Cluster Aggregate Inequality and Multi-level Hierarchical Clustering

We show that (1) in hierarchical clustering, many linkage functions satisfy a cluster aggregate inequality, which allows an exact O(N2) multi-level (using mutual nearest neighbor) implementation of the standard O(N3) agglomerative hierarchical clustering algorithm. (2) a desirable close friends cohesion of clusters can be translated into kNN consistency which is guaranteed by the multi-level algorithm; (3) For similarity-based linkage functions, the multi-level algorithm is naturally implemented as graph contraction. The effectiveness of our algorithms is demonstrated on a number of real life applications.