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Reconstruction of signals from undersampled and noisy measurements is a topicof considerable interest. Sharpness conditions directly control the recoveryperformance of restart schemes for first-order methods without the need forrestrictive assumptions such as strong convexity. However, they are challengingto apply in the presence of noise or approximate model classes (e.g.,approximate sparsity). We provide a first-order method: Weighted, Acceleratedand Restarted Primal-dual (WARPd), based on primal-dual iterations and a novelrestart-reweight scheme. Under a generic approximate sharpness condition, WARPdachieves stable linear convergence to the desired vector. Many problems ofinterest fit into this framework. For example, we analyze sparse recovery incompressed sensing, low-rank matrix recovery, matrix completion, TVregularization, 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 provideexplicit approximate sharpness constants. Numerical experiments show that WARPdcompares favorably with specialized state-of-the-art methods and is ideallysuited for solving large-scale problems. We also present a noise-blind variantbased on the Square-Root LASSO decoder. Finally, we show how to unroll WARPd asneural networks. This approximation theory result provides lower bounds forstable and accurate neural networks for inverse problems and sheds light onarchitecture choices. Code and a gallery of examples are made available onlineas a MATLAB package.

WARPd: A linearly convergent first-order method for inverse problems with approximate sharpness conditions