Constructive Approximation

, Volume 37, Issue 1, pp 1–18 | Cite as

Constructive Representation of Functions in Low-Rank Tensor Formats

Article

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.

Keywords

TT-format QTT-format Tensor decompositions Multivariate functions Explicit representations 

Mathematics Subject Classification

15A69 41A30 

Notes

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.

References

  1. 1.
    Beylkin, G., Mohlenkamp, M.J.: Numerical operator calculus in higher dimensions. Proc. Natl. Acad. Sci. USA 99, 10246–10251 (2002) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Beylkin, G., Mohlenkamp, M.J.: Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26, 2133–2159 (2005) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bieri, M., Schwab, C.: Sparse high order FEM for elliptic sPDEs. Comput. Methods Appl. Mech. Eng. 198, 1149–1170 (2009) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Buhmann, M.: Multivariate cardinal interpolation with radial-basis functions. Constr. Approx. 6, 225–255 (1990) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Buhmann, M.: Radial basis functions. Acta Numer. 9, 1–38 (2000) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bungartz, H.-J., Griebel, M., Röschke, D., Zenger, C.: Pointwise convergence of the combination technique for Laplace’s equation. East-West J. Numer. Math. 2, 21–45 (1994) MathSciNetMATHGoogle Scholar
  7. 7.
    Caroll, J.D., Chang, J.J.: Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart–Young decomposition. Psychometrika 35, 283–319 (1970) CrossRefGoogle Scholar
  8. 8.
    de Lathauwer, L., de Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    de Silva, V., Lim, L.-H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30, 1084–1127 (2008) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Espig, M., Grasedyck, L., Hackbusch, W.: Black box low tensor rank approximation using fibre-crosses. Constr. Approx. 30, 557–597 (2009) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Falco, A., Hackbusch, W.: On minimal subspaces in tensor representation. Preprint 70, MPI MIS, Leipzig (2010) Google Scholar
  12. 12.
    Garcke, J., Griebel, M., Thess, M.: Data mining with sparse grids. Computing 67, 225–253 (2001) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Grasedyck, L.: Polynomial approximation in hierarchical Tucker format by vector-tensorization. DFG-SPP1324 Preprint 43, Philipps-Univ., Marburg (2010) Google Scholar
  14. 14.
    Hackbusch, W., Khoromskij, B.N.: Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. I. Separable approximation of multi-variate functions. Computing 76, 177–202 (2006) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15, 706–722 (2009) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Harshman, R.A.: Foundations of the PARAFAC procedure: models and conditions for an explanatory multimodal factor analysis. UCLA Working Papers in Phonetics, 16, pp. 1–84 (1970) Google Scholar
  17. 17.
    Hastad, J.: Tensor rank is NP-complete. J. Algorithms 11, 644–654 (1990) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Khoromskaia, V.: Numerical solution of the Hartree–Fock equation by multilevel tensor-structured methods. PhD thesis, TU, Berlin (2010) Google Scholar
  19. 19.
    Khoromskij, B.N.: Tensor-structured preconditioners and approximate inverse of elliptic operators in ℝd. Constr. Approx. 599–620 (2009) Google Scholar
  20. 20.
    Khoromskij, B.N.: Fast and accurate tensor approximation of multivariate convolution with linear scaling in dimension. J. Comput. Appl. Math. 234, 3122–3139 (2010) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Khoromskij, B.N.: \(\mathcal{O}(d \log n)\)—quantics approximation of Nd tensors in high-dimensional numerical modeling. Constr. Approx. 34, 257–280 (2011) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Khoromskij, B.N., Khoromskaia, V.: Multigrid accelerated tensor approximation of function related multidimensional arrays. SIAM J. Sci. Comput. 31, 3002–3026 (2009) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Khoromskij, B.N., Khoromskaia, V., Chinnamsetty, S.R., Flad, H.-J.: Tensor decomposition in electronic structure calculations on 3D Cartesian grids. J. Comput. Phys. 228, 5749–5762 (2009) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51, 455–500 (2009) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Oseledets, I.V.: Lower bounds for separable approximations of the Hilbert kernel. Mat. Sb. 198, 137–144 (2007) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Oseledets, I.V.: Approximation of matrices with logarithmic number of parameters. Dokl. Math. 428, 23–24 (2009) Google Scholar
  27. 27.
    Oseledets, I.V.: Compact matrix form of the d-dimensional tensor decomposition. Preprint 2009-01, INM RAS, Moscow (2009) Google Scholar
  28. 28.
    Oseledets, I.V.: Approximation of 2d×2d matrices using tensor decomposition. SIAM J. Matrix Anal. Appl. 31, 2130–2145 (2010) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33, 2295–2317 (2011) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Oseledets, I.V., Tyrtyshnikov, E.E.: Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput. 31, 3744–3759 (2009) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Oseledets, I.V., Tyrtyshnikov, E.E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432, 70–88 (2010) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Östlund, S., Rommer, S.: Thermodynamic limit of density matrix renormalization. Phys. Rev. Lett. 75, 3537–3540 (1995) CrossRefGoogle Scholar
  33. 33.
    Sloan, I., Wozniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high dimensional integrals. J. Complex. 14, 1–33 (1998) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Tucker, L.R.: Some mathematical notes on three-mode factor analysis. Psychometrika 31, 279–311 (1966) MathSciNetCrossRefGoogle Scholar
  35. 35.
    Vidal, G.: Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003) CrossRefGoogle Scholar
  36. 36.
    Wang, X., Sloan, I.H.: Why are high-dimensional finance problems often of low effective dimension? SIAM J. Sci. Comput. 27, 159–183 (2006) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Institute of Numerical MathematicsRussian Academy of SciencesMoscowRussia

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