MiS Preprint Repository

This paper investigates best rank-($r_1,...,r_d$) Tucker tensor approximation of higher-order tensors arising from the discretization of linear operators and functions in $\mathbb{R}^{d} $. 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 $\frac{1}{|x-y|} $, $e^{-\alpha |x-y|} $, $\frac{e^{-|x-y| }}{|x-y|} $ and $\frac{\operatorname*{erf}(|x|)}{|x|}$ with $x,y \in \mathbb{R}^{d} $. Numerical results for tri-linear decompositions illustrate exponential convergence in the Tucker rank, and robustness of the orthogonal ALS iteration.