  Theoretical guarantees for the robust solution of inverse problems have
important implications for applications. To achieve both guarantees and high
reconstruction quality, we propose to learn a pixel-based ridge regularizer
with a data-dependent and spatially-varying regularization strength. For this
architecture, we establish the existence of solutions to the associated
variational problem and the stability of its solution operator. Further, we
prove that the reconstruction forms a maximum-a-posteriori approach.
Simulations for biomedical imaging and material sciences demonstrate that the
approach yields high-quality reconstructions even if only a small
instance-specific training set is available.
