A Novel Data Clustering Method Based on Smooth Non-negative Matrix Factorization

  • Chengcai LengEmail author
  • Hai Zhang
  • Guorong Cai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11010)


Non-negative matrix factorization (NMF) is a very popular dimensionality reduction method that has been widely used in computer vision and data clustering. However, NMF does not consider the intrinsic geometric information of a data set and also does not produce smooth and stable solutions. To resolve these problems, we propose a Graph regularized Lp Smooth Non-negative Matrix Factorization (GSNMF) method by incorporating graph regularization with Lp smooth constraint. The graph regularization can discover the hidden semantics and simultaneously respect the intrinsic geometric structure information of a data set. The Lp smooth constraint can combine the merits of isotropic (L2-norm) and anisotropic (L1-norm) diffusion smoothing, and produce a smooth and more accurate solution to the optimization problem. Experimental results on some data sets demonstrate that the proposed method outperforms related state-of-the-art NMF methods.


Graph regularization Smooth Non-negative Matrix Factorization (SNMF) Data clustering 



This work is supported by the National Natural Science Foundation of China under Grant Nos. 61702251, 61363049, 11571011, 61501286, the State Scholarship Fund of China Scholarship Council (CSC) under Grant No. 201708360040, the Natural Science Foundation of Jiangxi Province under Grant No. 20161BAB212033, the Natural Science Basic Research Plan in Shaanxi Province of China under Program No. 2018JM6030, the Key Research and Development Program in Shaanxi Province of China under Grant No. 2018GY-008, the Doctor Scientific Research Starting Foundation of Northwest University under Grant No. 338050050 and Youth Academic Talent Support Program of Northwest University.


  1. 1.
    Lu, G.F., Wang, Y., Zou, J.: Low-rank matrix factorization with adaptive graph regularizer. IEEE Trans. Image Process. 25(5), 2196–2205 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Shang, F.H., Jiao, L.C., Wang, F.: Graph dual regularization non-negative matrix factorization for co-clustering. Pattern Recognit. 45(6), 2237–2250 (2012)CrossRefGoogle Scholar
  3. 3.
    Wang, D., Gao, X.B., Wang, X.M.: Semi-supervised nonnegative matrix factorization via constraint propagation. IEEE Trans. Cybern. 46(1), 233–244 (2016)CrossRefGoogle Scholar
  4. 4.
    He, W., Zhang, H.Y., Zhang, L.P.: Sparsity-regularized robust non-negative matrix factorization for hyperspectral unmixing. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 9(9), 4267–4279 (2016)CrossRefGoogle Scholar
  5. 5.
    Fan, F., Ma, Y., Li, C., Mei, X.G., Huang, J., Ma, J.Y.: Hyperspectral image denoising with superpixel segmentation and low-rank representation. Inf. Sci. 397, 48–68 (2017)CrossRefGoogle Scholar
  6. 6.
    Li, Z.C., Liu, J., Lu, H.Q.: Structure preserving non-negative matrix factorization for dimensionality reduction. Comput. Vis. Image Underst. 117(9), 1175–1189 (2013)CrossRefGoogle Scholar
  7. 7.
    Tenenbaum, J.B., Silva, V.D., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)CrossRefGoogle Scholar
  8. 8.
    Roweis, S., Saul, L.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)CrossRefGoogle Scholar
  9. 9.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Advances in Neural Information Processing Systems, Vancouver, BC, Canada, vol. 14, pp. 585–591 (2001)Google Scholar
  10. 10.
    Cai, D., He, X. F., Han, J. W.: Isometric projection. In: Proceeding of the National Conference on Artificial Intelligence, Vancouver, BC, Canada, vol. 1, pp. 528–533 (2007)Google Scholar
  11. 11.
    Najafi, A., Joudaki, A., Fatemizadeh, E.: Nonlinear dimensionality reduction via path-based isometric mapping. IEEE Trans. Pattern Anal. Mach. Intell. 38(7), 1452–1464 (2016)CrossRefGoogle Scholar
  12. 12.
    Wang, Y.X., Zhang, Y.J.: Nonnegative matrix factorization: a comprehensive review. IEEE Trans. Knowl. Data Eng. 25(6), 1336–1353 (2013)CrossRefGoogle Scholar
  13. 13.
    Lee, D.D., Seung, H.S.: Learning the parts of objects by non-negative matrix factorization. Nature 401(6755), 788–791 (1999)CrossRefGoogle Scholar
  14. 14.
    Zeng, K., Yu, J., Li, C.H., You, J., Jin, T.S.: Image clustering by hyper-graph regularized non-negative matrix factorization. Neurocomputing 138, 209–217 (2014)CrossRefGoogle Scholar
  15. 15.
    Li, G.P., Zhang, X.Y., Zheng, S.Y., Li, D.Y.: Semi-supervised convex nonnegative matrix factorizations with graph regularized for image representation. Neurocomputing 237, 1–11 (2017)CrossRefGoogle Scholar
  16. 16.
    Leng, C.C., Cai, G.R., Yu, D.D., Wang, Z.Y.: Adaptive total-variation for non-negative matrix factorization on manifold. Pattern Recognit. Lett. 98, 68–74 (2017)CrossRefGoogle Scholar
  17. 17.
    Cai, D., He, X.F., Han, J.W., Huang, T.S.: Graph regularized nonnegative matrix factorization for data representation. IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1548–1560 (2011)CrossRefGoogle Scholar
  18. 18.
    Liu, H.F., Wu, Z.H., Li, X.L., Cai, D., Huang, T.S.: Constrained nonnegative matrix factorization for image representation. IEEE Trans. Pattern Anal. Mach. Intell. 34(7), 1299–1311 (2012)CrossRefGoogle Scholar
  19. 19.
    Lee, D.D., Seung, H.S.: Algorithms for non-negative matrix factorization. Adv. Neural. Inf. Process. Syst. 13, 556–562 (2000)Google Scholar
  20. 20.
    Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  21. 21.
    Xu, W., Liu, X., Gong, Y.H.: Document clustering based on non-negative matrix factorization. In: Proceedings of the Annual International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR 2003), Toronto, Canada, pp. 267–273 (2003)Google Scholar
  22. 22.
    Cai, D., He, X.F., Han, J.W.: Document clustering using locality preserving indexing. IEEE Trans. Knowl. Data Eng. 17(12), 1624–1637 (2005)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of MathematicsNorthwest UniversityXi’anChina
  2. 2.Department of Computing ScienceUniversity of AlbertaEdmontonCanada
  3. 3.School of Mathematics and Information SciencesNanchang Hangkong UniversityNanchangChina
  4. 4.Faculty of Information Technology, State Key Laboratory of Quality Research in Chinese MedicinesMacau University of Science and TechnologyMacauPeople’s Republic of China
  5. 5.College of Computer EngineeringJimei UniversityXiamenChina

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