From Aggregation Operators to Soft Learning Vector Quantization and Clustering Algorithms

Abstract

This chapter proposes a general framework for developing soft learning vector quantization (LVQ) and clustering algorithms by using gradient descent to minimize a reformulation function based on admissible aggregation operators. This approach establishes a link between competitive LVQ models and operators developed over the years to perform aggregation on fuzzy sets. For mean type aggregation operators, the development of LVQ and clustering algorithms reduces to the selection of admissible generator functions. The chapter studies the properties of soft LVQ and clustering algorithms derived using nonlinear generator functions. Another family of soft LVQ and clustering algorithms was developed by minimizing admissible reformulation functions based on ordered weighted aggregation operators. In addition to its use in the development of soft LVQ and clustering algorithms, the proposed formulation provides the basis for exploring the structure of the data by identifying outliers in the feature set. A major study is currently under way, which aims at the evaluation of a broad variety of soft LVQ and clustering algorithms on segmentation of magnetic resonance images of the brain.