A Split-Complex Valued Gradient-Based Descent Neuro-Fuzzy Algorithm for TS System and Its Convergence

In order to broaden the study of the most popular and general Takagi–Sugeno (TS) system, we propose a complex-valued neuro-fuzzy inference system which realises the zero-order TS system in the complex-valued network architecture and develop it. In the complex domain, boundedness and analyticity cannot be achieved together. The splitting strategy is given by computing the gradients of the real-valued error function with respect to the real and the imaginary parts of the weight parameters independently. Specifically, this system has four layers: in the Gaussian layer, the L-dimensional complex-valued input features are mapped to a Q-dimensional real-valued space, and in the output layer, complex-valued weights are employed to project it back to the complex domain. Hence, split-complex valued gradients of the real-valued error function are obtained, forming the split-complex valued neuro-fuzzy (split-CVNF) learning algorithm based on gradient descent. Another contribution of this paper is that the deterministic convergence of the split-CVNF algorithm is analysed. It is proved that the error function is monotone during the training iteration process, and the sum of gradient norms tends to zero. By adding a moderate condition, the weight sequence itself is also proved to be convergent.