Structured multi-way arrays and their applications

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-γ 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-γ 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.

Keywords

Tensor / rank / singular value decomposition (SVD) / higher-order singular value decomposition / approximation

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