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SymNMF: nonnegative low-rank approximation of a similarity matrix for graph clustering

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  • Da Kuang
  • Sangwoon Yun
  • Haesun Park

Abstract

Nonnegative matrix factorization (NMF) provides a lower rank approximation of a matrix by a product of two nonnegative factors. NMF has been shown to produce clustering results that are often superior to those by other methods such as K-means. In this paper, we provide further interpretation of NMF as a clustering method and study an extended formulation for graph clustering called Symmetric NMF (SymNMF). In contrast to NMF that takes a data matrix as an input, SymNMF takes a nonnegative similarity matrix as an input, and a symmetric nonnegative lower rank approximation is computed. We show that SymNMF is related to spectral clustering, justify SymNMF as a general graph clustering method, and discuss the strengths and shortcomings of SymNMF and spectral clustering. We propose two optimization algorithms for SymNMF and discuss their convergence properties and computational efficiencies. Our experiments on document clustering, image clustering, and image segmentation support SymNMF as a graph clustering method that captures latent linear and nonlinear relationships in the data. Copyright Springer Science+Business Media New York 2015

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  • Da Kuang & Sangwoon Yun & Haesun Park, 2015. "SymNMF: nonnegative low-rank approximation of a similarity matrix for graph clustering," Journal of Global Optimization, Springer, vol. 62(3), pages 545-574, July.
    • Handle: RePEc:spr:jglopt:v:62:y:2015:i:3:p:545-574
    DOI: 10.1007/s10898-014-0247-2
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    References listed on IDEAS

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    Cited by:

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    2. Saeedmanesh, Mohammadreza & Geroliminis, Nikolas, 2016. "Clustering of heterogeneous networks with directional flows based on “Snake” similarities," Transportation Research Part B: Methodological, Elsevier, vol. 91(C), pages 250-269.
    3. Srinivas Eswar & Ramakrishnan Kannan & Richard Vuduc & Haesun Park, 2021. "ORCA: Outlier detection and Robust Clustering for Attributed graphs," Journal of Global Optimization, Springer, vol. 81(4), pages 967-989, December.
    4. Radu-Alexandru Dragomir & Alexandre d’Aspremont & Jérôme Bolte, 2021. "Quartic First-Order Methods for Low-Rank Minimization," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 341-363, May.
    5. Rundong Du & Barry Drake & Haesun Park, 2019. "Hybrid clustering based on content and connection structure using joint nonnegative matrix factorization," Journal of Global Optimization, Springer, vol. 74(4), pages 861-877, August.
    6. Arnaud Vandaele & François Glineur & Nicolas Gillis, 2018. "Algorithms for positive semidefinite factorization," Computational Optimization and Applications, Springer, vol. 71(1), pages 193-219, September.
    7. Rundong Du & Da Kuang & Barry Drake & Haesun Park, 2017. "DC-NMF: nonnegative matrix factorization based on divide-and-conquer for fast clustering and topic modeling," Journal of Global Optimization, Springer, vol. 68(4), pages 777-798, August.

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