提出可自适应调整的分布式主成分分析方法，提升抗异常节点能力。
Scale-Calibrated Median-of-Means for Robust Distributed Principal Component Analysis
- 基于欧氏空间与流形乘积结构设计校准的中位数平均估计器
- 在单细胞数据上验证其对特征值间隔驱动的不确定性具有鲁棒性
- 适合处理存在异常节点的大规模分布式数据，如基因组学分析
分布式主成分分析（PCA）生成各节点的均值向量和主子空间估计。稳健聚合这些异质对象需要均值误差与子空间误差之间的相对尺度。本文研究一种基于欧氏空间与格拉斯曼流形乘积几何结构的尺度校准中位数平均估计器。节点级PCA展开显示，均值分量具线性影响，而子空间分量为特征值间隔加权的协方差扰动。我们证明了局部约化结果：所提乘积流形中位数平均估计器渐近等价于节点影响误差的缩放空间中位数。该结果导出固定节点下的非高斯极限、增长节点下的高斯极限（含有限块偏差）及显式的尺度相关协方差公式。本文提出鲁棒块尺度校准与推断最优校准规则，建立高概率中位数平均界，刻画逐因子坏节点影响，并证明节点自助法有效性。模拟与大规模单细胞RNA-seq数据表明，尺度校准能自适应特征值间隔驱动的子空间不确定性，提供稳健的分布式PCA汇总结果。
原文摘要 · Abstract (English)
Distributed principal component analysis (PCA) produces node-level estimates of both a mean vector and a principal subspace. Robustly aggregating these heterogeneous objects requires a relative scale between mean error and subspace error. We study a scale-calibrated median-of-means estimator for this problem using the product geometry of Euclidean space and the Grassmann manifold. A node-level PCA expansion shows that the mean component has the usual linear influence, whereas the subspace component is an eigengap-weighted covariance perturbation. We prove a local reduction showing that the proposed product-manifold median-of-means estimator is asymptotically equivalent to a scaled spatial median of node influence errors. This yields fixed-node non-Gaussian limits, growing-node Gaussian limits with finite-block bias, and an explicit scale-dependent covariance formula. We propose robust block-scale and inference-optimal calibration rules, establish high-probability median-of-means bounds, characterize factorwise bad-node influence, and prove node-bootstrap validity. Simulations and large-scale single-cell RNA-seq data show that scale calibration adapts to eigengap-driven subspace uncertainty and provides a robust distributed PCA summary.
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