Quaternion Gradient and Hessian

TL;DR

This paper introduces new definitions of quaternion gradient and Hessian using GHR calculus, enabling efficient optimization directly in the quaternion domain and simplifying algorithm derivation.

Contribution

It proposes the generalized HR calculus for quaternion derivatives, allowing product and chain rules, and aligns quaternion derivatives with real counterparts.

Findings

Simplifies derivation of quaternion LMS algorithms

Enables quaternion least squares and Newton methods

Maintains correspondence with real-valued derivatives

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.