Kernels in the sense of machine learning are two-placed functions whose values can be represented as inner products in some Hilbert space. By this, classification algorithms based on and restricted to linear models can be carried over to non-linear methods by replacing the Gram matrix by a kernel map. Following this strategy in the last decades powerful kernel-based methods like support vector machines, kernel principal component analysis and kernel Fischer discriminant were developped and successfully applied to classification and learning problems in the fields of computer vision, bioinformatics and data mining.

It is interesting that in literature the mechanism of the so-called kernel trick is often motivated heuristically by arguments pointing out that kernels act as a similarity measures without providing any axiomatic framework for this aspect. According to this similarity argument two different data points being similar are mapped close together in the Hilbert space by which the data points are arranged in a more separable manner.

This is the point where fuzzy equivalence relations come in which---based on a relaxed concept for transitivity---provide an axiomatic framework for similarity as a generalization of the classical concept of an equivalence relation. It is worth pointing out that for classical relations the concepts of an equivalence relation and a kernel map are equivalent. Recently it could be demonstrated that all kernels mapping to the unit interval with constant one in its diagonal can be represented as fuzzy equivalence relations in a way that is commonly used in fuzzy systems for representing fuzzy rule bases. At first glance fuzzy systems and kernel-based methods are totally different paradigms. Fuzzy systems are used to model explicit knowledge of features and dependencies, e.g., by means of rule bases. In contrast, kernel-based methods act implicitly in an Hilbert space induced by a feature map for which only the existence has to be guaranteed, that is no explicit knowledge of the feature map is required.

The goal of this contribution is to present recent results and to discuss the synthesis of knowledge-based and kernel-based paradigms.