Cluster Trees on Manifolds

TL;DR

This paper studies the estimation of cluster trees for densities on manifolds embedded in high-dimensional space, showing convergence rates depend only on the manifold dimension, not the ambient space.

Contribution

It introduces a modified $k$-nearest neighbor algorithm for manifold-supported densities and proves dimension-dependent convergence rates, also extending results to kernel density estimators.

Findings

Convergence rates depend only on the manifold dimension $d$, not the ambient dimension $D$.

Modified $k$-nearest neighbor algorithm achieves optimal rates under mild assumptions.

Spatially adaptive algorithms improve rates in the known manifold case.

Abstract

In this paper we investigate the problem of estimating the cluster tree for a density $f$ supported on or near a smooth $d$-dimensional manifold $M$ isometrically embedded in $\mathbb{R}^D$. We analyze a modified version of a $k$-nearest neighbor based algorithm recently proposed by Chaudhuri and Dasgupta. The main results of this paper show that under mild assumptions on $f$ and $M$, we obtain rates of convergence that depend on $d$ only but not on the ambient dimension $D$. We also show that similar (albeit non-algorithmic) results can be obtained for kernel density estimators. We sketch a construction of a sample complexity lower bound instance for a natural class of manifold oblivious clustering algorithms. We further briefly consider the known manifold case and show that in this case a spatially adaptive algorithm achieves better rates.