Abstract
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 l m ≤ n. Finally, some numerical illustrations for spectral norm computation are presented.
Original language | English |
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Pages (from-to) | 400-425 |
Journal | SIAM Journal on Matrix Analysis and Applications |
Volume | 39 |
Issue number | 1 |
Early online date | 8 Mar 2018 |
DOIs | |
Publication status | Published - Mar 2018 |
Keywords
- orthogonal tensor
- rank-one approximation
- spectral norm
- nuclear norm
- Hurwitz problem