Best rank one approximation of real symmetric tensors can be chosen symmetric

TL;DR

This paper proves that the best rank-one approximation of a real symmetric tensor can always be chosen symmetric, and such an approximation is unique under certain conditions, simplifying tensor approximation tasks.

Contribution

It establishes that symmetric tensors have symmetric best rank-one approximations and characterizes their uniqueness outside a specific algebraic variety.

Findings

Symmetric best rank-one approximation always exists for real symmetric tensors.

Such approximation can be chosen symmetric, simplifying computations.

Uniqueness holds outside a certain algebraic variety.

Abstract

We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.