Approximate multiplication of tensor matrices based on the individual filtering of factors

Algorithms are proposed for the approximate calculation of the matrix product $\tilde{\mathbf C}\approx\mathbf C=\mathbf A\cdot\mathbf B$, where the matrices $\mathbf A$ and $\mathbf B$ are given by their tensor decompositions in either canonical or Tucker format of rank $r$. The matrix $\mathbf C$ is not calculated as a full array; instead, it is first represented by a similar decomposition with a redundant rank and is then reapproximated (compressed) within the prescribed accuracy to reduce the rank. The available reapproximation algorithms as applied to the above problem require that an array containing $r^{2d}$ elements be stored, where d is the dimension of the corresponding space. Due to the memory and speed limitations, these algorithms are inapplicable even for the typical values $d=3$ and $r\sim30$. In this paper, methods are proposed that approximate the mode factors of $\mathbf C$ using individually chosen accuracy criteria. As an application, the three-dimensional Coulomb potential is calculated. It is shown that the proposed methods are efficient if r can be as large as several hundreds and the reapproximation (compression) of $\mathbf C$ has low complexity compared to the preliminary calculation of the factors in the tensor decomposition of $\mathbf C$ with a edundant rank.