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Enabling quaternion derivatives: the generalized HR calculus

Cheong Took, C (2015) Enabling quaternion derivatives: the generalized HR calculus Royal Society Open Science, 2 (8).

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Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.

Item Type: Article
Subjects : Computing
Divisions : Surrey research (other units)
Authors :
Cheong Took,
Date : 26 August 2015
DOI : 10.1098/rsos.150255
Copyright Disclaimer : © 2015 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License, which permits unrestricted use, provided the original author and source are credited.
Uncontrolled Keywords : Generalized HR calculus, Non-analytic quaternion function, Quaternion derivatives, Quaternion least mean square, Nonlinear quaternion functions
Depositing User : Symplectic Elements
Date Deposited : 17 May 2017 13:58
Last Modified : 25 Jan 2020 00:36

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