Fuzzy clustering: More than just fuzzification

The initial idea of extending the classical k-means clustering technique to an algorithm that uses membership degrees instead of crisp assignments of data objects to clusters led to the invention of a large variety of new fuzzy clustering algorithms. However, most of these algorithms are concerned with cluster shapes or outliers and could have been defined without any problems in the context of crisp assignments of data objects to clusters. In this paper, we demonstrate that the use of membership degrees for these algorithms - although it is not necessary from the theoretical point of view - is essential for these algorithms to function in practice. With crisp assignments of data objects to clusters these algorithms would get stuck most of the time in a local minimum of their underlying objective function, leading to undesired clustering results. In other contributions it was shown that the use of membership degrees can avoid this problem of local minima but it also introduces new problems, especially for clusters with varying density and for high-dimensional data, at least if fuzzy clustering is carried out with the simple standard fuzzifier.