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

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.

25 pages, 7 figures

References in corpus (4)

• The topography of multivariate normal mixtures
• Asymptotic normality of plug-in level set estimates
• A Uniqueness Theorem for Clustering
• Asymptotics and optimal bandwidth selection for highest density region estimation

Cited by in corpus (13)

• Nonparametric ridge estimation
• A Population Background for Nonparametric Density-Based Clustering
• Learning by Unsupervised Nonlinear Diffusion
• Quickshift++: Provably Good Initializations for Sample-Based Mean Shift
• DTM-based Filtrations
• A comparison of bandwidth selectors for mean shift clustering
• Feature Selection For High-Dimensional Clustering
• Nonparametric Inference For Density Modes
• Density Level Sets: Asymptotics, Inference, and Visualization
• A Comprehensive Approach to Mode Clustering
• Statistical Inference Using Mean Shift Denoising
• Analysis of a Mode Clustering Diagram
• The functional mean-shift algorithm for mode hunting and clustering in infinite dimensions