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

Abstract

Algorithms are proposed for the approximate calculation of the matrix product \( \tilde C \) ≈ C = A · B, where the matrices A and B are given by their tensor decompositions in either canonical or Tucker format of rank r. The matrix 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 r2d 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 ∼ 30. In this paper, methods are proposed that approximate the mode factors of 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 C has low complexity compared to the preliminary calculation of the factors in the tensor decomposition of C with a redundant rank.