Clusters and water flows: a novel approach to modal clustering through Morse theory

TL;DR

This paper introduces a new theoretical framework for density-based clustering using Morse theory, providing an explicit formulation for ideal population clusters to enhance understanding of nonparametric clustering methods.

Contribution

It offers a novel theoretical approach to density-based clustering by applying Morse theory to define ideal population clusters explicitly.

Findings

Provides an explicit formulation for population-level clusters.

Bridges the gap between theoretical and practical clustering methods.

Enhances understanding of the foundations of density-based clustering.

Abstract

The problem of finding groups in data (cluster analysis) has been extensively studied by researchers from the fields of Statistics and Computer Science, among others. However, 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, unlike the situation with other statistical problems as regression or classification, for some of the cluster methodologies it is quite difficult to specify a 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 the usual nonparametric approach to clustering, which understands clusters as regions of high density, by presenting an explicit formulation for the ideal population clustering.