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

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.