Optimal orthogonal approximations to symmetric tensors cannot always be chosen symmetric

TL;DR

This paper demonstrates that, unlike matrices and rank-one tensor approximations, optimal orthogonal low-rank approximations of symmetric tensors of higher rank cannot always be symmetric, highlighting a fundamental limitation.

Contribution

It proves that for higher-rank symmetric tensor approximations, optimal orthogonal solutions may not preserve symmetry, contrasting with classical results for matrices and rank-one tensors.

Findings

Optimal orthogonal approximations of symmetric tensors of rank > 1 are not necessarily symmetric.

Classical symmetry preservation results do not extend to higher-rank orthogonal tensor approximations.

The study covers four common notions of tensor orthogonality in the literature.

Abstract

We study the problem of finding orthogonal low-rank approximations of symmetric tensors. In the case of matrices, the approximation is a truncated singular value decomposition which is then symmetric. Moreover, for rank-one approximations of tensors of any dimension, a classical result proven by Banach in 1938 shows that the optimal approximation can always be chosen to be symmetric. In contrast to these results, this article shows that the corresponding statement is no longer true for orthogonal approximations of higher rank. Specifically, for any of the four common notions of tensor orthogonality used in the literature, we show that optimal orthogonal approximations of rank greater than one cannot always be chosen to be symmetric.