Spectrally approximating large graphs with smaller graphs

How does coarsening affect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to depend on standard graph-theoretic properties, such as the degree and eigenvalue distributions, as well as on the ratio between the coarsened and actual graph sizes. Our results carry implications for learning methods that utilize coarsening. For the particular case of spectral clustering, they imply that coarse eigenvectors can be used to derive good quality assignments even without refinement---this phenomenon was previously observed, but lacked formal justification.


Published in:
Proceedings of International Conference in Machine Learning 2018
Presented at:
International Conference in Machine Learning (ICML), Stockholmsmässan, Sweden, July 10-15, 2018
Year:
Jul 10 2018
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 Record created 2018-05-11, last modified 2018-09-13

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