arXiv Analytics

Sign in

arXiv:1407.5178 [math.OC]AbstractReferencesReviewsResources

Properties and Applications of a Restricted HR Gradient Operator

Mengdi Jiang, Yi Li, Wei Liu

Published 2014-07-19Version 1

For quaternionic signal processing algorithms, the gradients of a quaternion-valued function are required for gradient-based methods. Given the non-commutativity of quaternion algebra, the definition of the gradients is non-trivial. The HR gradient operator provides a viable framework and has found a number of applications. However, the applications so far have been mainly limited to real-valued quaternion functions and linear quaternion-valued functions. To generalize the operator to nonlinear quaternion functions, we define a restricted version of the HR operator. The restricted HR gradient operator comes in two versions, the left and the right ones. We then present a detailed analysis of the properties of the operators, including several different product rules and chain rules. Using the new rules, we derive explicit expressions for the derivatives of a class of regular nonlinear quaternion-valued functions, and prove that the restricted HR gradients are consistent with the gradients in real domain.

Comments: This paper is about a quaternion-valued gradient operator and its properties and applications
Categories: math.OC
Related articles: Most relevant | Search more
arXiv:0803.2091 [math.OC] (Published 2008-03-14, updated 2008-07-04)
Properties and applications of dual reduction
arXiv:1304.7892 [math.OC] (Published 2013-04-30)
Metric Regularity of the Sum of Multifunctions and Applications
arXiv:1101.1019 [math.OC] (Published 2011-01-05)
Symmetry in variational principles and applications