用可调参数的矩阵均值改进分布式PCA，提升结果稳定性与准确性。
A Generalized Mean Approach for Distributed-PCA
- 引入可调节参数β的矩阵β-均值，融合局部投影矩阵与特征值信息。
- 在不同β值下表现稳定，尤其在β=-1（调和平均）时抗噪能力更强。
- 适合处理大规模分布式数据的主成分分析，对特征值扰动敏感度低。
主成分分析（PCA）是广泛使用的降维技术。随着数据规模持续增长，分布式PCA（DPCA）成为研究热点。其核心挑战在于如何高效聚合多台机器上的计算结果，避免计算开销过大。Fan等（2019）提出一种开创性方法，通过平均本地秩-r投影矩阵估计前r个主成分空间，但未利用特征值信息。本文提出一种新方法——β-DPCA，通过矩阵β-均值聚合本地结果，融合特征值信息。矩阵β-均值可通过调整β值实现灵活且鲁棒的聚合：β=1对应算术平均，β=-1为调和平均，β→0趋近几何平均。该方法与矩阵β-散度（矩阵Bregman散度子类）相关联，支持其鲁棒性。我们还研究了β-DPCA在特征值扰动下的特征向量排序稳定性。数值实验验证了所提方法的有效性。
原文摘要 · Abstract (English)
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.
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