Spectral clustering based on the graph p-Laplacian

Authors:
Thomas Bühler;Matthias Hein
Affiliations:
Saarland University, Saarbrücken, Germany;Saarland University, Saarbrücken, Germany
Venue:
ICML '09 Proceedings of the 26th Annual International Conference on Machine Learning
Year:
2009

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Abstract

We present a generalized version of spectral clustering using the graph p-Laplacian, a nonlinear generalization of the standard graph Laplacian. We show that the second eigenvector of the graph p-Laplacian interpolates between a relaxation of the normalized and the Cheeger cut. Moreover, we prove that in the limit as p → 1 the cut found by thresholding the second eigenvector of the graph p-Laplacian converges to the optimal Cheeger cut. Furthermore, we provide an efficient numerical scheme to compute the second eigenvector of the graph p-Laplacian. The experiments show that the clustering found by p-spectral clustering is at least as good as normal spectral clustering, but often leads to significantly better results.