The Mean Partition Theorem in consensus clustering. (July 2018) - Jain, Brijnesh J.
Highlights: Mean and Expected Partition Theorem in consensus clustering. Necessary and sufficient conditions of optimality. Equivalence of mean partitions and multiple optimal alignments. Relationship between mean partitions and cluster stability. Relationship between mean partitions and diversity. Abstract: 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.