Positive definite kernel functions on fuzzy sets

Abstract

Abstract-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.

Cross product kernels for fuzzy set similarity

“…Kernels on fuzzy sets is a new area of research on the realm of similarity measures between fuzzy sets. Some of those kernels are the intersection kernel on fuzzy sets which is defined in terms of the T-norm operator and it turns out to be positive definite [3]. The non-singleton Takagi-Sugeno-Kang fuzzy kernel, which can be casted as an intersection kernel.…”

Fuzzy Set Similarity using a Distance-Based Kernel on Fuzzy Sets

“…where F(Ω) denotes the set of fuzzy sets on Ω, that is 16) We say that the value X(x), x ∈ Ω, is the membership degree of x in the fuzzy set X.…”

“…Another advantage is that it is possible to optimize the values β i from data. We denoted by k ∩ + k lin , the kernel resulting from the convex combination of linear kernels on crisp variables and the intersection kernel on fuzzy sets over the fuzzy variables age, menopause, tumor-size and inv-nodes [16]. For the intersection kernel on fuzzy sets, we used the minimum T-norm operator.…”

“…The main contribution of this work is to show how to engineer kernels on fuzzy sets [14,15,16], using some well-known distances between fuzzy sets. The advantage of our approach is that it is possible to have a geometrical interpretation of the similarity measure between fuzzy sets.…”

Granular computing, computational intelligence, and the analysis of non-geometric input spaces

“…From a more operative perspective, instead, IGs typically play the role of computational components in some data-driven inference mechanism; as discussed in the previous sections. Nonetheless, more recently researchers (Ha et al 2013;Guevara et al 2014;Rizzi et al 2013) realized that IGs could be considered also as a particular type of (non-geometric) patterns. This perspective opens the way to a multitude of future research works.…”