Beyond classical consensus clustering: The least squares approach to multiple solutions

Clustering is one of the most important unsupervised learning problems and it consists of finding a common structure in a collection of unlabeled data. However, due to the ill-posed nature of the problem, different runs of the same clustering algorithm applied to the same data-set usually produce different solutions. In this scenario choosing a single solution is quite arbitrary. On the other hand, in many applications the problem of multiple solutions becomes intractable, hence it is often more desirable to provide a limited group of "good" clusterings rather than a single solution. In the present paper we propose the least squares consensus clustering. This technique allows to extrapolate a small number of different clustering solutions from an initial (large) ensemble obtained by applying any clustering algorithm to a given data-set. We also define a measure of quality and present a graphical visualization of each consensus clustering to make immediately interpretable the strength of the consensus. We have carried out several numerical experiments both on synthetic and real data-sets to illustrate the proposed methodology. (C) 2011 Elsevier B.V. All rights reserved.