# Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems

arXiv: https://arxiv.org/abs/1902.05404 · HTML: https://mlsanity.com/paper/1902.05404/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2019.

## Built on by (most cited first, 7 of 7)

- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigation](https://mlsanity.com/paper/2605.24041/) (2026)
- [Learned reconstruction methods for inverse problems: sample error estimates](https://mlsanity.com/paper/2312.14078/) (2023)
- [Statistical inverse learning problems with random observations](https://mlsanity.com/paper/2312.15341/) (2023)
- [Statistical Inverse Problems in Hilbert Scales](https://mlsanity.com/paper/2208.13289/) (2022)
- [A statistical perspective on algorithm unrolling models for inverse problems](https://mlsanity.com/paper/2311.06395/) (2023)
- [Statistical inverse learning and ℓ1-regularization](https://mlsanity.com/paper/2607.07468/) (2026)

## Abstract (arXiv)

We study a non-linear statistical inverse learning problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization, MOR) approach to reconstruct the estimator of the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the ansatz of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.