Journal of Global Optimization

, Volume 62, Issue 3, pp 545–574 | Cite as

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

  • Da Kuang
  • Sangwoon Yun
  • Haesun ParkEmail author


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.


Symmetric nonnegative matrix factorization Low-rank approximation Graph clustering Spectral clustering 



The work of the first and third authors was supported in part by the National Science Foundation (NSF) Grants CCF-0808863 and the Defense Advanced Research Projects Agency (DARPA) XDATA program Grant FA8750-12-2-0309. The work of the second author was supported by the TJ Park Science Fellowship of POSCO TJ Park Foundation. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF, the DARPA, or the NRF.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Computational Science and EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of Mathematics EducationSungkyunkwan UniversitySeoulKorea
  3. 3.Korea Institute for Advanced StudySeoulKorea

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