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Four algorithms to solve symmetric multi-type non-negative matrix tri-factorization problem

Author

Listed:
  • Rok Hribar

    (Jožef Stefan Institute
    Jožef Stefan International Postgraduate School)

  • Timotej Hrga

    (University of Ljubljana)

  • Gregor Papa

    (Jožef Stefan Institute)

  • Gašper Petelin

    (Jožef Stefan Institute
    Jožef Stefan International Postgraduate School)

  • Janez Povh

    (University of Ljubljana
    Institute of Mathematics, Physics and Mechanics)

  • Nataša Pržulj

    (Institute of Mathematics, Physics and Mechanics
    University College London
    Barcelona Supercomputing Center)

  • Vida Vukašinović

    (Jožef Stefan Institute)

Abstract

In this paper, we consider the symmetric multi-type non-negative matrix tri-factorization problem (SNMTF), which attempts to factorize several symmetric non-negative matrices simultaneously. This can be considered as a generalization of the classical non-negative matrix tri-factorization problem and includes a non-convex objective function which is a multivariate sixth degree polynomial and a has convex feasibility set. It has a special importance in data science, since it serves as a mathematical model for the fusion of different data sources in data clustering. We develop four methods to solve the SNMTF. They are based on four theoretical approaches known from the literature: the fixed point method (FPM), the block-coordinate descent with projected gradient (BCD), the gradient method with exact line search (GM-ELS) and the adaptive moment estimation method (ADAM). For each of these methods we offer a software implementation: for the former two methods we use Matlab and for the latter Python with the TensorFlow library. We test these methods on three data-sets: the synthetic data-set we generated, while the others represent real-life similarities between different objects. Extensive numerical results show that with sufficient computing time all four methods perform satisfactorily and ADAM most often yields the best mean square error (MSE). However, if the computation time is limited, FPM gives the best MSE because it shows the fastest convergence at the beginning. All data-sets and codes are publicly available on our GitLab profile.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:82:y:2022:i:2:d:10.1007_s10898-021-01074-3
    DOI: 10.1007/s10898-021-01074-3
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    References listed on IDEAS

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    1. Shota Saito & Yoshito Hirata & Kazutoshi Sasahara & Hideyuki Suzuki, 2015. "Tracking Time Evolution of Collective Attention Clusters in Twitter: Time Evolving Nonnegative Matrix Factorisation," PLOS ONE, Public Library of Science, vol. 10(9), pages 1-17, September.
    2. Andrej Čopar & Blaž Zupan & Marinka Zitnik, 2019. "Fast optimization of non-negative matrix tri-factorization," PLOS ONE, Public Library of Science, vol. 14(6), pages 1-15, June.
    3. P. Tseng, 2001. "Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization," Journal of Optimization Theory and Applications, Springer, vol. 109(3), pages 475-494, June.
    4. 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.
    5. Peter Dickinson & Luuk Gijben, 2014. "On the computational complexity of membership problems for the completely positive cone and its dual," Computational Optimization and Applications, Springer, vol. 57(2), pages 403-415, March.
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