鲁棒信号恢复的回弹惩罚
The springback penalty for robust signal recovery
摘要：
我们提出了一种新的惩罚，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。它具有各种理论和计算的优势，基准凸$\ell_1$惩罚和它的许多非凸代理人，已在文献中得到很好的研究。我们建立了精确和稳定的恢复理论的恢复模型，分别使用稀疏和近稀疏信号的回弹惩罚，并推导出一个易于实现的凸差分算法。特别是，我们展示了其理论上的优越性，一些现有的模型与尖锐的恢复约束的测量噪声的水平是大的或测量量是有限的一些情况下。我们还证明了它的数值鲁棒性，无论传感矩阵的变化的相干性。回弹惩罚是特别有利的情况下，不完整和不准确的测量数据收集的相干隐藏或静态传感硬件，由于其理论上保证恢复与严重的测量，计算的易处理性和数值鲁棒性病态的传感矩阵。
英文摘要：
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.
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