Consistent procedures for cluster tree estimation and pruning

TL;DR

This paper introduces two consistent procedures for estimating and pruning the cluster tree of a density function from samples, with finite-sample guarantees and analysis of sample complexity.

Contribution

It proposes two novel algorithms for cluster tree estimation with proven convergence rates and a pruning method to remove spurious clusters under mild conditions.

Findings

Finite-sample convergence rates established for both algorithms

Algorithms are proven to be consistent in estimating the true cluster tree

A pruning procedure effectively removes spurious clusters while preserving salient ones

Abstract

For a density $f$ on ${\mathbb R}^d$, a {\it high-density cluster} is any connected component of $\{x: f(x) \geq \lambda\}$, for some $\lambda > 0$. The set of all high-density clusters forms a hierarchy called the {\it cluster tree} of $f$. We present two procedures for estimating the cluster tree given samples from $f$. The first is a robust variant of the single linkage algorithm for hierarchical clustering. The second is based on the $k$-nearest neighbor graph of the samples. We give finite-sample convergence rates for these algorithms which also imply consistency, and we derive lower bounds on the sample complexity of cluster tree estimation. Finally, we study a tree pruning procedure that guarantees, under milder conditions than usual, to remove clusters that are spurious while recovering those that are salient.