Computing the dominant eigenpair of an essentially nonnegative tensor via a homotopy method

TL;DR

This paper introduces a homotopy method for efficiently computing the dominant eigenpair of essentially nonnegative tensors, with proven convergence and demonstrated numerical effectiveness.

Contribution

It establishes the uniqueness of the dominant eigenpair for irreducible tensors and develops a homotopy algorithm applicable to both reducible and irreducible cases.

Findings

Proves the uniqueness of the dominant eigenpair for irreducible tensors.

Develops a convergent homotopy method for eigenpair computation.

Numerical results show the efficiency of the proposed algorithm.

Abstract

The theory of eigenvalues and eigenvectors is one of the fundamental and essential components in tensor analysis. Computing the dominant eigenpair of an essentially nonnegative tensor is an important topic in tensor computation because of the critical applications in network resource allocations. In this paper, we consider the aforementioned topic and there are two main contributions. First, we show that an irreducible essentially nonnegative tensor has a unique positive dominant eigenvalue with a unique positive normalized eigenvector. Second, we present a homotopy method to compute the dominant eigenpair and prove that it converges to the desired dominant eigenpair whether the given tensor is irreducible or reducible based on an approximation technique. Finally, we implement the method using a prediction-correction approach for path following and some numerical results are reported to illustrate the efficiency of the proposed algorithm.