Low rank Tucker-type tensor approximation to classical potentials

Abstract

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 ℝ ^d . Super-convergence of the best rank-(r ₁,..., r _d ) Tucker-type 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 is considered, including inner, outer and Hadamard products. Furthermore, we focus on fast convolution of higher-order tensors represented by the Tucker/canonical models. Optimized versions of the orthogonal alternating least-squares (ALS) algorithm is presented taking into account the different formats of input data. We propose and test numerically the mixed CT-model, which is based on the additive splitting of a tensor as a sum of canonical and Tucker-type representations. It allows to stabilize the ALS iteration in the case of “ill-conditioned” tensors. The best rank-(r ₁,..., r _d ) Tucker decomposition is applied to 3D tensors generated by classical potentials, for example \(\tfrac{1}{{\left| {x - y} \right|}}, e^{ - \alpha \left| {x - y} \right|} , \tfrac{{e^{ - \left| {x - y} \right|} }}{{\left| {x - y} \right|}}\) and \(\tfrac{{erf(|x|)}}{{|x|}}\) with x, y ∈ ℝ ^d . Numerical results for tri-linear decompositions illustrate exponential convergence in the Tucker rank, and robustness of the orthogonal ALS iteration.