The Annals of Statistics

Generalized cluster trees and singular measures

Yen-Chi Chen

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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 set created by the ordering of points to construct a cluster tree so that it is well defined even 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 pointwise 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 are points that contribute to cluster tree topology but cannot be defined using density gradient. Building on the analysis of the KDE and DCPs, we prove the topological consistency of an estimated $\alpha $-tree.

Article information

Ann. Statist., Volume 47, Number 4 (2019), 2174-2203.

Received: November 2016
Revised: February 2018
First available in Project Euclid: 21 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties
Secondary: 62G05: Estimation 62G07: Density estimation

Cluster tree kernel density estimator level set singular measure critical points topological data analysis


Chen, Yen-Chi. Generalized cluster trees and singular measures. Ann. Statist. 47 (2019), no. 4, 2174--2203. doi:10.1214/18-AOS1744.

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Supplemental materials

  • Supplementary proofs: Generalized cluster trees and singular measures. This document contains all proofs to the theorems and lemmas in this paper.