The Mean Partition Theorem of Consensus Clustering

TL;DR

This paper introduces the Mean Partition Theorem for consensus clustering, providing new theoretical insights and extending previous optimality conditions to broader applications like alignment, motif detection, and stability analysis.

Contribution

It presents a stronger necessary condition of optimality for consensus clustering and extends it to the Expected Partition Theorem, enhancing theoretical understanding.

Findings

Equivalence of mean partition and optimal multiple alignment

Construction of profiles and motifs from consensus clustering

Relationship between consensus clustering and cluster stability

Abstract

To devise efficient solutions for approximating a mean partition in consensus clustering, Dimitriadou et al. [3] presented a necessary condition of optimality for a consensus function based on least square distances. We show that their result is pivotal for deriving interesting properties of consensus clustering beyond optimization. For this, we present the necessary condition of optimality in a slightly stronger form in terms of the Mean Partition Theorem and extend it to the Expected Partition Theorem. To underpin its versatility, we show three examples that apply the Mean Partition Theorem: (i) equivalence of the mean partition and optimal multiple alignment, (ii) construction of profiles and motifs, and (iii) relationship between consensus clustering and cluster stability.