A Performance Guarantee for Spectral Clustering
This work offers a theoretical foundation for spectral clustering, addressing a central question in graph partitioning for researchers in machine learning and optimization.
The paper provides a theoretical performance guarantee for spectral clustering by identifying conditions under which it finds the global solution to the minimum ratio cut problem, based on intra- and inter-cluster connectivities and perturbation bounds for graph Laplacian eigenvalues.
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 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 inter-cluster 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 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.