A new proof of the generalized Hamiltonian–Real calculus

The recently introduced generalized Hamiltonian–Real (GHR) calculus comprises, for the first time, the product and chain rules that makes it a powerful tool for quaternion-based optimization and adaptive signal processing. In this paper, we introduce novel dual relationships between the GHR calculus...

Descripción completa

Detalles Bibliográficos
Autores principales: Xu, Dongpo, Gao, Hua, Mandic, Danilo P.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: The Royal Society 2016
Materias:
Engineering
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5043305/
http://dx.doi.org/10.1098/rsos.160211
_version_ 1782456730058227712
author Xu, Dongpo
Gao, Hua
Mandic, Danilo P.
author_facet Xu, Dongpo
Gao, Hua
Mandic, Danilo P.
author_sort Xu, Dongpo
collection PubMed
description The recently introduced generalized Hamiltonian–Real (GHR) calculus comprises, for the first time, the product and chain rules that makes it a powerful tool for quaternion-based optimization and adaptive signal processing. In this paper, we introduce novel dual relationships between the GHR calculus and multivariate real calculus, in order to provide a new, simpler proof of the GHR derivative rules. This further reinforces the theoretical foundation of the GHR calculus and provides a convenient methodology for generic extensions of real- and complex-valued learning algorithms to the quaternion domain.
format Online
Article
Text
id pubmed-5043305
institution National Center for Biotechnology Information
language English
publishDate 2016
publisher The Royal Society
record_format MEDLINE/PubMed
spelling pubmed-50433052016-10-04 A new proof of the generalized Hamiltonian–Real calculus Xu, Dongpo Gao, Hua Mandic, Danilo P. R Soc Open Sci Engineering The recently introduced generalized Hamiltonian–Real (GHR) calculus comprises, for the first time, the product and chain rules that makes it a powerful tool for quaternion-based optimization and adaptive signal processing. In this paper, we introduce novel dual relationships between the GHR calculus and multivariate real calculus, in order to provide a new, simpler proof of the GHR derivative rules. This further reinforces the theoretical foundation of the GHR calculus and provides a convenient methodology for generic extensions of real- and complex-valued learning algorithms to the quaternion domain. The Royal Society 2016-09-14 /pmc/articles/PMC5043305/ http://dx.doi.org/10.1098/rsos.160211 Text en © 2016 The Authors. http://creativecommons.org/licenses/by/4.0/ Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
spellingShingle Engineering
Xu, Dongpo
Gao, Hua
Mandic, Danilo P.
A new proof of the generalized Hamiltonian–Real calculus
title A new proof of the generalized Hamiltonian–Real calculus
title_full A new proof of the generalized Hamiltonian–Real calculus
title_fullStr A new proof of the generalized Hamiltonian–Real calculus
title_full_unstemmed A new proof of the generalized Hamiltonian–Real calculus
title_short A new proof of the generalized Hamiltonian–Real calculus
title_sort new proof of the generalized hamiltonian–real calculus
topic Engineering
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5043305/
http://dx.doi.org/10.1098/rsos.160211
work_keys_str_mv AT xudongpo anewproofofthegeneralizedhamiltonianrealcalculus
AT gaohua anewproofofthegeneralizedhamiltonianrealcalculus
AT mandicdanilop anewproofofthegeneralizedhamiltonianrealcalculus
AT xudongpo newproofofthegeneralizedhamiltonianrealcalculus
AT gaohua newproofofthegeneralizedhamiltonianrealcalculus
AT mandicdanilop newproofofthegeneralizedhamiltonianrealcalculus

Ejemplares similares