Oseledets I., Tyrtyshnikov E., Zamarashkin N. Tensor-train ranks for matrices and their inverses, Computational Methods in Applied Mathematics, 11 (3), с. 375-384, 2011.
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.
tensor ranks, tensor-train decomposition, QTT-ranks, inverse matrices, multilevel matrices, Toeplitz matrices, banded matrices