TR2017-001

Signed Laplacian for Spectral Clustering Revisited

    •  Knyazev, A., "Signed Laplacian for Spectral Clustering Revisited", arXiv, January 2017.
      BibTeX arXiv
      • @article{Knyazev2017jan,
      • author = {Knyazev, Andrew},
      • title = {Signed Laplacian for Spectral Clustering Revisited},
      • journal = {arXiv},
      • year = 2017,
      • month = jan,
      • url = {https://arxiv.org/abs/1701.01394}
      • }
  • Research Area:

    Signal Processing

Abstract:

Classical spectral clustering is based on a spectral decomposition of a graph Laplacian, obtained from a graph adjacency matrix representing positive graph edge weights describing similarities of graph vertices. In signed graphs, the graph edge weights can be negative to describe disparities of graph vertices, for example, negative correlations in the data. Negative weights lead to possible negative spectrum of the standard graph Laplacian, which is cured by defining a signed Laplacian. We revisit comparing the standard and signed Laplacians and argue that the former is more natural than the latter, also showing that the negative spectrum is actually beneficial, for spectral clustering of signed graphs.

 

  • Related News & Events

    •  NEWS    Andrew Knyazev (MERL) presents at the Schlumberger-Tufts U. Computational and Applied Math Seminar
      Date: April 10, 2018
      Research Area: Machine Learning
      Brief
      • Andrew Knyazev, Distinguished Research Scientist of MERL, has accepted an invitation to speak about his work on Big Data and spectral graph partitioning at the Schlumberger-Tufts U. Computational and Applied Math Seminar. A primary focus of this seminar series is on mathematical and computational aspects of remote sensing. A partial list of the topics of interest includes: numerical solution of large scale PDEs (a.k.a. forward problems); theory and numerical methods of inverse and ill-posed problems; imaging; related problems in numerical linear algebra, approximation theory, optimization and model reduction. The seminar meets on average once a month, the location alternates between Schlumberger's office in Cambridge, MA and the Tufts Medford Campus.

        Abstract: Data clustering via spectral graph partitioning requires constructing the graph Laplacian and solving the corresponding eigenvalue problem. We consider and motivate using negative edge weights in the graph Laplacian. Preconditioned iterative solvers for the Laplacian eigenvalue problem are discussed and preliminary numerical results are presented.
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