Statistical Inverse Problems in Hilbert Scales
This work provides theoretical improvements for inverse problems in statistics, but it is incremental as it generalizes prior results in related settings.
The paper tackles the Tikhonov regularization scheme in Hilbert scales for nonlinear statistical inverse problems with general noise, establishing high-probability error estimates and explicit convergence rates in terms of sample size for oversmoothing and regular cases.
In this paper, we study the Tikhonov regularization scheme in Hilbert scales for the nonlinear statistical inverse problem with a general noise. The regularizing norm in this scheme is stronger than the norm in Hilbert space. We focus on developing a theoretical analysis for this scheme based on the conditional stability estimates. We utilize the concept of the distance function to establish the high probability estimates of the direct and reconstruction error in Reproducing kernel Hilbert space setting. Further, the 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 appropriate source condition. Our results improve and generalize previous results obtained in related settings.