The Nystr&#x00F6;m Kernel Conjugate Gradient Algorithm Based on <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-Means Sampling

Abstract

The kernel conjugate gradient (KCG) algorithms have been proposed to improve the convergence rate and the filtering accuracy of kernel adaptive filters (KAFs) efficiently. However, sparsification is necessary in the KCG algorithms to curb the growth of network structure for online applications. To this end, a novel online kernel conjugate gradient algorithm under the mean square error criterion is proposed to approximate the kernel matrix in KAFs by combining k-means sampling into the Nystrom method in a fixed-dimensional feature space, namely a Nystrom kernel conjugate gradient algorithm based on k-means sampling (NysKCG-KM). The approximation accuracy of the kernel matrix as well as the filtering performance of NysKCG-KM are therefore guaranteed by k-means sampling. The proposed NysKCG-KM with no requirement of sparsification can achieve dramatically better filtering accuracy than the kernel least mean square with sparsification, and approach the filtering accuracy of the kernel recursive least squares with sparsification. Monte Carlo simulations using both the synthetic and real-world data validate the superiorities of the proposed NysKCG-KM.

Keywords

• Stochastic gradient descent
• kernel conjugate gradient
• recursive method
• Nyström approximation
• k-means sampling%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">k</italic>-means sampling
• online learning