Spectral Clustering from a Geometric Viewpoint

Spectral methods have received attention as powerful theoretical and practical approaches to a number of machine learning problems. The methods are based on the solution of the eigenproblem of a similarity matrix formed from distance kernels. In this article we discuss three problems that are endemic in current implementations of spectral clustering: (1) the need to use another clustering method such as k-means as a final step, (2) the determination of the number of clusters, and (3) the failure of spectral clustering on multi-scale examples. These three problems are manifest even when the clusters are separated connected components. We advocate the use of the LU -factorization to solve (1), and treat clustering as a geometry problem to attack the second two problems. Specifically, the ideas of persistence and reconstruction of the Laplace-Beltrami operator are introduced as solutions to (2) and (3). We show that these suggested solutions are robust in a series of illustrative examples.