A New Convergence Proof of Fuzzy c-Means

In this letter, we give a new, more direct derivation of the convergence properties of the fuzzy c-means (FCM) algorithm, using the equivalence between the original and reduced FCM criterion. From the point of view of the reduced criterion, the FCM algorithm is simply a steepest descent algorithm with variable steplength. We prove that steplength adjustment follows from the majorization principle for steplength. By applying the majorization principle we give a straightforward proof of global convergence. Further convergence properties follow immediately using known results of optimization theory.