Weak and strong convergence analysis of fully complex-valued high-order TSK model

The higher-order Takagi-Sugeno-Kang (TSK) model, renowned for its interpretability, adaptability, robustness, and ease of training, has been extensively utilized in fuzzy inference and modeling. However, there has been a noticeable scarcity of studies exploring its counterparts in the complex-valued domain, particularly employing fully complex-valued mechanisms. Therefore, this paper introduced an adaptive fully complex-valued fuzzy inference system (AFCFIS). Leveraging Wirtinger calculus, the paper found partial derivatives and updated the network weights according to the gradient descent method, which was easily solved due to the fully complex-valued learning mechanism. Furthermore, the paper provided convergence results of the proposed algorithm under mild conditions. Finally, numerical simulations verified the convergence of AFCFIS, and demonstrated its good performance in both real and complex domain tasks, as well as both regression and classification tasks.