Hierarchical clustering is a popular method for analyzing data which
associates a tree to a dataset. Hartigan consistency has been used extensively
as a framework to analyze such clustering algorithms from a statistical point
of view. Still, as we show in the paper, a tree which is Hartigan consistent
with a given density can look very different than the correct limit tree.
Specifically, Hartigan consistency permits two types of undesirable
configurations which we term over-segmentation and improper nesting. Moreover,
Hartigan consistency is a limit property and does not directly quantify
difference between trees.
In this paper we identify two limit properties, separation and minimality,
which address both over-segmentation and improper nesting and together imply
(but are not implied by) Hartigan consistency. We proceed to introduce a merge
distortion metric between hierarchical clusterings and show that convergence in
our distance implies both separation and minimality. We also prove that uniform
separation and minimality imply convergence in the merge distortion metric.
Furthermore, we show that our merge distortion metric is stable under
perturbations of the density.
Finally, we demonstrate applicability of these concepts by proving
convergence results for two clustering algorithms. First, we show convergence
(and hence separation and minimality) of the recent robust single linkage
algorithm of Chaudhuri and Dasgupta (2010). Second, we provide convergence
results on manifolds for topological split tree clustering