Stability of Density-Based Clustering

Alessandro Rinaldo, Aarti Singh, Rebecca Nugent, Larry Wasserman.

Year: 2012, Volume: 13, Issue: 32, Pages: 905−948


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