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. We conclude by thoroughly analyzing density-connectivity and its properties -- a task that has been elusive thus far in the literature due to the lack of formal tools. Our code recreates every experiment below: https://github.com/Andrew-Draganov/dc_dist

Spectral clustering is one of the most advantageous clustering approaches. However, standard Spectral Clustering is sensitive to noisy input data and has a high runtime complexity. Tackling one of these problems often exacerbates the other. As real-world datasets are often large and compromised by noise, we need to improve both robustness and runtime at once. Thus, we propose Spectral Clustering - Accelerated and Robust (SCAR), an accelerated, robustified spectral clustering method. In an iterative approach, we achieve robustness by separating the data into two latent components: cleansed and noisy data. We accelerate the eigendecomposition - the most time-consuming step - based on the Nyström method. We compare SCAR to related recent state-of-the-art algorithms in extensive experiments. SCAR surpasses its competitors in terms of speed and clustering quality on highly noisy data.