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 121 articles , 1 standard article )

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  1. Bai, Minru; Zhao, Jing; Zhang, ZhangHui: A descent cautious BFGS method for computing US-eigenvalues of symmetric complex tensors (2020)
  2. Beik, Fatemeh Panjeh Ali; Najafi-Kalyani, Mehdi; Reichel, Lothar: Iterative Tikhonov regularization of tensor equations based on the Arnoldi process and some of its generalizations (2020)
  3. Brandoni, D.; Simoncini, V.: Tensor-train decomposition for image recognition (2020)
  4. Dong, Bo; Jiang, Nan; Chu, Moody T.: Nonlinear power-like iteration by polar decomposition and its application to tensor approximation (2020)
  5. Guan, Hong-Bo; Li, Dong-Hui: Linearized methods for tensor complementarity problems (2020)
  6. Hajarian, Masoud: Conjugate gradient-like methods for solving general tensor equation with Einstein product (2020)
  7. Heyouni, Mohammed; Saberi-Movahed, Farid; Tajaddini, Azita: A tensor format for the generalized Hessenberg method for solving Sylvester tensor equations (2020)
  8. Hong, David; Kolda, Tamara G.; Duersch, Jed A.: Generalized canonical polyadic tensor decomposition (2020)
  9. Huang, Baohua; Ma, Changfeng: Global least squares methods based on tensor form to solve a class of generalized Sylvester tensor equations (2020)
  10. Hu, Shenglong: An inexact augmented Lagrangian method for computing strongly orthogonal decompositions of tensors (2020)
  11. Lv, Changqing; Ma, Changfeng: A modified CG algorithm for solving generalized coupled Sylvester tensor equations (2020)
  12. Malik, Osman Asif; Becker, Stephen: Guarantees for the Kronecker fast Johnson-Lindenstrauss transform using a coherence and sampling argument (2020)
  13. Marconi, Jacopo; Tiso, Paolo; Braghin, Francesco: A nonlinear reduced order model with parametrized shape defects (2020)
  14. Najafi-Kalyani, Mehdi; Beik, Fatemeh Panjeh Ali; Jbilou, Khalide: On global iterative schemes based on Hessenberg process for (ill-posed) Sylvester tensor equations (2020)
  15. Usevich, Konstantin; Dreesen, Philippe; Ishteva, Mariya: Decoupling multivariate polynomials: interconnections between tensorizations (2020)
  16. Zhang, Xin; Ni, Qin; Ge, Zhili: A convergent Newton algorithm for computing Z-eigenvalues of an almost nonnegative irreducible tensor (2020)
  17. Benson, Austin R.; Gleich, David F.: Computing tensor (Z)-eigenvectors with dynamical systems (2019)
  18. Che, Maolin; Wei, Yimin: Randomized algorithms for the approximations of Tucker and the tensor train decompositions (2019)
  19. Chen, Chiyu; Zhang, Liping: Finding Nash equilibrium for a class of multi-person noncooperative games via solving tensor complementarity problem (2019)
  20. Chen, Liping; Han, Lixing; Yin, Hongxia; Zhou, Liangmin: A homotopy method for computing the largest eigenvalue of an irreducible nonnegative tensor (2019)

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