Incremental Method for Spectral Clustering of Increasing Orders

The smallest eigenvalues and the associated eigenvectors (i.e.,eigenpairs) of a graph Laplacian matrix have been widelyused for spectral clustering and community detection. How-ever, in real-life applications the number of clusters or com-munities (say, K) is generally unknown a-priori. Conse-quently, the majority of the existing methods either chooseK heuristically or they repeat the clustering method withdifferent choices of K and accept the best clustering result.The first option, more often, yields suboptimal result, whilethe second option is computationally expensive. In this work,we propose an incremental method for constructing the eigen-spectrum of the graph Laplacian matrix. This method lever-ages the eigenstructure of graph Laplacian matrix to obtaintheK-th eigenpairs of the Laplacian matrix given a collectionof all the K − 1 smallest eigenpairs. Our proposed methodadapts the Laplacian matrix such that the batch eigenvaluedecomposition problem transforms into an efficient sequentialleading eigenpair computation problem. As a practical appli-cation, we consider user-guided spectral clustering. Specif-ically, we demonstrate that users can utilize the proposedincremental method for effective eigenpair computation anddetermining the desired number of clusters based on multipleclustering metrics.