Likelihood Optimization of Probabilistic Fuzzy C-Means based on k-NN and Local Search

Fuzzy C-Means (FCM) is a widely used clustering algorithm that offers robustness, efficiency, and simplicity. However, its parameters (the membership matrix) and hyperparameter (the fuzzification coefficient) can be interpreted in probabilistic terms, thus opening the door for the application of a probabilistic methodology for clustering that is alternative to classical soft clustering methods like Gaussian Mixture Models (GMM). In particular, the Probabilistic FCM (PFCM) is inspired by FCM but adopts a different objective function that is based on negative log-likelihood (NLL). Based on the hypothesis that the optimal values of prototypes are close to data points, this paper describes an optimization algorithm using k-NN and local search that significantly reduces both the computation time and the NLL in comparison to a standard optimization based on the basin-hopping algorithm. Furthermore, a preliminary experimental comparison of PFCM with FCM and GMM shows that prototypes resulting from PFCM are always located in datadense regions, thus providing more significant clustering of data.