A unifying view of modal clustering

Two important non-parametric approaches to clustering emerged in the 1970s: clustering by level sets or cluster tree as proposed by Hartigan, and clustering by gradient lines or gradient flow as proposed by Fukunaga and Hostetler. In a recent paper, we draw a connection between these two approaches, in particular, by showing that the gradient flow provides a way to move along the cluster tree. Here, we argue the case that these two approaches are fundamentally the same. We do so 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.