Analysis of perturbations of singular values in concatenated matrices
DOI:
https://doi.org/10.3842/umzh.v77i8.9103Keywords:
numerical analysis, perturbation analysis, singular values, concatenated matrices, clustering, Singular Value Decomposition, matrix theoryAbstract
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.
References
1. D. Chandler, The rotation of eigenvectors by a perturbation, J. Math. Anal. and Appl. (US), 6 (1963). DOI: https://doi.org/10.1016/0022-247X(63)90001-5
2. C. Eckart, G. Young, The approximation of one matrix by another of lower rank, Psychometrika, 1, № 3, 211–218 (1936). DOI: https://doi.org/10.1007/BF02288367
3. A. Gramfort, M. Luessi, E. Larson, D. A. Engemann, D. Strohmeier, C. Brodbeck, R. T. Hlushchuk, M. Hämäläinen, Meg and eeg data analysis with mne-python, Front. Neuroscie, 7(267), 1–13 (2013).
4. R. A. Horn, C. R. Johnson, Matrix analysis, Cambridge University Press (1985). DOI: https://doi.org/10.1017/CBO9780511810817
5. P. Kairouz, H. B. McMahan, B. Avent, A. Bellet, M. Bennis, A. N. Bhagoji et al., Advances and open problems in federated learning, Found. and Trends Machine Learning, 14, № 1–2, 1–210 (2021). DOI: https://doi.org/10.1561/2200000083
6. H. B. McMahan, E. Moore, D. Ramage, S. Hampson, B. Aguera y Arcas, Communication-efficient learning of deep networks from decentralized data, Proc. 20th International Conference on Artificial Intelligence and Statistics (AISTATS), vol. 54, Proc. Machine Learning Research, 1273–1282 (2017).
7. E. Schmidt, Zur Theorie der linearen und nichtlinearen Integralgleichungen: I. Entwicklung willkürlicher Funktionen nach systemen Vorgeschriebener, Math. Ann., 63, № 4, 433–476 (1907). DOI: https://doi.org/10.1007/BF01449770
8. G. W. Stewart, Perturbation theory for the singular value decomposition, Digital Repository at the University of Maryland (1998).
9. G. W. Stewart, Ji-Guang Sun, Matrix perturbation theory, Academic Press, San Diego (1990).
10. D. Tse, P. Viswanath, Fundamentals of wireless communication, Cambridge University Press (2005). DOI: https://doi.org/10.1017/CBO9780511807213
11. H. Weyl, Das asymptotische Verteilungsgesetz der eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71, 441–479 (1912). DOI: https://doi.org/10.1007/BF01456804
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Максим Шамрай
This work is licensed under a Creative Commons Attribution 4.0 International License.
