Generalized hard cluster analysis

Abstract

In this paper, we generalize the hard clustering paradigm. While in this paradigm a data set is subdivided into disjoint clusters, we allow different clusters to have a nonempty intersection. The concept of hard clustering is then analysed in this general setting, and we show which specific properties hard clusterings possess in comparison to more general clusterings. We also introduce the concept of equivalent clusterings and show that in the case of hard clusterings equivalence and equality coincide. However, if more general clusterings are considered, these two concepts differ, and this implies the undesired fact that equivalent clusterings can have different representations in the traditional view on clustering. We show how a matrix representation can solve this representation problem.

Keywords:

• hard clustering
• matrix representation
• transitivity
• (m, n)-set
• equivalent clusterings

2010 AMS Subject Classification :

• 62H30

CR Category :

• I.5.3

Acknowledgements

Financial support from BOF (Special Research Fund of Ghent University) is gratefully acknowledged. I am grateful to Bart Wyns for his useful comments.