| The motivation for this proposal is based on a recently revealed interrelationship
between kernels as used in machine learning and fuzzy equivalence relations. This result shows
that normalized kernels can be represented as fuzzy equivalence relations in a way that is commonly
used in fuzzy systems for representing fuzzy relations and fuzzy rule bases. Driven by
the same geometric imagination which led to this insight new mathematical conjectures are stated
whose verification would be of fundamental interest concerning questions of optimization and determining
estimates and bounds in the context of kernel-based methods. In addition to this basic
and fundamental theoretical point of view the revealed relationship also opens up a new way of
looking at the problem of incorporating prior knowledge for the design of kernel-based methods.
The novel aspect is that by this it becomes possible to take advantage from knowledge in terms of
fuzzy sets and relations when designing a kernel. The goal is to explore thoroughly this relationship
and its impact on the design of kernel-based methods from a mathematical as well as from a
machine learning point of view. |