Prototype-less Fuzzy Clustering

Abstract

Abstract-In contrast to standard fuzzy clustering, which optimizes a set of prototypes, one for each cluster, this paper studies fuzzy clustering without prototypes. Starting from an objective function that only involves the distances between data points and the membership degrees of the data points to the different clusters, an iterative update rule is derived. The properties of the resulting algorithm are then examined, especially w.r.t. to schemes that focus on a constrained neighborhood for each data point. Corresponding experimental results are reported that demonstrate the merits of this approach.

An Extended Objective Function for Prototype-less Fuzzy Clustering

“…Furthermore it may be worthwhile to investigate other neighborhood schemes than those discussed in [7], since they may provide a simple and effective way to lower the computational costs.…”

“…The scheme has some relation to the reformulation approach [14], which, if it is used to eliminate the update of the prototype parameters rather than the update of the membership degrees, leads to a similar, but more complex objective function, and to fuzzy k-nearest neighbors algorithms [18], from which one may also derive a candidate update rule for prototype-less fuzzy clustering. However, as discussed in [7], the latter does not lead to useful results, as it tends to equalize the membership degrees.…”

“…Choosing α = 0 yields a fairly standard objective function, which was explored in detail in [7], in particular w.r.t. local neighborhood schemes.…”

“…local neighborhood schemes. However, the goal of this paper is to investigate the influence of a non-vanishing α (this parameter is not present in [7]). As our investigation shows (see the experiments in Section IV), negative values are particularly interesting.…”

“…The properties of the resulting algorithm are then examined, especially w.r.t. to an additional parameter of the objective function (compared to the one proposed in [7]) that can be seen as a more flexible alternative to the fuzzifier. Corresponding experimental results are reported that demonstrate the merits of our approach.…”