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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 matrix with is 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 tensors of order , also
called the best rank-one approximation ratio in the literature, is
investigated. The exact value is not known for most configurations of . 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 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 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
is equivalent to the admissibility of the triple to the Hurwitz
problem. Some implications for higher-order tensors are then given. For
instance, real orthogonal tensors of order
do exist, but only when . In the complex case, the situation is
more drastic: unitary tensors of size with exist only when . Finally, some numerical illustrations
for spectral norm computation are presented
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