Consistency of mean partitions in consensus clustering

This article studies the asymptotic behavior of mean partitions in consensus clustering. We show that the mean partition approach is consistent and asymptotic normal under mild assumptions. To derive both results, we represent partitions as points of some geometric space, called orbit space. Then we draw on results from the theory of Fréchet means and stochastic programming. The asymptotic properties hold for continuous extensions of standard cluster criteria (indices). The results justify consensus clustering using finite but sufficiently large sample sizes. Furthermore, the orbit space framework provides a mathematical foundation for studying further statistical, geometrical, and analytical properties of sets of partitions.