Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time

By Ivan V. Oseledets, Dmitry V. Savostyanov, E. E. Tyrtyshnikov. 2008

198 citations recorded by the source corpus.

Subfields: Atomic and Molecular Physics, and Optics, Computational Mathematics, Computational Theory and Mathematics

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, positive): “A first extension of cross approximation to tensors was proposed in [209] for approximating third-order tensors by a Tucker decomposition.”
Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis (2014, background): “been derived in [99]–[101] and are illustrated in Figure 11, while cross-approximation for the TT format has been derived in [102].”
Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions (2009, methodology): “In the two-dimensional case this observation probably belongs to Van Loan and Pitsianis [18] and the generalization to the d-dimensional case is easy (it was considered, for example, in [17, 12]).”
Factorization strategies for third-order tensors (2011, background): “A common thread in such applications is the need to compress, sort, and/or otherwise manipulate the data by taking advantage of its multidimensional structure (see for example the recent article [34]).”
Krylov-Type Methods for Tensor Computations (2010, unknown): “An important thread of research in this field is the computation of low multilinear rank approximations of tensors [21,6,7,1,33,5,9,28,15,25,17,20].”
Tensor-Train Decomposition (2011, background): “It is suitable for “small” dimensions, especially for the three-dimensional case [22, 30, 23].”
Mathematik in den Naturwissenschaften Leipzig Tensor-Structured Galerkin Approximation of parametric and stochastic Elliptic PDEs (2010, methodology): “Several heuristic methods for computing the corresponding rank structured approximations in different classes of input/output tensors were discussed in the literature [7, 26, 36, 23, 29, 18].”
How to find a good submatrix (2010, methodology): “This problem arises for example in maximization of two-dimensional separable function on a grid, or as an essential part of the Cross3D algorithm for computation of Tucker approximation of three dimensional tensor in linear time [4].”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, background): “…2016, Wang et al., 2015, Wetzstein et al., 2012], • Tucker decompositions and their applications [Caiafa and Cichocki, 2013, 2015, Jeon et al., 2015, 2016, Oseledets et al., 2008, Zhao et al., 2016, Zhe et al., 2016a], • TT/HT decompositions and their applications in numerical methods for…”
MACH: Fast Randomized Tensor Decompositions (2009, background): “Another approach to approximating the Tucker decomposition for the case of a three-way tensor is presented in [36].”
Quasi-Newton Methods on Grassmannians and Multilinear Approximations of Tensors (2009, methodology): “Yet a third approach to obtain low multilinear rank tensor approximations are the cross methods in [48, 47, 25].”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, methodology): “5 On tensor-structured solution of multidimensional equations Tensor numerical methods are proved to be efficient for data-sparse representation of functions and operators in the Hartree-Fock and Kohn-Sham models in electronic structure calculations [7, 39, 59, 57, 23, 76, 47].”
Tensor Ring Decomposition (2016, background): “…(CPD) [3], [4], [5], [6] can approximate an observed tensor by a sum of rank-one tensors, which requires O ( dnr ) parameters. attempts The Tucker decomposition [7], [8], [9] to approximate tensors by a core tensor and several factor matrices, which requires O ( dnr + r d ) parameters.”
Wedderburn Rank Reduction and Krylov Subspace Method for Tensor Approximation. Part 1: Tucker Case (2010, methodology): “Such recompression was done in [4] using the Tucker-ALS algorithm [22, 7], in [20] by the Tucker-ALS with the initial guess obtained from the coarser grids, in [11] by the Cross3D algorithm [25], in [26] by the individual cross approximation of canonical factors and in [32] by the cross approximatio”