A Generalized Mean Approach for Distributed-PCA
提出一种利用特征值信息的矩阵β均值方法（β-DPCA），通过灵活调整β值（如算术、调和、几何均值）来稳健聚合分布式节点的主成分分析结果，解决分布式PCA中计算开销大的问题。
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. 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 use 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→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 perturbations for β-DPCA. The performance of our proposal is evaluated through numerical studies. Supplementary materials for this article are available online.