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器，并对其弱凸性模量设定预设上界。此类正则化器可生成最小化凸能量的变分去噪器。它们参数极少（少于15,000个），且因其模仿手工设计的稀疏促进正则化器而具备信号处理可解释性。通过数值实验，我们证明此类去噪器优于凸正则化方法及流行的BM3D去噪器。此外，所学习的正则化器可部署于可证明收敛的迭代方案中，以解决逆问题。在CT和MRI重建中，该正则化器展现出良好泛化能力，并在性能、参数量、收敛保证与可解释性之间实现了优于其他数据驱动方法的平衡。