Good Clusterings Have Large Volume

Abstract

In “Good Clusterings Have Large Volume,” S. Borgwardt and F. Happach study the strong contrast between the favorable performance of clustering algorithms in practice and their bad theoretical worst cases. They explain this contrast using polyhedral theory. Classical clustering algorithms can be interpreted as the search for vertices in the so-called bounded-shape partition polytopes, following a "random" direction. The vertices correspond to clusterings with extraordinary separation properties, and the quality of this separation depends on the volume of the normal cone of the vertex—the larger the better. This gives rise to a new quality criterion for clusterings and explains why good clusterings are also the most likely to be found. The authors further characterize the edges of the polytopes, which allows for an explicit description of the normal cone, and in turn the computation of a best separation between the clusters of a given clustering.