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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


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Online
ISSN
1609-9389
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Volume 11, Issue 3

Issues

Tensor-Train Ranks for Matrices and Their Inverses

Ivan Oseledets
Eugene Tyrtyshnikov
Nickolai Zamarashkin

Abstract

We show that the recent tensor-train (TT) decompositions of matrices come up from its recursive Kronecker-product representations with a systematic use of common bases. The names TTM and QTT used in this case stress the relation with multilevel matrices or quantization that increases artificially the number of levels. Then we investigate how the tensor-train ranks of a matrix can be related to those of its inverse. In the case of a banded Toeplitz matrix, we prove that the tensor-train ranks of its inverse are bounded above by 1+(l+u)^2, where l and u are the bandwidths in the lower and upper parts of the matrix without the main diagonal.

Keywords: tensor ranks; tensor-train decomposition; QTT-ranks; inverse matrices; multilevel matrices; Toeplitz matrices; banded matrices

About the article

Received: 2011-05-06

Revised: 2011-08-16

Accepted: 2011-09-21

Published in Print:


Citation Information: Computational Methods in Applied Mathematics Comput. Methods Appl. Math., Volume 11, Issue 3, Pages 394–403, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.2478/cmam-2011-0022.

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© Institute of Mathematics, NAS of Belarus. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. BY-NC-ND 4.0

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