Stability of Density-Based Clustering

TL;DR

This paper investigates the stability of density-based clustering methods by analyzing how estimates of density level sets and cluster trees vary with the kernel bandwidth, providing theoretical insights into their instability.

Contribution

It introduces two measures of instability for density level set and cluster tree estimates and studies their theoretical properties as functions of the bandwidth.

Findings

Instability measures depend on bandwidth choice

Theoretical bounds for instability are derived

Guidelines for selecting stable clustering parameters

Abstract

High density clusters can be characterized by the connected components of a level set $L(\lambda) = \{x:\ p(x)>\lambda\}$ of the underlying probability density function $p$ generating the data, at some appropriate level $\lambda\geq 0$. The complete hierarchical clustering can be characterized by a cluster tree ${\cal T}= \bigcup_{\lambda} L(\lambda)$. In this paper, we study the behavior of a density level set estimate $\widehat L(\lambda)$ and cluster tree estimate $\widehat{\cal{T}}$ based on a kernel density estimator with kernel bandwidth $h$. We define two notions of instability to measure the variability of $\widehat L(\lambda)$ and $\widehat{\cal{T}}$ as a function of $h$, and investigate the theoretical properties of these instability measures.