Robust adaptive sparse system identification by using weighted ℓ ₁ balls and Moreau envelopes

This paper presents a novel approach to the time-recursive sparse system identification task by revisiting the classical Wiener-Hopf equation. The proposed methodology is built on the concept of the Moreau envelope of a convex function. The objective of employing such a convex analytic tool is twofold: i) it penalizes the deviations from the Wiener-Hopf equation, which are often met in practice due to outliers, model inaccuracies, etc, and ii) it fortifies the method against strongly correlated input signal samples. The resulting algorithm enjoys a clear geometrical description; the apriori information on sparsity is exploited by the introduction of a sequence of weighted ℓ ₁ balls, and the recursions are obtained by simple relations based on the generic tool of projections onto closed convex sets. The method is tested against the state-of-the-art batch and time-recursive techniques, and in several scenarios, which also include signal recovery tasks. The proposed design shows a competitive performance in cases where the model is corrupted by Gaussian noise, and excels in scenarios of non-Gaussian heavy-tailed noise processes, albeit at a higher complexity.