A unifying Perron-Frobenius theorem for nonnegative tensors via multi-homogeneous maps

TL;DR

This paper presents a unified Perron-Frobenius theorem for nonnegative tensors using multi-homogeneous maps, generalizing and improving previous results, and introduces a power method with convergence analysis.

Contribution

It introduces a general tensor eigenvalue problem via shape partitions, unifies existing theories, and develops a new power method with convergence guarantees.

Findings

Unified Perron-Frobenius theorem for nonnegative tensors

General power method with proven convergence

Improved assumptions over previous results

Abstract

We introduce the concept of shape partition of a tensor and formulate a general tensor eigenvalue problem that includes all previously studied eigenvalue problems as special cases. We formulate irreducibility and symmetry properties of a nonnegative tensor $T$ in terms of the associated shape partition. We recast the eigenvalue problem for $T$ as a fixed point problem on a suitable product of projective spaces. This allows us to use the theory of multi-homogeneous order-preserving maps to derive a new and unifying Perron-Frobenius theorem for nonnegative tensors which either implies earlier results of this kind or improves them, as weaker assumptions are required. We introduce a general power method for the computation of the dominant tensor eigenpair, and provide a detailed convergence analysis.