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Stable Clusterings and the Cones of Outer Normals

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Operations Research Proceedings 2017

Part of the book series: Operations Research Proceedings ((ORP))

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Abstract

We consider polytopes that arise in the cluster analysis of a finite set of data points. These polytopes encode all possible clusterings and their vertices correspond to clusterings that admit a power diagram, which is a polyhedral separation of the underlying space where each cluster has its own cell. We study the edges of these polytopes and show that they encode cyclical transfers of elements between clusters. We use this characterization to obtain a relation between power diagrams and the volume of the cones of outer normals of the respective clustering. This allows us to derive a new stability criterion for clusterings, which can be used to measure the dependability of the clustering for decision-making. Further, the results explain why many popular clustering algorithms work so well in practice.

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References

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Acknowledgements

I am grateful to my master’s thesis advisor Steffen Borgwardt for introducing me to this exciting topic and for the continuous support.

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Authors and Affiliations

  1. Technische Universität München, Munich, Germany

    Felix Happach

Authors
  1. Felix Happach
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Correspondence to Felix Happach .

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Editors and Affiliations

  1. Department of Information Systems, Freie Universität Berlin, Berlin, Germany

    Natalia Kliewer

  2. Management Science Group, Otto-von-Guericke Universität Magdeburg, Magdeburg, Germany

    Jan Fabian Ehmke

  3. Department of Mathematics, Freie Universität Berlin, Berlin, Germany

    Ralf Borndörfer

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Happach, F. (2018). Stable Clusterings and the Cones of Outer Normals. In: Kliewer, N., Ehmke, J., Borndörfer, R. (eds) Operations Research Proceedings 2017. Operations Research Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-319-89920-6_6

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