Locally Adaptive Regularization of Linear Statistical Inverse Problems

This paper is concerned with a novel regularization technique for solving linear ill-posed operator equations in Hilbert spaces from data that is corrupted by white noise. As fit-to-data measures we employ extreme-value statistics of projections of residuals on a given set of sub-spaces in the image-space of the operator. We show that the proposed regularization technique exhibits local adaptive behaviour and chooses the amount of regularization in a data-driven way. This also leads to honest confidence-regions. We prove consistency and convergence rates in the framework of Bregmam-divergences.