Robust Eigenvectors of Symmetric Tensors
Tommi Muller, Elina Robeva, Konstantin Usevich
TL;DR
This work shows that, for real symmetric tensors, eigenvectors that appear in a symmetric decomposition become robust fixed points of the tensor power method when the tensor order is sufficiently large. The authors derive a main theorem linking robustness to the Jacobian of the power map, and they apply it to families of equiangular tensors, proving that their generating vectors are eigenvectors and, in many cases, robust. They develop and analyze equiangular sets and ETFs, regular simplex tensors, and Mercedes-Benz frames, providing explicit bounds and region-of-attraction results for even and odd tensor orders. The results offer a pathway to reliably decompose certain high-order tensors using the tensor power method, while also highlighting open questions about uniqueness and stability under perturbations in broader equiangular configurations.
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.