Fuzzy clustering with $$\hbox {L}_0$$ regularization

The Fuzzy K-Means algorithm extends the well-known classical K-Means algorithm by replacing the standard allocation matrix with the membership degree one. Consistently with the fuzzy approach to clustering, this allows for obtaining a soft assignment of the units to the clusters. The units are assigned to the clusters with membership degree taking values in the unit interval. In practice, what we get from Fuzzy K-Means is a fuzzy partition where even the units clearly belonging to only one cluster generally present non-zero membership degrees to all the clusters. In order to overcome this drawback, a generalization of Fuzzy K-Means is proposed where an L0 regularization term for the membership degree matrix is introduced. This makes it possible to obtain a sparse membership degree matrix, where the units that clearly belong to one cluster have membership degrees strictly equal to one to the cluster involved and zero to the other clusters, without compromising the soft membership degrees of the units with unclear assignments. The adequacy of the proposal is evaluated by means of simulation and real-case studies.