Structured multi-way arrays and their applications

Kong, Xu; Jiang, Yaolin
April 2013
Frontiers of Mathematics in China;Apr2013, Vol. 8 Issue 2, p345
Academic Journal
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. By 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 (SVD) method for tensors proposed by J. Salmi et al. [IEEE Trans Signal Process, 2009, 57(12): 4719-4733] 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.


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