| This proposal focuses on a recently revealed interrelationship between kernels as
used in machine learning, on the one hand, and fuzzy equivalence relations on the other hand.
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 measure 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 axiomtic 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. |