A Note On Spectral Clustering

Let G = (V,E) be an undirected graph, λk the kth smallest eigenvalue of the normalized Laplacian matrix LG of G, and ρ(k) (ρ̂(k)) the smallest value of the maximal conductance over all k disjoint subsets Z1, . . . , Zk (that form a partition) of V . Oveis Gharan and Trevisan [3] proved the existence of a k-way partition (P1, . . . , Pk) of V with ρ̂(k) 6 kρ(k). The k-way (approximate) partitioning problem asks to partition a graph into k clusters such that the conductance of each cluster is (approximately) bounded by ρ̂(k). Peng et al. [4] gave the first rigorous analysis of approximation algorithms for the k-way partitioning problem that are based on clustering suitably normalized eigenvectors of LG with the help of an approximate k-means algorithm. Their analysis relies on the following gap assumption: Υ , λk+1 ρ̂(k) > Ω(k). We strengthen the analysis in two directions. First, we improve the approximation guarantee by a factor of Θ(k) and second we require only a weaker gap assumption: Ψ , λk+1 ρ̂avr(k) > Ω(k), (1) where ρ̂avr(k) is the minimal average conductance over all k-way partitions achieving ρ̂(k). Furthermore, for graphs G that satisfy the gap assumption (1) with k = w(1), our improved analysis gives an algorithm running in time O(nk) that on input a suitable spectral embedding of V outputs with constant probability a k-way partition of V with identical approximation guarantees as in [4]. This speeds up the algorithm in [4] by a O(2)-factor. This work has been funded by the Cluster of Excellence “Multimodal Computing and Interaction” within the Excellence Initiative of the German Federal Government.