An axiomatic approach to soft learning vector quantization and clustering

This paper presents an axiomatic approach to soft learning vector quantization (LVQ) and clustering based on reformulation. The reformulation of the fuzzy c-means (FCM) algorithm provides the basis for reformulating entropy-constrained fuzzy clustering (ECFC) algorithms. This analysis indicates that minimization of admissible reformulation functions using gradient descent leads to a broad variety of soft learning vector quantization and clustering algorithms. According to the proposed approach, the development of specific algorithms reduces to the selection of a generator function. Linear generator functions lead to the FCM and fuzzy learning vector quantization (FLVQ) algorithms while exponential generator functions lead to ECFC and entropy-constrained learning vector quantization (ECLVQ) algorithms. The reformulation of LVQ and clustering algorithms also provides the basis for developing uncertainty measures that can identify feature vectors equidistant from all prototypes. These measures are employed by a procedure developed to make soft LVQ and clustering algorithms capable of identifying outliers in the data set. This procedure is evaluated by testing the algorithms generated by linear and exponential generator functions on speech data.