Robust Eigenvectors of Symmetric Tensors

TL;DR

This paper demonstrates that for high-order symmetric tensors, certain eigenvectors are robust fixed points of the tensor power method, enabling their decomposition through this iterative approach.

Contribution

The paper establishes conditions under which eigenvectors are robust fixed points for symmetric tensors, extending the tensor power method's applicability.

Findings

Eigenvectors generating symmetric decompositions are robust fixed points.

New classes of tensors, like equiangular tight frame decomposable tensors, are shown to be decomposable via the tensor power method.

The results generalize the robustness of eigenvectors in higher-order symmetric tensors.

Abstract

The tensor power method generalizes the matrix power method to higher order arrays, or tensors. Like in the matrix case, the fixed points of the tensor power method are the eigenvectors of the tensor. While every real symmetric matrix has an eigendecomposition, the vectors generating a symmetric decomposition of a real symmetric tensor are not always eigenvectors of the tensor. In this paper we show that whenever an eigenvector is a generator of the symmetric decomposition of a symmetric tensor, then (if the order of the tensor is sufficiently high) this eigenvector is robust, i.e., it is an attracting fixed point of the tensor power method. We exhibit new classes of symmetric tensors whose symmetric decomposition consists of eigenvectors. Generalizing orthogonally decomposable tensors, we consider equiangular tight frame decomposable and equiangular set decomposable tensors. Our main result implies that such tensors can be decomposed using the tensor power method.