On the cohesion and separability of average-link for hierarchical agglomerative clustering

Average-link is widely recognized as one of the most popular and effective methods for building hierarchical agglomerative clustering. The available theoretical analyses show that this method has a much better approximation than other popular heuristics, as single-linkage and complete-linkage, regarding variants of Dasgupta's cost function [STOC 2016]. However, these analyses do not separate average-link from a random hierarchy and they are not appealing for metric spaces since every hierarchical clustering has a 1/2 approximation with regard to the variant of Dasgupta's function that is employed for dissimilarity measures [Moseley and Yang 2020]. In this paper, we present a comprehensive study of the performance of average-link in metric spaces, regarding several natural criteria that capture separability and cohesion and are more interpretable than Dasgupta's cost function and its variants. We also present experimental results with real datasets that, together with our theoretical analyses, suggest that average-link is a better choice than other related methods when both cohesion and separability are important goals.

The computational complexity of some explainable clustering problems

We study the computational complexity of some explainable clustering problems in the framework proposed by [Dasgupta et al., ICML 2020], where explainability is achieved via axis-aligned decision trees. We consider the $k$-means, $k$-medians, $k$-centers and the spacing cost functions. We prove that the first three are hard to optimize while the latter can be optimized in polynomial time.

Minimization of Gini impurity via connections with the k-means problem

The Gini impurity is one of the measures used to select attribute in Decision Trees/Random Forest construction. In this note we discuss connections between the problem of computing the partition with minimum Weighted Gini impurity and the $k$-means clustering problem. Based on these connections we show that the computation of the partition with minimum Weighted Gini is a NP-Complete problem and we also discuss how to obtain new algorithms with provable approximation for the Gini Minimization problem.