Hierarchical Clustering Algorithms on Poisson and Cox Point Processes

Abstract: Clustering is a widely used technique in unsupervised learning to identify groups within a dataset based on the similarities between its elements. This paper introduces three new hierarchical clustering models, Clustroid Hierarchical Nearest Neighbor ($\mathrm{CHN}^2$), Single Linkage Hierarchical Nearest Neighbor ($\mathrm{SHN}^2$), and Hausdorff (Complete Linkage) Hierarchical Nearest Neighbor ($\mathrm{H}^2\mathrm{N}^2$), all designed for datasets with a countably infinite number of points. These algorithms proceed through multiple levels of clustering and construct clusters by connecting nearest-neighbor points or clusters, but differ in the distance metrics they employ (clustroid, single linkage, or Hausdorff, respectively). Each method is first applied to the homogeneous Poisson point process on the Euclidean space, where it defines a phylogenetic forest, which is a factor of the point process and therefore unimodular. The results established for the $\mathrm{CHN}^2$ algorithm include the almost-sure finiteness of the clusters and bounds on the mean cluster size at each level of the algorithm. The mean size of the typical cluster is shown to be infinite. Moreover, the limiting structure of all three algorithms is examined as the number of levels tends to infinity, and properties such as the one-endedness of the limiting connected components are derived. In the specific case of $\mathrm{SHN}^2$ on the Poisson point process, the limiting graph is shown to be a subgraph of the Minimal Spanning Forest. The $\mathrm{CHN}^2$ algorithm is also extended beyond the Poisson setting, to certain stationary Cox point processes. Similar finite-cluster properties are shown to hold in these cases. It is also shown that efficient detection of Cox-triggered aggregation can be achieved through this clustering algorithm.