标题： $φ$-PCA 框架：一种统一且保持效率的方法及鲁棒变体 显示英文标题

标题： The $φ$-PCA Framework: A Unified and Efficiency-Preserving Approach with Robust Variants

摘要： 主成分分析（PCA）是多元统计中的基本工具，但其对异常值的敏感性和在分布式环境中的局限性限制了其在现代大规模应用中的有效性。 为解决这些挑战，我们引入了$\phi$-PCA框架，该框架提供了稳健和分布式PCA的统一公式。 $\phi$-PCA方法类保留了标准PCA的渐近效率，同时通过使用适当的$\phi$函数聚合多个局部估计，增强了顺序鲁棒性，从而在污染条件下实现了更精确的特征子空间估计。 值得注意的是，调和均值PCA（HM-PCA），对应于选择$\phi(u)=u^{-1}$，实现了最优的顺序鲁棒性，推荐用于实际应用。 理论结果进一步表明，鲁棒性随着分区数量的增加而提高，这一现象在稳健或分布式PCA的文献中很少被探讨。 总体而言，$\phi$-PCA所依据的分区分聚原则为开发适用于稳健和分布式数据分析的稳健且保持效率的方法提供了通用策略。 显示英文摘要

摘要： Principal component analysis (PCA) is a fundamental tool in multivariate statistics, yet its sensitivity to outliers and limitations in distributed environments restrict its effectiveness in modern large-scale applications. To address these challenges, we introduce the $\phi$-PCA framework which provides a unified formulation of robust and distributed PCA. The class of $\phi$-PCA methods retains the asymptotic efficiency of standard PCA, while aggregating multiple local estimates using a proper $\phi$ function enhances ordering-robustness, leading to more accurate eigensubspace estimation under contamination. Notably, the harmonic mean PCA (HM-PCA), corresponding to the choice $\phi(u)=u^{-1}$, achieves optimal ordering-robustness and is recommended for practical use. Theoretical results further show that robustness increases with the number of partitions, a phenomenon seldom explored in the literature on robust or distributed PCA. Altogether, the partition-aggregation principle underlying $\phi$-PCA offers a general strategy for developing robust and efficiency-preserving methodologies applicable to both robust and distributed data analysis.