  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.
