The HR-Calculus: Enabling Information Processing with Quaternion Algebra
This work addresses a foundational gap for researchers in machine learning, signal processing, and control who need to handle quaternion-valued data, though it appears incremental as it builds on existing HR-calculus foundations.
The paper tackles the lack of adaptive information processing techniques for quaternion-valued signals by revising the HR-calculus, providing mathematical tools like gradient operators and derivative rules to enable such processing in the quaternion domain.
From their inception, quaternions and their division algebra have proven to be advantageous in modelling rotation/orientation in three-dimensional spaces and have seen use from the initial formulation of electromagnetic filed theory through to forming the basis of quantum filed theory. Despite their impressive versatility in modelling real-world phenomena, adaptive information processing techniques specifically designed for quaternion-valued signals have only recently come to the attention of the machine learning, signal processing, and control communities. The most important development in this direction is introduction of the HR-calculus, which provides the required mathematical foundation for deriving adaptive information processing techniques directly in the quaternion domain. In this article, the foundations of the HR-calculus are revised and the required tools for deriving adaptive learning techniques suitable for dealing with quaternion-valued signals, such as the gradient operator, chain and product derivative rules, and Taylor series expansion are presented. This serves to establish the most important applications of adaptive information processing in the quaternion domain for both single-node and multi-node formulations. The article is supported by Supplementary Material, which will be referred to as SM.