2,234 research outputs found
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.Comment: 22 pages, 10 figure
Fast Approximate Spectral Clustering for Dynamic Networks
Spectral clustering is a widely studied problem, yet its complexity is
prohibitive for dynamic graphs of even modest size. We claim that it is
possible to reuse information of past cluster assignments to expedite
computation. Our approach builds on a recent idea of sidestepping the main
bottleneck of spectral clustering, i.e., computing the graph eigenvectors, by
using fast Chebyshev graph filtering of random signals. We show that the
proposed algorithm achieves clustering assignments with quality approximating
that of spectral clustering and that it can yield significant complexity
benefits when the graph dynamics are appropriately bounded
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