In this paper we study variants of the widely used spectral clustering that partitions a graph into $k$ clusters by (1) embedding the vertices of a graph into a low-dimensional space using the bottom eigenvectors of the Laplacian matrix and (2) grouping the embedded points into $k$ clusters via $k$-means algorithms. We show that, for a wide class of graphs, spectral clustering gives a good approximation of the optimal clustering. While this approach was proposed in the early 1990s and has comprehensive applications, prior to our work similar results were known only for graphs generated from stochastic models. We also give a nearly linear time algorithm for partitioning well-clustered graphs based on computing a matrix exponential and approximate nearest neighbor data structures.
|Number of pages||34|
|Journal||SIAM Journal on Computing|
|Publication status||Published - 30 Mar 2017|
|Event||Conference on Learning Theory 2015 - University Pierre and Marie Curie, Paris, France|
Duration: 3 Jul 2015 → 6 Jul 2015