Pseudo-Boolean clustering

This paper proposes to cluster any finite data set by optimizing with respect to a cluster score function, assigning a positive real worth to each data subset and thereby dealt with in pseudo-Boolean form, so that the multilinear extension (MLE) allows to evaluate fuzzy data subsets or clusters as well. A fuzzy clustering being a collection of fuzzy clusters over which every data point has to distribute a unitary membership mass, the objective function (to be maximized) is global worth, obtained through summation over constituents fuzzy clusters of their own worth as given by the score function MLE. Optimization then proceeds by means of pseudo-Boolean techniques, leading to a local-search algorithm. Also, any fuzzy clustering is shown to admit some hard one (or partition of the data set) that does at least as good, and concavity of the objective function is interpreted in terms of the underlying clustering problem.