A constructive arbitrary-degree Kronecker product decomposition of tensors. We propose the tensor Kronecker product singular value decomposition (TKPSVD) that decomposes a real k-way tensor A into a linear combination of tensor Kronecker products with an arbitrary number of d factors A=∑Rj=1σjA(d)j⊗⋯⊗A(1)j. We generalize the matrix Kronecker product to tensors such that each factor A(i)j in the TKPSVD is a k-way tensor. The algorithm relies on reshaping and permuting the original tensor into a d-way tensor, after which a polyadic decomposition with orthogonal rank-1 terms is computed. We prove that for many different structured tensors, the Kronecker product factors A(1)j,…,A(d)j are guaranteed to inherit this structure. In addition, we introduce the new notion of general symmetric tensors, which includes many different structures such as symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors.

References in zbMATH (referenced in 12 articles , 1 standard article )

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  1. Ehrlacher, Virginie; Ruiz, Maria Fuente; Lombardi, Damiano: SOTT: greedy approximation of a tensor as a sum of tensor trains (2022)
  2. Batselier, Kim; Cichocki, Andrzej; Wong, Ngai: MERACLE: constructive layer-wise conversion of a tensor train into a MERA (2021)
  3. Koiran, Pascal: Orthogonal tensor decomposition and orbit closures from a linear algebraic perspective (2021)
  4. Hu, Shenglong: An inexact augmented Lagrangian method for computing strongly orthogonal decompositions of tensors (2020)
  5. Fu, Tao-Ran; Fan, Jin-Yan: Successive partial-symmetric rank-one algorithms for almost unitarily decomposable conjugate partial-symmetric tensors (2019)
  6. Guan, Yu; Chu, Moody T.; Chu, Delin: SVD-based algorithms for the best rank-1 approximation of a symmetric tensor (2018)
  7. Wang, Xuezhong; Che, Maolin; Wei, Yimin: Partial orthogonal rank-one decomposition of complex symmetric tensors based on the Takagi factorization (2018)
  8. Batselier, Kim; Chen, Zhongming; Wong, Ngai: A tensor network Kalman filter with an application in recursive MIMO Volterra system identification (2017)
  9. Batselier, Kim; Wong, Ngai: A constructive arbitrary-degree Kronecker product decomposition of tensors. (2017)
  10. Boralevi, Ada; Draisma, Jan; Horobeţ, Emil; Robeva, Elina: Orthogonal and unitary tensor decomposition from an algebraic perspective (2017)
  11. Batselier, Kim; Wong, Ngai: Symmetric tensor decomposition by an iterative eigendecomposition algorithm (2016)
  12. Batselier, Kim; Liu, Haotian; Wong, Ngai: A constructive algorithm for decomposing a tensor into a finite sum of orthonormal rank-1 terms (2015)