Stable and consistent density-based clustering

We present a multiscale, consistent approach to density-based clustering that satisfies stability theorems -- in both the input data and in the parameters -- which hold without distributional assumptions. The stability in the input data is with respect to the Gromov--Hausdorff--Prokhorov distance on metric probability spaces and interleaving distances between (multi-parameter) hierarchical clusterings we introduce. We prove stability results for standard simplification procedures for hierarchical clusterings, which can be combined with our approach to yield a stable flat clustering algorithm. We illustrate the stability of the approach with computational examples. Our framework is based on the concepts of persistence and interleaving distance from Topological Data Analysis.