# A Population Background for Nonparametric Density-Based Clustering

arXiv: https://arxiv.org/abs/1408.1381 · HTML: https://mlsanity.com/paper/1408.1381/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Aug 2014.

## Built on by (most cited first, 14 of 14)

- [Moving Up the Cluster Tree with the Gradient Flow](https://mlsanity.com/paper/2109.08362/) (2021)
- [Tight Bounds on the Hardness of Learning Simple Nonparametric Mixtures](https://mlsanity.com/paper/2203.15150/) (2022)
- [Space Partitioning and Regression Mode Seeking via a Mean-Shift-Inspired Algorithm](https://mlsanity.com/paper/2104.10103/) (2021)
- [Skeleton Clustering: Dimension-Free Density-based Clustering](https://mlsanity.com/paper/2104.10770/) (2021)
- [Clustering by Hill-Climbing: Consistency Results](https://mlsanity.com/paper/2202.09023/) (2022)
- [An Asymptotic Equivalence between the Mean-Shift Algorithm and the Cluster Tree](https://mlsanity.com/paper/2111.10298/) (2021)
- [Scalable Posterior Uncertainty for Flexible Density-Based Clustering](https://mlsanity.com/paper/2603.03188/) (2026)
- [Nonparametric clustering for image segmentation](https://mlsanity.com/paper/2101.08345/) (2021)
- [On Integral Theorems and their Statistical Properties](https://mlsanity.com/paper/2107.10947/) (2021)
- [Bump hunting through density curvature features](https://mlsanity.com/paper/2208.00174/) (2022)
- [Bagged k-Distance for Mode-Based Clustering Using the Probability of Localized Level Sets](https://mlsanity.com/paper/2210.09786/) (2022)
- [Bayesian taut splines for estimating the number of modes](https://mlsanity.com/paper/2307.05825/) (2023)
- [Graph Max Shift: A Hill-Climbing Method for Graph Clustering](https://mlsanity.com/paper/2411.18794/) (2024)
- [Stable Density Ridges: Consistency and Convergence of Subspace Constrained Mean Shift](https://mlsanity.com/paper/2608.05112/) (2026)

## Abstract (arXiv)

Despite its popularity, it is widely recognized that the investigation of some theoretical aspects of clustering has been relatively sparse. One of the main reasons for this lack of theoretical results is surely the fact that, whereas for other statistical problems the theoretical population goal is clearly defined (as in regression or classification), for some of the clustering methodologies it is difficult to specify the population goal to which the data-based clustering algorithms should try to get close. This paper aims to provide some insight into the theoretical foundations of clustering by focusing on two main objectives: to provide an explicit formulation for the ideal population goal of the modal clustering methodology, which understands clusters as regions of high density; and to present two new loss functions, applicable in fact to any clustering methodology, to evaluate the performance of a data-based clustering algorithm with respect to the ideal population goal. In particular, it is shown that only mild conditions on a sequence of density estimators are needed to ensure that the sequence of modal clusterings that they induce is consistent.