A Performance Guarantee for Spectral Clustering

TL;DR

This paper provides a theoretical performance guarantee for spectral clustering, identifying conditions under which it finds the optimal graph partition by analyzing spectral properties and intra/inter-cluster connectivities.

Contribution

It introduces a condition based on intra- and inter-cluster connectivities that certifies when spectral clustering solves the minimum ratio cut problem globally.

Findings

A certifiable condition for optimality of spectral clustering.

A deterministic perturbation bound for the graph Laplacian's eigenvectors.

Spectral clustering guarantees under specific connectivity conditions.

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 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.