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

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 φ-PCA framework which provides a unified formulation of robust and distributed PCA. The class of φ-PCA methods retains the asymptotic efficiency of standard PCA, while aggregating multiple local estimates using a proper φ function enhances ordering-robustness, leading to more accurate eigensubspace estimation under contamination. Notably, the harmonic mean PCA (HM-PCA), corresponding to the choice φ(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 φ-PCA offers a general strategy for developing robust and efficiency-preserving methodologies applicable to both robust and distributed data analysis.