Separation theorem and maximal margin classification for fuzzy number spaces

The theory of machine learning in metric space is a new research topic and has drawn much attention in recent years. The theoretical foundation of this topic is the question under which conditions two sample sets can be separated in this space. In this paper, motivated by developing a new support vector machine (SVM) in fuzzy number space, we present a necessary and sufficient condition of separating two finite classes of samples by a hyper-plane in n-dimensional fuzzy number space. We also present an attainable expression of maximal margin of the separating hyper-planes which includes some cases of the classes of infinite samples in n-dimensional fuzzy number space. These results generalize and improve the corresponding conclusions for the theory of SVM in Hilbert space to fuzzy number space.