Non-constant bounded holomorphic functions of hyperbolic numbers - Candidates for hyperbolic activation functions
This work addresses a bottleneck in hyperbolic number neural networks by providing new activation function options, though it appears incremental as it extends known concepts to a different number system.
The paper tackles the problem of finding non-constant bounded holomorphic functions for hyperbolic number neural networks, which are unavailable in complex numbers due to the Liouville theorem, and shows that such functions exist for hyperbolic numbers, expanding candidate activation functions.
The Liouville theorem states that bounded holomorphic complex functions are necessarily constant. Holomorphic functions fulfill the socalled Cauchy-Riemann (CR) conditions. The CR conditions mean that a complex $z$-derivative is independent of the direction. Holomorphic functions are ideal for activation functions of complex neural networks, but the Liouville theorem makes them useless. Yet recently the use of hyperbolic numbers, lead to the construction of hyperbolic number neural networks. We will describe the Cauchy-Riemann conditions for hyperbolic numbers and show that there exists a new interesting type of bounded holomorphic functions of hyperbolic numbers, which are not constant. We give examples of such functions. They therefore substantially expand the available candidates for holomorphic activation functions for hyperbolic number neural networks. Keywords: Hyperbolic numbers, Liouville theorem, Cauchy-Riemann conditions, bounded holomorphic functions