Tropical Spectral Theory of Tensors

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. 1. Background A tensor of order m and rank n is an array A = (ai1···im) of elements of a field K (which we shall take to be R or R = R ∪ {∞}), where 1 ≤ i1, . . . , im ≤ n. In ordinary arithmetic, given x ∈ R , we define (Ax)i := n ∑ i2,...,im=1 aii2···imxi2 · · ·xim . An H-eigenpair [Qi05] of a tensor is defined as follows. Define x = (x i )i. Then an H-eigenpair is a pair (x, λ) ∈ P × R such that Ax = λx. Let A be a n× n matrix with entries in the tropical semiring (R,⊕,⊙). An eigenvalue of A is a number λ such that A⊙ v = λ⊙ v. The nature of tropical eigenpairs is understood in the setting of matrices ([ST13],[Tra14]) but a survey of the literature shows no prior research on tropical eigenpairs of tensors. Definition 1.1. A tropical H-eigenpair for a tensor (ai1···im) ∈ R n of order m and rank n is a pair (x, λ) ∈ R/R(1, 1, . . . , 1)× R such that