A Reduced Gaussian Kernel Least-Mean-Square Algorithm for Nonlinear Adaptive Signal Processing

Abstract

The purpose of kernel adaptive filtering (KAF) is to map input samples into reproducing kernel Hilbert spaces and use the stochastic gradient approximation to address learning problems. However, the growth of the weighted networks for KAF based on existing kernel functions leads to high computational complexity. This paper introduces a reduced Gaussian kernel that is a finite-order Taylor expansion of a decomposed Gaussian kernel. The corresponding reduced Gaussian kernel least-mean-square (RGKLMS) algorithm is derived. The proposed algorithm avoids the sustained growth of the weighted network in a nonstationary environment via an implicit feature map. To verify the performance of the proposed algorithm, extensive simulations are conducted based on scenarios involving time-series prediction and nonlinear channel equalization, thereby proving that the RGKLMS algorithm is a universal approximator under suitable conditions. The simulation results also demonstrate that the RGKLMS algorithm can exhibit a comparable steady-state mean-square-error performance with a much lower computational complexity compared with other algorithms.

Appendix

To prove that the reduced Gaussian kernel function expressed in Eq. (7) is a kernel, the characteristics of kernels are used. The following proposition is given [4]. Let $K_1$ and $K_2$ be kernels over $X\times X$, where $X\subseteq R$; let $\alpha \in R^{+}$; let $f(\cdot )$ be a real-valued function on the space X; and let $\varPhi X \rightarrow R^m$. Then, the following propositions can be established:

Proposition 1

$K=K_1+K_2$ is a kernel if $K_1$ and $K_2$ are kernels.

Proposition 2

Due to the characteristics of kernels, $K=\alpha K_1 (k>0)$ is a kernel.

Proposition 3

$K=f(x_i)f(x_j)$.

Hypothesis I Each order term of Eq. (7) is a kernel. Based on Proposition 1, Eq. (7) is a kernel if Hypothesis I is satisfied. This is equivalent to saying that for any $p=\{1,2,\ldots ,P\}$, the function $\sum _{p=1}^{P}\frac{(2a)^p}{p!}x_i^p x_j^p \hbox {exp}(-a(x_i^2+x_j^2))$ is a kernel. Therefore, we need to identify only the pth-order term as a kernel, where $p=\{1,2,\ldots ,P\}$, because a sum of kernels is itself a kernel.

Hypothesis II The kernel parameter a is assumed to be larger than 0.

If Hypothesis I is satisfied, then the coefficients $\frac{(2a)^2}{p!}$ of each order are larger than 0. Based on Proposition 2, Hypothesis II can be said to be satisfied if the function $x_i^p x_j^p \hbox {exp}(-a(x_i^2+x_j^2))$ is a kernel for any $p=\{1,2,\ldots ,P\}$. Clearly, based on Proposition 3, $x_i^p x_j^p \hbox {exp}(-a(x_i^2+x_j^2))=x_i^p x_j^p \hbox {exp}(-ax_i^2) \hbox {exp}(-ax_j^2)=f(x_i)f(x_j)$ holds. Therefore, the reduced Gaussian kernel function given in Eq. (7) is a kernel.