Generalized Cluster Trees and Singular Measures

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.