A generalized fuzzy clustering regularization model with optimality tests and model complexity analysis

In this paper, we propose a generalized fuzzy clustering regularization (GFCR) model and then study its theoretical properties. GFCR unifies several fuzzy clustering algorithms, such as fuzzy c-means (FCM), maximum entropy clustering (MEC), fuzzy clustering based on Fermi-Dirac entropy, and fuzzy bidirectional associative clustering network, etc. The proposed GFCR becomes an alternative model of the generalized FCM (GFCM) that was recently proposed by Yu and Yang. To advance theoretical study, we have the following three considerations. 1) We give an optimality test to monitor if GFCR converges to a local minimum. 2) We relate the GFCR optimality tests to Occam's razor principle, and then analyze the model complexity for fuzzy clustering algorithms. 3) We offer a general theoretical method to evaluate the performance of fuzzy clustering algorithms. Finally, some numerical experiments are used to demonstrate the validity of our theoretical results and complexity analysis.