Fuzzy clustering: Consistency of entropy regularization

We introduce in this paper anew formulation of the regularized fuzzyc-means (FCM) algorithm which allows us to set automatically the actual number of clusters. The approach is based on the minimization of an objective function which mixes, via, a particular parameter, a classic al FCM term and an entropy regularizer. The method uses a new exponential form of the fuzzy memberships which ensures the consistency of their bounds and makes it possible to interpret the mixing parameter as the variance (or scale) of the clusters. This variance closely related to the number of clusters, provides us with a more intuitive and an easy to set parameter. We will discuss the proposed approach from the regularization point-of-view and we will demonstrate its validity both analytically and experimental ly. We conducted preliminary experiments both on simple toy examples as well as challenging image segmentation problems.