Quaternion Gradient and Hessian

Abstract

The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions often require the calculation of the gradient and Hessian, however, real functions of quaternion variables are essentially non-analytic. To address this issue, we propose new definitions of quaternion gradient and Hessian, based on the novel generalized HR (GHR) calculus, thus making possible efficient derivation of optimization algorithms directly in the quaternion field, rather than transforming the problem to the real domain, as is current practice. In addition, unlike the existing quaternion gradients, the GHR calculus allows for the product and chain rule, and for a one-to-one correspondence of the proposed quaternion gradient and Hessian with their real counterparts. Properties of the quaternion gradient and Hessian relevant to numerical applications are elaborated, and the results illuminate the usefulness of the GHR calculus in greatly simplifying the derivation of the quaternion least mean squares, and in quaternion least square and Newton algorithm. The proposed gradient and Hessian are also shown to enable the same generic forms as the corresponding real- and complex-valued algorithms, further illustrating the advantages in algorithm design and evaluation.

23 pages

References in corpus (3)

• The Complex Gradient Operator and the CR-Calculus
• The Theory of Quaternion Matrix Derivatives
• Quaternion Derivatives: The GHR Calculus

Cited by in corpus (15)

• The Theory of Quaternion Matrix Derivatives
• Quaternion Derivatives: The GHR Calculus
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• A general framework for constrained convex quaternion optimization
• Quaternion Non-negative Matrix Factorization: definition, uniqueness and algorithm
• A General Framework for Hypercomplex-valued Extreme Learning Machines
• Hypercomplex-Valued Recurrent Correlation Neural Networks
• Compressing deep quaternion neural networks with targeted regularization
• Quaternion Convolutional Neural Networks: Current Advances and Future Directions
• Convex Quaternion Optimization for Signal Processing: Theory and Applications
• Properties and Applications of a Restricted HR Gradient Operator
• Solutions of quaternion-valued differential equations with or without commutativity
• Online Learning Algorithms for Quaternion ARMA Model