A New Scheme for the Tensor Representation


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.


  1. 1.

    Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)

  2. 2.

    Chinnamsetty, S.R., Espig, M., Khoromskij, B.N., Hackbusch, W., Flad, H.-J.: Tensor product approximation with optimal rank in quantum chemistry. J. Chem. Phys. 127, 084110 (2007), 14 pages

  3. 3.

    Dahmen, W., Prössdorf, S., Schneider, R.: Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution. Adv. Comput. Math. 1, 259–335 (1993)

  4. 4.

    Espig, M.: Approximation mit Elementartensorsummen. Dissertation, Universität Leipzig (2008)

  5. 5.

    Espig, M., Grasedyck, L., Hackbusch, W.: Black box low tensor rank approximation using fibre-crosses. Preprint 60, Max-Planck-Institut für Mathematik, Leipzig (2008)

  6. 6.

    Grasedyck, L.: Hierarchical singular value decomposition of tensors. Preprint 27, Max-Planck-Institut für Mathematik, Leipzig (2009)

  7. 7.

    Grasedyck, L., Hackbusch, W.: Construction and arithmetics of ℋ-matrices. Computing 70, 295–334 (2003)

  8. 8.

    Hackbusch, W., Khoromskij, B.N.: Tensor-product approximation to operators and functions in high dimensions. J. Complex. 23, 697–714 (2007)

  9. 9.

    Khoromskij, B.N.: Structured rank-(r 1,…,r d) decomposition of function-related tensors in ℝd. Comp. Methods Appl. Math. 6, 194–220 (2006)

  10. 10.

    Khoromskij, B.N., Khoromskaia, V.: Low rank Tucker-type tensor approximation to classical potentials. Cent. Eur. J. Math. 5(3), 1–18 (2007)

  11. 11.

    Tucker, L.R.: Implications of factor analysis of three-way matrices for measurement of change. In: Harris, C.W. (ed.) Problems in Measuring Change, pp. 122–137. University of Wisconsin Press, Madison (1963)

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Correspondence to W. Hackbusch.

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Dedicated to the 60th birthday of Wolfgang Dahmen.

Communicated by Reinhard Hochmuth.

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Hackbusch, W., Kühn, S. A New Scheme for the Tensor Representation. J Fourier Anal Appl 15, 706–722 (2009).

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  • Multilinear algebra
  • Tensor representation
  • Singular value decomposition

Mathematics Subject Classification (2000)

  • 15A69