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Computational Mathematics and Mathematical Physics

, Volume 46, Issue 10, pp 1641–1650 | Cite as

Minimization methods for approximating tensors and their comparison

  • I. V. Oseledets
  • D. V. Savost’yanov
Article

Abstract

Application of various minimization methods to trilinear approximation of tensors is considered. These methods are compared based on numerical calculations. For the Gauss-Newton method, an efficient implementation is proposed, and the local rate of convergence is estimated for the case of completely symmetric tensors.

Keywords

trilinear approximation minimization methods nonlinear approximation 

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References

  1. 1.
    R. A. Harshman, “Foundations of the Parafac Procedure: Models and Conditions for an Explanatory Multimodal Factor Analysis,” UCLA Working Papers in Phonetics 16, 1–84 (1970).Google Scholar
  2. 2.
    P. Comon, “Tensor Decomposition: State of the Art and Applications,” in IMA Conf. Math. Signal Proc., Warwick, UK, 2000; http://www.i3s.fr/:_comon/FichiersPs/ima2000.ps.
  3. 3.
    I. Ibraghimov, “Application of the Three-Way Decomposition for Matrix Compression,” Numer. Lin. Algebra Appl. 9, 551–565 (2002).zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    J. D. Caroll and J. J. Chang, “Analysis of Individual Differences in Multidimensional Scaling via N-Way Generalization of Eckart-Young Decomposition,” Psychometrica 35, 283–319 (1970).CrossRefGoogle Scholar
  5. 5.
    R. Bro, “PARAFAC: Tutorial and Applications,” Chemom. Intel. Lab. Systems 38, 149–171 (1997).CrossRefGoogle Scholar
  6. 6.
    J.-H. Wang, P. K. Hopke, T. M. Hancewicz, and S. L. Zhang, “Application of Modified Least Squares Regression to Spectroscopic Image Analysis,” Analys. Chim. Acta 476, 93–109 (2003).CrossRefGoogle Scholar
  7. 7.
    J. P. Dedieu and M. Schub, “Newton’s Method for Overdetermined Systems of Equations,” Math. Comput. 69(281), 1099–1115 (2000).zbMATHGoogle Scholar
  8. 8.
    T. Zhang and G. H. Golub, “Rank-One Approximation to High-Order Tensors,” SIAM J. Matrix Anal. Appl. 23, 534–550 (2001).zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    H. B. Nielsen, “Damping Parameter in Marquardt’s Method,” Inform. Math. Model., Denmark (1999); http://www/imm.dtu.dk/:_hbn/publ/TR9905.ps.
  10. 10.
    L. R. Tucker, “Some Mathematical Notes on Three-Mode Factor Analysis,” Psychometrica 31, 279–311 (1966).MathSciNetCrossRefGoogle Scholar
  11. 11.
    I. V. Oseledets and D. V. Savost’yanov, “A Fast Algorithm for Simultaneous Reduction of Matrixes to a Triangular Form and Approximation of Tensors,” in Matrix Methods for Solving Large-Scale Problems (Institut Vychislitel’noi Matematiki, RAN, Moscow, 2005), pp. 101–116 [in Russian].Google Scholar
  12. 12.
    I. V. Oseledets and D. V. Savost’yanov, “Methods for Decomposition of Tensors,” in Matrix Methods for Solving Large-Scale Problems (Institut Vychislitel’noi Matematiki, RAN, Moscow, 2005), pp. 51–64 [in Russian].Google Scholar

Copyright information

© MAIK “Nauka/Interperiodica” 2006

Authors and Affiliations

  • I. V. Oseledets
    • 1
  • D. V. Savost’yanov
    • 1
  1. 1.Institute of Numerical MathematicsRussian Academy of SciencesMoscowRussia

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