An Asymptotic Equivalence between the Mean-Shift Algorithm and the Cluster Tree

TL;DR

This paper demonstrates that the mean-shift algorithm and the cluster tree approach to clustering are asymptotically equivalent, unifying two classical nonparametric clustering methods under standard density assumptions.

Contribution

It introduces two natural methods to derive partitions from the cluster tree and proves their asymptotic equivalence to the gradient flow-based partition.

Findings

Mean-shift and cluster tree methods are asymptotically equivalent.

Two natural partitions from the cluster tree converge to the gradient flow partition.

The results unify two classical clustering approaches under standard assumptions.

Abstract

Two important nonparametric approaches to clustering emerged in the 1970's: clustering by level sets or cluster tree as proposed by Hartigan, and clustering by gradient lines or gradient flow as proposed by Fukunaga and Hosteler. In a recent paper, we argue the thesis that these two approaches are fundamentally the same by showing that the gradient flow provides a way to move along the cluster tree. In making a stronger case, we are confronted with the fact the cluster tree does not define a partition of the entire support of the underlying density, while the gradient flow does. In the present paper, we resolve this conundrum by proposing two ways of obtaining a partition from the cluster tree -- each one of them very natural in its own right -- and showing that both of them reduce to the partition given by the gradient flow under standard assumptions on the sampling density.