On Kernel Adaptive Filtering Algorithms With Sparse Mechanism

As a new class of adaptive filters(AF),kernel adaptive filter(KAF)makes the filters' learning and generalization capabilities further improved by means of the kernel method.However,KAFs in the application process often lead to a large amount of computation,while has the higher requirements with respect to the storage device.In order to address this issue,the researchers have proposed different types of sparsification methods.As one of the most popular sparse methods,the online vector quantization(VQ)strategy has been widely applied in KAFs to curb their linearly increasing network structure,thereby generating a family of quantized kernel adaptive filters(QKAF).This paper discusses the deficiencies of QKAF basing the representative quantized kernel least mean square(QKLMS)algorithm,and then gives some suggestions and measures for further improving the performance of it.The researches will provide a solid application support for the development of nonlinear adaptive filter theory,while they can promote the real-time application of KAFs.The works of this paper focus on the following aspects.(1)Improvement of the structure is considered.To simultaneously improve the convergence rate and filtering accuracy of QKLMS,a convex combination structure is proposed,thereby generating a convex combination of quantized kernel least mean square(CC-QKLMS)algorithm.Due to the application of the online VQ method,CC-QKLMS can naturally avoid the problem of linearly growing network structure.In addition,the combined parameter used here is the kernel width,so that the proposed convex combination structure can be easily extended to other KAFs as long as the Gaussian kernel is adopted.(2)Improvement of the update process is considered.In fact,QKLMS only uses the current prediction error to update the coefficients in the process of update,thus it ignores the difference between the current input and the closest center in the dictionary.Therefore,the gradient descent method is used to update the corresponding coefficient of the closest center in the dictionary,generating modified quantized kernel least mean square(M-QKLMS).It can be easily found that a kernel-based weighting operation is introduced during the update process of M-QKLMS,which reflects the difference between the current input and the closest center in the dictionary.As a result,M-QKLMS takes advantage of more information hidden in the input,and hence improve the filtering accuracy.(3)Improvement of the cost function is considered.The filtering performance of QKAFs based on the mean square error(MSE)criterion often degenerates in some extent under the non-Gaussian noise environments.In order to improve the ability of QKAF to deal with complex noises,the quantized kernel maximum correntropy(QKMC)algorithm is proposed basing the maximum correntropy criterion(MCC).As a similar version of QKLMS,QKMC can maintain good performance when faced with complex noise environments such as impulse noise.Theoretical analysis proves that QKMC can achieve higher filtering accuracy than QKLMS.(4)Combination of the update process and the cost function,QKMC based on bilateral gradient(QKMC-BG)is proposed.In the process of update,QKMC-BG will adjust both the coefficients of the closest center and the current desired output.Therefore,the QKMC-BG considers the case that,for two closest centers in the input space,their corresponding desired output may have a big difference,which needs some necessary adjustments in the output space.As a fixed budget version of QKMC-BG,QKMC-BG with fixed budget(QKMC-BG-FB)can fix the final network size in advance and will not cause large loss in filtering accuracy.