Tensor norm and maximal singular vectors of nonnegative tensors — A Perron–Frobenius theorem, a Collatz–Wielandt characterization and a generalized power method

Abstract

We study the ℓp1,…,pm-singular value problem for nonnegative 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.