Gromov-Hausdorff stability of linkage-based hierarchical clustering methods

On the Properties of α-Unchaining Single Linkage Hierarchical Clustering

“…In [9] and [10] we proved the chaining, unchaining and stability properties of SL(α) and SL * (α). Herein, we complete the work by analyzing which of the abstract characteristic properties of SL are also satisfied by these two methods and which properties are lost by adding the unchaining condition.…”

“…In particular, SL(α) is no longer stable in the Gromov-Hausdorff sense. In fact, as we proved in [10], there is no stable solution to this chaining effect in the range of almost-standard linkage-based HC methods using SL . Now, the question is when should we use SL(α) and SL * (α).…”

“…In [10] the methods defined by applying this algorithm for some linkage function are called standard linkage-based HC methods. Let us now recall the definition of SL(α) and SL * (α).…”

“…This construction is generalized in [10] to define the class of almost-standard linkage-based HC methods. Remark 2.2.…”

“…The authors prove that if two metric spaces are close (in the Gromov-Hausdorff metric) then the corresponding ultrametric spaces obtained as output of the algorithm are also close. In [10] we studied Gromov-Hausdorff stability of linkage-based HC methods defining the following conditions.…”