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On Representing and Generating Kernels by Fuzzy Equivalence Relations

Bernhard Moser; 7(93):2603−2620, 2006.


Kernels are two-placed functions that can be interpreted as inner products in some Hilbert space. It is this property which makes kernels predestinated to carry linear models of learning, optimization or classification strategies over to non-linear variants. Following this idea, various kernel-based methods like support vector machines or kernel principal component analysis have been conceived which prove to be successful for machine learning, data mining and computer vision applications. When applying a kernel-based method a central question is the choice and the design of the kernel function. This paper provides a novel view on kernels based on fuzzy-logical concepts which allows to incorporate prior knowledge in the design process. It is demonstrated that kernels mapping to the unit interval with constant one in its diagonal can be represented by a commonly used fuzzy-logical formula for representing fuzzy rule bases. This means that a great class of kernels can be represented by fuzzy-logical concepts. Apart from this result, which only guarantees the existence of such a representation, constructive examples are presented and the relation to unlabeled learning is pointed out.

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