The Perron-Frobenius Theorem for Multi-homogeneous Maps

TL;DR

This paper extends Perron-Frobenius theory to order-preserving multi-homogeneous maps, unifying spectral analysis of nonnegative tensors and proposing a generalized power method with convergence guarantees.

Contribution

It introduces a new class of maps, proves Perron-Frobenius theorems for them, and develops a generalized power method with convergence analysis.

Findings

Established weak and strong Perron-Frobenius theorems for multi-homogeneous maps.

Provided a Collatz-Wielandt principle for the maximal eigenvalue.

Proposed and analyzed a generalized power method for eigenvector computation.

Abstract

We introduce the notion of order-preserving multi-homogeneous mapping which allows to study Perron-Frobenius type theorems and nonnegative tensors in unified fashion. We prove a weak and strong Perron-Frobenius theorem for these maps and provide a Collatz-Wielandt principle for the maximal eigenvalue. Additionally, we propose a generalization of the power method for the computation of the maximal eigenvector and analyse its convergence. We show that the general theory provides new results and strengthens existing results for various spectral problems for nonnegative tensors.