Fully Adaptive Density-Based Clustering

Based on the work of Hartigan, the clusters of a distribution are often defined to be the connected components of a density level set. Unfortunately, this definition depends on the user-specified level, and in general finding a reasonable level is a difficult task. In addition, the definition is not rigorous for discontinuous densities, since the topological structure of a density level set may be changed by modifying the density on a set of measure zero. In this work, we address these issues by first modifying the notion of density level sets in a way that makes the level sets independent of the actual choice of the density. We then propose a simple algorithm for estimating the smallest level at which the modified level sets have more than one connected component. For this algorithm we provide a finite sample analysis, which is then used to show that the algorithm consistently estimates both the smallest level and the corresponding connected components. We further establish rates of convergence for the two estimation problems, and last but not least, we present a simple strategy for determining the width-parameter of the involved density estimator in a data-depending way. The resulting algorithm turns out to be adaptive, that is, it achieves the optimal rates achievable by our analysis without knowing characteristics of the underlying distribution.