Skip to main content
SpringerLink
Log in

Optimization problems in contracted tensor networks

  • Published:
Computing and Visualization in Science

Abstract

We discuss the calculus of variations in tensor representations with a special focus on tensor networks and apply it to functionals of practical interest. The survey provides all necessary ingredients for applying minimization methods in a general setting. The important cases of target functionals which are linear and quadratic with respect to the tensor product are discussed, and combinations of these functionals are presented in detail. As an example, we consider the representation rank compression in tensor networks. For the numerical treatment, we use the nonlinear block Gauss–Seidel method. We demonstrate the rate of convergence in numerical tests.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bebendorf M.: Approximation of boundary element matrices. Numer. Math. 86(4), 565–589 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beylkin G., Mohlenkamp M.J.: Numerical operator calculus in higher dimensions. Proc. Natl. Acad. Sci. USA 99(16), 10246–10251 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beylkin G., Mohlenkamp M.J.: Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26(6), 2133–2159 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Espig, M., Schuster, M., Killaitis, A., Waldren, N., Wähnert, P., Handschuh, S., Auer, H.: TensorCalculus library. http://gitorious.org/tensorcalculus (2011)

  5. Falcó, A., Hackbusch, W.: On Minimal Subspaces in Tensor Representations. MIS preprint 70 (2010)

  6. Grasedyck L.: Hierarchical singular value decomposition of tensors. SIAM J Matix Anal. Appl. 31, 2029–2054 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hackbusch W.: Tensor Spaces and Numerical Tensor Calculus. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

  9. Holtz, S., Rohwedder, T., Schneider, R.: The alternating linear scheme for tensor optimisation in the TT format. Preprint 71 (2010)

  10. Huckle, T., Waldherr, K., Schulte-Herbrüggen, T.: Computations in Quantum Tensor Networks. Preprint (2010)

  11. Khoromskij B.N.: \({\mathcal{O}(d \log {N})}\) -quantics approximation of N−d tensors in high-dimensional numerical modeling. Const. Approx. 34, 1–24 (2010)

    MathSciNet  Google Scholar 

  12. Landsberg, J.M., Qi, Y., Ye, K.: On the geometry of tensor network states. arXiv:1105.4449 [math.AG] (2011)

  13. Ortega J.M, Rheinboldt W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Society for Industrial Mathe- matics, Academic Press, Inc., San Diego (1970)

    MATH  Google Scholar 

  14. Oseledets, I.V.: Compact matrix form of the d-dimensional tensor decomposition. INM RAS preprint 01 (2009)

  15. Oseledets I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Oseledets, I.V., Khoromskij, B.N.: DMRG and QTT approach to high-dimensional quantum molecular dynamics. MIS preprint 69 (2010)

  17. Schuch, N., Cirac, I., Pérez-García, D.: PEPS as ground states: degeneracy and topology. arXiv:1001.3807v2 [quant-ph] (2010)

  18. Singh, S., Pfeifer, R.N.C., Vidal, G.: Tensor network decompositions in the presence of a global symmetry. arXiv:0907.2994v1 [cond-mat.str-el] (2009)

  19. Tyrtyshnikov E.E., Oseledets I.V.: Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput. 31, 3744–3759 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Uschmajew, A.: Local convergence of the alternating least squares algorithm for canonical tensor approximation. Preprint 103 (2011)

  21. Vidal, G.: Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91(14). doi:10.1103/PhysRevLett.91.147902 (2003)

Download references

Author information

Authors and Affiliations

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

    Mike Espig, Wolfgang Hackbusch & Stefan Handschuh

  2. Technical University Berlin, Berlin, Germany

    Reinhold Schneider

Authors
  1. Mike Espig
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wolfgang Hackbusch
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefan Handschuh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Reinhold Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Handschuh.

Additional information

Communicated by: Gabriel Wittum.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Espig, M., Hackbusch, W., Handschuh, S. et al. Optimization problems in contracted tensor networks. Comput. Visual Sci. 14, 271–285 (2011). https://doi.org/10.1007/s00791-012-0183-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-012-0183-y

Keywords

Search

Navigation