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首页> 外文期刊>Neural Networks and Learning Systems, IEEE Transactions on >Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms
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Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms

机译：四元数动态系统的优化：梯度，粗麻和学习算法

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摘要

The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions typically require the calculation of the gradient and Hessian. However, real functions of quaternion variables are essentially nonanalytic, which are prohibitive to the development of quaternion-valued learning systems. To address this issue, we propose new definitions of quaternion gradient and Hessian, based on the novel generalized Hamilton-real (GHR) calculus, thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field, rather than using the isomorphism with 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 novel quaternion gradient and Hessian with their real counterparts. Properties of the quaternion gradient and Hessian relevant to numerical applications are also introduced, opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms. The proposed GHR calculus is shown to yield the same generic algorithm forms as the corresponding real- and complex-valued algorithms. Advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks.
机译：四元数变量的实数标量函数（例如均方误差或数组输出功率）的优化为许多实际应用奠定了基础。解决方案通常需要计算梯度和Hessian。然而，四元数变量的实际函数本质上是非解析的，这阻碍了四元数值学习系统的发展。为了解决这个问题，我们基于新颖的广义汉密尔顿实数（GHR）演算，提出了四元数梯度和Hessian的新定义，从而使直接在四元数域中通用优化算法的有效推导成为可能，而不是将同构用于实际领域，按照当前的做法。此外，与现有的四元数梯度不同，GHR演算允许乘积和链规则，并且允许新颖的四元数梯度和Hessian与它们的真实对应物一一对应。还介绍了四元数梯度的性质和与数值应用有关的Hessian，为四元数优化研究开辟了一条新途径，并大大简化了学习算法的推导。所提出的GHR演算显示出与相应的实值和复值算法相同的通用算​​法形式。与四元数信号处理和神经网络中的说明性仿真相比，所提出的框架的优点得到了阐明。

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