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.

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  1. Bai, Xueli; He, Hongjin; Ling, Chen; Zhou, Guanglu: A nonnegativity preserving algorithm for multilinear systems with nonsingular (\mathcalM)-tensors (2021)
  2. Beik, Fatemeh P. A.; Najafi-Kalyani, Mehdi: A preconditioning technique in conjunction with Krylov subspace methods for solving multilinear systems (2021)
  3. Ceruti, Gianluca; Lubich, Christian; Walach, Hanna: Time integration of tree tensor networks (2021)
  4. Che, Maolin; Wei, Yimin; Yan, Hong: Randomized algorithms for the low multilinear rank approximations of tensors (2021)
  5. Chen, Can; Surana, Amit; Bloch, Anthony M.; Rajapakse, Indika: Multilinear control systems theory (2021)
  6. 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)
  7. Huang, Baohua: Numerical study on Moore-Penrose inverse of tensors via Einstein product (2021)
  8. Iavernaro, F.; Mazzia, F.; Mukhametzhanov, M. S.; Sergeyev, Ya. D.: Computation of higher order Lie derivatives on the infinity computer (2021)
  9. Ke, Yifen; Ma, Changfeng; Zhang, Huai: A modified LM algorithm for tensor complementarity problems over the circular cone (2021)
  10. Liang, Maolin; Zheng, Bing; Zheng, Yutao; Zhao, Ruijuan: A two-step accelerated Levenberg-Marquardt method for solving multilinear systems in tensor-train format (2021)
  11. Li, Dong-Hui; Chen, Cui-Dan; Guan, Hong-Bo: A lower dimensional linear equation approach to the m-tensor complementarity problem (2021)
  12. Tokcan, Neriman; Gryak, Jonathan; Najarian, Kayvan; Derksen, Harm: Algebraic methods for tensor data (2021)
  13. Tolle, Kevin; Marheineke, Nicole: Extended group finite element method (2021)
  14. Wang, Qing-Wen; Xu, Xiangjian; Duan, Xuefeng: Least squares solution of the quaternion Sylvester tensor equation (2021)
  15. Xiao, Chuanfu; Yang, Chao; Li, Min: Efficient alternating least squares algorithms for low multilinear rank approximation of tensors (2021)
  16. Zeng, Chao; Jiang, Tai-Xiang; Ng, Michael K.: An approximation method of CP rank for third-order tensor completion (2021)
  17. Zeng, Chao; Ng, Michael K.: Incremental CP tensor decomposition by alternating minimization method (2021)
  18. Zhang, Liping; Chen, Chiyu: A Newton-type algorithm for the tensor eigenvalue complementarity problem and some applications (2021)
  19. Bai, Minru; Zhao, Jing; Zhang, ZhangHui: A descent cautious BFGS method for computing US-eigenvalues of symmetric complex tensors (2020)
  20. Beik, Fatemeh P. A.; Jbilou, Khalide; Najafi-Kalyani, Mehdi; Reichel, Lothar: Golub-Kahan bidiagonalization for ill-conditioned tensor equations with applications (2020)

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