Positive definite kernel functions on fuzzy sets

Abstract

Embedding non-vectorial data into a vector space is very common in machine learning, aiming to perform tasks such as classification, regression or clustering. Fuzzy datasets or datasets whose observations are fuzzy sets, are examples of non-vectorial data and, several 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 vector space. To overcome such limitation, we propose the embedding of fuzzy data into a proper Hilbert space of functions called the Reproducing Kernel Hilbert Space (RKHS). This embedding is possible by using a positive definite kernel function on fuzzy sets. We present a formulation of a real-valued kernels on fuzzy sets, in particular, 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 give some examples of kernels on fuzzy sets that can be easily constructed from the previous ones.