Deep Unfolding of Tail-Based Methods for Robust Sparse Recovery Under Noise and Model Mismatch

In this article, we introduce a deep unfolding framework for Tail-iterative soft thresholding algorithm (ISTA) and Tail-fast ISTA (FISTA), extending classical sparse recovery algorithms into learned architectures and improving upon existing unfolding techniques. By combining the interpretability of iterative solvers with the adaptability of model-based networks, our approach achieves efficient and robust recovery of sparse signals. Tail-based methods incorporate an iterative support estimation step, where the support and target estimations are refined alternately, providing a key advantage over traditional approaches. We integrate this into our architecture, enhancing both recovery performance and noise robustness. We compare the proposed methods against classical solvers, including FISTA and Tail-FISTA, as well as deep unfolding techniques, LISTA and DU-FISTA, across various sparsity levels, dynamic ranges (DRs), and both noiseless and noisy conditions. In noiseless cases, our methods achieve slightly lower performance than classical solvers but with significantly reduced computational costs. Under heavy noise and a high number of nonzero elements, where classical methods struggle, our learned approaches remain resilient and achieve improved recovery rates. To evaluate generalization, we also tested our methods on data generated with a perturbed sensing matrix. In this case, under noisy scenarios, our proposed methods outperform classical sparse recovery algorithms. The proposed framework is general and applies to any linear sparse recovery task in compressed sensing (CS), offering computational efficiency, robustness to noise, and adaptability to real-world data, showcasing the advantages of deep unfolding techniques with iterative support estimation.