Tropical Spectral Theory of Tensors

TL;DR

This paper extends tropical spectral theory to tensors, establishing the existence and uniqueness of eigenvalues, linking them to hypergraph cycles, and providing an efficient linear programming method for computation.

Contribution

It introduces tropical eigenpairs for tensors, defines H-cycles on hypergraphs, and connects eigenvalues to minimal cycle lengths, advancing spectral theory in hypergraph contexts.

Findings

Eigenvalues are unique and exist for tropical tensors.

Eigenvalues correspond to minimal normalized H-cycle lengths.

Eigenvalues can be computed efficiently using linear programming.

Abstract

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.