Stability of Data-Dependent Ridge-Regularization for Inverse Problems
This provides a theoretically grounded approach for inverse problems in domains like biomedical imaging and material sciences, though it appears incremental as it builds on existing ridge regularization methods.
The authors tackled the problem of achieving both theoretical guarantees and high reconstruction quality in inverse problems by proposing a data-dependent ridge regularizer with spatially varying regularization strength. They proved solution existence, stability, and maximum-a-posteriori properties, and demonstrated high-quality reconstructions with small instance-specific training sets in biomedical imaging and material sciences.
Theoretical guarantees for the robust solution of inverse problems have important implications for applications. To achieve both guarantees and high reconstruction quality, we propose learning 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.