Recovering an unknown but structured signal from its measurements is a challenging problem with significant applications in fields such as imaging restoration, wireless communications, and signal processing. In this paper, we consider the inherent problem stems from the prior knowledge about the signal's structure, such as sparsity which is critical for signal recovery models. We investigate three constrained optimization models that effectively address this challenge, each leveraging distinct forms of structural priors to regularize the solution space. Our theoretical analysis demonstrates that these models exhibit robustness to noise while maintaining stability with respect to tuning parameters that is a crucial property for practical applications, when the parameter selection is often nontrivial. By providing theoretical foundations, our work supports their practical use in scenarios where measurement imperfections and model uncertainties are unavoidable. Furthermore, under mild conditions, we establish tradeoff between the sample complexity and the mismatch error.

翻译：从测量数据中恢复未知但具有结构的信号是一个具有挑战性的问题，在图像复原、无线通信和信号处理等领域具有重要应用。本文考虑了源于信号结构先验知识的内在问题，例如对信号恢复模型至关重要的稀疏性。我们研究了三种能够有效应对这一挑战的约束优化模型，每种模型都利用不同形式的结构先验来正则化解空间。我们的理论分析表明，这些模型在保持对调谐参数稳定性的同时，表现出对噪声的鲁棒性——这是一个对实际应用至关重要的性质，因为参数选择通常并非易事。通过提供理论基础，我们的工作支持了这些模型在测量缺陷和模型不确定性不可避免的场景中的实际应用。此外，在温和条件下，我们建立了样本复杂度与失配误差之间的权衡关系。