A Regularized Nonnegative Third Order Tensor decomposition Using a Primal-Dual Projected Gradient Algorithm: Application to 3D Fluorescence Spectroscopy

  • Karima El QateEmail author
  • Mohammed El Rhabi
  • Abdelilah Hakim
  • Eric Moreau
  • Nadàge Thirion-Moreau
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11010)


This paper investigates the use of Primal-Dual optimization algorithms on multidimensional signal processing problems. The data blocks interpreted in a tensor way can be modeled by means of multi-linear decomposition. Here we will focus on the Canonical Polyadic Decomposition (CPD), and we will present an application to fluorescence spectroscopy using this decomposition. In order to estimate the factors or latent variables involved in these decompositions, it is usual to use criteria optimization algorithms. A classical cost function consists of a measure of the modeling error (fidelity term) to which a regularization term can be added if necessary. Here, we consider one of the most efficient optimization methods, Primal-Dual Projected Gradient.

The effectiveness and the robustness of the proposed approach are shown through numerical examples.


Constrained optimization Nonnegative tensor decomposition Primal-Dual Regularization Projected gradient 


  1. 1.
    Harshman, R.A.: Foundations of the PARAFAC procedure: models and conditions for an “explanatory” multi-modal factor analysis. UCLA Working Papers in Phonetics, vol. 16, pp. 1–84 (1970)Google Scholar
  2. 2.
    Carroll, J.D., Chang, J.J.: Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition. Psychometrika 35(3), 283–319 (1970)CrossRefGoogle Scholar
  3. 3.
    Harshman, R.A., Lundy, M.E.: PARAFAC: parallel factor analysis. Comput. Stat. Data Anal. 18(1), 39–72 (1994)CrossRefGoogle Scholar
  4. 4.
    Harshman, R.A., Lundy, M.E.: The PARAFAC model for three-way factor analysis and multidimensional scaling. Res. Methods Multimode Data Anal. 46, 122–215 (1984)Google Scholar
  5. 5.
    Mocks, J.: Topographic components model for event-related potentials and some biophysical considerations. IEEE Trans. Biomed. Eng. 35(6), 482–484 (1988)CrossRefGoogle Scholar
  6. 6.
    Tendeiro, J., Dosse, M.B., Berge, T., Jos, M.F.: First and second-order derivatives for CP and INDSCAL. Chemom. Intell. Lab. Syst. 106(1), 27–36 (2011)CrossRefGoogle Scholar
  7. 7.
    Acar, E., Dunlavy, D.M., Kolda, T.G., et al.: Scalable tensor factorizations with missing data. In: Proceedings of the 2010 SIAM International Conference on Data Mining. Society for Industrial and Applied Mathematics, pp. 701–712 (2010)Google Scholar
  8. 8.
    Kindermann, S., Navasca, C.: News algorithms for tensor decomposition based on a reduced functional. Numer. Linear Algebr. Appl. 21(3), 340–374 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Coble, P.G.: Characterization of marine and terrestrial dom in seawater using excitation-emission matrix spectroscopy. Mar. Chem. 52, 325–346 (1996)CrossRefGoogle Scholar
  10. 10.
    Smilde, A., Bro, R., Geladi, P.: Multi-way Analysis: Applications in the Chemical Sciences. Wiley, Hoboken (2005)Google Scholar
  11. 11.
    Royer, J.-P., Thirion-Moreau, N., Comon, P.: Computing the polyadic decomposition of nonnegative third order tensors. EURASIP Signal Process. 91(9), 2159–2171 (2011)CrossRefGoogle Scholar
  12. 12.
    Hitchcock, F.L.: The expression of a tensor or a polyadic as a sum of products. J. Math. Phys. 6, 165–189 (1927)CrossRefGoogle Scholar
  13. 13.
    Cichocki, A., Zdunek, R., Phan, A.H., Amari, S.I.: Non Negative Matrix and Tensor Factorizations: Application to Exploratory Multi-way Data Analysis and Blind Separation. Wiley, Hoboken (2009)CrossRefGoogle Scholar
  14. 14.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging (2010)Google Scholar
  15. 15.
    Zhu, M., Chan, T.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration, Technical report, UCLA CAM Report 08–34 (2008)Google Scholar
  16. 16.
    Ekeland, I., Roger, T.: Convex analysis and variational problems, vol. 28. SIAM (1999)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Karima El Qate
    • 1
    Email author
  • Mohammed El Rhabi
    • 2
  • Abdelilah Hakim
    • 1
  • Eric Moreau
    • 3
  • Nadàge Thirion-Moreau
    • 3
  1. 1.LAMAI, FSTG MarrakechUniversity of Cady AyyadMarrakeshMorocco
  2. 2.Applied Mathematics and Computer Science DepartmentEcole des Ponts ParisTech (ENPC)ParisFrance
  3. 3.Aix Marseille Université, Université de Toulon, CNRS UMR 7020, LISMarseilleFrance

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