Fast and Simple Spectral Clustering in Theory and Practice

TL;DR

This paper introduces a fast spectral clustering algorithm that uses a logarithmic number of eigenvector computations, enabling nearly-linear time clustering with accuracy comparable to traditional methods.

Contribution

The paper proposes a simple, efficient spectral clustering algorithm based on a vertex embedding with O(log(k)) vectors, significantly reducing computational complexity.

Findings

Algorithm runs in nearly-linear time.

Achieves similar clustering accuracy to traditional methods.

Effective on both synthetic and real-world datasets.

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, and the 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.