Computing tensor Z-eigenvectors with dynamical systems

TL;DR

This paper introduces a novel dynamical systems approach for computing Z-eigenvectors of general tensors, overcoming limitations of traditional algebraic methods like the higher-order power method.

Contribution

The paper presents a new framework based on numerical integration of dynamical systems to find Z-eigenvectors, applicable to general tensors beyond specific stochastic models.

Findings

The method successfully computes Z-eigenvectors that algebraic methods cannot.

It is applicable to a broad class of tensors, including those from stochastic processes.

The approach offers a new perspective on tensor eigenvector computation.

Abstract

We present a new framework for computing Z-eigenvectors of general tensors based on numerically integrating a dynamical system that can only converge to a Z-eigenvector. Our motivation comes from our recent research on spacey random walks, where the long-term dynamics of a stochastic process are governed by a dynamical system that must converge to a Z-eigenvector of a transition probability tensor. Here, we apply the ideas more broadly to general tensors and find that our method can compute Z-eigenvectors that algebraic methods like the higher-order power method cannot compute.