Tensor-Train Ranks for Matrices and Their Inverses

By Ivan V. Oseledets, E. E. Tyrtyshnikov, Nickolai L. Zamarashkin. 2011

32 citations recorded by the source corpus.

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.

Subfields: Computational Mathematics, Computational Mechanics, Computational Theory and Mathematics, Numerical Methods & Scientific Computing

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In Computational Mathematics

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Citations in the corpus

Efficient Tensor Completion for Color Image and Video Recovery: Low-Rank Tensor Train (2016, background): “Realizing the computational efficiency of low TT rank tensors, there has been numerous works in applying it to numerical linear algebra [29]–[31].”