Thomas K. Paul and Tokunbo Ogunfunmi, A Kernel Adaptive Algorithm for Quaternion-Valued Inputs, Neural Networks and Learning Systems, IEEE Transactions on (Volume:26, Issue: 10), IEEE TNNLS October 2015, Page(s): 2422 – 2439, DOI: 10.1109/TNNLS.2014.2383912,
Abstract: 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.
Source: Email from ieeetnnls_AT_gmail.com, 05 Oct. 2015, http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7006723