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Spectra of the Laplacian (Matrix)

  1. Mar 6, 2013 #1
    Hi all

    I was reading this paper about spectral clustering:
    Ng et all. (Nips 2001) http://ai.stanford.edu/~ang/papers/nips01-spectral.pdf

    Short question:
    What is the connection between the eigenvalues of a Laplacian?

    Long question:

    I got stuck on why the process makes sense.
    Substantially the idea is that some data cannot be simply represented by distribution, but rather as vertices of a graph, where something simpler than graph-cut can be employed.

    The "something simpler" is to construct a Laplacian matrix from the graph and then use
    the eigenvectors of this matrix as new data and then perform a traditional clustering
    algorithm. In this way the data are more "manegeable".

    The part where I got stuck is why the eigenvectors are more manegeable?
    In a survey about methods for spectral clustering (von Luxburg 2007), when they talk about this paper they say something like: "hence the graph laplacian measures the variation of
    a function along a graph: the eingenvalue is low if close points have close function value".
    this anyway does not explain why a classification using eigenvectors is easier?

    In other words, does the eigenvalues capture the local "shape variation"?
    This is what I gather from
    and reading other stuff about the eigenvalues of the laplace-beltrami
  2. jcsd
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