Finding the Spectral Radius of a Nonnegative Tensor

TL;DR

This paper introduces strictly nonnegative tensors, explores their spectral properties, and develops a convergent power method algorithm for computing the spectral radius of general nonnegative tensors, supported by numerical validation.

Contribution

It defines strictly nonnegative tensors, establishes their spectral properties, and proposes a new convergent algorithm for spectral radius computation in nonnegative tensors.

Findings

Spectral radius of strictly nonnegative tensors is always positive.

A power method converges globally for weakly irreducible nonnegative tensors.

The algorithm effectively computes spectral radii for general nonnegative tensors.

Abstract

In this paper, we introduce a new class of nonnegative tensors --- strictly nonnegative tensors. A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa. We show that the spectral radius of a strictly nonnegative tensor is always positive. We give some sufficient and necessary conditions for the six well-conditional classes of nonnegative tensors, introduced in the literature, and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors. We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility. We show that for a nonnegative tensor T, there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible; and the spectral radius of T can be obtained from those spectral radii of the induced tensors. In this way, we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption. The preliminary numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.