Analysis of perturbations of singular values in concatenated matrices - Shamrai, M. (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 512.5 Concatenating matrices is a common technique for uncovering shared structures in the data by using singular-valued decomposition (SVD) and low-rank approximations. The main analyzed question is to determine the relationship between the singular-valued spectrum of the concatenated matrix and the spectra of its individual components. In the present work, we develop a perturbation technique that extends the classical results, such as Weyl’s inequality, to the case of concatenated matrices. We establish the analytic bounds, which give quantitative characteristics of the stability of singular values under small perturbations in submatrices. The obtained results demonstrate that if the submatrices are close in a certain norm, then the predominant singular values of the concatenated matrix remain stable, which enables us to control the compromise between the accuracy and compression. The accumulated results also lay the theoretical basis for the improved matrix clustering and compression strategies with applications in the numerical linear algebra, signal processing, and data-driven modeling.