Approximate Spectral Clustering: Efficiency and Guarantees

Abstract: Approximate Spectral Clustering (ASC) is a popular and successful heuristic for partitioning the nodes of a graph $G$ into clusters for which the ratio of outside connections compared to the volume (sum of degrees) is small. ASC consists of the following two subroutines: i) compute an approximate Spectral Embedding via the Power method; and ii) partition the resulting vector set with an approximate $k$-means clustering algorithm. The resulting $k$-means partition naturally induces a $k$-way node partition of $G$. We give a comprehensive analysis of ASC building on the work of Peng et al.~(SICOMP'17), Boutsidis et al.~(ICML'15) and Ostrovsky et al.~(JACM'13). We show that ASC i) runs efficiently, and ii) yields a good approximation of an optimal $k$-way node partition of $G$. Moreover, we strengthen the quality guarantees of a structural result of Peng et al. by a factor of $k$, and simultaneously weaken the eigenvalue gap assumption. Further, we show that ASC finds a $k$-way node partition of $G$ with the strengthened quality guarantees.