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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2410.00397 |
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| _version_ | 1866910626948841472 |
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| author | Jou, Zhi-Yu Huang, Su-Yun Hung, Hung Eguchi, Shinto |
| author_facet | Jou, Zhi-Yu Huang, Su-Yun Hung, Hung Eguchi, Shinto |
| contents | Principal component analysis (PCA) is a widely used technique for dimension reduction. As datasets continue to grow in size, distributed-PCA (DPCA) has become an active research area. A key challenge in DPCA lies in efficiently aggregating results across multiple machines or computing nodes due to computational overhead. Fan et al. (2019) introduced a pioneering DPCA method to estimate the leading rank-$r$ eigenspace, aggregating local rank-$r$ projection matrices by averaging. However, their method does not utilize eigenvalue information. In this article, we propose a novel DPCA method that incorporates eigenvalue information to aggregate local results via the matrix $β$-mean, which we call $β$-DPCA. The matrix $β$-mean offers a flexible and robust aggregation method through the adjustable choice of $β$ values. Notably, for $β=1$, it corresponds to the arithmetic mean; for $β=-1$, the harmonic mean; and as $β\to 0$, the geometric mean. Moreover, the matrix $β$-mean is shown to associate with the matrix $β$-divergence, a subclass of the Bregman matrix divergence, to support the robustness of $β$-DPCA. We also study the stability of eigenvector ordering under eigenvalue perturbation for $β$-DPCA. The performance of our proposal is evaluated through numerical studies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00397 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Generalized Mean Approach for Distributed-PCA Jou, Zhi-Yu Huang, Su-Yun Hung, Hung Eguchi, Shinto Machine Learning Principal component analysis (PCA) is a widely used technique for dimension reduction. As datasets continue to grow in size, distributed-PCA (DPCA) has become an active research area. A key challenge in DPCA lies in efficiently aggregating results across multiple machines or computing nodes due to computational overhead. Fan et al. (2019) introduced a pioneering DPCA method to estimate the leading rank-$r$ eigenspace, aggregating local rank-$r$ projection matrices by averaging. However, their method does not utilize eigenvalue information. In this article, we propose a novel DPCA method that incorporates eigenvalue information to aggregate local results via the matrix $β$-mean, which we call $β$-DPCA. The matrix $β$-mean offers a flexible and robust aggregation method through the adjustable choice of $β$ values. Notably, for $β=1$, it corresponds to the arithmetic mean; for $β=-1$, the harmonic mean; and as $β\to 0$, the geometric mean. Moreover, the matrix $β$-mean is shown to associate with the matrix $β$-divergence, a subclass of the Bregman matrix divergence, to support the robustness of $β$-DPCA. We also study the stability of eigenvector ordering under eigenvalue perturbation for $β$-DPCA. The performance of our proposal is evaluated through numerical studies. |
| title | A Generalized Mean Approach for Distributed-PCA |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2410.00397 |