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

by   Ery Arias-Castro, et al.

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.


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