Robust Kernel Adaptive Filtering Algorithms Based On Conjugate Gradient Method

Adaptive Filter(AF)has become a powerful tool in the fields of signal processing and automatic control because that it can effectively run and track the changes of the statistical characteristics of the input data in an unknown environment.However,many adaptive filter systems present complex input-output nonlinear relationship,which makes the linear structure-based AF limited in practical application,or even cannot work properly.The kernel method transforms the nonlinear problem into a linear one,then develops a series of kernel adaptive filters(KAFs).Due to its simple calculation,stochastic gradient descent(SGD)method is widely adopted to find the optimal solutions in KAFs.However,with a predefined step size for each update,the SGD method may result in slow convergence rates and poor filtering performance.Secondly,practical applications such as radar detection,seismic monitoring and biomedical engineering show that the actual input data often contains a large number of interference signals or noises with pulse characteristic,and most of their statistical characteristics meet the non-Gaussian distribution.Therefore,the performance of the traditional adaptive filter based on the Gaussian distribution model will degrade seriously.In addition,with the growth of input,the linearly growing network topology increases the computational and memory burdens on KAFs.In view of the above problems,the current research of KAFs mainly include three modules: robust error criterion,optimization method and sparsification strategy.Common optimization methods include the SGD method,adaptive optimization method,Newton method,conjugate gradient(CG)method and so on.The CG method can ensure the low computational complexity and space complexity when the filtering accuracy is high.For non-Gaussian noises,robust error criterion such as the low-order error criterion,the high-order error criterion,the log-order error criterion,and the information theoretic learning(ITL)error entropy criterion are proposed.Sparsification processing methods,such as vector quantization criteria,suppress the linearly growing radial basis function network in KAFs,effectively,but these data sparsification methods still have the problem of sublinear growth.However,the structure sparsification effectively inhibits the network growth by prefixing the network size.This thesis will also start from three perspectives: error criterion,optimization method and sparsification strategy,and its work is focused on the following aspects:(1)Improvement from the perspective of the error criterion and optimization method:The CG has lower computational and spatial complexity,to this end,the mean p-power error(MPE)criterion is integrated into the conjugate gradient optimization method,and a weighted least square problem based on MPE criterion is constructed,and the conjugate gradient least mean square p-power(CGLMP)algorithm is proposed to deal with nonGaussian noises.Furthermore,a robust kernel conjugate gradient least mean square ppower(KCGLMP)algorithm is derived for the input-output nonlinear relationship.(2)Optimization algorithm from error criterion and sparsification point of view: The Cauchy loss criterion has a strong theoretical support in robustness research,a new robust random Fourier feature Cauchy conjugate gradient(RFFCCG)algorithm is therefore proposed in this thesis.The random Fourier mapping(RFM),which is mapping the original input data into a fixed-dimensional space,can not only significantly solve the network space subgrowth problem of traditional sparsification method,but also greatly improve the ability of the algorithm and process non-stationary situations.