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A unified global convergence analysis of multiplicative update rules for nonnegative matrix factorization

Author

Listed:
  • Norikazu Takahashi

    (Okayama University)

  • Jiro Katayama

    (Kyushu University)

  • Masato Seki

    (Okayama University)

  • Jun’ichi Takeuchi

    (Kyushu University)

Abstract

Multiplicative update rules are a well-known computational method for nonnegative matrix factorization. Depending on the error measure between two matrices, various types of multiplicative update rules have been proposed so far. However, their convergence properties are not fully understood. This paper provides a sufficient condition for a general multiplicative update rule to have the global convergence property in the sense that any sequence of solutions has at least one convergent subsequence and the limit of any convergent subsequence is a stationary point of the optimization problem. Using this condition, it is proved that many of the existing multiplicative update rules have the global convergence property if they are modified slightly so that all variables take positive values. This paper also proposes new multiplicative update rules based on Kullback–Leibler, Gamma, and Rényi divergences. It is shown that these three rules have the global convergence property if the same modification as above is made.

Suggested Citation

  • Norikazu Takahashi & Jiro Katayama & Masato Seki & Jun’ichi Takeuchi, 2018. "A unified global convergence analysis of multiplicative update rules for nonnegative matrix factorization," Computational Optimization and Applications, Springer, vol. 71(1), pages 221-250, September.
    • Handle: RePEc:spr:coopap:v:71:y:2018:i:1:d:10.1007_s10589-018-9997-y
    DOI: 10.1007/s10589-018-9997-y
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    References listed on IDEAS

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    1. GILLIS, Nicolas & GLINEUR, François, 2008. "Nonnegative factorization and the maximum edge biclique problem," LIDAM Discussion Papers CORE 2008064, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Norikazu Takahashi & Ryota Hibi, 2014. "Global convergence of modified multiplicative updates for nonnegative matrix factorization," Computational Optimization and Applications, Springer, vol. 57(2), pages 417-440, March.
    3. Daniel D. Lee & H. Sebastian Seung, 1999. "Learning the parts of objects by non-negative matrix factorization," Nature, Nature, vol. 401(6755), pages 788-791, October.
    4. Michael W. Berry & Murray Browne, 2005. "Email Surveillance Using Non-negative Matrix Factorization," Computational and Mathematical Organization Theory, Springer, vol. 11(3), pages 249-264, October.
    5. Jingu Kim & Yunlong He & Haesun Park, 2014. "Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework," Journal of Global Optimization, Springer, vol. 58(2), pages 285-319, February.
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    Cited by:

    1. Takehiro Sano & Tsuyoshi Migita & Norikazu Takahashi, 2022. "A novel update rule of HALS algorithm for nonnegative matrix factorization and Zangwill’s global convergence," Journal of Global Optimization, Springer, vol. 84(3), pages 755-781, November.

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