Partitioning Well-Clustered Graphs: Spectral Clustering Works!

author: Luca Zanetti, Department of Computer Science, University of Bristol
published: Aug. 20, 2015,   recorded: July 2015,   views: 1622


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In this work we study the widely used spectral clustering algorithms, i.e. partition a graph into k clusters via (1) embedding the vertices of a graph into a low-dimensional space using the bottom eigenvectors of the Laplacian matrix, and (2) partitioning embedded points via k-means algorithms. We show that, for a wide class of well-clustered graphs, spectral clustering algorithms can give a good approximation of the optimal clustering. To the best of our knowledge, it is the first theoretical analysis of spectral clustering algorithms for a wide family of graphs, even though such approach was proposed in the early 1990s and has comprehensive applications. We also give a nearly-linear time algorithm for partitioning well-clustered graphs, which is based on heat kernel embeddings and approximate nearest neighbor data structures.

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