Clustering via Hilbert space

Abstract

We discuss novel clustering methods that are based on mapping data points to a Hilbert space by means of a Gaussian kernel. The first method, support vector clustering (SVC), searches for the smallest sphere enclosing data images in Hilbert space. The second, quantum clustering (QC), searches for the minima of a potential function defined in such a Hilbert space. In SVC, the minimal sphere, when mapped back to data space, separates into several components, each enclosing a separate cluster of points. A soft margin constant helps in coping with outliers and overlapping clusters. In QC, minima of the potential define cluster centers, and equipotential surfaces are used to construct the clusters. In both methods, the width of the Gaussian kernel controls the scale at which the data are probed for cluster formations. We demonstrate the performance of the algorithms on several data sets.