On The Fixed-dimensional Feature Spaces Based Robust Adaptive Filters

As a powerful tool for statistic signal processing,the adaptive filter has been successfully applied in the fields of information processing,automatic control,targets tracking,and biomedical science.For these real-world applications,the statistics of the modeling environments are not purely Gaussian but generally non-Gaussian,i.e.,light-tailed(e.g.,uniform and binary)and heavy-tailed(e.g.,Laplace and ?-stable).In addition,the modeling data of system is usually nonlinear,large-scaled,multivariate,and non-stationary.From the aspect of system environment,the normalization,variable step size,and nonquadratic cost functions were introduced in adaptive filters for robust learning,and the design of cost function is the most fascinating strategy.From the aspect of data,the random features approximation method was used to approximate the kernel space as the fixed-dimensional feature space,thus reducing the computation cost and storage requirement,efficiently.However,all these two strategies based adaptive filters cannot provide the desirable filtering accuracy,computational and storage efficiencies,and tracking capability in both Gaussian and non-Gaussian environments,simultaneously.To this end,we propose different fixed-dimensional robust adaptive filters in original data space,random features space,and complex-valued features space to address afore-mentioned issues.The main works of this paper are as follows.(1)In the original data space,to combat both Gaussian and non-Gaussian noises,a novel generalized logarithmic cost function is presented,then the robust least mean logarithmic square(RLMLS)algorithm is proposed by minimizing this cost function and applying the gradient descend method.For theoretical analysis,the transient performance,steady-state performance,and tracking performance of the RLMLS algorithm are derived in the mean square sense.To improve the convergence rate and filtering accuracy of the RLMLS algorithm further,the variable step-size RLMLS(VSSRLMLS)algorithm is proposed using a novel variable step-size method.The proposed RLMLS and VSSRLMLS algorithms can provide the robustness against both Gaussian and non-Gaussian noises,and the performance improvement over the traditional filters.(2)To address the concave issue of correntropy in convex optimization,we transform the correntropy to a half-quadratic function by applying the half-quadratic(HQ)optimization method.Then,the original maximum correntropy problem is transformed to the weighted least-squares problem,leading to the correntropic HQ(CHQ)optimization method.The developed CHQ method can be extended to various local similarity measures and is global convex.By using the random Fourier features to approximate Gaussian kernel,the random Fourier features mapping(RFFM)is developed to project the input signal into the fixed-dimensional random Fourier features space(RFFS).Thus,the random Fourier features kernel correntropy conjugate gradient(RFFKCCG)algorithm is proposed by using the conjugate gradient method to solve the weighted least-squares problem.The RFFKCCG algorithm with a linear filter structure can reduce the computational and space complexities of kernel adaptive filters significantly and guarantee the filtering accuracy in both impulsive and non-impulsive noise environments.(3)Based on the traditional random Fourier features method,by applying the complexification of the real reproducing kernel Hilbert spaces(RKHSs),the complex RFFM(CRFFM)is presented to project the complex-valued data into the fixed-dimensional complex RFFS.To combat complex non-Gaussian noises,the complex Cauchy cost function is proposed,and its important properties and proofs are provided.Then,by using the CRFFM to project the input signal into the complex RFFS and minimizing the complex Cauchy cost function,the complex random Fourier features recursive complex Cauchy(CRFFRCC)algorithm is proposed.In complex non-Gaussian noise environments,the CRFFRCC algorithm can provide complexity reduction and performance improvement over traditional complex adaptive filters,respectively.In addition,the CRFFRCC algorithm can provide effective tracking performance in non-stationary systems.