Nonlinear Tikhonov regularization in Hilbert scales for inverse learning

Abstract

In this paper, we study Tikhonov regularization scheme in Hilbert scales for a nonlinear statistical inverse problem with general noise. The regularizing norm in this scheme is stronger than the norm in the Hilbert space. We focus on developing a theoretical analysis for this scheme based on conditional stability estimates. We utilize the concept of the distance function to establish high probability estimates of the direct and reconstruction errors in the Reproducing Kernel Hilbert space setting. Furthermore, explicit rates of convergence in terms of sample size are established for the oversmoothing case and the regular case over the regularity class defined through an appropriate source condition. Our results improve upon and generalize previous results obtained in related settings.