Correspondences Between Fuzzy Equivalence Relations and Kernels: Theoretical Results and Potential Applications

Kernels have proven useful for machine learning, data mining, and computer vision as they provide a means to derive non-linear variants of learning, optimization or classification strategies from linear ones. 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 that allows to incorporate prior knowledge in the design process. It is demonstrated that kernels that map to the unit interval and have constantly 1 in their diagonals can be represented by a commonly used fuzzy-logical formula for representing fuzzy relations. This means that a large and important 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.