously attains strong robustness against outliers and high estimation accuracy.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Adversarial Attacks & Robustness
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• Subspace recovery in Winsorized PCA
• Robustness to outliers analysis
• Perturbation bounds for contaminated data

Innovation

• Winsorized PCA for subspace recovery
• Capping extreme values mitigates outliers
• Perturbation bounds ensure subspace accuracy