<title>Fuzzy clustering: critical analysis of the contextual mechanisms employed by three neural network models</title>

Abstract

According to the following definition, taken from the literature, a fuzzy clustering mechanism allows the same input pattern to belong to multiple categories to different degrees. Many clustering neural network (NN) models claim to feature fuzzy properties, but several of them (like the Fuzzy ART model) do not satisfy this definition. Vice versa, we believe that Kohonen's Self-Organizing Map, SOM, satisfies the definition provided above, even though this NN model is well-known to (robustly) perform topologically ordered mapping rather than fuzzy clustering. This may sound as a paradox if we consider that several fuzzy NN models (such as the Fuzzy Learning Vector Quantization, FLVQ, which was first called Fuzzy Kohonen Clustering Network, FKCN) were originally developed to enhance Kohonen's models (such as SOM and the vector quantization model, VQ). The fuzziness of SOM indicates that a network of processing elements (PEs) can verify the fuzzy clustering definition when it exploits local rules which are biologically plausible (such as the Kohonen bubble strategy). This is equivalent to state that the exploitation of the fuzzy set theory in the development of complex systems (e.g., clustering NNs) may provide new mathematical tools (e.g., the definition of membership function) to simulate the behavior of those cooperative/competitive mechanisms already identified by neurophysiological studies. When a biologically plausible cooperative/competitive strategy is pursued effectively, neighboring PEs become mutually coupled to gain sensitivity to contextual effects. PEs which are mutually coupled are affected by vertical (inter-layer) as well as horizontal (intra-layer) connections. To summarize, we suggest to relate the study of fuzzy clustering mechanisms to the multi-disciplinary science of complex systems, with special regard to the investigation of the cooperative/competitive local rules employed by complex systems to gain sensitivity to contextual effects in cognitive tasks. In this paper, the FLVQ model is critically analyzed in order to stress the meaning of a fuzzy learning mechanism. This study leads to the development of a new NN model, termed the fuzzy competitive/cooperative Kohonen (FCCK) model, which replaces FLVQ. Then, the architectural differences amongst three NN algorithms and the relationships between their fuzzy clustering properties are discussed. These models, which all perform on-line learning, are: (1) SOM; (2) FCCK; and (3) improved neural-gas (INC).