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

• URL: http://arxiv.org/abs/2510.13159v1
• Date: Wed, 15 Oct 2025 05:21:11 GMT
• Title: The $φ$-PCA Framework: A Unified and Efficiency-Preserving Approach with Robust Variants
• Authors: Hung Hung, Zhi-Yu Jou, Su-Yun Huang, Shinto Eguchi,
• Abstract summary: We introduce the $phi$-PCA framework which provides a unified formulation of robust and distributed PCA.<n>We show that the partition-aggregation principle underlying $phi$-PCA offers a general strategy for developing robust and efficiency-preserving methodologies.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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