How Many Clusters Are There? – An Essay on the Basic Notions of Clustering

Abstract

This lecture starts with two remarks:

• 1 Many people, papers, books claim “the number of clusters has to be given in advance”

• 1 Even good texts do not provide a formal definition of “cluster” or “clustering”

Starting with remark 1 we show, that any formal definition of “clustering” uniquely defines the number of clusters. For remark 2 we prove that relations called “similarities” define nicely “clustering”. We also show that every clustering is defined by some similarity, although it may be given only implicitly. Finding clusters given by some similarity is the same problem as the “maximal clique problem”. Hence it is NP-complete. It is also very clear that the very old techniques of “reduction of partial automate” is the same. In both areas algorithms are known, for maximal cliques even good heuristics. Coming back to remark 1. The problem that people really address, is the case where the desired similarity is not given explicitly, but by some of its properties. Among these, there may be the number of clusters. Very many of the known algorithms have the following structure:

• 1 Generate some similarity Pi

• 1 Is it “good”?

  Yes: Stop here and use Pi for clustering

  No: generate Pi+1 and continue.

For generating new similarities many techniques are used: distances, probabilities, etc. and also fuzzy theory. Finally we show that using “representatives” for clustering is not the same as using similarities.