Abstract
Vector quantization (VQ) is a fundamental and omnipresent mechanism of data compression with various conceptual underpinnings and diversified algorithmic realizations. The objective of this study is to investigate the concept of VQ in the setting of fuzzy sets by forming a coherent algorithmic framework referred to as a fuzzy VQ (FVQ). Given the nature of the framework of VQ in which fuzzy sets are involved, we may refer to the discussed processes of FVQ as a fuzzy
granulation and fuzzy
degranulation. In comparison to the winner-takes-all strategy encountered in VQ where a result of decoding typically arises as a single element of the codebook, in the FVQ we exploit an efficient usage of all components of the codebook (fuzzy sets) in the reconstruction of the original data. In this study, we present a complete development scheme of the FVQ and elaborate on its essential features. Its main design phases involve: (a) an encoding in which we encode data in terms of the elements of the given codebook; (b) a decoding during which we reconstruct the original data; and (c) a development of the codebook. The mechanisms of encoding and decoding are created as a result of some well-formed optimization tasks. The buildup of the codebook is completed through a mechanism of global optimization realized in the form of the particle swarm optimization (PSO). We offer a collection of experiments using synthetic data by focusing on and quantifying the role of fuzzy sets in VQ. While FVQ outperforms VQ (which seems to be an intuitively appealing finding), we also show that this improvement could be achieved through a careful optimization of the elements of the granulation scheme. It is also shown that without optimization of the FVQ scheme, the enhancements could not be possible or may become very much limited. A series of experiments involving synthetic data and data sets coming from the Machine Learning repository is included as well.