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 68 articles , 1 standard article )

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  1. Massarenti, Alex; Mella, Massimiliano; Staglianò, Giovanni: Effective identifiability criteria for tensors and polynomials (2018)
  2. Zhu, Wei; Wang, Bao; Barnard, Richard; Hauck, Cory D.; Jenko, Frank; Osher, Stanley: Scientific data interpolation with low dimensional manifold model (2018)
  3. Brezinski, Claude; Redivo-Zaglia, Michela: The simplified topological $\varepsilon$-algorithms: software and applications (2017)
  4. Che, Mao-Lin; Wei, Yi-Min: An inequality for the Perron pair of an irreducible and symmetric nonnegative tensor with application (2017)
  5. Chen, Bilian; He, Simai; Li, Zhening; Zhang, Shuzhong: On new classes of nonnegative symmetric tensors (2017)
  6. Hackbusch, Wolfgang; Kressner, Daniel; Uschmajew, André: Perturbation of higher-order singular values (2017)
  7. Hackbusch, Wolfgang; Uschmajew, André: On the interconnection between the higher-order singular values of real tensors (2017)
  8. Han, Lixing: A homotopy method for solving multilinear systems with M-tensors (2017)
  9. Hashemi, Behnam; Trefethen, Lloyd N.: Chebfun in three dimensions (2017)
  10. Insuasty, Edwin; Van den Hof, Paul M.J.; Weiland, Siep; Jansen, Jan-Dirk: Flow-based dissimilarity measures for reservoir models: a spatial-temporal tensor approach (2017)
  11. Kressner, Daniel; Periša, Lana: Recompression of Hadamard products of tensors in Tucker format (2017)
  12. Teng, Peiyuan: Accurate calculation of the geometric measure of entanglement for multipartite quantum states (2017)
  13. Zhao, Na; Yang, Qingzhi; Liu, Yajun: Computing the generalized eigenvalues of weakly symmetric tensors (2017)
  14. Beik, Fatemeh Panjeh Ali; Saberi Movahed, Farid; Ahmadi-Asl, Salman: On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations. (2016)
  15. Bigoni, Daniele; Engsig-Karup, Allan P.; Marzouk, Youssef M.: Spectral tensor-train decomposition (2016)
  16. Chen, Yannan; Qi, Liqun; Wang, Qun: Computing extreme eigenvalues of large scale Hankel tensors (2016)
  17. Chen, Zhongming; Qi, Liqun: A semismooth Newton method for tensor eigenvalue complementarity problem (2016)
  18. De Sterck, Hans; Howse, Alexander: Nonlinearly preconditioned optimization on Grassmann manifolds for computing approximate Tucker tensor decompositions (2016)
  19. Fan, H.-Y.; Zhang, L.; Chu, E.K.-w.; Wei, Y.: Q-less QR decomposition in inner product spaces (2016)
  20. Yang, Yuning; Feng, Yunlong; Huang, Xiaolin; Suykens, Johan A.K.: Rank-1 tensor properties with applications to a class of tensor optimization problems (2016)

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