Perturbation Analysis of Singular Values in Concatenated Matrices

• URL: http://arxiv.org/abs/2505.01427v2
• Date: Sun, 29 Jun 2025 16:31:24 GMT
• Title: Perturbation Analysis of Singular Values in Concatenated Matrices
• Authors: Maksym Shamrai,
• Abstract summary: How does the singular value spectrum and singular perturbationd matrix relate to the spectra of its individual components?<n>We setup analytical bounds that quantify stability of values under small perturbations in submatrices.<n>Results demonstrate that if submatrices are close in a norm, dominant singular values of the singular matrix remain stable enabling controlled trade-offs between accuracy and compression.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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