Nonlinear Adaptive Filtering with Dimension Reduction in the Wavelet Domain

Recent advances in adaptive filter theory and the hardware for signal acquisition have led to the realization that purely linear algorithms are often not adequate in these domains. Nonlinearities in the input space have become apparent with today's real world problems. Algorithms that process the data must keep pace with the advances in signal acquisition. Recently kernel adaptive (online) filtering algorithms have been proposed that make no assumptions regarding the linearity of the input space. Additionally, advances in wavelet data compression/dimension reduction have also led to new algorithms that are appropriate for producing a hybrid nonlinear filtering framework. In this paper we utilize a combination of wavelet dimension reduction and kernel adaptive filtering. We derive algorithms in which the dimension of the data is reduced by a wavelet transform. We follow this by kernel adaptive filtering algorithms on the reduced-domain data to find the appropriate model parameters demonstrating improved minimization of the mean-squared error (MSE). Another important feature of our methods is that the wavelet filter is also chosen based on the data, on-the-fly. In particular, it is shown that by using a few optimal wavelet coefficients from the constructed wavelet filter for both training and testing data sets as the input to the kernel adaptive filter, convergence to the near optimal learning curve (MSE) results. We demonstrate these algorithms on simulated and a real data set from food processing.