Towards a theory of holistic clustering

In this note, cluster theory is presented from a rather abstract point of view, basic known results are reviewed from this view point, and some new results which motivated the proposed approach, as well as some new problems which naturally arise in this context, are presented. Towards a Theory of Holistic Clustering A.W.M. Dress Bielefeld, Germany Abstract: In this note, cluster theory is presented from a rather abstract point of view, basic known results are reviewed from this view point, and some new results which motivated the proposed approach, as well as some new problems which naturally arise in this context, are presented. In this note, cluster theory is presented from a rather abstract point of view, basic known results are reviewed from this view point, and some new results which motivated the proposed approach, as well as some new problems which naturally arise in this context, are presented. 1 A Local-Global Approach to Clustering Clustering procedures (as well as many other mathematical techniques) can be viewed as methods for extracting globally relevant features from locally distributed information. A rather natural, simple and sufficiently general conceptual framework for describing clustering procedures is, therefore, the following one: We assume that, for any given (generally finite) set X, we can form the set Inj (X) comprising all possible structures defined on X which encode the information regarding X we are seeking: this could be the set of all tree structures or the set of all ultrametrics definable on X, or just the set P(P(X)) of all subsets of the power set P(X) of X. In addition, we assume that, for any pair (X, Y) consisting of a set X and a subset Y of X, information regarding X implies information regarding Y which is expressed in form of a map res = resx-+Y : Inf (X)--+ Inj(Y) : i I-t ily called the restriction map (relative to X andY), and we assume consistency of restriction by requiring that, for all Z ~ Y ~ X and i E Inf(X), we have