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Constructive Representation of Functions in Low-Rank Tensor Formats

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Constructive Approximation Aims and scope

Abstract

In this paper, we obtain explicit representations of several multivariate functions in the Tensor Train (TT) format and explicit TT-representations of tensors that stem from the tensorization of univariate functions on grids. Previous results contained only estimates on the number of parameters (tensor ranks), and this paper fills this gap by providing explicit low-parametric representations for these functions and tensors.

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Notes

  1. Canonical ranks depend on the field: for the complex field the rank is equal to 2, but for the real field the upper bound is d, which is assumed to be tight.

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Acknowledgements

I am thankful to the anonymous referees. Their comments helped to improve the paper a lot.

Supported by RFBR grants, 12-01-00546-a, 12-01-33013-mol-a-ved, 11-01-12137-ofi-m-2011, 11-01-00549-a, by Rus. Gov. Contracts Π1112, 14.740.11.0345, 14.740.11.1067, 16.740.12.0727 by Rus. President grant MK-140.2011.1, by Priority Research Program OMN-3, by Dmitriy Zimin Dynasty Foundation. Part of this work was done during the stay of the author in Max-Planck Institute for Mathematics in Sciences, Leipzig.

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Correspondence to I. V. Oseledets.

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Communicated by Wolfgang Dahmen.

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Oseledets, I.V. Constructive Representation of Functions in Low-Rank Tensor Formats. Constr Approx 37, 1–18 (2013). https://doi.org/10.1007/s00365-012-9175-x

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