As is well known, the smallest possible ratio between the spectral norm and
the Frobenius norm of an m×n matrix with m≤n is 1/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 n1×⋯×nd 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 n1≤⋯≤nd. 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/n1⋯nd−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 n1,…,nd 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 ℓ×m×n
is equivalent to the admissibility of the triple [ℓ,m,n] to the Hurwitz
problem. Some implications for higher-order tensors are then given. For
instance, real orthogonal n×⋯×n tensors of order d≥3
do exist, but only when n=1,2,4,8. In the complex case, the situation is
more drastic: unitary tensors of size ℓ×m×n with ℓ≤m≤n exist only when ℓm≤n. Finally, some numerical illustrations
for spectral norm computation are presented