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Split Gradient Method for Informed Non-negative Matrix Factorization

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Latent Variable Analysis and Signal Separation (LVA/ICA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9237))

Abstract

Recently, some informed Non-negative Matrix Factorization (NMF) methods were introduced in which some a priori knowledge (i.e., expert’s knowledge) were taken into account in order to improve the separation process. This knowledge was expressed as known components of one factor, namely the profile matrix. Also, the sum-to-one property of the profile matrix was taken into account by an appropriate sequential normalization. However, our previous approach was unable to check both constraints at the same time. In this work, a new parametrization is proposed which takes into consideration both constraints simultaneously by incorporating a new unconstrained matrix. From this parameterization, new updates rules are introduced which are based on the framework of the Split Gradient Method by Lantéri et al. The cost function is defined in terms of a weighted Frobenius norm and the developed rules involve a new shift in order to ensure the non-negativity property. Simulations on a noisy source apportionment problem show the relevance of the proposed method.

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Notes

  1. 1.

    Actually, Lantéri et al. summed the rows of G to 1 and the rows of F to the sums of the rows of X. When expressed in the source apportionment application, it can be equivalently written as Eqs. (2) and (3).

  2. 2.

    Please note that in our previous work [7, 8], we were computing the sums of the MERs estimated on each column of G.

References

  1. Comon, P., Jutten, C.: Handbook of Blind Source Separation, Independent Component Analysis and Applications. Academic Press, Oxford (2010)

    Google Scholar 

  2. Févotte, C., Idier, J.: Algorithms for nonnegative matrix factorization with the beta-divergence. Neural Comput. 23(9), 2421–2456 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Heinz, D.C., Chang, C.: Fully constrained least squares linear mixture analysis for material quantification in hyperspectral imagery. IEEE Trans. Geosci. Remote Sens. 39, 529–545 (2001)

    Article  Google Scholar 

  4. Ho, N.D.: Non negative matrix factorization algorithms and applications. Ph.D. thesis, Université Catholique de Louvain (2008)

    Google Scholar 

  5. Lantéri, H., Theys, C., Richard, C., Févotte, C.: Split gradient method for nonnegative matrix factorization. In: Proceedings of EUSIPCO (2010)

    Google Scholar 

  6. Lee, D., Seung, H.: Learning the parts of objects by non negative matrix factorization. Nature 401(6755), 788–791 (1999)

    Article  Google Scholar 

  7. Limem, A., Delmaire, G., Puigt, M., Roussel, G., Courcot, D.: Non-negative matrix factorization under equality constraints–a study of industrial source identification. Appl. Numer. Math. 85, 1–15 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  8. Limem, A., Puigt, M., Delmaire, G., Roussel, G., Courcot, D.: Bound constrained weighted NMF for industrial source apportionment. In: Proceedings of MLSP (2014)

    Google Scholar 

  9. Ma, W.K., Bioucas-Dias, J.M., Chan, T.H., Gillis, N., Gader, P., Plaza, A.J., Ambikapathi, A., Chi, C.Y.: A signal processing perspective on hyperspectral unmixing. IEEE Sig. Proc. Mag. 31, 67–81 (2014)

    Article  Google Scholar 

  10. Peng, C., Wong, K., Rockwood, A., Zhang, X., Jiang, J., Keyes, D.: Multiplicative algorithms for constrained non-negative matrix factorization. In: 12th IEEE International Conference on Data Mining, ICDM 2012, 10–13 December 2012, Brussels, Belgium, pp. 1068–1073 (2012)

    Google Scholar 

  11. Plouvin, M., Limem, A., Puigt, M., Delmaire, G., Roussel, G., Courcot, D.: Enhanced NMF initialization using a physical model for pollution source apportionment. In: Proceedings of ESANN, pp. 261–266 (2014)

    Google Scholar 

  12. Viana, M., Pandolfi, A., Minguillo, M.C., Querol, X., Alastuey, A., Monfort, E., Celades, I.: Inter-comparison of receptor models for PM source apportionment: case study in an industrial area. Atmos. Environ. 42, 3820–3832 (2008)

    Article  Google Scholar 

  13. Vincent, E., Araki, S., Bofill, P.: The 2008 signal separation evaluation campaign: A community-based approach to large-scale evaluation. In: Adali, T., Jutten, C., Romano, J.M.T., Barros, A.K. (eds.) ICA 2009. LNCS, vol. 5441, pp. 734–741. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

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Acknowledgements

This work was funded by the “ECUME” project granted by the DREAL Nord Pas de Calais Agency.

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Authors and Affiliations

  1. LISIC, ULCO, Université Lille Nord de France, Calais, France

    Robert Chreiky, Gilles Delmaire, Matthieu Puigt & Gilles Roussel

  2. University of Balamand, Koura, Lebanon

    Robert Chreiky & Antoine Abche

  3. UCEIV, ULCO, Université Lille Nord de France, Dunkerque, France

    Dominique Courcot

Authors
  1. Robert Chreiky
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  2. Gilles Delmaire
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  3. Matthieu Puigt
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  4. Gilles Roussel
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  5. Dominique Courcot
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  6. Antoine Abche
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Corresponding author

Correspondence to Gilles Delmaire .

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Editors and Affiliations

  1. Inria, Villers-les-Nancy, France

    Emmanuel Vincent

  2. Tel Aviv University, Tel-Aviv, Israel

    Arie Yeredor

  3. Technical University of Libere, Liberec, Czech Republic

    Zbyněk Koldovský

  4. The Czech Academy of Sciences, Prague, Czech Republic

    Petr Tichavský

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Chreiky, R., Delmaire, G., Puigt, M., Roussel, G., Courcot, D., Abche, A. (2015). Split Gradient Method for Informed Non-negative Matrix Factorization. In: Vincent, E., Yeredor, A., Koldovský, Z., Tichavský, P. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2015. Lecture Notes in Computer Science(), vol 9237. Springer, Cham. https://doi.org/10.1007/978-3-319-22482-4_44

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  • DOI: https://doi.org/10.1007/978-3-319-22482-4_44

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22481-7

  • Online ISBN: 978-3-319-22482-4

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