Reconstruction of signals from undersampled and noisy measurements is a topic of considerable interest. Sharpness conditions directly control the recovery performance of restart schemes for first-order methods without the need for restrictive assumptions such as strong convexity. However, they are challenging to apply in the presence of noise or approximate model classes (e.g., approximate sparsity). We provide a first-order method: Weighted, Accelerated and Restarted Primal-dual (WARPd), based on primal-dual iterations and a novel restart-reweight scheme. Under a generic approximate sharpness condition, WARPd achieves stable linear convergence to the desired vector. Many problems of interest fit into this framework. For example, we analyze sparse recovery in compressed sensing, low-rank matrix recovery, matrix completion, TV regularization, minimization of $\|Bx\|_{l^1}$ under constraints ($l^1$-analysis problems for general $B$), and mixed regularization problems. We show how several quantities controlling recovery performance also provide explicit approximate sharpness constants. Numerical experiments show that WARPd compares favorably with specialized state-of-the-art methods and is ideally suited for solving large-scale problems. We also present a noise-blind variant based on the Square-Root LASSO decoder. Finally, we show how to unroll WARPd as neural networks. This approximation theory result provides lower bounds for stable and accurate neural networks for inverse problems and sheds light on architecture choices. Code and a gallery of examples are made available online as a MATLAB package.
翻译:对未得到充分采样和噪音测量的信号进行重建是一个引起极大兴趣的议题。 快速条件直接控制了一流方法的恢复性能,而不需要严格的假设,例如强烈的混凝土。 然而,在出现噪音或大致模型类(例如,大致的宽度)时,很难应用这些条件。 我们提供了一种一流方法:根据原始双向迭代和新颖的重现计划,加权、加速和重新启动原始版(WARPd),直接控制了一流方法的恢复性能。在一般的锐利状态下,WARPd实现了与理想矢量的稳定的线性趋同。许多问题都适合这一框架。例如,我们分析压缩感测、低级矩阵恢复、矩阵完成、电视规范化、将美元减到最小化,一般B$1美元的分析问题,以及混合的正规化问题。我们发现,控制恢复业绩的量也提供了明确的直线性常数。 数字实验表明,WARPDS实现了稳定的线性趋近线性趋近线性趋近。我们最后的压式的压式压式压式压式压式压式网络提供了一种不甚高的甚高的压式压式的压式的压式压式的压式平压式的压式的压式的压式的压式的压式的压式的压式压式的压式的压压式的压式的压式的压式的压式的压式的压式的压式的压式结构。