Good Clusterings Have Large Volume
Abstract
The clustering of a data set is one of the core tasks in data analytics. Many clustering algorithms exhibit a strong contrast between a favorable performance in practice and bad theoretical worstcases. Prime examples are leastsquares assignments and the popular $k$means algorithm. We are interested in this contrast and study it through polyhedral theory. Several popular clustering algorithms can be connected to finding a vertex of the socalled boundedshape partition polytopes. The vertices correspond to clusterings with extraordinary separation properties, in particular allowing the construction of a separating power diagram, defined by its socalled sites, such that each cluster has its own cell. First, we quantitatively measure the space of all sites that allow construction of a separating power diagram for a clustering by the volume of the normal cone at the corresponding vertex. This gives rise to a new quality criterion for clusterings, and explains why good clusterings are also the most likely to be found by some classical algorithms. Second, we characterize the edges of the boundedshape partition polytopes. Through this, we obtain an explicit description of the normal cones. This allows us to compute measures with respect to the new quality criterion, and even compute "most stable" sites, and thereby "most stable" power diagrams, for the separation of clusters. The hardness of these computations depends on the number of edges incident to a vertex, which may be exponential. However, the computational effort is rewarded with a wealth of information that can be gained from the results, which we highlight through some proofofconcept computations.
 Publication:

arXiv eprints
 Pub Date:
 June 2017
 DOI:
 10.48550/arXiv.1706.08996
 arXiv:
 arXiv:1706.08996
 Bibcode:
 2017arXiv170608996B
 Keywords:

 Mathematics  Optimization and Control;
 91C20;
 90C90;
 51M20;
 90C20