A Performance Guarantee for Spectral Clustering

Abstract

The two-step spectral clustering method, which consists of the Laplacian eigenmap and a rounding step, is a widely used method for graph partitioning. It can be seen as a natural relaxation to the NP-hard minimum ratio cut problem. In this paper, we study the following central question: When is spectral clustering able to find the global solution to the minimum ratio cut problem? First, we provide a condition that naturally depends on the intra- and intercluster connectivities of a given partition under which we may certify that this partition is the solution to the minimum ratio cut problem. Then, we develop a deterministic two-to-infinity norm perturbation bound for the invariant subspace of the graph Laplacian that corresponds to the $k$ smallest eigenvalues. Finally, by combining these two results we give a condition under which spectral clustering is guaranteed to output the global solution to the minimum ratio cut problem, which serves as a performance guarantee for spectral clustering.