Generalized Cluster Trees and Singular Measures

TL;DR

This paper extends the analysis of $ ext{alpha}$-cluster trees to singular measures, establishing convergence rates, behavior of KDE, and topological consistency, including the novel concept of dimensional critical points.

Contribution

It introduces a framework for $ ext{alpha}$-trees under singular measures, analyzes KDE behavior, and proves topological consistency, highlighting the role of dimensional critical points.

Findings

KDE is pointwise consistent after rescaling for singular measures.

Estimated $ ext{alpha}$-trees are topologically consistent despite non-uniform KDE convergence.

Dimensional critical points uniquely occur at singular measures and influence cluster topology.

Abstract

In this paper, we study the $\alpha$-cluster tree ($\alpha$-tree) under both singular and nonsingular measures. The $\alpha$-tree uses probability contents within a level set to construct a cluster tree so that it is well-defined for singular measures. We first derive the convergence rate for a density level set around critical points, which leads to the convergence rate for estimating an $\alpha$-tree under nonsingular measures. For singular measures, we study how the kernel density estimator (KDE) behaves and prove that the KDE is not uniformly consistent but pointwisely consistent after rescaling. We further prove that the estimated $\alpha$-tree fails to converge in the $L_\infty$ metric but is still consistent under the integrated distance. We also observe a new type of critical points--the dimensional critical points (DCPs)--of a singular measure. DCPs occur only at singular measures, and similar to the usual critical points, DCPs contribute to cluster tree topology as well. Building on the analysis of the KDE and DCPs, we prove the topological consistency of an estimated $\alpha$-tree.