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On Rank Selection in Non-Negative Matrix Factorization Using Concordance

Author

Listed:
  • Paul Fogel

    (Mazars, Tour Exaltis, 61 Rue Henri-Régnault, 92400 Courbevoie, France
    These authors contributed equally to this work.)

  • Christophe Geissler

    (Mazars, Tour Exaltis, 61 Rue Henri-Régnault, 92400 Courbevoie, France
    These authors contributed equally to this work.)

  • Nicolas Morizet

    (Mazars, Tour Exaltis, 61 Rue Henri-Régnault, 92400 Courbevoie, France)

  • George Luta

    (Department of Biostatistics, Bioinformatics and Biomathematics, Georgetown University, 3700 O St NW, Washington, DC 20057, USA)

Abstract

The choice of the factorization rank of a matrix is critical, e.g., in dimensionality reduction, filtering, clustering, deconvolution, etc., because selecting a rank that is too high amounts to adjusting the noise, while selecting a rank that is too low results in the oversimplification of the signal. Numerous methods for selecting the factorization rank of a non-negative matrix have been proposed. One of them is the cophenetic correlation coefficient ( c c c ), widely used in data science to evaluate the number of clusters in a hierarchical clustering. In previous work, it was shown that c c c performs better than other methods for rank selection in non-negative matrix factorization (NMF) when the underlying structure of the matrix consists of orthogonal clusters. In this article, we show that using the ratio of c c c to the approximation error significantly improves the accuracy of the rank selection. We also propose a new criterion, c o n c o r d a n c e , which, like c c c , benefits from the stochastic nature of NMF; its accuracy is also improved by using its ratio-to-error form. Using real and simulated data, we show that c o n c o r d a n c e , with a CUSUM-based automatic detection algorithm for its original or ratio-to-error forms, significantly outperforms c c c . It is important to note that the new criterion works for a broader class of matrices, where the underlying clusters are not assumed to be orthogonal.

Suggested Citation

  • Paul Fogel & Christophe Geissler & Nicolas Morizet & George Luta, 2023. "On Rank Selection in Non-Negative Matrix Factorization Using Concordance," Mathematics, MDPI, vol. 11(22), pages 1-18, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:22:p:4611-:d:1278024
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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. Paul Fogel & Yann Gaston-Mathé & Douglas Hawkins & Fajwel Fogel & George Luta & S. Stanley Young, 2016. "Applications of a Novel Clustering Approach Using Non-Negative Matrix Factorization to Environmental Research in Public Health," IJERPH, MDPI, vol. 13(5), pages 1-14, May.
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