Non-negative Matrix Factoriaztion on Manifold (Graph)


Matrix factorization techniques have been frequently applied in information retrieval, computer vision and pattern recognition. Among them, Non-negative Matrix Factorization (NMF) have received considerable attentions due to its psychological and physiological interpretation of naturally occurring data whose representation may be parts-based in human brain. On the other hand, from geometric perspective the data is usually sampled from a low dimensional manifold embedded in high dimensional ambient space. One hopes then to find a compact representation which uncovers the hidden semantics and simultaneously respects the intrinsic geometric structure. In this paper, we propose a novel algorithm, called {\em Graph Regularized Non-negative Matrix Factorization} (GNMF), for this purpose. In GNMF, an affinity graph is constructed to encode the geometrical information and we seek a matrix factorization which respects the graph structure. Our empirical study shows the encouraging results of the proposed algorithm in comparisons to the state-of-the-art algorithms on on real world problems.


  • GNMF: Graph-regularized NMF (F-norm formulation). GNMF_Multi is required.
  • GNMF_KL: Graph-regularized NMF (Divergence formulation) GNMF_KL_Multi is required.
  • LCCF: Locally Consistant Concept Factorization. LCCF_Multi is required.

    Data sets

    If you find these algoirthms useful, we appreciate it very much if you can cite our following works:


    1. Deng Cai, Xiaofei He, Jiawei Han, Thomas Huang, "Graph Regularized Non-negative Matrix Factorization for Data Representation", IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 33, No. 8, pp. 1548-1560, 2011.
      Bibtex source | PDF

    2. Deng Cai, Xiaofei He, Xuanhui Wang, Hujun Bao and Jiawei Han, "Locality Preserving Nonnegative Matrix Factorization", Proc. 2009 Int. Joint Conference on Artificial Intelligence (IJCAI'09), Pasadena, CA, July 2009.
      Bibtex source

    3. Deng Cai, Xiaofei He, Xiaoyun Wu and Jiawei Han, "Non-negative Matrix Factorization on Manifold", Proc. 2008 Int. Conf. on Data Mining (ICDM'08), Pisa, Italy, Dec. 2008.
      Bibtex source

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