Authors : Dr. Mitat Uysal

Volume/Issue : Volume 10 - 2025, Issue 6 - June

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Google Scholar : https://tinyurl.com/4mnhx2cv

DOI : https://doi.org/10.38124/ijisrt/25jun139

Note : A published paper may take 4-5 working days from the publication date to appear in PlumX Metrics, Semantic Scholar, and ResearchGate.

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Abstract : Clustering is a fundamental task in machine learning and data analysis, enabling the discovery of inherent patterns within data. Nonnegative Matrix Factorization (NMF) has emerged as a powerful tool for clustering due to its ability to learn parts-based, interpretable representations. This article explores the theoretical foundations of clustering and NMF, their synergy, algorithmic formulations, and practical implementations. Experimental validation on synthetic data demonstrates the effectiveness of NMF-based clustering without using libraries such as sklearn or tensorflow.

Keywords : NMF,Clustering,Machine Learning,Objective Function,Frobenius Norm.

References :

1. Lee, D. D., & Seung, H. S. (1999). Learning the parts of objects by non-negative matrix factorization. Nature, 401(6755), 788–791.
2. Xu, W., Liu, X., & Gong, Y. (2003). Document clustering based on non-negative matrix factorization. In Proceedings of the 26th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval (pp. 267–273).
3. Kim, H., & Park, H. (2008). Nonnegative matrix factorization based on alternating nonnegativity constrained least squares and active set method. IEEE Transactions on Pattern Analysis and Machine Intelligence, 30(9), 1600–1615.
4. Ding, C., Li, T., & Jordan, M. I. (2010). Convex and semi-nonnegative matrix factorizations. SIAM Journal on Matrix Analysis and Applications, 30(2), 852–874.
5. Berry, M. W., Browne, M., Langville, A. N., Pauca, V. P., & Plemmons, R. J. (2007). Algorithms and applications for approximate nonnegative matrix factorization. Computational Statistics & Data Analysis, 52(1), 155–173.
6. Brunet, J. P., Tamayo, P., Golub, T. R., & Mesirov, J. P. (2004). Metagenes and molecular pattern discovery using matrix factorization. Proceedings of the National Academy of Sciences, 101(12), 4164–4169.
7. Gillis, N. (2014). The why and how of nonnegative matrix factorization. SIAM Review, 57(1), 167–197.
8. Cichocki, A., Zdunek, R., & Amari, S. I. (2006). Hierarchical ALS algorithms for nonnegative matrix and 3D tensor factorization. Neurocomputing, 69(7–9), 544–555.
9. Hoyer, P. O. (2004). Non-negative matrix factorization with sparseness constraints. Journal of Machine Learning Research, 5, 1457–1469.
10. Sra, S., Nowozin, S., & Wright, S. J. (2012). Optimization for Machine Learning. MIT Press.
11. Lin, C. J. (2007). Projected gradient methods for nonnegative matrix factorization. Journal of Machine Learning Research, 8, 595–614.
12. Liu, Y., et al. (2013). A novel initialization strategy for NMF based on clustering and SVD. Neurocomputing, 106, 1–6.
13. Zafeiriou, S., Tefas, A., & Pitas, I. (2006). Learning discriminant nonnegative features for face recognition. IEEE Transactions on Image Processing, 15(1), 269–282.
14. Yang, Z., et al. (2010). Non-negative matrix factorization framework for face recognition. Pattern Recognition, 43(4), 1515–1527.
15. Pauca, V. P., et al. (2004). Nonnegative matrix factorization for spectral data analysis. Linear Algebra and Its Applications, 391, 141–157.
16. Fevotte, C., & Idier, J. (2011). Algorithms for nonnegative matrix factorization with the β-divergence. IEEE Transactions on Signal Processing, 59(7), 2925–2934.
17. Wang, Y. X., & Zhang, Y. J. (2013). Nonnegative matrix factorization: A comprehensive review. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(3), 554–566.
18. Yoo, C. D., & Choi, S. (2009). Semi-supervised nonnegative matrix factorization. Neurocomputing, 72(10–12), 2466–2476.
19. Lin, J., & Jeon, G. (2014). A novel initialization for NMF based on the density and orthogonality of basis vectors. Pattern Recognition, 47(3), 1201–1211.
20. Cai, D., et al. (2008). Non-negative matrix factorization on manifold. IEEE Transactions on Knowledge and Data Engineering, 20(12), 1724–1739.
21. Banerjee, A., et al. (2005). Clustering with Bregman divergences. Journal of Machine Learning Research, 6, 1705–1749.
22. Ding, C., & He, X. (2005). On the equivalence of nonnegative matrix factorization and spectral clustering. In Proceedings of the 22nd International Conference on Machine Learning (pp. 225–232).
23. Li, T., et al. (2007). Solving consensus and semi-supervised clustering problems using nonnegative matrix factorization. In Proceedings of the Seventh IEEE International Conference on Data Mining (pp. 577–582).
24. Zhang, Z., et al. (2012). Graph-regularized nonnegative matrix factorization for data representation. Pattern Recognition, 45(3), 1061–1070.
25. Liu, J., & Wu, J. (2006). A hybrid clustering method based on K-means and particle swarm optimization. Pattern Recognition Letters, 27(6), 597–603.