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 × 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 Froben...
| Main Authors: | , , , |
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| Format: | Journal article |
| Language: | English |
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Society for Industrial and Applied Mathematics
2018
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| _version_ | 1797091591149584384 |
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| author | Li, Z Nakatsukasa, Y Soma, T Uschmajew, A |
| author_facet | Li, Z Nakatsukasa, Y Soma, T Uschmajew, A |
| author_sort | Li, Z |
| collection | OXFORD |
| description | 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 lm ≤ n. Finally, some numerical illustrations for spectral norm computation are presented. |
| first_indexed | 2024-03-07T03:35:13Z |
| format | Journal article |
| id | oxford-uuid:bc0e21db-6dc3-437d-8f26-fce435161f9e |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T03:35:13Z |
| publishDate | 2018 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:bc0e21db-6dc3-437d-8f26-fce435161f9e2022-03-27T05:21:32ZOn orthogonal tensors and best rank-one approximation ratioJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:bc0e21db-6dc3-437d-8f26-fce435161f9eEnglishSymplectic Elements at OxfordSociety for Industrial and Applied Mathematics2018Li, ZNakatsukasa, YSoma, TUschmajew, AAs 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 lm ≤ n. Finally, some numerical illustrations for spectral norm computation are presented. |
| spellingShingle | Li, Z Nakatsukasa, Y Soma, T Uschmajew, A On orthogonal tensors and best rank-one approximation ratio |
| title | On orthogonal tensors and best rank-one approximation ratio |
| title_full | On orthogonal tensors and best rank-one approximation ratio |
| title_fullStr | On orthogonal tensors and best rank-one approximation ratio |
| title_full_unstemmed | On orthogonal tensors and best rank-one approximation ratio |
| title_short | On orthogonal tensors and best rank-one approximation ratio |
| title_sort | on orthogonal tensors and best rank one approximation ratio |
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