Low Rank Tucker-Type Tensor Approximation to Classical Potentials

Boris N. Khoromskij and Venera Khoromskaia

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Submission date: 29. Sep. 2006 (revised version: October 2006)
Pages: 28
published in: Central European journal of mathematics, 5 (2007) 3, p. 523-550 
DOI number (of the published article): 10.2478/s11533-007-0018-0
Bibtex
MSC-Numbers: 65F30, 65F50, 65N35, 65F10
Keywords and phrases: kronecker products, Tucker decomposition, classical potentials
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Abstract:
This paper investigates best rank-() Tucker tensor approximation of higher-order tensors arising from the discretization of linear operators and functions in . Super-convergence of the Tucker decomposition with respect to the relative Frobenius norm is proven. Dimensionality reduction by the two-level Tucker-to-canonical approximation is discussed. Tensor-product representation of basic multi-linear algebra operations are considered, including inner, outer and Hadamard products. We also focus on fast convolution of higher-order tensors represented either by the Tucker or via the canonical models. Special versions of the orthogonal alternating least-squares (ALS) algorithm are implemented corresponding to the different formats of input data. We propose and test numerically the novel mixed CT-model, which is based on the additive splitting of a tensor as a sum of canonical and Tucker-type representations. This model allows to stabilise the ALS iteration in the case of ``ill-conditioned'' tensors.

The orthogonal Tucker decomposition is applied to 3D tensors generated by classical potentials, for example , , and with . Numerical results for tri-linear decompositions illustrate exponential convergence in the Tucker rank, and robustness of the orthogonal ALS iteration.