Tropical Spectral Theory of Tensors
This work provides foundational theory for tropical tensor eigenvalues, relevant to researchers in tropical geometry and optimization.
The paper generalizes tropical spectral theory from matrices to tensors, proving existence and uniqueness of eigenvalues and linking them to H-cycles in hypergraphs, with efficient computation via linear programming.
We introduce and study tropical eigenpairs of tensors, a generalization of the tropical spectral theory of matrices. We show the existence and uniqueness of an eigenvalue. We associate to a tensor a directed hypergraph and define a new type of cycle on a hypergraph, which we call an H-cycle. The eigenvalue of a tensor turns out to be equal to the minimal normalized weighted length of H-cycles of the associated hypergraph. We show that the eigenvalue can be computed efficiently via a linear program. Finally, we suggest possible directions of research.