## Abstract

Algorithms are proposed for the approximate calculation of the matrix product C̃ ≈ 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.

Original language | English |
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Pages (from-to) | 1662-1677 |

Number of pages | 16 |

Journal | Computational Mathematics and Mathematical Physics |

Volume | 49 |

Issue number | 10 |

DOIs | |

Publication status | Published - 1 Oct 2009 |

## Keywords

- Canonical decomposition
- Coulomb potential
- Data compression
- Fast recompression
- Low-parameter representations
- Low-rank matrices
- Multidimensional arrays
- Multidimensional operators
- Skeleton approximation
- Tucker decomposition