Approximate multiplication of tensor matrices based on the individual filtering of factors |
| |
Authors: | D. V. Savostyanov E. E. Tyrtyshnikov |
| |
Affiliation: | 1.Institute of Numerical Mathematics,Russian Academy of Sciences,Moscow,Russia |
| |
Abstract: | Algorithms are proposed for the approximate calculation of the matrix product $
tilde C
$
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 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 ∼ 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. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|