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The Cauchy loss (CL) is a high-order loss function which has been successfully used to overcome large outliers in kernel adaptive filters. The squared error in the CL is then transformed into a reproducing kernel Hilbert space (RKHS) to generate the Cauchy kernel loss (CKL). However, due to its non-convexity, the CKL optimized by the stochastic gradient descent (SGD) suffers from poor performance in the presence of non-Gaussian noise. To improve the performance of CKL, the kernel conjugate gradient (KCG) method combining the half-quadratic (HQ) method twice is used to transform it into a globally convex function. A novel Cauchy kernel loss conjugate gradient (CKCG) algorithm is therefore proposed in the transformed CKL. Simulations on nonlinear system identification in non-Gaussian noises confirm the superiorities of the proposed CKCG from the aspects of robustness and filtering accuracy.

The kernel least mean square (KLMS) using the stochastic gradient descent (SGD) method has been proposed successfully in a Gaussian noise. However, KLMS suffers from performance degradation in terms of convergence rate and filtering accuracy. To this end, an adaptive gradient (Adagrad) optimization algorithm is used to find the solution to the least mean square of error, and the Nystrom method is applied into KLMS to reduce its computational and spatial complexity, generating an online Nystrom kernel Adagrad least mean square (NKALMS) algorithm. To further improve the filtering accuracy of NKALMS, another online Nystrom kernel Adagrad least mean square algorithm using k-means sampling (NKALMS-K) is developed in this paper. Monte Carlo simulations on time series prediction demonstrate that the NKALMS-K algorithm outperforms KLMS and its extensions in the presence of Gaussian noise from the aspects of accuracy and computational efficiency.