Data-Driven Regularization with Weak Convexity for Robust Image Reconstruction

We describe a practical framework for data-driven regularization in image reconstruction. It combines model expressivity with the guarantees of variational methods. The approach is based on a weakly convex ridge regularizer, defined as the composition of a convolutional filter bank and pointwise potentials constrained to be weakly convex. These potentials are implemented as learnable splines with a strict control of their weak-convexity modulus. The resulting denoisers outperform classic convex regularization techniques as well as competitive benchmarks such as BM3D, while they still correspond to the minimization of a convex energy. The learned regularizers further extend to general inverse problems with provable convergence to critical points. Overall, this framework shows that a controlled relaxation of convexity enables the design of learnable priors that achieve strong empirical performance while preserving mathematical guarantees.