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A Block Coordinate Descent-Based Projected Gradient Algorithm for Orthogonal Non-Negative Matrix Factorization

Author

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  • Soodabeh Asadi

    (Institute for Data Science, School of Engineering, University of Applied Sciences and Arts Northwestern Switzerland, 5210 Windisch, Switzerland)

  • Janez Povh

    (Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva ulica 6, SI-1000 Ljubljana, Slovenia
    Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia)

Abstract

This article uses the projected gradient method (PG) for a non-negative matrix factorization problem (NMF), where one or both matrix factors must have orthonormal columns or rows. We penalize the orthonormality constraints and apply the PG method via a block coordinate descent approach. This means that at a certain time one matrix factor is fixed and the other is updated by moving along the steepest descent direction computed from the penalized objective function and projecting onto the space of non-negative matrices. Our method is tested on two sets of synthetic data for various values of penalty parameters. The performance is compared to the well-known multiplicative update (MU) method from Ding (2006), and with a modified global convergent variant of the MU algorithm recently proposed by Mirzal (2014). We provide extensive numerical results coupled with appropriate visualizations, which demonstrate that our method is very competitive and usually outperforms the other two methods.

Suggested Citation

  • Soodabeh Asadi & Janez Povh, 2021. "A Block Coordinate Descent-Based Projected Gradient Algorithm for Orthogonal Non-Negative Matrix Factorization," Mathematics, MDPI, vol. 9(5), pages 1-22, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:540-:d:510526
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
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    Cited by:

    1. Rok Hribar & Timotej Hrga & Gregor Papa & Gašper Petelin & Janez Povh & Nataša Pržulj & Vida Vukašinović, 2022. "Four algorithms to solve symmetric multi-type non-negative matrix tri-factorization problem," Journal of Global Optimization, Springer, vol. 82(2), pages 283-312, February.

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