On orthogonal tensors and best rank-one approximation ratio - Li, Z; Nakatsukasa, Y; Soma, T; Uschmajew, A (Society for Industrial and Applied Mathematics, 2018)
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.