Towards an Axiomatic Approach to Hierarchical Clustering of Measures
This work addresses the foundational challenge of clustering measures in a theoretical, axiomatic framework, which is incremental as it builds on existing clustering concepts but without metric dependencies.
The paper tackles the problem of hierarchical clustering for probability measures by proposing axioms that allow users to define clusters for elementary measures without relying on metrics, similarity, or dissimilarity, resulting in a unique clustering notion for a large set of distributions based on additivity and continuity axioms.
We propose some axioms for hierarchical clustering of probability measures and investigate their ramifications. The basic idea is to let the user stipulate the clusters for some elementary measures. This is done without the need of any notion of metric, similarity or dissimilarity. Our main results then show that for each suitable choice of user-defined clustering on elementary measures we obtain a unique notion of clustering on a large set of distributions satisfying a set of additivity and continuity axioms. We illustrate the developed theory by numerous examples including some with and some without a density.