Complex valued neural network with Möbius activation function

In this work, we propose a new type of activation function for a complex valued neural network (CVNN). This activation function is a special Möbius transformation classified as reflection. It is bounded outside of the unit disk and has partial continuous derivatives but not differentiable since it does not satisfy the Cauchy–Riemann equalities. However, the fixed points set of this function is a circle. Therefore, we employ this function to a specific complex valued Hopfield neural network (CVHNN) and increase the number of fixed points of the CVHNN. Using of this activation function leads us also to guarantee the existence of fixed points of the CVHNN. It is shown that the fixed points are all asymptotically stable states of the CVHNN which indicates that the information capacity is enlarged.

► We propose a new type of activation function for complex valued neural Network (CVNN).
► It is a Möbius transformation that has infinite number fixed points lie on a circle.
► We show that fixed points of CVNN and the activation function correspond each others.
► It is proved that the fixed points are all asymptotically stable states of the CVNN.
► Consequently, the information capacity is enlarged.