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Quaternion-Valued Nonlinear Adaptive Filtering

<12>It has always been considered that functions differentiable in the "quaternion" sense to capture the rate of change are limited to linear functions, implying that nonlinear quaternion functions cannot be used in neural networks. This paper is the first demonstration that this is not the case: Quaternions can behave as complex numbers, meaning that quaternion nonlinear functions can be differentiated in the “complex” sense. This is a major breakthrough, and has provided the missing jigsaw piece in neural networks theory to justify the use of quaternion functions for nonlinear modelling. Consequently other works have now emerged involving quaternion nonlinear functions.