Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms.

Area of Science:

• Quaternion calculus
• Optimization theory
• Machine learning

Background:

• Real scalar functions of quaternion variables are crucial for applications like signal processing and neural networks.
• Current methods for optimizing these functions are computationally intensive and hinder the development of quaternion-valued learning systems.
• Existing quaternion gradients lack essential calculus rules like the product and chain rule.

Purpose of the Study:

• To propose novel definitions for quaternion gradient and Hessian using generalized Hamilton-real (GHR) calculus.
• To enable efficient derivation of optimization algorithms directly within the quaternion field.
• To overcome limitations of existing quaternion gradient definitions and facilitate quaternion-valued learning systems.

Main Methods:

• Development of the generalized Hamilton-real (GHR) calculus.
• Introduction of new definitions for quaternion gradient and Hessian.
• Derivation of optimization algorithms directly in the quaternion field.
• Exploration of properties relevant to numerical applications.

Main Results:

• The GHR calculus allows for product and chain rules for quaternion functions.
• A one-to-one correspondence between GHR quaternion gradient/Hessian and their real counterparts is established.
• The proposed GHR calculus yields generic algorithm forms similar to real- and complex-valued algorithms.
• Simplified derivations for learning algorithms in quaternion optimization.

Conclusions:

• The GHR calculus provides an efficient framework for quaternion optimization.
• This approach simplifies derivations and enhances the development of quaternion-valued learning systems.
• The proposed methods demonstrate advantages in quaternion signal processing and neural network simulations.