On representing and generating kernels by fuzzy equivalence relations

B. Moser. On representing and generating kernels by fuzzy equivalence relations. Journal Machine Learning Research, volume 7, pages 2603-2620, 12, 2006.

  • Bernhard Moser
JournalJournal Machine Learning Research
Abstract Kernels are two-placed functions that can be interpreted as inner products of some Hilbert space. By this from linear models of learning, optimization or classification strategies non-linear variants can be derived. Following this idea various kernel-based methods like support vector machines or kernel principal component analysishave been conceived which prove to be successful for machine learning, data mining and computer vision applications. A central question when applying a kernel-based method 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 1 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. Beside this result which only guarantees the existence of such a representation, constructive examples are presented.