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A constructive arbitrary‐degree Kronecker product decomposition of tensors
Author(s) -
Batselier Kim,
Wong Ngai
Publication year - 2017
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2097
Subject(s) - kronecker product , kronecker delta , mathematics , tensor product , tensor product of hilbert spaces , toeplitz matrix , tensor (intrinsic definition) , cartesian tensor , singular value decomposition , symmetric tensor , pure mathematics , algebra over a field , hankel matrix , tensor contraction , matrix (chemical analysis) , tensor density , mathematical analysis , tensor field , exact solutions in general relativity , algorithm , physics , quantum mechanics , materials science , composite material
Summary We generalize the matrix Kronecker product to tensors and propose the tensor Kronecker product singular value decomposition that decomposes a real k ‐way tensor A into a linear combination of tensor Kronecker products with an arbitrary number of d factors. We show how to construct A = ∑ j = 1 Rσ jA j ( d ) ⊗ ⋯ ⊗ A j ( 1 ) , where each factorA j ( i )is also a k ‐way tensor, thus including matrices ( k =2) as a special case. This problem is readily solved by reshaping and permuting A into a d ‐way tensor, followed by a orthogonal polyadic decomposition. Moreover, we introduce the new notion of general symmetric tensors (encompassing symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors, etc.) and prove that when A is structured then its factorsA j ( 1 ) , ⋯ , A j ( d )will also inherit this structure.