改进梯度下降算法，让信号恢复更稳定、抗噪更强。
From sparse recovery to plug-and-play priors, understanding trade-offs for stable recovery with generalized projected gradient descent
- 提出广义投影梯度法，统一稀疏恢复与深度先验方法。
- 引入归一化幂等正则化，显著提升恢复稳定性。
- 适用于图像逆问题，适合研究鲁棒信号恢复的学者。
本文研究从噪声性、欠定观测中恢复未知低维向量的问题。聚焦广义投影梯度下降（GPGD）框架，该框架统一了传统稀疏恢复方法与基于学习的深度投影先验方法。我们扩展了先前的收敛性结果，使其对模型误差和投影误差具有鲁棒性。利用这些理论结果，探索如何更好地控制稳定性和鲁棒性常数。为降低测量噪声带来的恢复误差，考虑了广义反投影策略，以适应结构化噪声（如稀疏异常值）。为提高GPGD的稳定性，提出一种用于深度投影先验学习的归一化幂等正则化方法。在稀疏恢复和图像逆问题场景下进行数值实验，揭示了可实现的可辨识性与稳定性之间的权衡。
原文摘要 · Abstract (English)
We consider the problem of recovering an unknown low-dimensional vector from noisy, underdetermined observations. We focus on the Generalized Projected Gradient Descent (GPGD) framework, which unifies traditional sparse recovery methods and modern approaches using learned deep projective priors. We extend previous convergence results to robustness to model and projection errors. We use these theoretical results to explore ways to better control stability and robustness constants. To reduce recovery errors due to measurement noise, we consider generalized back-projection strategies to adapt GPGD to structured noise, such as sparse outliers. To improve the stability of GPGD, we propose a normalized idempotent regularization for the learning of deep projective priors. We provide numerical experiments in the context of sparse recovery and image inverse problems, highlighting the trade-offs between identifiability and stability that can be achieved with such methods.
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