Tensor norm and maximal singular vectors of non-negative tensors - a Perron-Frobenius theorem, a Collatz-Wielandt characterization and a generalized power method

TL;DR

This paper extends Perron-Frobenius theory to non-negative tensors, providing a characterization of maximal singular values and a convergent power method for computing singular vectors.

Contribution

It introduces a generalized Perron-Frobenius theorem, a Collatz-Wielandt characterization, and a higher order power method with proven linear convergence for non-negative tensors.

Findings

Established a Perron-Frobenius theorem for non-negative tensors.

Provided a Collatz-Wielandt characterization of the maximal singular value.

Developed a higher order power method with asymptotic linear convergence.

Abstract

We study the l^{p_1,...,p_m} singular value problem for non-negative tensors. We prove a general Perron-Frobenius theorem for weakly irreducible and irreducible nonnegative tensors and provide a Collatz-Wielandt characterization of the maximal singular value. Additionally, we propose a higher order power method for the computation of the maximal singular vectors and show that it has an asymptotic linear convergence rate.