A Closed-Form Solution for Kernel Adaptive Filtering

Abstract

Unlike the conventional kernel adaptive filtering (KAF) approach of using a fixed kernel to define the Reproducing Kernel Hilbert Space (RKHS), this paper embeds the statistics of the input data in the kernel definition, obtaining a closed-form solution for nonlinear adaptive filtering. We call this solution the Functional Wiener Filter (FWF), and it is formally an extension of Parzen's work on the autocorrelation RKHS to nonlinear functional spaces. We present a method for approximating the FWF in an explicit, finite-dimensional RKHS to model time series directly from realizations, which is less computationally demanding at test time than other KAF methods. We show that FWF outperforms KAF on a synthetic dataset that meets the conditions of the theory, and is comparable to other KAF algorithms for both a chaotic and real-world time series. We demonstrate how the difference equation learned by the FWF can be extracted, leading to possible applications in system identification.