Connecting the Dots–Density-Connectivity Distance unifies DBSCAN, k-Center and Spectral Clustering

Abstract

Despite the popularity of density-based clustering, its procedural definition makes it difficult to analyze compared to clustering methods that minimize a loss function. In this paper, we reformulate DBSCAN through a clean objective function by introducing the density-connectivity distance (dc-dist), which captures the essence of density-based clusters by endowing the minimax distance with the concept of density. This novel ultrametric allows us to show that DBSCAN, k-center, and spectral clustering are equivalent in the space given by the dc-dist, despite these algorithms being perceived as fundamentally different in their respective literatures. We also verify that finding the pairwise dc-dists gives DBSCAN clusterings across all epsilon-values, simplifying the problem of parameterizing density-based clustering.