Real eigenstructure of regular simplex tensors

TL;DR

This paper analyzes the real eigenstructure of regular simplex tensors, revealing both robust and non-robust eigenvectors and their relation to the tensor's symmetric decomposition.

Contribution

It provides a comprehensive analysis of the eigenstructure of regular simplex tensors, including robustness and the relation to their symmetric decomposition.

Findings

Regular simplex tensors have both robust and non-robust eigenvectors.

Eigenvectors only partially coincide with the generators of the symmetric decomposition.

The paper discusses the full real eigenstructure and robustness of eigenvectors.

Abstract

We are concerned with the eigenstructure of supersymmetric tensors. Like in the matrix case, normalized tensor eigenvectors are fixed points of the tensor power iteration map. However, unless the given tensor is orthogonally decomposable, some of these fixed points may be repelling and therefore be undetectable by any numerical scheme. In this paper, we consider the case of regular simplex tensors whose symmetric decomposition is induced by an overcomplete, equiangular set of $n+1$ vectors from $\mathbb R^n$. We discuss the full real eigenstructure of such tensors, including the robustness analysis of all normalized eigenvectors. As it turns out, regular simplex tensors exhibit robust as well as non-robust eigenvectors which, moreover, only partly coincide with the generators from the symmetric tensor decomposition.