The Mean Partition Theorem in consensus clustering

This article presents the Mean Partition Theorem of consensus clustering. We show that the Mean Partition Theorem is a fundamental result that connects to different, but not obviously related branches such as: (i) optimization, (ii) statistical consistency, (iii) optimal multiple alignment, (iv) profiles and motifs, (v) cluster stability, (vi) diversity, and (vii) Condorcet's Jury Theorem. All proofs rest on the orbit space framework. The implications are twofold: First, the Mean Partition Theorem plays a far-reaching and central role in consensus clustering. Second, orbit spaces constitute a convenient representation for gaining insight into partition spaces.