Enabling quaternion derivatives: the generalized HR calculus
File(s)
Author(s)
Xu, D
Jahanchahi, C
Took, CC
Mandic, DP
Type
Journal Article
Abstract
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.
Date Issued
2015-08-26
Date Acceptance
2015-07-28
Citation
Royal Society Open Science, 2015, 2 (8)
ISSN
2054-5703
Publisher
The Royal Society
Journal / Book Title
Royal Society Open Science
Volume
2
Issue
8
Copyright Statement
© 2015 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
License URL
Subjects
Science & Technology
Multidisciplinary Sciences
Science & Technology - Other Topics
generalized HR calculus
non-analytic quaternion function
nonlinear quaternion functions
quaternion derivatives
quaternion least mean square
GRADIENT OPERATOR
REGULAR FUNCTIONS
LMS ALGORITHM
COMPLEX
SIGNAL
NETWORKS
Publication Status
Published
Article Number
150255