

Preprint 105/2006
Low Rank Tucker-Type Tensor Approximation to Classical Potentials
Boris N. Khoromskij and Venera Khoromskaia
Contact the author: Please use for correspondence this email.
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
Download full preprint: PDF (2110 kB)
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.