Efficient MATLAB computations with sparse and factored tensors. The term tensor refers simply to a multidimensional or N-way array, and we consider how specially structured tensors allow for efficient storage and computation. First, we study sparse tensors, which have the property that the vast majority of the elements are zero. We propose storing sparse tensors using coordinate format and describe the computational efficiency of this scheme for various mathematical operations, including those typical to tensor decomposition algorithms. Second, we study factored tensors, which have the property that they can be assembled from more basic components. We consider two specific types: A Tucker tensor can be expressed as the product of a core tensor (which itself may be dense, sparse, or factored) and a matrix along each mode, and a Kruskal tensor can be expressed as the sum of rank-1 tensors. We are interested in the case where the storage of the components is less than the storage of the full tensor, and we demonstrate that many elementary operations can be computed using only the components. All of the efficiencies described in this paper are implemented in the Tensor Toolbox for MATLAB.

References in zbMATH (referenced in 186 articles , 1 standard article )

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  1. Al Daas, Hussam; Ballard, Grey; Benner, Peter: Parallel algorithms for tensor train arithmetic (2022)
  2. Chen, Juefei; Wei, Yimin; Xu, Yanwei: Tensor CUR decomposition under T-product and its perturbation (2022)
  3. Dong, Shuyu; Gao, Bin; Guan, Yu; Glineur, François: New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition (2022)
  4. Huang, Guang-Xin; Chen, Qi-Xing; Yin, Feng: Preconditioned TBiCOR and TCORS algorithms for solving the Sylvester tensor equation (2022)
  5. Khosravi Dehdezi, Eisa; Karimi, Saeed: A rapid and powerful iterative method for computing inverses of sparse tensors with applications (2022)
  6. Mao, Xianpeng; Yang, Yuning: Best sparse rank-1 approximation to higher-order tensors via a truncated exponential induced regularizer (2022)
  7. Poythress, J. C.; Ahn, Jeongyoun; Park, Cheolwoo: Low-rank, orthogonally decomposable tensor regression with application to visual stimulus decoding of fMRI data (2022)
  8. Saberi-Movahed, Farid; Tajaddini, Azita; Heyouni, Mohammed; Elbouyahyaoui, Lakhdar: Some iterative approaches for Sylvester tensor equations. II: A tensor format of simpler variant of GCRO-based methods (2022)
  9. Saberi-Movahed, Farid; Tajaddini, Azita; Heyouni, Mohammed; Elbouyahyaoui, Lakhdar: Some iterative approaches for Sylvester tensor equations. I: A tensor format of truncated loose simpler GMRES (2022)
  10. Bai, Xueli; He, Hongjin; Ling, Chen; Zhou, Guanglu: A nonnegativity preserving algorithm for multilinear systems with nonsingular (\mathcalM)-tensors (2021)
  11. Beik, Fatemeh P. A.; Najafi-Kalyani, Mehdi: A preconditioning technique in conjunction with Krylov subspace methods for solving multilinear systems (2021)
  12. Ceruti, Gianluca; Lubich, Christian; Walach, Hanna: Time integration of tree tensor networks (2021)
  13. Che, Maolin; Wei, Yimin; Yan, Hong: Randomized algorithms for the low multilinear rank approximations of tensors (2021)
  14. Che, Maolin; Wei, Yimin; Yan, Hong: An efficient randomized algorithm for computing the approximate Tucker decomposition (2021)
  15. Chen, Can; Surana, Amit; Bloch, Anthony M.; Rajapakse, Indika: Multilinear control systems theory (2021)
  16. Corizzo, Roberto; Ceci, Michelangelo; Fanaee-T, Hadi; Gama, Joao: Multi-aspect renewable energy forecasting (2021)
  17. Dehdezi, Eisa Khosravi: Iterative methods for solving Sylvester transpose tensor equation (\mathcalA\star_N\mathcalX\star_M\mathcalB+\mathcalC\star_M\mathcalX^T\star_N\mathcalD=\mathcalE) (2021)
  18. De Sterck, Hans; He, Yunhui: On the asymptotic linear convergence speed of Anderson acceleration, Nesterov acceleration, and nonlinear GMRES (2021)
  19. Eswar, Srinivas; Hayashi, Koby; Ballard, Grey; Kannan, Ramakrishnan; Matheson, Michael A.; Park, Haesun: PLANC. Parallel low-rank approximation with nonnegativity constraints (2021)
  20. He, Hongjin; Bai, Xueli; Ling, Chen; Zhou, Guanglu: An index detecting algorithm for a class of TCP ((\mathcalA,q)) equipped with nonsingular (\mathcalM)-tensors (2021)

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