Generalised blurring mean-shift algorithms for nonparametric clustering

Gaussian blurring mean-shift (GBMS) is a nonparametric clustering algorithm, having a single bandwidth parameter that controls the number of clusters. The algorithm iteratively shrinks the data set under the application of a mean-shift update, stops in just a few iterations and yields excellent clusterings. We propose several families of generalised GBMS (GGBMS) algorithms based on explicit, implicit and exponential updates, and depending on a step-size parameter. We give conditions on the step size for the convergence of these algorithms and show that the convergence rate for Gaussian clusters ranges from sublinear to linear, cubic and even higher order depending on the update and step size. We show that the algorithms are related to spectral clustering if using a random-walk matrix with modified eigenvalues and updated after each iteration, and show the relation with methods developed for surface smoothing in the computer graphics literature. Detailed experiments in toy problems and image segmentation show that, while all the GGBMS algorithms can achieve essentially the same result (for appropriate settings of the bandwidth and step size), they significantly differ in runtime, with slightly over-relaxed explicit updates being fastest in practice.