Abstract: Spectral clustering is a popular and effective algorithm designed to find $k$ clusters in a graph $G$.In the classical spectral clustering algorithm, the vertices of $G$ are embedded into $\mathbb{R}^k$ using $k$ eigenvectors of the graph Laplacian matrix.However, computing this embedding is computationally expensive and dominates the running time of the algorithm.In this paper, we present a simple spectral clustering algorithm based on a vertex embedding with $O(\log(k))$ vectors computed by the power method.The vertex embedding is computed in nearly-linear time with respect to the size of the graph, andthe algorithm provably recovers the ground truth clusters under natural assumptions on the input graph.We evaluate the new algorithm on several synthetic and real-world datasets, finding that it is significantly faster than alternative clustering algorithms,while producing results with approximately the same clustering accuracy.