Winsorized PCA在去除异常值影响的同时，能稳定恢复真实数据子空间。
Subspace Recovery in Winsorized PCA: Insights into Accuracy and Robustness
- 通过截断极端值提升主成分分析的鲁棒性，避免异常点干扰。
- 样本量增大、异常比例降低时，子空间估计与真实子空间一致。
- 首次给出子空间统计量的断裂点下界，证明其对异常值强容忍度。
本文研究了基于截尾化（Winsorized）主成分分析（WPCA）的子空间恢复理论性质。尽管截尾化广泛应用于多元分析以抑制异常值影响，但其在子空间恢复方面的理论研究仍较少。本文详细分析了WPCA的准确性：当样本数增加且异常比例下降时，样本子空间可一致收敛至真实总体子空间。同时建立了扰动界，确保污染数据下的WPCA子空间与纯净数据下的子空间保持接近。进一步将经典断裂点概念扩展至子空间统计量，推导出WPCA断裂点的下界。分析表明，WPCA在温和假设下兼具一致性与强鲁棒性。通过一个简单示例数值验证了扰动界与断裂点上界的特性，凸显截尾化在子空间恢复中的有效性。
原文摘要 · Abstract (English)
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.
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