Se p 20 15 A Note On Spectral Clustering ∗ Pavel Kolev Kurt

Let G = (V,E) be an undirected graph, λk the kth smallest eigenvalue of the normalized Laplacian matrix of G, and ρ(k) the smallest value of the maximal conductance over all k-way partitions S1, . . . , Sk of V . Peng et al. [4] gave the first rigorous analysis of k-clustering algorithms that use spectral embedding and k-means clustering algorithms to partition the vertices of a graph G into k disjoint subsets. Their analysis builds upon a gap parameter Υ = ρ(k)/λk+1 that was introduced by Oveis Gharan and Trevisan [2]. In their analysis Peng et al. [4] assume a gap assumption Υ > Ω(APR ·k3), where APR > 1 is the approximation ratio of a k-means clustering algorithm. We exhibit an error in one of their Lemmas and provide a correction. With the correction the proof by Peng et al. [4] requires a stronger gap assumption Υ > Ω(APR · k). Our main contribution is to improve the analysis in [4] by an O(k) factor. We demonstrate that a gap assumption Ψ > Ω(APR · k) suffices, where Ψ = ρavr(k)/λk+1 and ρavr(k) is the value of the average conductance of a partition S1, . . . , Sk of V that yields ρ(k). This work has been funded by the Cluster of Excellence “Multimodal Computing and Interaction” within the Excellence Initiative of the German Federal Government.