The $ϕ$-PCA Framework: A Unified and Efficiency-Preserving Approach with Robust Variants
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| Format: | Preprint |
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2025
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| _version_ | 1866917015402315776 |
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| author | Hung, Hung Jou, Zhi-Yu Huang, Su-Yun Eguchi, Shinto |
| author_facet | Hung, Hung Jou, Zhi-Yu Huang, Su-Yun Eguchi, Shinto |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_13159 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $ϕ$-PCA Framework: A Unified and Efficiency-Preserving Approach with Robust Variants Hung, Hung Jou, Zhi-Yu Huang, Su-Yun Eguchi, Shinto Methodology Statistics Theory Machine Learning 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. |
| title | The $ϕ$-PCA Framework: A Unified and Efficiency-Preserving Approach with Robust Variants |
| topic | Methodology Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2510.13159 |