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.
Highlights
Quaternions have become a standard tool in many modern areas, including image processing [1,2], aerospace and satellite tracking [3], modelling of wind profile in renewable energy [4] and in the processing of polarized waves [5,6]
We present some generalized HR (GHR) derivatives of nonlinear quaternion functions enabled by the GHR calculus
The widely linear quaternion least mean square (QLMS) (WL-QLMS) algorithm is based on the quaternion widely linear model y(n) = wT(n)p(n) which deals with the generality of quaternion signals [8,22,35], where p = (xT(n), xiT(n), xjT(n), xkT(n))T is the augmented input vector and w = (hT(n), gT(n), uT(n), vT(n))T is the associated weight vector
Summary
Quaternions have become a standard tool in many modern areas, including image processing [1,2], aerospace and satellite tracking [3], modelling of wind profile in renewable energy [4] and in the processing of polarized waves [5,6]. The objective functions in practical applications are 2 typically based on the mean square error (MSE), a real function of quaternion variables, and are not analytic according to standard quaternion analysis [10,11,12] This is a major obstacle to a more widespread use of quaternions in learning systems. An important consequence of this property is that within the GHR calculus, the choice of the left/right GHR derivative is irrelevant for practical applications of quaternion optimization; this is currently a major source of confusion in the quaternion community Another consequence of the novel product rule is that it enables the calculation of the GHR derivatives for general functions of quaternion variables, and it is generic—if one function within the product is real-valued, this novel product rule degenerates into the traditional product rule, as shown in corollary 4.11.
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