研究拼接矩阵的奇异值稳定性，为数据压缩与聚类提供理论依据。
Perturbation Analysis of Singular Values in Concatenated Matrices
- 基于扰动分析扩展经典不等式，推导拼接矩阵奇异值的稳定性边界。
- 当子矩阵范数接近时，拼接矩阵主奇异值保持稳定，支持精度与压缩的可控权衡。
- 适用于数值线性代数、信号处理等领域中的矩阵压缩与聚类任务。
矩阵拼接是通过奇异值分解（SVD）和低秩逼近揭示数据共享结构的常用方法。核心问题在于：拼接矩阵的奇异值谱如何与各组成部分的谱相关？本文提出一种扰动分析技术，将经典结果如Weyl不等式推广至拼接矩阵，建立了子矩阵微小扰动下奇异值稳定性的解析界。结果表明，若子矩阵在范数上接近，拼接矩阵的主导奇异值保持稳定，从而实现精度与压缩率之间的可控权衡。该理论为改进矩阵聚类与压缩策略提供了基础，可应用于数值线性代数、信号处理及数据驱动建模等领域。
原文摘要 · Abstract (English)
Concatenating matrices is a common technique for uncovering shared structures in data through singular value decomposition (SVD) and low-rank approximations. The fundamental question arises: How does the singular value spectrum of the concatenated matrix relate to the spectra of its individual components? In the present work, we develop a perturbation technique that extends classical results such as Weyl's inequality to concatenated matrices. We setup analytical bounds that quantify stability of singular values under small perturbations in submatrices. The results demonstrate that if submatrices are close in a norm, dominant singular values of the concatenated matrix remain stable enabling controlled trade-offs between accuracy and compression. These provide a theoretical basis for improved matrix clustering and compression strategies with applications in the numerical linear algebra, signal processing, and data-driven modeling.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。