Structure of feature spaces related to fuzzy similarity relations as kernels

Abstract

Fuzzy similarity relations have been considered as kernels in the domain of machine learning in previous works. As a supplementary of this consideration, this paper focuses on construction of feature space related to fuzzy similarity relations as indefinite kernels. First we generalize the Mercer Theorem from positive definite kernels to fuzzy similarity relations. With this generalization we construct the feature space related to a fuzzy similarity relation as a particular Krein space. The proposed feature space is a naturally generalization of the existing Pseudo-Euclidean space from finite to infinite universe of discourses. At the end of this paper we develop an open problem to set up the connection between transitivity and the positive definiteness of a fuzzy similarity relation. Results in this paper complete the mathematical foundation for further applications of fuzzy similarity relations under the framework of kernel tricks.