On orthogonal tensors and best rank-one approximation ratio

TL;DR

This paper investigates the minimal spectral-to-Frobenius norm ratio for tensors, characterizes when orthogonal tensors attain this bound, and explores their existence related to classical algebraic problems, with implications for higher-order tensors.

Contribution

It establishes the conditions under which orthogonal and unitary tensors attain the minimal spectral-to-Frobenius ratio and links their existence to the Hurwitz problem, extending understanding of tensor approximation ratios.

Findings

Orthogonal tensors attain the minimal ratio iff they are orthogonal up to scaling.

Existence of real orthogonal tensors of size    is limited to dimensions 1, 2, 4, 8.

Complex unitary tensors exist only when            .

Abstract

As is well known, the smallest possible ratio between the spectral norm and the Frobenius norm of an $m \times n$ matrix with $m \le n$ is $1/\sqrt{m}$ and is (up to scalar scaling) attained only by matrices having pairwise orthonormal rows. In the present paper, the smallest possible ratio between spectral and Frobenius norms of $n_1 \times \dots \times n_d$ tensors of order $d$, also called the best rank-one approximation ratio in the literature, is investigated. The exact value is not known for most configurations of $n_1 \le \dots \le n_d$. Using a natural definition of orthogonal tensors over the real field (resp., unitary tensors over the complex field), it is shown that the obvious lower bound $1/\sqrt{n_1 \cdots n_{d-1}}$ is attained if and only if a tensor is orthogonal (resp., unitary) up to scaling. Whether or not orthogonal or unitary tensors exist depends on the dimensions $n_1,\dots,n_d$ and the field. A connection between the (non)existence of real orthogonal tensors of order three and the classical Hurwitz problem on composition algebras can be established: existence of orthogonal tensors of size $\ell \times m \times n$ is equivalent to the admissibility of the triple $[\ell,m,n]$ to the Hurwitz problem. Some implications for higher-order tensors are then given. For instance, real orthogonal $n \times \dots \times n$ tensors of order $d \ge 3$ do exist, but only when $n = 1,2,4,8$. In the complex case, the situation is more drastic: unitary tensors of size $\ell \times m \times n$ with $\ell \le m \le n$ exist only when $\ell m \le n$. Finally, some numerical illustrations for spectral norm computation are presented.