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Lobachevskii Journal of Mathematics

, Volume 40, Issue 11, pp 1863–1872 | Cite as

Nonnegative Tensor Train Factorization with DMRG Technique

  • E. M. ShcherbakovaEmail author
Article
  • 8 Downloads

Abstract

Tensor train is one of the modern decompositions used as low-rank tensor approximations of multidimensional arrays. If the original data is nonnegative we sometimes want the approximant to keep this property. In this work new methods for nonnegative tensor train factorization are proposed. Low-rank approximation approach helps to speed up the computations whereas DMRG technique allows to adapt nonnegative TT ranks for better accuracy. The performance analysis of the proposed algorithms as well as comparison with other nonnegative TT factorization method are presented.

Keywords and phrases

nonnegative tensor train multiplicative updates DMRG 

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Notes

Acknowledgments

The author thanks E. E. Tyrtyshnikov for useful advices and participation in discussions of the results.

Funding

This article contains the results of the project performed in the framework of the implementation of the programs of the Central Competences of the National Technological Database “Center for Big Data Storage and Analysis” (project “Tensor methods for processing and analysis of Big Data”) of Lomonosov MSU with the Project Support Funding of the National Technological Reporting dated December 11, 2018, no. 13/1251/2018.

References

  1. 1.
    I. Oseledets, “Tensor-Train Decomposition,” SIAM J. Sci. Comput. 33, 2295–2317 (2011).MathSciNetCrossRefGoogle Scholar
  2. 2.
    I. Oseledets and E. Tyrtyshnikov, “TT-cross approximation for multidimensional arrays,” Linear Algebra Appl. 432, 70–88 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    N. Lee, A. Phan, F. Cong, and A. Cichocki, “Nonnegative tensor train decompositions for multi-domain feature extraction and clustering,” Lect. Notes Comput. Sci. 9949, 87–95 (2016).CrossRefGoogle Scholar
  4. 4.
    D. Lee and H. Seung, “Learning the parts of objects by nonnegative matrix factorization,” Nature (London, U.K.) 401, 788–791 (1999).CrossRefGoogle Scholar
  5. 5.
    D. Lee and H. Seung, “Algorithms for non-negative matrix factorization,” Adv. Neural Inform. Process. Syst. 13 (2001).Google Scholar
  6. 6.
    S. Vavasis, “On the complexity of nonnegative matrix factorization,” SIAM J. Optimiz. 20, 1364–1377 (2007).MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Cohen and U. Rothblum, “Nonnegative ranks, decompositions and factorizations of nonnegative matrices,” Linear Algebra Appl. 190, 149–168 (1993).MathSciNetCrossRefGoogle Scholar
  8. 8.
    G. Zhou, A. Cichocki, and S. Xie, “Fast nonnegative matrix/tensor factorization based on low-rank approximation,” IEEE Trans. Signal Process. 60, 2928–2940 (2012).MathSciNetCrossRefGoogle Scholar
  9. 9.
    S. Goreinov, E. Tyrtyshnikov, and N. Zamarashkin, “A theory of pseudo-skeleton approximations,” Linear Algebra Appl. 261, 1–21 (1997).MathSciNetCrossRefGoogle Scholar
  10. 10.
    E. Tyrtyshnikov, “Incomplete cross approximation in the mosaic-skeleton method,” Computing 64, 367–380 (2000).MathSciNetCrossRefGoogle Scholar
  11. 11.
    S. Goreinov and E. Tyrtyshnikov, “The maximal-volume concept in approximation by low-rank matrices,” Contemp. Math. 208, 47–51 (2001).MathSciNetCrossRefGoogle Scholar
  12. 12.
    S. Goreinov, I. Oseledets, D. Savostyanov, E. Tyrtyshnikov, and N. Zamarashkin, “How to find a good submatrix,” in Matrix Methods: Theory, Algorithms and Applications (World Scientific, Singapore, 2010), pp. 247–256.CrossRefGoogle Scholar
  13. 13.
    I. Oseledets et al., TT-Toolbox (TT=Tensor Train), Vers. 2.2.2 (Inst. Numerical Mathematics, Moscow, 2009–2013). https://github.com/oseledets/TT-Toolbox.Google Scholar
  14. 14.
    TDALAB-MATLAB Toolbox for High-order Tensor Data Decompositions and Analysis. Current Vers. 1.1 (2013). https://github.com/andrewssobral/TDALAB.
  15. 15.
    F. Cong, Q.-H. Lin, L.-D. Kuang, X.-F Gong, P. Astikainen, and T. Ristaniemi, “Tensor decomposition of EEG signals: a brief review,” J. Neurosci. Methods 248, 59–69 (2015).CrossRefGoogle Scholar
  16. 16.
    F. Cong, A. Phan, Q. Zhao, T. Huttunen-Scott, J. Kaartinen, T. Ristaniemi, H. Lyytinen, and A. Cichocki, “Benefits of multi-domain feature of mismatch negativity extracted by nonnegative tensor factorization from low-density array EEG,” Int. J. Neural Syst. 22 (6) (2012).Google Scholar
  17. 17.
    E. Tyrtyshnikov and E. Shcherbakova, “Nonnegative tensor train factorizations and some applications,” Lect. Notes Comput. Sci. (in press).Google Scholar
  18. 18.
    D. Savostyanov and I. Oseledets, “Fast adaptive interpolation of multi-dimensional arrays in tensor train format,” in Proceedings of the 7th International Workshop on Multidimensional (nD) Systems, nDS 2011 (2011), pp. 1–8.Google Scholar
  19. 19.
    E. Tyrtyshnikov and E. Shcherbakova, “Nonnegative matrix factorization methods based on low-rank cross approximations,” Comput. Math. Math. Phys. 59 (8) (2019, in press).Google Scholar

Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia
  2. 2.Marchuk Institute of Numerical Mathematics of Russian Academy of SciencesMoscowRussia

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