title: Enabling quaternion derivatives: the generalized HR calculus.
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.
authors.affiliation: School of Mathematics and Statistics , Northeast Normal University , Changchun 130024, People's Republic of China.
authors.forename: Dongpo
authors.initials: D
authors.lastname: Xu
authors.affiliation: Department of Electrical and Electronic Engineering , Imperial College London , London SW7 2AZ, UK.
authors.forename: Cyrus
authors.initials: C
authors.lastname: Jahanchahi
authors.affiliation: Department of Computing , University of Surrey , Guildford GU2 7XH, UK.
authors.forename: Clive C
authors.initials: CC
authors.lastname: Took
authors.affiliation: Department of Electrical and Electronic Engineering , Imperial College London , London SW7 2AZ, UK.
authors.forename: Danilo P
authors.initials: DP
authors.lastname: Mandic
keywords: generalized HR calculus; non-analytic quaternion function; nonlinear quaternion functions; quaternion derivatives; quaternion least mean square