Advertisement

Error Analysis of TT-Format Tensor Algorithms

  • Dario FasinoEmail author
  • Eugene E. Tyrtyshnikov
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 30)

Abstract

The tensor train (TT) decomposition is a representation technique for arbitrary tensors, which allows efficient storage and computations. For a d-dimensional tensor with d ≥ 2, that decomposition consists of two ordinary matrices and d − 2 third-order tensors. In this paper we prove that the TT decomposition of an arbitrary tensor can be computed (or approximated, for data compression purposes) by means of a backward stable algorithm based on computations with Householder matrices. Moreover, multilinear forms with tensors represented in TT format can be computed efficiently with a small backward error.

Keywords

TT-format Backward stability Tensor compression Multilinear algebra 

Notes

Acknowledgements

The first author acknowledges the support received by INDAM-GNCS, Italy, for his research. The work of the second author was supported by the Russian Scientific Foundation project 14-11-00806.

References

  1. 1.
    Bachmayr, M., Kazeev, V.: Stability of low-rank tensor representations and structured multilevel preconditioning for elliptic PDEs. ArXiv preprint (2018). http://arxiv.org/pdf/1802.09062.pdf
  2. 2.
    Golub, G.H., Van Loan, C.: Matrix Computations, 4th edn. The John Hopkins University Press, Baltimore (2013)zbMATHGoogle Scholar
  3. 3.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia (2002)CrossRefGoogle Scholar
  4. 4.
    Jeannerod, C.-P., Rump, S.M.: Improved error bounds for inner products in floating-point arithmetic. SIAM J. Matrix Anal. Appl. 34, 338–344 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Lee, N., Cichocki, A.: Estimating a few extreme singular values and vectors for large-scale matrices in tensor train format. SIAM J. Matrix Anal. Appl. 36(3), 994–1014 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Oseledets, I.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Oseledets, I., Dolgov, S.V.: Solution of linear systems and matrix inversion in the TT-format. SIAM J. Sci. Comput. 34(5), A2718–A2739 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Oseledets, I., Tyrtyshnikov, E.: Recursive decomposition of multidimensional tensors. Dokl. Math. 80(1), 460–462 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Oseledets, I., Tyrtyshnikov, E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432(1), 70–88 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Computer Science and PhysicsUniversity of UdineUdineItaly
  2. 2.Institute of Numerical Mathematics of Russian Academy of SciencesMoscowRussia

Personalised recommendations