Initialising Kernel Adaptive Filters via Probabilistic Inference

Within kernel methods, kernel adaptive filters (KAFs) [1] are state-of-the-art nonlinear models for time series that build on the properties of reproducing kernel Hilbert spaces (RKHS) [2], in order to provide accurate predictions at a low computational cost. In the same way that support vectors play a fundamental role in support vector machines [3], KAFs rely on a subset of observed input samples referred to as centres, where new inputs are compared to these centres through a kernel function to compute the prediction. This procedure involves a number of parameters: those of the kernel, those related to the selection of the set of centres (dictionary), and those controlling the tradeoff between historical data and new observations. By adapting these model parameters, algorithms, such as kernel least mean square (KLMS) [4], [5] provide an efficient way to improve signal estimation over time as more data become available. Specifically, KLMS applies the least-mean-square rationale to the "kernelised" input (i.e., transformed by the kernel function), thus allowing for an efficient online implementation based on gradient steepest descent for updating the model parameters (i.e., the filter weights only). The main drawback of KAFs is the lack of a principled approach to tune filter weights, kernel parameters and the dictionary.