# Convex regularization in statistical inverse learning problems

arXiv: https://arxiv.org/abs/2102.09526 · HTML: https://mlsanity.com/paper/2102.09526/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 36 references that are in the map.
- First posted on arXiv 18 Feb 2021.

## Builds on (most cited first, 2 of 2)

- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Large Noise in Variational Regularization](https://mlsanity.com/paper/1602.00520/) (2016)

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

- [Two-layer neural networks with values in a Banach space](https://mlsanity.com/paper/2105.02095/) (2021)
- [Learning sparsity-promoting regularizers for linear inverse problems](https://mlsanity.com/paper/2412.16031/) (2024)
- [A Lipschitz spaces view of infinitely wide shallow neural networks](https://mlsanity.com/paper/2410.14591/) (2024)
- [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)
- [The learned range test method for the inverse inclusion problem](https://mlsanity.com/paper/2411.00463/) (2024)
- [Statistical inverse learning and ℓ1-regularization](https://mlsanity.com/paper/2607.07468/) (2026)

## Abstract (arXiv)

We consider a statistical inverse learning problem, where the task is to estimate a function f based on noisy point evaluations of Af, where A is a linear operator. The function Af is evaluated at i.i.d. random design points un, n=1,...,N generated by an unknown general probability distribution. We consider Tikhonov regularization with general convex and p-homogeneous penalty functionals and derive concentration rates of the regularized solution to the ground truth measured in the symmetric Bregman distance induced by the penalty functional. We derive concrete rates for Besov norm penalties and numerically demonstrate the correspondence with the observed rates in the context of X-ray tomography.