Kernel LMS algorithm with forward-backward splitting for dictionary learning

Nonlinear adaptive filtering with kernels has become a topic of high interest over the last decade. A characteristics of kernel-based techniques is that they deal with kernel expansions whose number of terms is equal to the number of input data, making them unsuitable for online applications. Kernel-based adaptive filtering algorithms generally rely on a two-stage process at each iteration: a model order control stage that limits the increase in the number of terms by including only valuable kernels into the so-called dictionary, and a filter parameter update stage. It is surprising to note that most existing strategies for dictionary update can only incorporate new elements into the dictionary. This unfortunately means that they cannot discard obsolete kernel functions, within the context of a time-varying environment in particular. Recently, to remedy this drawback, it has been proposed to associate an ℓ₁-norm regularization criterion with the mean-square error criterion. The aim of this paper is to provide theoretical results on the convergence of this approach.