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

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 nona...

Celý popis

Uložené v:

Podrobná bibliografia
Vydané v:	IEEE transaction on neural networks and learning systems Ročník 27; číslo 2; s. 249 - 261
Hlavní autori:	Dongpo Xu, Yili Xia, Mandic, Danilo P.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	United States IEEE 01.02.2016 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Predmet:	Algebra Algorithms Backpropagation Calculus Computer simulation Dynamical systems generalized Hamilton-real (GHR) calculus Heuristic algorithms Learning Learning systems Mathematical analysis Mathematical models Mean square errors nonlinear adaptive filtering Optimization quaternion gradient quaternion least mean square (QLMS) quaternion optimization Quaternions real-time recurrent learning Signal processing algorithms Backpropagation quaternion gradient quaternion least mean square (QLMS) generalized Hamilton-real (GHR) calculus nonlinear adaptive filtering quaternion optimization real-time recurrent learning
ISSN:	2162-237X, 2162-2388, 2162-2388
On-line prístup:	Získať plný text
Tagy:	Pridať tag Žiadne tagy, Buďte prvý, kto otaguje tento záznam!

Popis
Shrnutí:	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.
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23
ISSN:	2162-237X 2162-2388 2162-2388
DOI:	10.1109/TNNLS.2015.2440473