Incremental Eigenpair Computation for Graph Laplacian Matrices: Theory and Applications

TL;DR

This paper introduces an incremental eigenpair computation method for graph Laplacian matrices, enabling efficient spectral clustering without predefining the number of clusters, thus improving adaptability and computational efficiency.

Contribution

The paper presents a novel incremental approach to compute eigenpairs of graph Laplacians, facilitating adaptive spectral clustering and reducing computational costs compared to existing methods.

Findings

Efficient sequential eigenpair computation for graph Laplacians.

Enhanced spectral clustering with unknown number of clusters.

Practical user-guided clustering applications.

Abstract

The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs) of a graph Laplacian matrix have been widely used in spectral clustering and community detection. However, in real-life applications the number of clusters or communities (say, $K$) is generally unknown a-priori. Consequently, the majority of the existing methods either choose $K$ heuristically or they repeat the clustering method with different choices of $K$ and accept the best clustering result. The first option, more often, yields suboptimal result, while the second option is computationally expensive. In this work, we propose an incremental method for constructing the eigenspectrum of the graph Laplacian matrix. This method leverages the eigenstructure of graph Laplacian matrix to obtain the $K$-th smallest eigenpair of the Laplacian matrix given a collection of all previously computed $K-1$ smallest eigenpairs. Our proposed method adapts the Laplacian matrix such that the batch eigenvalue decomposition problem transforms into an efficient sequential leading eigenpair computation problem. As a practical application, we consider user-guided spectral clustering. Specifically, we demonstrate that users can utilize the proposed incremental method for effective eigenpair computation and for determining the desired number of clusters based on multiple clustering metrics.