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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.

A recent paper titled "The Complex Gaussian Kernel LMS Algorithm" published by Bouboulis and Theodoridis introduced a complex version of the Gaussian kernel LMS (KLMS) algorithm. In this paper, we extend the concepts of complex and complexified RKHS spaces to develop suitable complex Kernel based adaptive algorithms using the Affine projection algorithm (KAPA) method. We apply the complex Gaussian kernel here, as well as develop APA-based algorithms using other suitable complex kernels. We evaluate the performance of the new algorithms using practical simulation applications. The complex KAPA algorithms are seen to outperform their LMS-based counterparts, particularly for applications where convergence rate is important.