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Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics

Author

Listed:
  • Ke-Lin Du

    (Department of Electrical and Computer Engineering, Concordia University, Montreal, QC H3G 1M8, Canada)

  • M. N. S. Swamy

    (Department of Electrical and Computer Engineering, Concordia University, Montreal, QC H3G 1M8, Canada)

  • Zhang-Quan Wang

    (College of Information Science and Technology, Zhejiang Shuren University, Hangzhou 310015, China)

  • Wai Ho Mow

    (Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong SAR, China)

Abstract

Compressed sensing is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Sparse coding represents a signal as a sparse linear combination of atoms, which are elementary signals derived from a predefined dictionary. Compressed sensing, sparse approximation, and dictionary learning are topics similar to sparse coding. Matrix completion is the process of recovering a data matrix from a subset of its entries, and it extends the principles of compressed sensing and sparse approximation. The nonnegative matrix factorization is a low-rank matrix factorization technique for nonnegative data. All of these low-rank matrix factorization techniques are unsupervised learning techniques, and can be used for data analysis tasks, such as dimension reduction, feature extraction, blind source separation, data compression, and knowledge discovery. In this paper, we survey a few emerging matrix factorization techniques that are receiving wide attention in machine learning, signal processing, and statistics. The treated topics are compressed sensing, dictionary learning, sparse representation, matrix completion and matrix recovery, nonnegative matrix factorization, the Nyström method, and CUR matrix decomposition in the machine learning framework. Some related topics, such as matrix factorization using metaheuristics or neurodynamics, are also introduced. A few topics are suggested for future investigation in this article.

Suggested Citation

  • Ke-Lin Du & M. N. S. Swamy & Zhang-Quan Wang & Wai Ho Mow, 2023. "Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics," Mathematics, MDPI, vol. 11(12), pages 1-50, June.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:12:p:2674-:d:1169553
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    References listed on IDEAS

    as
    1. Ke-Lin Du & Chi-Sing Leung & Wai Ho Mow & M. N. S. Swamy, 2022. "Perceptron: Learning, Generalization, Model Selection, Fault Tolerance, and Role in the Deep Learning Era," Mathematics, MDPI, vol. 10(24), pages 1-46, December.
    2. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    3. Scott Deerwester & Susan T. Dumais & George W. Furnas & Thomas K. Landauer & Richard Harshman, 1990. "Indexing by latent semantic analysis," Journal of the American Society for Information Science, Association for Information Science & Technology, vol. 41(6), pages 391-407, September.
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    Cited by:

    1. Ke-Lin Du & Rengong Zhang & Bingchun Jiang & Jie Zeng & Jiabin Lu, 2025. "Foundations and Innovations in Data Fusion and Ensemble Learning for Effective Consensus," Mathematics, MDPI, vol. 13(4), pages 1-49, February.
    2. Ke-Lin Du & Bingchun Jiang & Jiabin Lu & Jingyu Hua & M. N. S. Swamy, 2024. "Exploring Kernel Machines and Support Vector Machines: Principles, Techniques, and Future Directions," Mathematics, MDPI, vol. 12(24), pages 1-58, December.
    3. Zhiyong Zhou & Gui Wang, 2024. "The Capped Separable Difference of Two Norms for Signal Recovery," Mathematics, MDPI, vol. 12(23), pages 1-10, November.
    4. Ke-Lin Du & Rengong Zhang & Bingchun Jiang & Jie Zeng & Jiabin Lu, 2025. "Understanding Machine Learning Principles: Learning, Inference, Generalization, and Computational Learning Theory," Mathematics, MDPI, vol. 13(3), pages 1-56, January.

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