A Weighted Gaussian Kernel Least Mean Square Algorithm

• https://link.springer.com/article/10.1007/s00034-023-02337-y

Kernel methods, Least mean square, Reproducing kernel Hilbert space, Gaussian
kernel, Kernel adaptive filtering

In this work, a novel weighted kernel least mean square (WKLMS) algorithm is
proposed by introducing a weighted Gaussian kernel. The learning behavior of the
WKLMS algorithm is studied. Mean square error (MSE) analysis shows that the WKLMS
algorithm outperforms both the least mean square (LMS) and KLMS algorithms in
terms of transient state as well as steady-state responses. We study the effect
of the weighted Gaussian kernel on the associated kernel matrix, its eigenvalue
spread and distribution, and show how these parameters affect the convergence
behavior of the algorithm. Both of the transient and steady-state
mean-square-error (MSE) behaviors of the WKLMS algorithm are studied, and
a stability bound is derived. For a non-stationary environment, tracking analysis
for a correlated random walk channel is presented. We also prove that the
steady-state excess MSE (EMSE) of the WKLMS is Schur convex function of the
weight elements in its kernel weight matrix and hence it follows the majorization
of the kernel weight elements. This helps to decide which kernel weight matrix
can provide better MSE performance. Simulations results are provided to contrast
the performance of the proposed WKLMS with those of its counterparts KLMS and LMS
algorithms. The derived analytical results of the proposed WKLMS algorithm are
also validated via simulations for various step-size values.