Wedderburn Rank Reduction and Krylov Subspace Method for Tensor Approximation. Part 1: Tucker Case

By Sergei A. Goreinov, Ivan V. Oseledets, Dmitry V. Savostyanov. 2010

31 citations recorded by the source corpus.

New algorithms are proposed for the Tucker approximation of a 3-tensor accessed only through a tensor-by-vector-by-vector multiplication subroutine. In the matrix case, the Krylov methods are methods of choice to approximate the dominant column and row subspaces of a sparse or structured matrix given through a matrix-by-vector operation. Using the Wedderburn rank reduction formula, we propose a matrix approximation algorithm that computes the Krylov subspaces and can be generalized to 3-tensors. The numerical experiments show that on quantum chemistry data the proposed tensor methods outperform the minimal Krylov recursion of Savas and Elden.

Subfields: Computational Mathematics, Computational Theory and Mathematics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Recent developments include Newton-type methods on manifolds [71, 131, 132, 229], a Jacobi algorithm for symmetric tensors [130], generalizations of Krylov subspace methods [98, 228], and modifications of the HOSVD [259].”
Two-level Tucker-TT-QTT format for optimized tensor calculus (2012, methodology): “During the history of tensor methods, different formats were proposed and analyzed, such as canonical [8, 9, 10, 11], Tucker [12, 13, 14, 15], Matrix Product States [16, 17, 18, 19] and MCTDH [20] in quantum physics community, HT (Hierarchical Tucker) [21, 22] and TT (Tensor Train) in numerical alge”
Tensor-Train Decomposition (2011, methodology): “The situation is the same with the Tucker format, and several techniques have been proposed to evaluate the matrix-by-vector product quickly [13, 34, 31].”
Superfast Fourier Transform Using QTT Approximation (2012, background): “We refer also to [30, 10, 52, 35, 47, 14, 56, 28] for the detailed discussions on recent progress in this field.”
Fast truncation of mode ranks for bilinear tensor operations (2010, methodology): “This problem was approached in [25] using Tucker-ALS algorithm, in [26] by Tucker-ALS with initial guess obtained from the coarser grids, in [27] by Cross3D algorithm, in [28] by individual cross approximation of canonical factors, in [16] by cross approximation of Gram matrices of unfoldings and in”