# Statistical Inverse Problems in Hilbert Scales

arXiv: https://arxiv.org/abs/2208.13289 · HTML: https://mlsanity.com/paper/2208.13289/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 22 references that are in the map.
- First posted on arXiv 28 Aug 2022.

## Builds on (most cited first, 4 of 4)

- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems](https://mlsanity.com/paper/1902.05404/) (2019)
- [Inverse learning in Hilbert scales](https://mlsanity.com/paper/2002.10208/) (2020)
- [Stochastic Gradient Descent in Hilbert Scales: Smoothness, Preconditioning and Earlier Stopping](https://mlsanity.com/paper/2006.10840/) (2020)

## Abstract (arXiv)

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.