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A New Scheme for the Tensor Representation

Abstract

The paper presents a new scheme for the representation of tensors which is well-suited for high-order tensors. The construction is based on a hierarchy of tensor product subspaces spanned by orthonormal bases. The underlying binary tree structure makes it possible to apply standard Linear Algebra tools for performing arithmetical operations and for the computation of data-sparse approximations. In particular, a truncation algorithm can be implemented which is based on the standard matrix singular value decomposition (SVD) method.

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Correspondence to W. Hackbusch.

Additional information

Dedicated to the 60th birthday of Wolfgang Dahmen.

Communicated by Reinhard Hochmuth.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Hackbusch, W., Kühn, S. A New Scheme for the Tensor Representation. J Fourier Anal Appl 15, 706–722 (2009). https://doi.org/10.1007/s00041-009-9094-9

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Keywords

  • Multilinear algebra
  • Tensor representation
  • Singular value decomposition

Mathematics Subject Classification (2000)

  • 15A69