Structured multi-way arrays and their applications. (English) Zbl 1279.15022
Authors’ abstract: Based on the structure of the rank-1 matrix and the different unfolding ways of the tensor, we present two types of structured tensors which contain the rank-1 tensors as special cases. We study some properties of the ranks and the best rank-r approximations of the structured tensors. Using the upper-semicontinuity of the matrix rank, we show that for the structured tensors, there always exist the best rank-\(r\) approximations. This can help one to better understand the sequential unfolding singular value decomposition method for tensors proposed by J. Salmi, A. Richter and V. Koivunen [IEEE Trans. Signal Process. 57, No. 12, 4719–4733 (2009)] and offer a generalized way of low rank approximations of tensors. Moreover, we apply the structured tensors to estimate the upper and lower bounds of the best rank-1 approximations of the 3rd-order and 4th-order tensors, and to distinguish the well written and non-well written digits.
Reviewer: Jaydeb Sarkar (Bangalore)
MSC:
| 15A69 | Multilinear algebra, tensor calculus |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
Keywords:
singular value decomposition; rank-1 tensor; structured tensor; best rank-\(r\) approximationReferences:
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