This paper provides a sparse signal recovery algorithm, DU-PSISTA (Deep Unfolded-Periodic Sketched Iterative Shrinkage-Thresholding Algorithm), which aims to balance computational efficiency and accuracy for recovering high-dimensional sparse signals, and a convergence analysis under sufficient conditions. DU-PSISTA introduces a random matrix projection known as sketching to reduce the dimensionality of gradient computations and periodically alternates between the standard ISTA and the sketched variant. This hybrid structure enables flexible control over the trade-off between accuracy and computational complexity through a pre-configurable period parameter. The algorithm includes many parameters to be tuned such as step sizes and thresholding factors so that we incorporate deep unfolding that optimizes the parameters through data-driven training, enabling the algorithm to adaptively improve convergence speed and performance. We show that the proposed method achieves a linear-type contraction to a neighborhood of the true sparse signal with properly selected parameters. The analysis provides an interpretation for the effectiveness of the hybrid structure to improve recovery accuracy. Numerical experiments confirm that our method achieves comparable recovery performance to conventional deep unfolded ISTA while reducing computational complexity, especially when the period parameter and sketch size are properly selected. The results are also consistent with the theoretical insights.

翻译：本文提出了一种稀疏信号恢复算法——DU-PSISTA（深度展开-周期性草图迭代收缩阈值算法），旨在平衡高维稀疏信号恢复中的计算效率与精度，并给出了充分条件下的收敛性分析。该算法引入称为草图的随机矩阵投影来降低梯度计算的维度，并在标准ISTA与草图变体之间周期性交替。这种混合结构通过可预设的周期参数，灵活控制精度与计算复杂度之间的权衡。由于算法包含步长、阈值因子等诸多待调参数，我们融入深度展开技术，通过数据驱动训练优化参数，使算法能够自适应地提升收敛速度与性能。研究表明，在参数适当选择条件下，所提方法可实现对真实稀疏信号邻域的线性型收缩。该分析为混合结构改善恢复精度的有效性提供了理论解释。数值实验证实，当周期参数与草图规模选择适当时，本方法在降低计算复杂度的同时，实现了与传统深度展开ISTA相当的恢复性能，实验结果也与理论分析相吻合。