Analysis of perturbations of singular values in concatenated matrices
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 an...
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| Date: | 2026 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/9103 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513387983667200 |
|---|---|
| author | Shamrai, M. Шамрай, Максим |
| author_facet | Shamrai, M. Шамрай, Максим |
| author_sort | Shamrai, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-03-22T13:31:15Z |
| description | 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. |
| doi_str_mv | 10.3842/umzh.v77i8.9103 |
| first_indexed | 2026-03-24T03:43:53Z |
| format | Article |
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| id | umjimathkievua-article-9103 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:43:53Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0d/edf8ab9c653c70ce74cbca61a53c680d |
| spelling | umjimathkievua-article-91032026-03-22T13:31:15Z Analysis of perturbations of singular values in concatenated matrices Аналіз збурень сингулярних значень конкатенованих матриць Shamrai, M. Шамрай, Максим numerical analysis perturbation analysis singular values concatenated matrices clustering Singular Value Decomposition matrix theory Numerical Analysis Perturbation Analysis Matrix theory 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. УДК 512.5 Конкатенація матриць є поширеним прийомом для виявлення спільної структури в даних за допомогою сингулярного розкладу матриць (SVD) та низькорангової апроксимації. Основне питання полягає в тому як спектр сингулярних значень конкатенованої матриці пов’язаний зі спектрами її складових. У цій статті розширено техніку збурень, що узагальнює класичні результати, такі як нерівність Вейля, на випадок конкатенованих матриць. Встановлено аналітичні межі, які кількісно характеризують стабільність сингулярних значень при малих збуреннях її підматриць. Показано, що якщо підматриці близькі за певною нормою, домінантні сингулярні значення конкатенованої матриці залишаються стабільними, що дає змогу керувати компромісом між точністю та стисненням. Отримані результати забезпечують теоретичну основу для вдосконалених стратегій кластеризації та стиснення матриць із застосуваннями в чисельній лінійній алгебрі, обробці сигналів та моделях, побудованих на даних. Institute of Mathematics, NAS of Ukraine 2026-03-21 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/9103 10.3842/umzh.v77i8.9103 Ukrains’kyi Matematychnyi Zhurnal; Vol. 77 No. 8 (2025); 521–532 Український математичний журнал; Том 77 № 8 (2025); 521–532 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/9103/10557 Copyright (c) 2025 Максим Шамрай |
| spellingShingle | Shamrai, M. Шамрай, Максим Analysis of perturbations of singular values in concatenated matrices |
| title | Analysis of perturbations of singular values in concatenated matrices |
| title_alt | Аналіз збурень сингулярних значень конкатенованих матриць |
| title_full | Analysis of perturbations of singular values in concatenated matrices |
| title_fullStr | Analysis of perturbations of singular values in concatenated matrices |
| title_full_unstemmed | Analysis of perturbations of singular values in concatenated matrices |
| title_short | Analysis of perturbations of singular values in concatenated matrices |
| title_sort | analysis of perturbations of singular values in concatenated matrices |
| topic_facet | numerical analysis perturbation analysis singular values concatenated matrices clustering Singular Value Decomposition matrix theory Numerical Analysis Perturbation Analysis Matrix theory |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9103 |
| work_keys_str_mv | AT shamraim analysisofperturbationsofsingularvaluesinconcatenatedmatrices AT šamrajmaksim analysisofperturbationsofsingularvaluesinconcatenatedmatrices AT shamraim analízzburenʹsingulârnihznačenʹkonkatenovanihmatricʹ AT šamrajmaksim analízzburenʹsingulârnihznačenʹkonkatenovanihmatricʹ |