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In this paper, the authors propose a novel online scheme for the sparse adaptive filtering problem. It is based on a formulation of the adaptive filtering problem as a minimization of the sum of (possibly no smooth) convex functions. Their proposed scheme is a time-varying extension of the so-called Douglas-Rachford splitting method. It covers many existing adaptive filtering algorithms as special cases. They show several examples of special choices of the cost functions that reproduce those existing algorithms. Their scheme achieves a monotone decrease of an upper bound of the distance to the solution set of the minimization under certain conditions. They applied a simple algorithm that falls under their scheme to a sparse echo cancellation problem where it shows excellent convergence performance.
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