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Fuzzy c -means (FCM) is the fuzzy version of c -means clustering, in which memberships are fuzzified by introducing an additional parameter into the linear objective function of the weighted sum of distances between datapoints and cluster centers. Regularization of hard c -means clustering is another approach to fuzzification, in which regularization terms such as entropy and quadratic terms have been adopted. We generalized the fuzzification concept and propose a new approach to fuzzy clustering in which linear weights of hard c -means clustering are replaced by nonlinear ones through regularization. Numerical experiments demonstrated that the proposed algorithm has the characteristic features of the standard FCM algorithm and of regularization approaches. One of the proposed nonlinear weights makes it possible to both to attract data to clusters and to repulse different clusters. This feature derives different types of fuzzy classification functions in both probabilistic and possibilistic models.

A Regularization Approach to Fuzzy Clustering with Nonlinear Membership Weights