Quaternion Derivatives: The GHR Calculus

1. Introduction.

• Real-valued objectives in quaternion optimization are often non-analytic, motivating formal derivatives beyond those defined for analytic functions.The introduction contrasts componentwise real derivatives with HR derivatives using quaternion variables and their conjugates.Source: §1 · p. 1, §1.2 · pp. 3–4
• GHR calculus generalizes HR derivatives over a general orthogonal system and introduces rotation-based product and chain rules, alongside mean-value and Taylor theorems.For real-valued functions, the left and right HR derivative results are identical; the novel product rule reduces to the traditional rule when one factor is real-valued.Source: §1 · p. 2

1.4. The Equivalence Relations and Involutions

• Quaternion involutions let functions of four real components be expressed using quaternion variables and their involutions or conjugate involutions.The involutions support recovery of the four real components from quaternion representations.Source: §1.4 · pp. 5–6

1.5. Results Used to Introduce GHR Derivatives

• The component differentials are independent, allowing the authors to establish that a quaternion-valued linear combination vanishing across rotations has all four coefficients equal to zero.The result is stated for the left case, with the right case proved similarly.Source: §1.5 · pp. 6–7

2.1. Relation Between the HR Derivatives

• The left and right HR derivatives differ in operator placement, but give identical results when the function is real-valued.This equivalence is useful for real-valued objective functions, such as mean square error.Source: §2 · p. 8, §2.1 · p. 9

2.2. Higher Order HR Derivatives

• The HR calculus supports higher-order derivatives by taking the HR derivative of an HR derivative.The section considers second-order left derivatives.Source: §2.2 · p. 9
• Mixed second-order left HR derivatives are generally unequal, while real-valued functions have a stated commutative relation between left and right second-order derivatives.The section also notes a commutative property for second-order left derivatives.Source: §2.2 · pp. 9–10
• Examples show that the traditional product rule is not applicable to either the left or right HR derivative.For both examples, the derivative obtained by differentiating the product differs from the expression given by the traditional product rule.Source: §2.3 · p. 11

3. The Generalization of HR Calculus

• Changing to a general orthogonal basis defines GHR derivatives that include HR derivatives as a special case and support a novel product rule.The calculus can also be extended to other orthogonal systems.Source: §3 · pp. 11–12

3.1. The Novel Product Rule

• The left and right GHR product rules apply to quaternion-valued functions, while real-valued factors recover the traditional product rule.The traditional product rule is a special case of the novel rule; corresponding left and right HR product rules are also given.Source: §3.1 · pp. 13–14

3.2. The Chain Rule

• Left and right GHR chain rules cover quaternion-valued compositions and extend to complex- and real-valued functions of quaternion variables.The section also gives corresponding left and right HR chain rules.Source: §3.2 · pp. 14–16

3.3. Mean Value Theorem

• For continuous quaternion-valued functions with continuous HR derivatives along the segment between two inputs, the section gives left- and right-form mean value theorems.It also provides corresponding results for real-valued functions.Source: §3.3 · pp. 16–18

Taylor’s Theorem

• Under continuity and three-times continuous left HR differentiability along the segment, the section establishes left-form Taylor theorems for quaternion-valued and real-valued functions.The Taylor remainder is of order |λ|^3 as λ approaches zero; a separate center-form theorem is given for real-valued functions.Source: §3.4 · pp. 18–19
• A right-form Taylor theorem is also given for quaternion-valued functions under continuity and three-times continuous right HR differentiability along the segment.A corresponding real-valued right-form corollary is stated.Source: §3.4 · p. 20

• Theorem 3.20 expresses its Taylor expansion using HR derivatives and augments q through quaternion involutions, unlike Schwartz’s perpendicular-quaternion decomposition.Schwartz’s expansion uses a local coordinate system.Source: §3.4 · p. 20
• Theorem 3.20 allows terms such as qωq without restricting f or the fixed point to be real, while requiring real analyticity, as does.The passage contrasts these conditions with restrictions attributed to Schwartz’s expansion.Source: §3.4 · p. 20

3.5. Method of Steepest Descent

• For real-valued f(q), the steepest-descent direction is −∂f/∂q*, extending the corresponding result from R and C.The derivation uses the first-order Taylor expansion and sets q−q_k=αd_k, α>0, shrinking α to neglect the second-order term.Source: §3.5 · pp. 20–21
• The steepest-descent method then expresses an iterative rule for quaternion variables.Source: §3.5 · p. 21

4. Applications of the GHR calculus

• Elementary-function GHR derivatives generalize real derivatives, and the cited functions commute with q and satisfy f*(q) = f(q*), properties noted for neural networks and nonlinear adaptive filters.When q is real, the GHR derivative reduces to the real derivative.Source: §4.1 · p. 22

4.2. Derivation of the QLMS Algorithm

• The GHR-derived QLMS update differs from the update in because that derivation uses the traditional product rule, and it has the same generic form as CMLS.The paper states that the traditional product rule is not applicable for HR calculus.Source: §4.2 · p. 23

4.3. Derivation of the WL-QLMS Algorithm

• WL-QLMS uses a real-valued cost and a widely linear output combining h, g, u, and v with x and its i, j, and k components.The weight updates are given for all four parameter vectors.Source: §4.3 · pp. 23–24
• GHR calculus derives QNGD using the real-valued quadratic cost; when Φ(q) = q, QNGD reduces to QLMS, and Φ need not satisfy the odd-symmetry condition required in.The paper says augmented and widely linear QNGD variants can be derived in the same way.Source: §4.4 · pp. 24–25

5. Conclusions

• GHR calculus uses quaternion rotation to address noncommutativity and applies to both analytic and non-analytic quaternion functions.The conclusion describes the framework as relaxing derivative existence conditions and simplifying calculations through product and chain rules.Source: §5 · p. 25
• Quaternion involutions provide bases such as {q, qi, qj, qk} and their conjugates, supporting the quaternion mean value theorem and Taylor’s theorem.The paper says this framework extends real- and complex-valued optimization algorithms to quaternions.Source: §5 · p. 26
• Adaptive signal-processing examples illustrate the proposed framework’s effectiveness.Source: §5 · p. 26

Appendix A: The derivation of HR calculus

• The derivation represents f through four real-valued component functions of qa, qb, qc, and qd, then expresses its differential using those real variables.The appendix notes that the two differential expressions are identical because the component differentials are real.Source: Appendix A · p. 26
• For the left HR and left conjugate HR cases, the appendix links quaternion and real differentials, substitutes the component differentials, and identifies the resulting derivatives.Source: Appendix A · pp. 26–27
• For the right HR and right conjugate HR cases, the appendix applies rotation transformations before comparing differential expressions to obtain the derivatives.Source: Appendix A · pp. 27–28

Appendix B: The Proof of The Product Rule

• If f has left GHR derivatives and g is real-valued with left GHR derivatives, fg satisfies the traditional product rule.Source: Appendix B · p. 29
• The proof decomposes f and g into real components in the basis {1, iµ, jµ, kµ}, then applies the sum rule to the component products.The appendix identifies this as another orthogonal basis of H.Source: Appendix B · p. 29
• The appendix applies the lemma to the right-hand side of the theorem’s expression and groups derivative terms to complete the proof.Source: Appendix B · pp. 29–30

Appendix C: The Proof of The Chain Rule.

• The theorem proof uses Lemma 5.2 to establish its first equality, with the other equalities derived in a similar fashion.The text introduces the lemma as the tool for proving the chain rule.Source: Appendix C · pp. 30–32
• Lemma 5.2 gives a chain rule for partial derivatives of the quaternion-valued composite function f(g(q)), with ξ drawn from q’s four real components and ν nonzero.The proof treats g(q) through its real-valued components.Source: Appendix C · p. 30
• The proof applies quaternion rotation to express g’s real components, differentiates those expressions, then substitutes and combines terms to obtain the lemma’s first equality.The remaining equality is derived similarly.Source: Appendix C · p. 31

Appendix D: Fundamental Results Based on the GHR Derivatives.

• Table 1 collects important GHR derivative results for constant ν, ω, and λ, a quaternion-valued variable q, and µ as a quaternion constant or function.The appendix introduces examples to show how to use the table.Source: Appendix D · p. 32, Table 1 · p. 33
• One worked example sets g(q) = ωqν + λ and f(q) = g(q)g(q), then applies the product rule to calculate the GHR derivative.The example specifies ν, ω, and λ as constants.Source: Appendix D · p. 34
• Another example takes f(q) = g∗g for g(q) = ωqν + λ and uses the product rule to find its GHR derivative.The listed example defines the function and begins its product-rule calculation.Source: Appendix D · pp. 32, 34
• Further examples apply the derivative operator to identities involving 1 = (ωqν + λ)f(q) and g(q) = f(q)|g(q)|.The latter calculation invokes the product rule in Corollary 3.3.Source: Appendix D · p. 35

Dongpo Xu College of Science, Harbin Engineering

• The listed affiliations include the Department of Computing at the University of Surrey and the Department of Electrical and Electronic Engineering at Imperial College London.The block also contains a fragment naming Harbin and China.Source: Dongpo Xu College of Science, Harbin… · p. 38