The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics
二到无穷范数与奇异子空间几何及其高维统计应用AI 解读 ↓
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二到无穷范数与奇异子空间几何及其高维统计应用
- 研究问题
- 如何用二到无穷范数刻画矩阵扰动下奇异子空间的几何变化，并在奇异值有重数时控制奇异向量的逐元素误差？
- 方法与思路
- 作者从确定性Procrustes分析出发，提出矩阵分解并发展二到无穷范数工具，分析一般扰动及多种矩阵噪声模型。
- 材料陈述的贡献
- 摘要称得到奇异向量逐元素扰动界，并讨论协方差估计、奇异子空间恢复和多图推断中的应用及该范数的适用性。
- 阅读关注点
- 应核对子空间对齐方式、奇异值重数与谱间隔条件、各噪声模型的界，以及多图推断如何使用这些误差控制。
- 阅读边界
- 材料为OpenAlex索引摘要，稿件版本未注明；未提供误差界公式和条件，不能推断任意噪声下都成立或具有统一最优性。
原文摘要
The singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors. This paper provides a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two-to-infinity norm. Motivated by preliminary deterministic Procrustes analysis, we consider a general matrix perturbation setting in which we derive a new Procrustean matrix decomposition. Together with flexible machinery developed for the two-to-infinity norm, this allows us to conduct a refined analysis of the induced perturbation geometry with respect to the underlying singular vectors even in the presence of singular value multiplicity. Our analysis yields singular vector entrywise perturbation bounds for a range of popular matrix noise models, each of which has a meaningful associated statistical inference task. In addition, we demonstrate how the two-to-infinity norm is the preferred norm in certain statistical settings. Specific applications discussed in this paper include covariance estimation, singular subspace recovery, and multiple graph inference. Both our Procrustean matrix decomposition and the technical machinery developed for the two-to-infinity norm may be of independent interest.