Positive Definite Kernel Functions on Fuzzy Sets

Embedding non-vectorial data into a normed vectorial space is very common in machine learning, aiming to perform tasks such classification, regression, clustering and so on. Fuzzy datasets or datasets whose observations are fuzzy sets, are an example of non vectorial data and, many of fuzzy pattern recognition algorithms analyze them in the space formed by the set of fuzzy sets. However, the analysis of fuzzy data in such space has the limitation of not being a vectorial space. To overcome such limitation, in this work, we propose the embedding of fuzzy data into a proper Hilbert space of functions called the Reproducing Kernel Hilbert Space or RKHS. This embedding is possible using a positive definite kernel function defined on fuzzy sets. As a result, we present a formulation of a real-valued kernels on fuzzy sets, particularly, we define the intersection kernel and the cross product kernel on fuzzy sets giving some examples of them using T-norm operators. Also, we analyze the nonsingleton TSK fuzzy kernel and, finally, we gave several examples of kernels on fuzzy sets, that can be easily constructed from the previous ones