Non-Hermitian Coupling of Heterogeneous Datasets via the Bi-Orthogonal Extremal Trace Method

In this paper, we develop a non-Hermitian coupling method for integrating heterogeneous datasets. Using the completeness of the root-vector system of the operator associated with the coupled graph Laplacian, we construct a generalization of a bi-orthogonal basis. Decomposing the trace functional into regular and nilpotent parts, we elaborate an analogue of the gradient ascent and descent method for the extremal trace problem. The central point of the concept is that the extremal values coincide with the trace values and their difference yields an intrinsic characteristic of dataset heterogeneity. The results are illustrated by single-cell multi-omics integration, where the indicators distinguish stable systems from stratified populations.