Non-constant bounded holomorphic functions of hyperbolic numbers - Candidates for hyperbolic activation functions

TL;DR

This paper explores bounded holomorphic functions of hyperbolic numbers, revealing non-constant examples that could serve as novel activation functions in hyperbolic neural networks.

Contribution

It introduces non-constant bounded holomorphic functions of hyperbolic numbers, expanding potential activation functions for hyperbolic neural networks.

Findings

Existence of non-constant bounded holomorphic functions of hyperbolic numbers

Examples of such functions provided

Potential for new activation functions in hyperbolic neural networks

Abstract

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