Fuzzy-Membership-Kernel Learning Based on Takagi-Sugeno Models

A new class of kernels is constructed via incorporating fuzzy information into learning to enhance the performance of kernel-based algorithms. The fuzzy information is given in the form of Takagi-Sugeno (TS) models. First, a TS model is transformed into a structure that includes an inner product. Then an insensitive loss function is employed to identify the parameters in the consequent parts of TS fuzzy rules. By virtue of the kernel tick, the inner product is replaced with common kernels (e.g., Gaussian kernels) in the corresponding dual optimization problem. Consequently, the new kernels, called fuzzy membership kernels (FMKs), are constructed via TS models and common kernels. Note that some research focuses on transforming fuzzy rules into the weights of kernels, or simply, fuzzy weight kernels (FWK), for short. However, FWKs only consider the weights of training points and neglect the weights of test points. Compared with FWKs, the representations of FMKs show that the fuzzy information is incorporated into learning by adding weights to feature mappings of training and test points, respectively. Therefore, FMKs can overcome the shortcomings of FWKs. Finally, a function approximation problem is employed to validate the proposed approach.