Consistency of Single Linkage for High-Density Clusters

Abstract

High-density clusters are defined on a population with density f in r dimensions to be the maximal connected sets of form {x | f(x) ≥ c}. Single-linkage clustering is evaluated for consistency in detecting such high-density clusters—other standard hierarchical techniques, such as average and complete linkage, are hopelessly inconsistent for these clusters. The asymptotic consistency of single linkage closely depends on the percolation problem of Broadbent and Hammersley—if small spheres are removed at random from a solid, at which density of spheres will water begin to flow through the solid? If there is a single critical density such that no flow takes place below a certain density, and flow occurs through a single connected set above that density, then single linkage is consistent in separating high-density clusters (by disjoint single-linkage clusters that include a positive fraction of sample points in the respective clusters and pass arbitrarily close to all points in the respective clusters). The existence of a single critical point remains a conjecture. A weaker result is proved that shows that single-linkage clusters detect high-density clusters if there is a low enough valley separating them.

Key Words:

• Single linkage
• High-density clusters
• Percolation processes