We propose a new penalty, the springback penalty, for constructing models to recover an unknown signal from incomplete and inaccurate measurements. Mathematically, the springback penalty is a weakly convex function. It bears various theoretical and computational advantages of both the benchmark convex ${\ell }_1$ penalty and many of its non-convex surrogates that have been well studied in the literature. We establish the exact and stable recovery theory for the recovery model using the springback penalty for both sparse and nearly sparse signals, respectively, and derive an easily implementable difference-of-convex algorithm. In particular, we show its theoretical superiority to some existing models with a sharper recovery bound for some scenarios where the level of measurement noise is large or the amount of measurements is limited. We also demonstrate its numerical robustness regardless of the varying coherence of the sensing matrix. The springback penalty is particularly favorable for the scenario where the incomplete and inaccurate measurements are collected by coherence-hidden or -static sensing hardware due to its theoretical guarantee of recovery with severe measurements, computational tractability, and numerical robustness for ill-conditioned sensing matrices.

我们提出了一种新的惩罚，即回弹惩罚，用于构建模型以从不完整和不准确的测量中恢复未知信号。在数学上，回弹惩罚是一个弱凸函数。它具有基准凸的各种理论和计算优势${\ell }_1$惩罚和它的许多非凸替代物在文献中得到了很好的研究。我们分别使用稀疏和近稀疏信号的回弹惩罚为恢复模型建立了精确和稳定的恢复理论，并推导出了一种易于实现的凸差分算法。特别是，在测量噪声水平较大或测量量有限的某些情况下，我们展示了其相对于一些现有模型的理论优势，具有更清晰的恢复界限。无论传感矩阵的相干性如何变化，我们还证明了它的数值鲁棒性。