We propose to learn non-convex regularizers with a prescribed upper bound on their weak-convexity modulus. Such regularizers give rise to variational denoisers that minimize a convex energy. They rely on few parameters (less than 15,000) and offer a signal-processing interpretation as they mimic handcrafted sparsity-promoting regularizers. Through numerical experiments, we show that such denoisers outperform convex-regularization methods as well as the popular BM3D denoiser. Additionally, the learned regularizer can be deployed to solve inverse problems with iterative schemes that provably converge. For both CT and MRI reconstruction, the regularizer generalizes well and offers an excellent tradeoff between performance, number of parameters, guarantees, and interpretability when compared to other data-driven approaches.

翻译：我们提出学习一类非凸正则化子，并对其弱凸性模量施加预设上界。此类正则化子可生成最小化凸能量的变分去噪器。它们依赖少量参数（少于15,000），并具有信号处理解释性——可模拟手工设计的稀疏性促进正则化子。通过数值实验，我们证明此类去噪器优于凸正则化方法以及流行的BM3D去噪器。此外，所学习的正则化子可部署于具有可证明收敛性的迭代求解方案中，用于解决逆问题。在CT和MRI重建任务中，该正则化子展现出良好的泛化能力，并在性能、参数量、保证性与可解释性之间实现了卓越平衡，优于其他数据驱动方法。