# Quaternion Gradient and Hessian

arXiv: https://arxiv.org/abs/1406.3587 · HTML: https://mlsanity.com/paper/1406.3587/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jun 2014.

## Built on by (most cited first, 9 of 9)

- [A General Framework for Hypercomplex-valued Extreme Learning Machines](https://mlsanity.com/paper/2101.06166/) (2021)
- [Quaternion Convolutional Neural Networks: Current Advances and Future Directions](https://mlsanity.com/paper/2307.08663/) (2023)
- [Time Series Compression using Quaternion Valued Neural Networks and Quaternion Backpropagation](https://mlsanity.com/paper/2403.11722/) (2024)
- [Convex Quaternion Optimization for Signal Processing: Theory and Applications](https://mlsanity.com/paper/2305.06879/) (2023)
- [Quaternion recurrent neural network with real-time recurrent learning and maximum correntropy criterion](https://mlsanity.com/paper/2402.14227/) (2024)
- [Improving Quaternion Neural Networks with Quaternionic Activation Functions](https://mlsanity.com/paper/2406.16481/) (2024)
- [Asymptotically Stable Quaternion-valued Hopfield-structured Neural Network with Periodic Projection-based Supervised Learning Rules](https://mlsanity.com/paper/2510.16607/) (2025)
- [Quaternion Backpropagation](https://mlsanity.com/paper/2212.13082/) (2022)
- [Quaternion Sparse Decomposition for Multi-focus Color Image Fusion](https://mlsanity.com/paper/2505.02365/) (2025)

## Abstract (arXiv)

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.