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Generalized cross approximation for 3d-tensors

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Computing and Visualization in Science

Abstract

In this article we present a generalized version of the Cross Approximation for 3d-tensors. The given tensor \({a\in\mathbb{R}^{n\times n\times n}}\) is represented as a matrix of vectors and 2d adaptive Cross Approximation is applied in a nested way to get the tensor decomposition. The main focus lies on theoretical issues of the construction such as the desired interpolation property or the explicit formulas for the vectors in the decomposition. The computational complexity of the proposed algorithm is shown to be linear in n.

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References

  1. Bebendorf M.: Approximation of boundary element matrices. Numer. Math. 86, 565–589 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bebendorf, M.: Adaptive Cross Approximation of Multivariate Functions. SFB 611 Preprint 453, Constructive Approximation (to appear)

  3. Espig M., Grasedyck L., Hackbusch W.: Black box low tensor-rank approximation using fiber-crosses. Constr. Approx. 30, 557–597 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Goreinov S.A., Tyrtyshnikov E.E., Zamarashkin N.L.: A theory of pseudoskeleton approximations. Linear Algebra Appl. 261, 1–21 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hackbusch W., Kühn S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15(5), 706–722 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Khoromskij, B.N.: Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances. Preprint 21/2010, MPI MIS Leipzig, Germany (2010)

  7. Khoromskij B.N., Khoromskaia V.: Multigrid tensor approximation of function related tensors. SIAM J. Sci. Comput. 31(4), 3002–3026 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kolda T.G., Badar B.W.: Tensor decomposition and applications. SIAM Review 51(3), 455–500 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Oseledets I.V., Savostianov D.V., Tyrtyshnikov E.E.: Tucker dimensionality reduction of three-dimensional arrays in lienar time. SIAM J. Matrix Anal. Appl. 30(3), 939–956 (2008)

    Article  MathSciNet  Google Scholar 

  10. Oseledets I.V, Tyrtyshnikov E.E.: TT-Cross approximation for multidimensional arrays. Linear Algebra Appl. 432(1), 70–88 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Schneider J.: Error estimates for two-dimensional cross approximation. J. Approx. Theory 162(9), 1685–1700 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

    Kishore Kumar Naraparaju & Jan Schneider

Authors
  1. Kishore Kumar Naraparaju
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  2. Jan Schneider
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Correspondence to Jan Schneider.

Additional information

Communicated by Gabriel Wittum.

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Naraparaju, K.K., Schneider, J. Generalized cross approximation for 3d-tensors. Comput. Visual Sci. 14, 105–115 (2011). https://doi.org/10.1007/s00791-011-0166-4

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  • DOI: https://doi.org/10.1007/s00791-011-0166-4

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