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A novel update rule of HALS algorithm for nonnegative matrix factorization and Zangwill’s global convergence

Author

Listed:
  • Takehiro Sano

    (Okayama University)

  • Tsuyoshi Migita

    (Okayama University)

  • Norikazu Takahashi

    (Okayama University)

Abstract

Nonnegative Matrix Factorization (NMF) has attracted a great deal of attention as an effective technique for dimensionality reduction of large-scale nonnegative data. Given a nonnegative matrix, NMF aims to obtain two low-rank nonnegative factor matrices by solving a constrained optimization problem. The Hierarchical Alternating Least Squares (HALS) algorithm is a well-known and widely-used iterative method for solving such optimization problems. However, the original update rule used in the HALS algorithm is not well defined. In this paper, we propose a novel well-defined update rule of the HALS algorithm, and prove its global convergence in the sense of Zangwill. Unlike conventional globally-convergent update rules, the proposed one allows variables to take the value of zero and hence can obtain sparse factor matrices. We also present two stopping conditions that guarantee the finite termination of the HALS algorithm. The practical usefulness of the proposed update rule is shown through experiments using real-world datasets.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:84:y:2022:i:3:d:10.1007_s10898-022-01167-7
    DOI: 10.1007/s10898-022-01167-7
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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. GILLIS, Nicolas & GLINEUR, François, 2011. "Accelerated multiplicative updates and hierarchical als algorithms for nonnegative matrix factorization," LIDAM Discussion Papers CORE 2011030, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. 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.
    5. Berry, Michael W. & Browne, Murray & Langville, Amy N. & Pauca, V. Paul & Plemmons, Robert J., 2007. "Algorithms and applications for approximate nonnegative matrix factorization," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 155-173, September.
    6. 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.
    7. 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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