On orthogonal tensors and best rank-one approximation ratio
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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 n_1 ×...× 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 <...< 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/√(n_1 ... 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,...,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 ℓ× m × n is equivalent to the admissibility of the triple [ℓ,m,n] to 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 ℓ m < n. Finally, some numerical illustrations for spectral norm computation are presented.
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