Partitioning Well-Clustered Graphs: Spectral Clustering Works!

TL;DR

This paper demonstrates that spectral clustering effectively approximates optimal partitions in well-clustered graphs and introduces a nearly-linear time algorithm for such clustering.

Contribution

The paper extends spectral clustering theory to a broad class of graphs and provides a fast algorithm leveraging matrix exponentials and approximate nearest neighbors.

Findings

Spectral clustering approximates optimal clustering in well-clustered graphs.

A nearly-linear time algorithm for spectral clustering is proposed.

Theoretical guarantees are established beyond stochastic models.

Abstract

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.