# Variational Regularization in Inverse Problems and Machine Learning

arXiv: https://arxiv.org/abs/2112.04591 · HTML: https://mlsanity.com/paper/2112.04591/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 41 references that are in the map.
- First posted on arXiv 8 Dec 2021.

## Builds on (most cited first, 5 of 5)

- [In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1412.6614/) (2014)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Large Noise in Variational Regularization](https://mlsanity.com/paper/1602.00520/) (2016)
- [CLEAR: Covariant LEAst-square Re-fitting with applications to image restoration](https://mlsanity.com/paper/1606.05158/) (2016)
- [Bregman Distances in Inverse Problems and Partial Differential Equation](https://mlsanity.com/paper/1505.05191/) (2015)

## Built on by (most cited first, 2 of 2)

- [Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning](https://mlsanity.com/paper/2211.15223/) (2022)
- [Optimal Linear Baseline Models for Scientific Machine Learning](https://mlsanity.com/paper/2508.05831/) (2025)

## Abstract (arXiv)

This paper discusses basic results and recent developments on variational regularization methods, as developed for inverse problems. In a typical setup we review basic properties needed to obtain a convergent regularization scheme and further discuss the derivation of quantitative estimates respectively needed ingredients such as Bregman distances for convex functionals. In addition to the approach developed for inverse problems we will also discuss variational regularization in machine learning and work out some connections to the classical regularization theory. In particular we will discuss a reinterpretation of machine learning problems in the framework of regularization theory and a reinterpretation of variational methods for inverse problems in the framework of risk minimization. Moreover, we establish some previously unknown connections between error estimates in Bregman distances and generalization errors.