Spectral inequalities for nonnegative tensors and their tropical analogues

TL;DR

This paper generalizes spectral inequalities from nonnegative matrices to tensors, linking spectral radius to entropy maximization and introducing a tropical spectral radius with combinatorial formulas.

Contribution

It extends classical spectral inequalities to nonnegative tensors and introduces a tropical analogue with explicit combinatorial characterization.

Findings

Spectral radius of tensors linked to entropy maximization.

Tropical spectral radius characterized as a limit of classical spectral radius.

Explicit combinatorial formula derived for the tropical spectral radius.

Abstract

We extend some characterizations and inequalities for the eigenvalues of nonnegative matrices, such as Donsker-Varadhan, Friedland-Karlin, Karlin-Ost inequalities, to nonnegative tensors. Our approach involves a correspondence between nonnegative tensors, ergodic control and entropy maximization: we show in particular that the logarithm of the spectral radius of a tensor is given by en entropy maximization problem over a space of occupation measures. We study in particular the tropical analogue of the spectral radius, that we characterize as a limit of the classical spectral radius, and we give an explicit combinatorial formula for this tropical spectral radius.