Generalized Splitting 2D Flexible Activation Function

It is well known that in problems where both amplitude and phase recovery is essential like in signal processing for communications, or in problems of nonlinear signal distortions, like control, signal processing and imaging applications it is important to consider the complex nature (and thus the intimate relation between real and imaginary part) of the data. One of the main problem to design complex neural networks (CpxNN) consists in the definition of the complex Activation Functions (AF): to ensure the universal approximation network capabilities, the AFs should be bounded and differentiable. In the complex domain these characteristics are in contrast with Louiville’s theorem, which asserts that the only bounded and differentiable (analytic) function is the constant function. In this paper we investigate the use of 2D spline to define a new class of flexible activation functions, which are bounded and (locally) analytic suitable to define a new class of complex domain neural networks (CpxNN).