Uniqueness Theorem of Complex-Valued Neural Networks with Polar-Represented Activation Function

Several models of feed-forward complex-valued neural networks have been proposed, and those with split and polar-represented activation functions have been mainly studied. Neural networks with split activation functions are relatively easy to analyze, but complex-valued neural networks with polar-represented functions have many applications but are difficult to analyze. In previous research, Nitta proved the uniqueness theorem of complex-valued neural networks with split activation functions. Subsequently, he studied their critical points, which caused plateaus and local minima in their learning processes. Thus, the uniqueness theorem is closely related to the learning process. In the present work, we first define three types of reducibility for feed-forward complex-valued neural networks with polar-represented activation functions and prove that we can easily transform reducible complex-valued neural networks into irreducible ones. We then prove the uniqueness theorem of complex-valued neural networks with polar-represented activation functions.