Computing tensor eigenvalues via homotopy methods

TL;DR

This paper introduces new homotopy algorithms for computing tensor eigenvalues and eigenvectors, providing theoretical bounds and practical software to efficiently find all eigenpairs, including real and complex ones.

Contribution

It proposes two novel homotopy continuation algorithms for tensor eigenproblems, along with a heuristic-Newton method and a MATLAB package, advancing computational methods for tensor eigenvalues.

Findings

Algorithms successfully compute all isolated eigenpairs

The methods are effective for both real and complex eigenpairs

Numerical results demonstrate efficiency and accuracy

Abstract

We introduce the concept of mode-k generalized eigenvalues and eigenvectors of a tensor and prove some properties of such eigenpairs. In particular, we derive an upper bound for the number of equivalence classes of generalized tensor eigenpairs using mixed volume. Based on this bound and the structures of tensor eigenvalue problems, we propose two homotopy continuation type algorithms to solve tensor eigenproblems. With proper implementation, these methods can find all equivalence classes of isolated generalized eigenpairs and some generalized eigenpairs contained in the positive dimensional components (if there are any). We also introduce an algorithm that combines a heuristic approach and a Newton homotopy method to extract real generalized eigenpairs from the found complex generalized eigenpairs. A MATLAB software package TenEig has been developed to implement these methods. Numerical results are presented to illustrate the effectiveness and efficiency of TenEig for computing complex or real generalized eigenpairs.