Online dictionary learning for kernel LMS. Analysis and forward-backward splitting algorithm
This work addresses the problem of high computational costs in nonlinear system identification for researchers and practitioners in adaptive filtering, though it is incremental as it builds on existing dictionary learning strategies.
The paper tackles the computational burden of kernel adaptive filters by analyzing the need for online dictionary updates in time-varying environments and introduces a kernel least-mean-square algorithm with L1-norm regularization to automatically update the dictionary, showing improved performance in experiments.
Adaptive filtering algorithms operating in reproducing kernel Hilbert spaces have demonstrated superiority over their linear counterpart for nonlinear system identification. Unfortunately, an undesirable characteristic of these methods is that the order of the filters grows linearly with the number of input data. This dramatically increases the computational burden and memory requirement. A variety of strategies based on dictionary learning have been proposed to overcome this severe drawback. Few, if any, of these works analyze the problem of updating the dictionary in a time-varying environment. In this paper, we present an analytical study of the convergence behavior of the Gaussian least-mean-square algorithm in the case where the statistics of the dictionary elements only partially match the statistics of the input data. This allows us to emphasize the need for updating the dictionary in an online way, by discarding the obsolete elements and adding appropriate ones. We introduce a kernel least-mean-square algorithm with L1-norm regularization to automatically perform this task. The stability in the mean of this method is analyzed, and its performance is tested with experiments.