Abstract
The recently introduced generalized Hamiltonian–Real (GHR) calculus comprises, for the first time, the product and chain rules that makes it a powerful tool for quaternion-based optimization and adaptive signal processing. In this paper, we introduce novel dual relationships between the GHR calculus and multivariate real calculus, in order to provide a new, simpler proof of the GHR derivative rules. This further reinforces the theoretical foundation of the GHR calculus and provides a convenient methodology for generic extensions of real- and complex-valued learning algorithms to the quaternion domain.
Highlights
Quaternions were introduced by William Hamilton in 1843, as an associative but not commutative algebra over reals, and have been used in many fields, such as physics, computer graphics and signal processing.Because of the non-commutativity of quaternion product, definitions of quaternion derivative are very different from those for real and complex derivatives
Sudbery [1] establishes that only linear quaternion functions fulfil the requirements of traditional derivative definition
HR calculus to provide the powerful product and chain derivative rules that facilitate the calculation of 2 quaternion gradient and Hessian in quaternion optimization problems [3]
Summary
Quaternions were introduced by William Hamilton in 1843, as an associative but not commutative algebra over reals, and have been used in many fields, such as physics, computer graphics and signal processing. Because of the non-commutativity of quaternion product, definitions of quaternion derivative are very different from those for real and complex derivatives. Sudbery [1] establishes that only linear quaternion functions fulfil the requirements of traditional derivative definition. Such a derivative is too stringent for practical optimization problems, whereby the cost functions are often real-valued and non-analytic. HR calculus to provide the powerful product and chain derivative rules that facilitate the calculation of 2 quaternion gradient and Hessian in quaternion optimization problems [3]. The original proof of the product and chain rules for the GHR calculus is quite involved and difficult to understand for non-experts. Based on the so-introduced relationships, we illustrate the effectiveness of such an approach by providing a dual version of the widely linear quaternion least mean square (WL-QLMS) adaptive learning algorithm [4,5], and show that it produces the same output as the primal WL-QLMS, but with a reduced computational complexity
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