Research On Adaptive Filtering Algorithms Based On Kernel Approximation Technology

At present,the traditional kernel adaptive filtering algorithm has a large computational complexity due to the unrestricted growth of the weighted network.Nuclear approximation technology has the advantage of accelerating the training efficiency of nuclear learning.The Random Fourier Features(RFF)kernel adaptive filtering algorithm,which is introduced into the field of kernel adaptive filtering,fundamentally solves the shortcomings of weight network growth and greatly reduces the computational complexity,but there are still some shortcomings such as computational complexity,accuracy and convergence speed.Specific manifestations include:(1)For the application requirements of online filtering in embedded systems with limited computing resources,the complexity of kernel approximation method based on stochastic Fourier feature is high;(2)the lack of optimization method for stochastic Fourier feature parameters leads to low accuracy of the algorithm;(3)the inadequate control ability for weight updating in the convergence process leads to slow convergence speed and many other problems.Therefore,this paper focuses on the problems of high complexity,low precision and slow convergence speed of the current algorithm,and studies the adaptive filtering algorithm based on kernel approximation technology.The main research work completed in this paper is as follows:Because the kernel adaptive filtering algorithm based on stochastic Fourier feature needs to map the input to the higher dimension feature space to achieve a satisfactory steadystate accuracy,it leads to the problem of high time complexity,large computational and storage resources,which makes it difficult to apply to online adaptive filtering applications with limited computational resources.In order to solve the shortcomings of feature mapping methods from the perspective of kernel approximation technology,this paper proposes a minimum mean square algorithm based on reduced Gauss kernel function.A reduced Gauss kernel function is obtained by approximating the lower order terms of partial Taylor expansion of Gauss kernel,which reduces the dimension required for feature mapping and reduces the computational complexity of the kernel adaptive filtering algorithm.In addition,the convergence performance of the proposed algorithm is analyzed by using a unified non-linear filtering algorithm to analyze the energy conservation relationship of the framework.By studying the energy flow of each iteration,the energy relationship between adjacent iterations and the sufficient conditions of mean square convergence are given.Theupper and lower bounds of the theoretical values of steady-state mean square error are established,and the basis for setting some parameters of the algorithm is provided.The results of time series prediction and channel equalization simulation experiments show that the proposed method can achieve better accuracy and less computational complexity than existing algorithms,and is suitable for online non-linear filtering application scenarios with limited computational resources.Frequency parameters of stochastic Fourier feature play an important role in the accuracy of least mean square algorithm of stochastic Fourier feature kernel.The frequency parameters are random sampling values with a certain probability distribution.Because of the randomness of sampling,the accuracy of approximate kernels is not high,which results in the poor steady-state accuracy of the algorithm.Therefore,aiming at the problem of poor steady-state accuracy caused by the lack of optimization methods for sampling the frequency parameters of stochastic Fourier features,this paper studies the optimization methods for sampling the characteristic parameters of stochastic Fourier features.Firstly,a pre-training strategy based on training MSE is proposed to screen the sample set of frequency parameters under different dimensions.The optimal sample set of frequency parameters is obtained to improve the steady-state accuracy of the algorithm.The experimental results show that the steady-state accuracy of the proposed algorithm is greatly improved compared with that of the random sampling method.However,the preprocessing method for training the whole set of frequency parameters is very complex,so this paper further proposes a preprocessing method for evaluating the sampling value of the characteristic parameters directly.By measuring the sampling value of the frequency parameters of random Fourier features by using the nuclear polarization method,a group of features with the training samples are screened out in the preprocessing stage of the characteristic parameters.Sample sets of highly consistent parameters.The RFF assigned by the parameter sample set is called the polarization random Fourier feature.The results of simulation experiments show that the polarization preprocessing method of the characteristic parameter can effectively improve the minimum mean square algorithm of the random Fourier feature core.Aiming at the problem that the existing stochastic Fourier eigenvalue adaptive filtering algorithm can not guarantee good steady-state accuracy and convergence speed at the same time because of the fixed step size,the least mean square algorithm of polarized stochastic Fourier eigenvalue kernel is taken as the research object,and the optimization method of convergence performance is studied.A variable step size strategy is proposed.By dynamically adjusting the coefficients of the error energy term in the updating step size,the algorithm can increase the regulation of step size parameters when the expected error is large,so as to improve the dynamic convergence performance of the algorithm.Furthermore,the convergence performance of the minimum mean square algorithm for polarized random Fourier feature kernels with variable step size strategy is studied.Momentum technology and variable forgetting factor strategy are introduced to improve the weight convergence process of the algorithm,and a variable forgetting factor variable step polarized random Fourier feature kernels minimum mean square algorithm is proposed.The strategy of variable forgetting factor adjusts the influence of historical information on weight optimization process dynamically,improves the utilization efficiency of weight updating to historical information in non-stationary environment,and further improves the convergence speed and dynamic tracking ability of the algorithm.Extensive simulation experiments show that the convergence speed of the proposed variable forgetting factor variable step polarization random Fourier feature kernel least mean square algorithm is improved.