Kernel Least Mean Square Algorithm Research Articles

Kernel Adaptive Filtering Over Complex Networks.

This brief is concerned with the problem of kernel adaptive filtering for a complex network. First, a coupled kernel least mean square (KLMS) algorithm is developed for each node to uncover its nonlinear measurement function by using a series of input-output data. Subsequently, an upper bound is derived for the step-size of the coupled KLMS algorithm to guarantee the mean square convergence. It is shown that the upper bound is dependent on the coupling weights of the complex network. Especially, an optimal step size is obtained to achieve the fastest convergence speed and a suboptimal step size is presented for the purpose of practical implementations. Besides, a coupled kernel recursive least square (KRLS) algorithm is further proposed to improve the filtering performance. Finally, simulations are provided to verify the validity of the theoretical results.

Robust kernel adaptive filtering for nonlinear time series prediction

Recently, online learning algorithms in machine learning have been imposed much attention. As a typical family, kernel adaptive filtering algorithms receive particular interest due to their sequential learning-based features. However, the kernel least mean square (KLMS) algorithm is not suitable for nonlinear tasks corrupted by non-Gaussian noise, especially impulsive noise. This is because the derivation of the KLMS algorithm is on the basis of the mean square error (MSE) criterion which only captures information of second-order statistics. In this paper, motivated by tanh function, we develop its generalized variant by introducing a scale factor for better representation capability; then we incorporate kernel adaptive filter with the generalized tanh function to propose a robust sequential learning algorithm. Based on establishing the energy conservation relation, we derive a sufficient condition for ensuring the algorithm convergence. In addition, to perform the steady-state excess mean square error (EMSE) analysis, we use the pre-tuned dictionary strategy to model the unknown nonlinear system in form of a finite-order combination; by Taylor expansion, we arrive at a closed-form solution for predicting the steady-state behavior. To further improve the algorithm performance, we design an optimization scheme for scale factor. Simulations for nonlinear time series prediction show that the designed schemes yield better performance than some state-of-art algorithms. The steady-state EMSE analysis is validated to provide accurate prediction results.

Adaptive Sparse Quantization Kernel Least Mean Square Algorithm for Online Prediction of Chaotic Time Series

Kernel least mean square (KLMS) algorithm is a popular method for time series online prediction. It has the advantages of good robustness, low computational complexity, model simplicity and online learning ability. Unfortunately, as input data grows, the dictionary size increases and the computational complexity raises significantly. In addition, how to improve the adaptability in time-varying environments with noise is also one of the main challenges. Therefore, we propose an improved KLMS algorithm from sparse perspective in response to the above problems, called adaptive sparse quantization kernel least mean square (ASQ-KLMS) algorithm. In the new model, sequential outlier criterion for sparsification and weights adaptive adjustment are combined with coherence criterion and quantization to form ASQ-KLMS algorithm. Firstly, it makes full use of effective information and ignores the interference of abnormal information to obtain a more accurate and compact dictionary. Then, a good balance between algorithm efficiency and accuracy can be achieved by controlling the choice of parameters. In addition, it can adaptively adjust weights in time-varying environment. At last, the Lorenz chaotic time series, the ENSO chaotic time series and the Beijing $$\text {P}{{\text {M}}_{2.5}}$$ chaotic time series are used to prove the reliability of the ASQ-KLMS algorithm.

A Wideband 5G Cyclostationary Spectrum Sensing Method by Kernel Least Mean Square Algorithm for Cognitive Radio Networks

This brief proposes a new cyclostationary method on the basis of the Gaussian Kernel Least Mean Square (KLMS) algorithm for wideband spectrum sensing in the upcoming 5G mobile networks. To focus on a special fragment of the 5G network frequency spectrum, the KLMS algorithm uses a set of inaccurate cyclic frequencies of low complexity belonging to the signal of the primary user (PU). A suboptimal detector is designed on the basis of the KLMS weights and the performance of the proposed detector is measured and compared with other recent spectrum sensing techniques. The method is very easy to implement and less complex to compute than other spectrum-sensing counterparts. Analytical and simulation results have been verified by an experimental 5G communication setup which shows the effectiveness of the proposed method.

Kernel Least Mean Square Based on the Nyström Method

The kernel least mean square (KLMS) algorithm is the simplest algorithm in kernel adaptive filters. However, the network growth of KLMS is still an issue for preventing its online applications, especially when the length of training data is large. The Nystrom method is an efficient method for curbing the growth of the network size. In this paper, we apply the Nystrom method to the KLMS algorithm, generating a novel algorithm named kernel least mean square based on the Nystrom method (NysKLMS). In comparison with the KLMS algorithm, the proposed NysKLMS algorithm can reduce the computational complexity, significantly. The NysKLMS algorithm is proved to be convergent in the mean square sense when its step size satisfies some conditions. In addition, the theoretical steady-state excess mean square error of NysKLMS supported by simulations is calculated to evaluate the filtering accuracy. Simulations on system identification and nonlinear channel equalization show that the NysKLMS algorithm can approach the filtering performance of the KLMS algorithm by using much lower computational complexity, and outperform the KLMS with the novelty criterion, the KLMS with the surprise criterion, the quantized KLMS, the fixed-budget QKLMS, and the random Fourier features KLMS.

Chromatic dispersion and nonlinear phase noise compensation based on KLMS method

In this study, kernel least mean square (KLMS) algorithm with fractionally spaced equalizing structure is proposed for electrical compensation of chromatic dispersion (CD) and nonlinear phase noise (NLPN) in a dual polarization optical communications system with coherent detection. We consider single mode fiber channel. At the receiver, the additive optical noise is represented as additive white Gaussian noise. Phase modification is utilized at high signal powers to maintain the validity of Gaussian model of noise. We consider QAM and PSK modulations and evaluate the performance of the proposed method in terms of error rate, phase error, and error vector magnitude (EVM). The results are obtained in both linear and nonlinear regimes. In the linear region, the KLMS algorithm can compensate CD and NLPN effectively and outperforms the existing compensation methods such as LMS, minimum mean square error (MMSE), and time domain FIR filter. In nonlinear regime, where the input power is higher, NLPN is stronger which results in compensation performance degradation. However, KLMS still achieves better results than the above algorithms.

Stochastic Behavior Analysis of the Gaussian Kernel Least-Mean-Square Algorithm

The kernel least-mean-square (KLMS) algorithm is a popular algorithm in nonlinear adaptive filtering due to its simplicity and robustness. In kernel adaptive filters, the statistics of the input to the linear filter depends on the parameters of the kernel employed. Moreover, practical implementations require a finite nonlinearity model order. A Gaussian KLMS has two design parameters, the step size and the Gaussian kernel bandwidth. Thus, its design requires analytical models for the algorithm behavior as a function of these two parameters. This paper studies the steady-state behavior and the transient behavior of the Gaussian KLMS algorithm for Gaussian inputs and a finite order nonlinearity model. In particular, we derive recursive expressions for the mean-weight-error vector and the mean-square-error. The model predictions show excellent agreement with Monte Carlo simulations in transient and steady state. This allows the explicit analytical determination of stability limits, and gives opportunity to choose the algorithm parameters a priori in order to achieve prescribed convergence speed and quality of the estimate. Design examples are presented which validate the theoretical analysis and illustrates its application.

Unsupervised kernel least mean square algorithm for solving ordinary differential equations

In this paper a novel method is introduced based on the use of an unsupervised version of kernel least mean square (KLMS) algorithm for solving ordinary differential equations (ODEs). The algorithm is unsupervised because here no desired signal needs to be determined by user and the output of the model is generated by iterating the algorithm progressively. However, there are several new approaches in literature to solve ODEs but the new approach has more advantages such as simple implementation, fast convergence and also little error. Furthermore, it is also a KLMS with obvious characteristics. In this paper the ability of KLMS is used to estimate the answer of ODE. First a trial solution of ODE is written as a sum of two parts, the first part satisfies the initial condition and the second part is trained using the KLMS algorithm so as the trial solution solves the ODE. The accuracy of the method is illustrated by solving several problems. Also the sensitivity of the convergence is analyzed by changing the step size parameters and kernel functions. Finally, the proposed method is compared with neuro-fuzzy [21] approach.

Theoretical convergence analysis of complex Gaussian kernel LMS algorithm

With the vigorous expansion of nonlinear adaptive filtering with real-valued kernel functions, its counterpart complex kernel adaptive filtering algorithms were also sequentially proposed to solve the complex-valued nonlinear problems arising in almost all real-world applications. This paper firstly presents two schemes of the complex Gaussian kernel-based adaptive filtering algorithms to illustrate their respective characteristics. Then the theoretical convergence behavior of the complex Gaussian kernel least mean square (LMS) algorithm is studied by using the fixed dictionary strategy. The simulation results demonstrate that the theoretical curves predicted by the derived analytical models consistently coincide with the Monte Carlo simulation results in both transient and steady-state stages for two introduced complex Gaussian kernel LMS algorithms using non-circular complex data. The analytical models are able to be regard as a theoretical tool evaluating ability and allow to compare with mean square error (MSE) performance among of complex kernel LMS (KLMS) methods according to the specified kernel bandwidth and the length of dictionary.