## 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*n*_{1 }× ··· ×*n*tensors of order_{d}*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*. 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/√_{d}*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*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_{d}*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