Finding the spectral radius of a nonnegative irreducible symmetric tensor via DC programming

TL;DR

This paper develops a new iterative method based on DC programming to efficiently find the spectral radius and positive eigenvector of an irreducible nonnegative symmetric tensor, with proven convergence and acceleration techniques.

Contribution

It introduces a simplified, cost-effective iterative approach using DC programming for spectral radius computation of symmetric tensors, with convergence guarantees and acceleration.

Findings

Method converges Q-linearly to spectral radius and eigenvector.

Line search technique improves convergence speed.

Preliminary results show strong practical performance.

Abstract

The Perron-Frobenius theorem says that the spectral radius of an irreducible nonnegative tensor is the unique positive eigenvalue corresponding to a positive eigenvector. With this in mind, the purpose of this paper is to find the spectral radius and its corresponding positive eigenvector of an irreducible nonnegative symmetric tensor. By transferring the eigenvalue problem into an equivalent problem of minimizing a concave function on a closed convex set, which is typically a DC (difference of convex functions) programming, we derive a simpler and cheaper iterative method. The proposed method is well-defined. Furthermore, we show that both sequences of the eigenvalue estimates and the eigenvector evaluations generated by the method $Q$-linearly converge to the spectral radius and its corresponding eigenvector, respectively. To accelerate the method, we introduce a line search technique. The improved method retains the same convergence property as the original version. Preliminary numerical results show that the improved method performs quite well.