Local convergence of the fuzzy c-Means algorithms

Much understanding has recently been gained concerning global convergence properties of the fuzzy c-Means (FCM) family of clustering algorithms. These global convergence properties, which hold for all iteration sequences, guarantee that every FCM iteration sequence converges, at least along a subsequence, to a stationary point of an FCM objective function. In this paper we prove a local convergence property, that is, a property pertaining to iteration sequences started near a solution. Specifically, a simple result is proved which shows that whenever an FCM algorithm is started sufficiently near a minimizer of the corresponding objective function, then the iteration sequence must converge to that particular minimizer. The result guarantees that once captured by the local neighborhood of a minimizer, the succeeding iterate sequence will not escape—thus, infinite oscillation of such a sequence cannot occur. The rate of convergence of the sequence to such a point is also discussed.