A Quaternion Gradient Operator and Its Applications

Real functions of quaternion variables are typical cost functions in quaternion valued statistical signal processing, however, standard differentiability conditions in the quaternion domain do not permit direct calculation of their gradients. To this end, based on the isomorphism with real vectors and the use of quaternion involutions, we introduce the HR calculus as a convenient way to calculate derivatives of such functions. It is shown that the maximum change of the gradient is in the direction of the conjugate gradient, which conforms with the corresponding solution in the complex domain. Examples in some typical gradient based optimization settings support the result.