On the Extremal Trace Problem on Sets Homeomorphic to the Stiefel Manifold and Its Application to Multi-Omics Data Integration

Kukushkin, Maksim V.; Arbatskiy, Mikhail S.; Balandin, Dmitriy E.; Churov, Alexey V.

doi:10.3390/math14132390

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On the Extremal Trace Problem on Sets Homeomorphic to the Stiefel Manifold and Its Application to Multi-Omics Data Integration

¹

Russian Clinical Research Center of Gerontology, Pirogov Russian National Research Medical University, Ministry of Healthcare of the Russian Federation, 129226 Moscow, Russia

²

Institute of Applied Mathematics and Automation, Kabardino-Balkarian Scientific Center, Russian Academy of Sciences, 360000 Nalchik, Russia

^*

Author to whom correspondence should be addressed.

Mathematics 2026, 14(13), 2390; https://doi.org/10.3390/math14132390

Submission received: 25 May 2026 / Revised: 24 June 2026 / Accepted: 1 July 2026 / Published: 3 July 2026

(This article belongs to the Special Issue Advances in Biological Systems with Mathematics)

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Abstract

In this paper, we consider the extremal trace problem for the coupled Laplacian on the sets homeomorphic to the Stiefel manifold defined on the complex Euclidean space. The study is implemented via various mathematical methods, including topological and probabilistic approaches. A detailed, comprehensive classification of the stationary points is given, which itself deserves to be considered as a general method in the framework of the optimization theory. Finally, an application to biologically meaningful integration of heterogeneous datasets, in which the structure of molecular interactions serves as a significant constraint for the mathematical model, is proposed. The main advantage of the elaborated method in comparison with the previously used ones is the absence of any conditions on the structure of the initial heterogeneous datasets. This paper is a continuation of a series of papers by our research group devoted to the development of new mathematical methods for integrating multi-omics data.

Keywords: manifold alignment; unsupervised topological alignment for single-cell multi-omics integration; graph Laplacian; single-cell RNA-seq; single-cell epigenomics

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MDPI and ACS Style

Kukushkin, M.V.; Arbatskiy, M.S.; Balandin, D.E.; Churov, A.V. On the Extremal Trace Problem on Sets Homeomorphic to the Stiefel Manifold and Its Application to Multi-Omics Data Integration. Mathematics 2026, 14, 2390. https://doi.org/10.3390/math14132390

AMA Style

Kukushkin MV, Arbatskiy MS, Balandin DE, Churov AV. On the Extremal Trace Problem on Sets Homeomorphic to the Stiefel Manifold and Its Application to Multi-Omics Data Integration. Mathematics. 2026; 14(13):2390. https://doi.org/10.3390/math14132390

Chicago/Turabian Style

Kukushkin, Maksim V., Mikhail S. Arbatskiy, Dmitriy E. Balandin, and Alexey V. Churov. 2026. "On the Extremal Trace Problem on Sets Homeomorphic to the Stiefel Manifold and Its Application to Multi-Omics Data Integration" Mathematics 14, no. 13: 2390. https://doi.org/10.3390/math14132390

APA Style

Kukushkin, M. V., Arbatskiy, M. S., Balandin, D. E., & Churov, A. V. (2026). On the Extremal Trace Problem on Sets Homeomorphic to the Stiefel Manifold and Its Application to Multi-Omics Data Integration. Mathematics, 14(13), 2390. https://doi.org/10.3390/math14132390

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On the Extremal Trace Problem on Sets Homeomorphic to the Stiefel Manifold and Its Application to Multi-Omics Data Integration

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