Computing and Visualization in Science

, Volume 15, Issue 6, pp 331–344 | Cite as

A note on tensor chain approximation

  • Mike Espig
  • Kishore Kumar Naraparaju
  • Jan SchneiderEmail author


This paper deals with the approximation of \(d\)-dimensional tensors, as discrete representations of arbitrary functions \(f(x_1,\ldots ,x_d)\) on \([0,1]^d\), in the so-called tensor chain format. The main goal of this paper is to show that the construction of a tensor chain approximation is possible using skeleton/cross approximation type methods. The complete algorithm is described, computational issues are discussed in detail and the complexity of the algorithm is shown to be linear in \(d\). Some numerical examples are given to validate the theoretical results.


Cross approximation Skeleton decomposition Tensor chain format Singular value decomposition 

Mathematical Subject Classification

15A69 41A63 65F30 


  1. 1.
    Bader, B.W., Kolda, T.G.: Tensor decomposition and applications. SIAM Rev. 51(3), 455–500 (2009).Google Scholar
  2. 2.
    Ballani, J., Grasedyck, L., Kluge, M.: Black box approximation of tensors in hierarchical tucker format. MPI MIS Preprint: 57/2010, accepted for Lin. Alg. ApplGoogle Scholar
  3. 3.
    Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86, 565–589 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bebendorf, M.: Adaptive cross approximation of multivariate functions. Constr. Approx. 34, 149–179 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bebendorf, M.: Hierarchical Matrices. Lectures in Computational Science and Engineering, vol. 63. Springer, Berlin (2008)Google Scholar
  6. 6.
    Chiu, J., Demanet, L.: Sublinear randomized algorithms for skeleton decompositions. arXiv:1110.4193v2 [math.NA]
  7. 7.
    Ciafa, C.F., Cichocki, A.: Generalizing the column–row matrix decomposition to multi-way arrays. Linear Algebr. Appl. 433(3), 557–573 (2010)CrossRefGoogle Scholar
  8. 8.
    Dolgov, S., Khoromskij, B.N., Oseledets, I.V.: Fast solution of multi-dimensional parabolic problems in the TT/QTT-format with initial application to the Fokker–Planck equation. MPI MIS Preprint: 80/2011Google Scholar
  9. 9.
    de Lathauwer, L., de Moor, B., Vandewalle, J.: A multilinear singular value decompostion. SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    de Lathauwer, L., de Moor, B., Vandewalle, J.: On best rank-1 and rank-\((r_1, r_2, r_n)\) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Espig, M., Grasedyck, L., Hackbusch, W.: Black box low tensor–rank approximation using fiber-crosses. Constr. Approx. 30, 557–597 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Espig, M., Hackbusch, W., Handschuh, S., Schneider, R.: Optimization problems in contracted tensor networks. MPI MIS Preprint No. 66, Leipzig (2011)Google Scholar
  13. 13.
    Espig, M., Handschuh, S., Khachatryan, A., Naraparaju, K.K., Schneider J.: Construction of arbitrary tensor networks. In preparation at MPI MIS LeipzigGoogle Scholar
  14. 14.
    Friedland, S., Mehrmann, V., Miedlar, A., Nkengla, M.: Fast lower rank approximations of matrices and tensors. Electron. J. Linear Algebr. 22, 1031–1048 (2011)Google Scholar
  15. 15.
    Gantmacher, F.R.: Theory of Matrices. Chelsea, New York (1959)zbMATHGoogle Scholar
  16. 16.
    Goreinov, S.A.: On cross approximation of multi-index array. Dokaldy Math. 420(4), 404–406 (2008)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Goreinov, S.A., Oseledets, I.V., Savostyanov, D.V. et al.: How to find a good submatrix. Research Report 08–10, Kowloon Tong, Hong Kong: ICM HKBU (2008)Google Scholar
  18. 18.
    Goreinov, S.A., Tyrtyshnikov, E.E., Zamarashkin, N.L.: A theory of pseudo-skeleton approximations. Linear Algebr. Appl. 261, 1–21 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Goreinov, S.A., Zamarashkin, N.L., Tyrtyshnikov, E.E.: Pseudo-skeleton approximations by matrices of maximal volume. Math. Notes 62(4), 515–519 (1997)Google Scholar
  20. 20.
    Goreinov, S.A., Tyrtyshnikov, E.E.: The maximal-volume concept in approximation by low-rank matrices. Contemp. Math. 208, 47–51 (2001)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31, 2029–2054 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Golub, G., Van Loan, C.: Matrix Computations, 3rd edn. John Hopkins Univ. Press, Baltimore (1996)zbMATHGoogle Scholar
  23. 23.
    Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15(5), 706–722 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer Series in Computational Mathematics, vol. 42 (2012)Google Scholar
  25. 25.
    Khoromskaia, V., Andrae, D., Khoromskij, B.N.: Fast and accurate tensor calculation of the fock operator in a general basis. MPI MIS, Preprint 4/2012Google Scholar
  26. 26.
    Khoromskij, B.N.: \(o(d\log N)\)-quantics approximation of N–d tensors in high-dimensional numerical modeling. Constr. Approx. 34(2), 257–280 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Landsberg, J.M., Yang, Q., Ke, Y.: On the geometry of tensor network states. arXiv:1105.4449 [math.AG] (2011)
  28. 28.
    Mahoney, M.W., Maggioni, M., Drineas, P.: Tensor–CUR decompositions for tensor based data. SIAM J. Matrix Anal. Appl. 30(3), 957–987 (2008)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Micchelli, C.A., Pinkus, A.: Some problems in the approximation of functions of two variables and \(n\)-widths of integral operators. Jour. Approx. Theo. 24, 51–77 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Naraparaju, K.K., Schneider, J.: Generalized cross approximation for 3d-tensors. Comput. Vis. Sci. 14(3), 105–115 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Savostyanov, D.V., Oseledets, I.V.: Fast adaptive interpolation of multidimensional arrays in tensor train format. In: Proceedings of 7th International Workshop on Multidimensional Systems (nDS). IEEE (2011)Google Scholar
  32. 32.
    Tyrtyshnikov, E.E.: Incomplete cross approximation in the mosaic-skeleton method. Computing 64(4), 367–380 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Oseledets, I.V., Savostianov, D.V., Tyrtyshnikov, E.E.: Tucker dimensionality reduction of three-dimensional arrays in linear time. SIAM J. Matrix Anal. Appl. 30(3), 939–956 (2008)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Oseledets, I.V., Tyrtyshnikov, E.E.: TT-cross approximation for multidimensional arrays. Lin. Algebr. Appl. 432(1), 70–88 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Oseledets, I.V.: Tensor–train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)Google Scholar
  36. 36.
    Schneider, J.: Error estimates for two-dimensional cross approximation. J. Approx. Theory 162(9), 1685–1700 (2010) Google Scholar
  37. 37.
    Zhu, X., Lin, W.: Randomised pseudo-skeleton approximation and its application in electromagnetics. Electron. Lett. 47(10), 590–592 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Mike Espig
    • 1
  • Kishore Kumar Naraparaju
    • 2
  • Jan Schneider
    • 3
    Email author
  1. 1.RWTH Aachen UniversityAachenGermany
  2. 2.BITS-Pilani Hyderabad CampusHyderabadIndia
  3. 3.University of RostockRostockGermany

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