A better metric in kernel adaptive filtering

The metric in the reproducing kernel Hilbert space (RKHS) is known to be given by the Gram matrix (which is also called the kernel matrix). It has been reported that the metric leads to a decorrelation of the kernelized input vector because its autocorrelation matrix can be approximated by the (down scaled) squared Gram matrix subject to some condition. In this paper, we derive a better metric (a best one under the condition) based on the approximation, and present an adaptive algorithm using the metric. Although the algorithm has quadratic complexity, we present its linear-complexity version based on a selective updating strategy. Numerical examples validate the approximation in a practical scenario, and show that the proposed metric yields fast convergence and tracking performance.