# A Kernel Least Mean Square Algorithm Based on Randomized Feature Networks
## 1 Introduction
Moreover, in nonstationary situations, all selected samples must be evaluated by the significance criteria to determine whether the new sample should be added to centers and which samples should be pruned. During this process, all selected samples and their significant evaluation results must be saved. A large scale of centers not only brings huge storage costs but also increases the burden of computation. To overcome the above difficulties, a novel idea of constructing a nonlinear kernel adaptive filter is proposed in this paper. Because the growth of weight network mainly occurs because the accurate mapping relationship of kernel function cannot be obtained, if an approximate kernel mapping function can be obtained, then the iteratively calculated weight coefficients can avoid the growing structure of the weight network. Fortunately, it has been proven that the shift-invariant kernel can be uniformly approximated by the inner products of the randomized features [18; 19]. Furthermore, the combination of randomized features with simple linear learning approaches can compete favorably in terms of speed and accuracy with the state-of-the-art kernel-based algorithm [18]. The main advantage of explicitly mapping the input sample into the random Fourier feature space (a relatively low-dimensional space) is that the recursive calculation machine of the linear adaptive algorithm can be utilized in each iteration. In addition, the learning process can eliminate the dependence on the historical training data because the feature information of historical data can be stored in the weight coefficients. In fact, random Fourier feature mapping as an effective explicit mapping method has been widely used in classification and regression applications [20; 21; 22]. As far as we know, it has never been used in the nonlinear adaptive signal processing field. To introduce this method into the kernel adaptive filter, the KLMS algorithm is taken as an example due to its simplicity and effectiveness. A framework of the explicit mapping-based KLMS algorithm is given, and the randomized feature networks-based kernel least mean square (KLMS-RFN) algorithm is presented by introducing an explicit random projection strategy. To exhibit the proposed KLMS-RFN algorithm, the following efforts have been made: (1) within the steady-state analysis framework based on the energy conservation relation, a convergence analysis of the proposed algorithm including the mean square convergence condition and a steady-state error analysis is given; (2) the detailed computation complexity of the KLMS-RFN algorithm is presented; and (3) extensive simulations are conducted to demonstrate the performance of the proposed algorithm. The remainder of this article is organized as follows. In Section 2, the background information of the KLMS algorithm and the random feature map is introduced. In addition, the unified framework for constructing the explicit-mapped KLMS algorithm and the KLMS-RFN algorithm is presented. The convergence analysis of the proposed KLMS-RFN algorithm is shown in Section 3. The computation complexity is analyzed in Section 4. Simulation results are illustrated in Section 5. Finally, Section 6 presents the conclusion and discussions of proposed approach.