Name
Abstract | ||
|---|---|---|
As is well known, the smallest possible ratio between the spectral norm and the Frobenius norm of an m x n matrix with m <= n is 1/root 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) x center dot center dot center dot x 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) <= center dot center dot center dot <= 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/root n(1) center dot center dot center dot 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 x m x 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 x center dot center dot center dot x 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 x m x n with l <= m <= n exist only when l(m) <= n. Finally, some numerical illustrations for spectral norm computation are presented. |
Year | DOI | Venue |
|---|---|---|
2018 | 10.1137/17M1144349 | SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS |
Keywords | DocType | Volume |
orthogonal tensor,rank-one approximation,spectral norm,nuclear norm,Hurwitz problem | Journal | 39 |
Issue | ISSN | Citations |
1 | 0895-4798 | 2 |
PageRank | References | Authors |
0.39 | 8 | 4 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Zhening Li | 1 | 105 | 8.31 |
| Yuji Nakatsukasa | 2 | 97 | 17.74 |
| Tasuku Soma | 3 | 11 | 3.69 |
| André Uschmajew | 4 | 135 | 9.34 |