| Quaternion is a non exchanged numerical system. It is put forward by Hamilton who is a physicist. The theory in quaternion analysis plays an important role in color image processing and prediction of wind speed signal etc. In addition, the reproducing kernel Hilbert space (RKHS) is an ideal space frame in numerical analysis. It is because that the data of the original space can be mapped to high dimensional feature space by nonlinear mapping, then the inner product operation of high-dimensional feature space by the nonlinear transformation can be converted to the calculation of the kernel function. The theory of RKHS has been widely used in the fields on neural network, numerical computation, machine learning and so on. Therefore, the theory of. quaternion reproducing kernel Hilbert space (QRKHS) is of great significance. It can be concluded that this kind of technology has a bright prospect of application in the field of quaternion function's differential theory in QRKHS. However, the quaternion's product can be exchanged. The regular conclusion is invalid, so the study becomes more challenging.In this paper, we have constructed the differential theory in QRKHS on the basis of quaternion function's differential theory. We has applied the differential theory in QRKHS to the kernel least mean square algorithm and maximum correntropy algorithm. We solve three problems:1. We present Frechet-HR according to the Frechet differential and Taylor expansion in Hilbert space. The Frechet-GHR in QRKHS which is on the basis of the GHR appears in this paper.2. The Frechet-GHR is put forward in QRKHS according to the Frechet-HR, the product rule and chain rule according to the GHR in QRKHS. We find out some examples by the Frechet-GHR. We modified the product rule and example in reference [21].3. We study kernel least mean square algorithm (QKLMS1 and QKLMS2) and improve quaternion maximum correntropy algorithm (QKMC) by the product rule and chain rule of the Frechet-GHR. |
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