Skip to main content
Springer Nature Link
Log in

Nonlinear Least Squares Solver for Evaluating Canonical Tensor Decomposition

  • Conference paper
  • First Online:
Optimization and Applications (OPTIMA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12422))

Included in the following conference series:

  • 455 Accesses

Abstract

Nonlinear least squares iterative solver developed earlier by the author is applied for numerical solution of special system of multilinear equations arising in the problem of canonical tensor decomposition. The proposed algorithm is based on easily parallelizable computational kernels such as matrix-vector multiplications and elementary vector operations and therefore has a potential for a quite efficient implementation on modern high-performance computers. The results of numerical testing presented for certain examples of large scale dense and medium size sparse 3D tensors found in existing literature seem very competitive with respect to computational costs involved.

Supported by RFBR grant No.19-01-00666.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Acar, E., Dunlavy, D.M., Kolda, T.G.: A scalable optimization approach for fitting canonical tensor decompositions. J. Chemometr. 25(2), 67–86 (2011)

    Article  Google Scholar 

  2. Armijo, L.: Minimization of functions having Lipschitz continuous first partial derivatives. Pac. J. Math. 16(1), 1–3 (1966)

    Article  MathSciNet  Google Scholar 

  3. Bellavia, S., Gratton, S., Riccietti, E.: A Levenberg-Marquardt method for large nonlinear least-squares problems with dynamic accuracy in functions and gradients. Numer. Math. 140(3), 791–825 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bellavia, S., Morini, B.: Strong local convergence properties of adaptive regularized methods for nonlinear least squares. IMA J. Numer. Anal. 35(2), 947–968 (2015)

    Article  MathSciNet  Google Scholar 

  5. Brent, R.P.: Algorithms for matrix multiplication (No. STAN-CS-70-157). Stanford University CA Department of Computer Science (1970)

    Google Scholar 

  6. Kaporin, I.E.: Esimating global convergence of inexact Newton methods via limiting stepsize along normalized direction. Rep. 9329, July 1993, Department of Mathematics, Catholic University of Nijmegen, Nijmegen, The Netherlands, 8p. (1993)

    Google Scholar 

  7. Kaporin, I.E.: The use of preconditioned Krylov subspaces in conjugate gradient type methods for the solution of nonlinear least square problems. (Russian) Vestnik Mosk. Univ., Ser. 15 (Computational Math. and Cybernetics) N 3, 26–31 (1995)

    Google Scholar 

  8. Kaporin, I.E., Axelsson, O.: On a class of nonlinear equation solvers based on the residual norm reduction over a sequence of affine subspaces. SIAM J. Sci. Comput. 16(1), 228–249 (1994)

    Article  MathSciNet  Google Scholar 

  9. Kaporin, I.: Using Chebyshev polynomials and approximate inverse triangular factorizations for preconditioning the conjugate gradient method. Comput. Math. Math. Phys. 52(2), 169–193 (2012)

    Article  MathSciNet  Google Scholar 

  10. Kaporin, I.: Preconditioned subspace descent methods for the solution of nonlinear systems of equations. In: Jacimovic, M., et al. (eds.) Proceedings of X International Conference on Optimization and Applications (OPTIMA 2019). Communications in Computer and Information Science (CCIS). Springer Nature Switzerland AG (2019)

    Google Scholar 

  11. Kaporin, I.: Preconditioned subspace descent method for nonlinear systems of equations. Open Comput. Sci. 10(1), 71–81 (2020)

    Article  Google Scholar 

  12. Karimi, S., Vavasis, S.: Conjugate gradient with subspace optimization (2012). arXiv preprint arXiv:1202.1479v1

  13. Kazeev, V.A., Tyrtyshnikov, E.E.: Structure of the Hessian matrix and an economical implementation of Newton’s method in the problem of canonical approximation of tensors. Comput. Math. Math. Phys. 50(6), 927–945 (2010)

    Article  MathSciNet  Google Scholar 

  14. Oseledets, I.V., Savostyanov, D.V.: Minimization methods for approximating tensors and their comparison. Comput. Math. Math. Phys. 46(10), 1641–1650 (2006)

    Article  MathSciNet  Google Scholar 

  15. Sterck, H.D., Winlaw, M.: A nonlinearly preconditioned conjugate gradient algorithm for rank-R canonical tensor approximation. Numer. Linear Algebra Appl. 22(3), 410–432 (2015)

    Article  MathSciNet  Google Scholar 

  16. Sterck, H.D., Miller, K.: An adaptive algebraic multigrid algorithm for low-rank canonical tensor decomposition. SIAM J. Sci. Comput. 35(1), B1–B24 (2013)

    Article  MathSciNet  Google Scholar 

  17. Yuan, Y.X.: Recent advances in numerical methods for nonlinear equations and nonlinear least squares. Numer. Algebra Control Optim. 1(1), 15–34 (2011)

    Article  MathSciNet  Google Scholar 

  18. Yu, L., Barbot, J.P., Zheng, G., Sun, H.: Compressive sensing with chaotic sequence. IEEE Signal Process. Lett. 17(8), 731–734 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dorodnicyn Computing Center of FRC CSC RAS, Vavilova street 40, Moscow, Russia

    Igor Kaporin

Authors
  1. Igor Kaporin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor Kaporin .

Editor information

Editors and Affiliations

  1. Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia

    Nicholas Olenev

  2. Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia

    Yuri Evtushenko

  3. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  4. Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia

    Vlasta Malkova

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kaporin, I. (2020). Nonlinear Least Squares Solver for Evaluating Canonical Tensor Decomposition. In: Olenev, N., Evtushenko, Y., Khachay, M., Malkova, V. (eds) Optimization and Applications. OPTIMA 2020. Lecture Notes in Computer Science(), vol 12422. Springer, Cham. https://doi.org/10.1007/978-3-030-62867-3_14

Download citation

Publish with us

Policies and ethics