A Population Background for Nonparametric Density-Based Clustering

TL;DR

This paper explores the theoretical foundations of density-based clustering by defining an explicit population goal and introducing new loss functions to evaluate clustering performance, ensuring consistency under mild conditions.

Contribution

It provides a clear formulation of the ideal population goal for modal clustering and introduces general loss functions for assessing clustering accuracy.

Findings

Explicit population goal for modal clustering defined

New loss functions for clustering evaluation proposed

Consistency of modal clustering under mild density estimator conditions

Abstract

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.