Fuzzy Clustering: Consistency of Entropy Regularization

We introduce in this paper a new formulation of the regularized fuzzy C-means (FCM) algorithm which allows us to find automatically the actual number of clusters. The approach is based on the minimization of an objective function which mixes, via a particular parameter, a classical FCM term and a new entropy regularizer. The main contribution of the method is the introduction of 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 an 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 experimentally. We will show an extension of the method to non-linearly separable data. Finally, we will illustrate preliminary results both on simple toy examples as well as image segmentation and database categorization problems. ∗Accepted in International Conference on Computational Intelligence (Special Session on Fuzzy Clustering), Dortmund, Germany, September 2004.