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 x n matrix with m x n is 1= p 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 find 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= p n1 nd1 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 ` x m x 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 x 3 do exist, but only when n = 1; 2; 4; 8. In the complex case, the situation is more drastic: unitary tensors of size ` x m x n with ` x m x n exist only when `m x n. Finally, some numerical illustrations for spectral norm computation are presented. [ABSTRACT FROM AUTHOR]