2021/11/19 by Arias-Castro, Ery, Qiao, Wanli
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2111.10298
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.