Defining Data Clusters

Clustering has been exten­ sively studied in data analy­ sis. However, clustering theories have thus far been unsatisfactory to justify use of the many proposed algorithms. A fun­ damental question is this: How do we define a cluster in a set of data points? In “A Mathematical Theory for Clus­ tering in Metric Spaces,” Cheng­Shang Chang and his colleagues attempt to answer this question by considering a set of data points associated with a dis­ tance measure, or metric (IEEE Trans. Network Science and Eng., vol. 3, no. 1, 2016, pp. 2–16). They propose a new cohesion measure with regard to the distance measure. Using this cohesion measure, they define a cluster as a set of points cohesive to themselves, and provide various equivalent statements with intuitive explanations to support their definition. The authors then consider a second question: How do we use this definition to find clusters and good partitions of clusters? Chang and his colleagues offer two algorithms: hierarchical ag­ glomerative and partitional. Unlike standard hierarchical agglomerative algorithms, their algorithm stops specifically at partitions of clusters. Their partitional algorithm—dubbed the K­sets algorithm—is quite novel and interesting. Unlike the K­means algorithm (for data points in a Euclid­ ean space) that assigns points to the closest centroid, the K­sets algorithm assigns points to the closest set with regard to the triangular distance—the average difference of the three sides of a triangle formed by the point and two randomly selected points in the set. As such, the K­sets algorithm per­ forms well when a cluster can’t be rep­ resented by a single point. The authors also find that the du­ ality between a distance measure and a cohesion measure leads to a dual K­sets algorithm for clustering a set of data points with a cohesion measure. The dual K-sets algorithm converges in the same way as a sequential ver­ sion of the classical kernel K­means algorithm. The key difference is that a cohesion measure doesn’t have to be positive semidefinite.