标题： 理论与快速学习求解器 for $\ell^1$-TV 正则化 显示英文标题

标题： Theory and Fast Learned Solver for $\ell^1$-TV Regularization

摘要： $\ell^1$和总变差（TV）惩罚项在许多领域中已被成功应用，而$\ell^1$与 TV 惩罚项的结合可以进一步提高性能。 在本工作中，我们研究了信号恢复背景下$\ell^1$-TV 模型的数学理论和数值算法：我们推导了$\ell^1$-TV 模型在恢复具有稀疏性和梯度稀疏性信号时的样本复杂度。 同时，我们提出了一种用于正则化$\ell^1$-TV 问题的新算法（PGM-ISTA），并建立了其全局收敛性和参数选择准则。 此外，我们通过展开 PGM-ISTA 构建了一个快速学习求解器（LPGM-ISTA）。 实验结果表明，LPGM-ISTA 在恢复精度和计算效率方面表现出优越的性能。 显示英文摘要

摘要： The $\ell^1$ and total variation (TV) penalties have been used successfully in many areas, and the combination of the $\ell^1$ and TV penalties can lead to further improved performance. In this work, we investigate the mathematical theory and numerical algorithms for the $\ell^1$-TV model in the context of signal recovery: we derive the sample complexity of the $\ell^1$-TV model for recovering signals with sparsity and gradient sparsity. Also we propose a novel algorithm (PGM-ISTA) for the regularized $\ell^1$-TV problem, and establish its global convergence and parameter selection criteria. Furthermore, we construct a fast learned solver (LPGM-ISTA) by unrolling PGM-ISTA. The results for the experiment on ECG signals show the superior performance of LPGM-ISTA in terms of recovery accuracy and computational efficiency.