Neural Processing Letters

, Volume 31, Issue 1, pp 45–64 | Cite as

Convergence Analysis of Non-Negative Matrix Factorization for BSS Algorithm

  • Shangming YangEmail author
  • Zhang Yi


In this paper the convergence of a recently proposed BSS algorithm is analyzed. This algorithm utilized Kullback–Leibler divergence to generate non-negative matrix factorizations of the observation vectors, which is considered an important aspect of the BSS algorithm. In the analysis some invariant sets are constructed so that the convergence of the algorithm can be guaranteed in the given conditions. In the simulation we successfully applied the algorithm and its analysis results to the blind source separation of mixed images and signals.


BSS Convergence analysis Non-negative ICA Non-negative matrix factorization NMF KL divergence 


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  1. 1.
    Comon P (1994) Independent component analysis-a new concept? Signal Process 36: 287–314zbMATHCrossRefGoogle Scholar
  2. 2.
    Bell AJ, Sejnowski TJ (1995) An information-maximization approach to blind separation and blind deconvolution. Neural Comput 7: 1129–1159CrossRefGoogle Scholar
  3. 3.
    Comon P, Mourrain B (1996) Decomposition of quantics in sums of powers of linear forms. Signal Process 53: 93–107zbMATHCrossRefGoogle Scholar
  4. 4.
    Cardoso JF (1998) Blind signal separation: statistical principles. Proc IEEE 86: 2009–2025CrossRefGoogle Scholar
  5. 5.
    Lee TW (1998) Independent component analysis: theory and applications. Kluwer, BostonzbMATHGoogle Scholar
  6. 6.
    Amari S, Cichocki A, Yang H (1996) A new learning algorithm for blind source separation, advances in neural information processing system, vol 8. MIT Press, Cambridge, pp 757–763Google Scholar
  7. 7.
    Cichocki A, Zdunek R, Amari S (2006) New algorithms for non-negative matrix factorization in applications to blind source separation. ICASSP-2006, Toulouse, France, pp 621–625Google Scholar
  8. 8.
    Cichocki A, Zdunek R, Amari S (2006) Csiszar’s divergences for non-negative matrix factorization: family of new algorithms”. In: 6th international conference on independent component analysis and blind signal separation, Charleston SC, USA, Springer LNCS 3889, pp 32–39Google Scholar
  9. 9.
    Lee DD, Seung HS (1999) Learning of the parts of objects by non-negative matrix factorization. Nature 401: 788–791CrossRefGoogle Scholar
  10. 10.
    Lee DD, Seung HS (2001) Algorithms for non-negative matrix factorization, vol 13. NIPS, MIT Press, CambridgeGoogle Scholar
  11. 11.
    Hoyer P (2004) Non-negative matrix factorization with sparseness constraints. J Mach Learn Res 5: 1457–1469MathSciNetGoogle Scholar
  12. 12.
    Plumbly M, Oja E (2004) A “non-negative PCA” algorithm for independent component analysis. IEEE Trans Neural Netw 15(1): 66–76CrossRefGoogle Scholar
  13. 13.
    Oja E, Plumbley M (2004) Blind separation of positive sources by globally covergent graditent search. Nueral Comput 16(9): 1811–1925zbMATHCrossRefGoogle Scholar
  14. 14.
    Plumbley M (2002) Condotions for non-negative inpendent component analysis. IEEE Signal Process Lett 9(6): 177–180CrossRefGoogle Scholar
  15. 15.
    Plumbley M (2003) Algorithms for non-negative inpendent component analysis. IEEE Trans Neural Netw 4(3): 534–543CrossRefGoogle Scholar
  16. 16.
    Xu L, Oja E, Suen CY (1992) Modified hebbian leraning for curve and surface fitting. Neural Netw 5: 441–457CrossRefGoogle Scholar
  17. 17.
    Agiza HN (1999) On the analysis of stbility, bifurcation, chaos and chaos control of Kopel Map. Chaos Solutions Fractals 10: 1909–1916zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Cichocki A, Amari S, Siwek K, Tanaka T (2006) The ICALAB package: for image processing, version 1.2. RIKEN Brain Science Institute, Wako shi, Saitama, JapanGoogle Scholar
  19. 19.
    Cichocki A, Zdunek R (2006) The NMFLAB package: for signal processing, version 1.1. RIKEN Brain Science Institute, Wako shi, Saitama, JapanGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.School of Computer Science and EngineeringUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China
  2. 2.College of Computer ScienceSichuan UniversityChengduPeople’s Republic of China

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