On orthogonal tensors and best rank-one approximation ratio

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 `×m×n is equivalent to the admissibility of the triple [`,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 ` × m × n with ` ≤ m ≤ n exist only when `m ≤ n. Finally, some numerical illustrations for spectral norm computation are presented.