The use of quaternion data can provide benefit in applications like robotics and image recognition, and particularly for performing transforms in 3-D space. Here, we describe a kernel adaptive algorithm for quaternions. A least mean square (LMS)-based method was used, resulting in the derivation of the quaternion kernel LMS (Quat-KLMS) algorithm. Deriving this algorithm required describing the idea of a quaternion reproducing kernel Hilbert space (RKHS), as well as kernel functions suitable with quaternions.
A modified HR calculus for Hilbert spaces was used to find the gradient of cost functions defined on a quaternion RKHS. In addition, the use of widely linear (or augmented) filtering is proposed to improve performance. The benefit of the Quat-KLMS and widely linear forms in learning nonlinear transformations of quaternion data are illustrated with simulations.