Closed-form conditions for convergence of the Gaussian kernel-least-mean-square algorithm

In addition to the choice of the usual linear adaptive filter parameters, designing kernel adaptive filters requires the choice of the kernel and its parameters. One of our recent works has brought a new contribution to the discussion about kernel-based adaptive filtering by providing the first convergence analysis of the kernel-LMS algorithm with Gaussian kernel. A necessary and sufficient condition for convergence has been clearly established. Checking the stability of the algorithm can, unfortunately, be computationally expensive because one needs to calculate the extreme eigenvalues of a large matrix, for each set of candidate tuning parameters. The aim of this paper is to circumvent this drawback by examining two easy-to-handle conditions that allow to examine how the stability limit varies as a function of the step-size, the kernel bandwidth, and the filter length. One of them is a conjectured necessary and sufficient condition for convergence that allows to greatly simplify calculations.