基于截断${\ell }_1$惩罚的可学习Peaceman-Rachford方法

Learned Peaceman-Rachford Method Based on Capped ${\ell }_1$ Penalty

摘要: 近年来，模型与数据驱动的深度展开方法，将基于凸或非凸正则化的迭代优化算法展开为深度神经网络，在稀疏信号重建问题中取得了显著成效。本文围绕非凸截断${\ell }_1$ 惩罚正则化下的稀疏重建问题，提出了基于截断${\ell }_1$ 惩罚的可学习Peaceman-Rachford方法。该模型具有良好的数学理论保障，不仅在高概率下近似无假阳性，而且可分析其线性收敛速率。数值实验表明，本文提出的模型显著优于现有代表性模型。

Abstract: In recent years, model and data-driven deep unfolding methods have unfolded iterative optimization algorithms based on convex or nonconvex regularization into deep neural networks, achieving remarkable performance in sparse signal reconstruction problems. Focusing on the sparse reconstruction problem under nonconvex capped ${\ell }_1$ penalty regularization, this paper proposes a learned Peaceman-Rachford method based on capped ${\ell }_1$ penalty. The proposed model is supported by sound mathematical theory: it not only yields approximate no-false-positives results with high probability, but also allows for the analysis of its linear convergence rate. Numerical experiments demonstrate that the proposed model significantly outperforms existing representative models.