Shifted Power Method for Computing Tensor Eigenpairs

TL;DR

This paper introduces the shifted symmetric higher-order power method (SS-HOPM) for computing real symmetric tensor eigenpairs, guaranteeing convergence and characterizing which eigenpairs can be found, with extensions to complex eigenpairs.

Contribution

The paper proposes SS-HOPM, a novel algorithm for tensor eigenpair computation, with convergence guarantees and eigenpair characterization, extending existing power methods.

Findings

SS-HOPM guarantees convergence to tensor eigenpairs.

Characterization of eigenpairs that can be found by SS-HOPM.

Extension of the method to complex eigenpairs.

Abstract

Recent work on eigenvalues and eigenvectors for tensors of order m >= 3 has been motivated by applications in blind source separation, magnetic resonance imaging, molecular conformation, and more. In this paper, we consider methods for computing real symmetric-tensor eigenpairs of the form Ax^{m-1} = \lambda x subject to ||x||=1, which is closely related to optimal rank-1 approximation of a symmetric tensor. Our contribution is a shifted symmetric higher-order power method (SS-HOPM), which we show is guaranteed to converge to a tensor eigenpair. SS-HOPM can be viewed as a generalization of the power iteration method for matrices or of the symmetric higher-order power method. Additionally, using fixed point analysis, we can characterize exactly which eigenpairs can and cannot be found by the method. Numerical examples are presented, including examples from an extension of the method to finding complex eigenpairs.