# Learned reconstruction methods for inverse problems: sample error estimates

arXiv: https://arxiv.org/abs/2312.14078 · HTML: https://mlsanity.com/paper/2312.14078/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 70 references that are in the map.
- First posted on arXiv 21 Dec 2023.

## Builds on (most cited first, 30 of 36)

- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [A Review of Convolutional Neural Networks for Inverse Problems in Imaging](https://mlsanity.com/paper/1710.04011/) (2017)
- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [One Network to Solve Them All --- Solving Linear Inverse Problems using Deep Projection Models](https://mlsanity.com/paper/1703.09912/) (2017)
- [Learned convex regularizers for inverse problems](https://mlsanity.com/paper/2008.02839/) (2020)
- [Total Deep Variation for Linear Inverse Problems](https://mlsanity.com/paper/2001.05005/) (2020)
- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [A characterization of proximity operators](https://mlsanity.com/paper/1807.04014/) (2018)
- [End-to-end reconstruction meets data-driven regularization for inverse problems](https://mlsanity.com/paper/2106.03538/) (2021)
- [Learning The Invisible: A Hybrid Deep Learning-Shearlet Framework for Limited Angle Computed Tomography](https://mlsanity.com/paper/1811.04602/) (2018)
- [Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem](https://mlsanity.com/paper/2211.08875/) (2022)
- [Learning the optimal Tikhonov regularizer for inverse problems](https://mlsanity.com/paper/2106.06513/) (2021)
- [Bayesian Imaging With Data-Driven Priors Encoded by Neural Networks: Theory, Methods, and Algorithms](https://mlsanity.com/paper/2103.10182/) (2021)
- [Multi-Scale Learned Iterative Reconstruction](https://mlsanity.com/paper/1908.00936/) (2019)
- [Deep learning architectures for nonlinear operator functions and nonlinear inverse problems](https://mlsanity.com/paper/1912.11090/) (2019)
- [A Perspective on Deep Imaging](https://mlsanity.com/paper/1609.04375/) (2016)
- [Deep neural networks for inverse problems with pseudodifferential operators: an application to limited-angle tomography](https://mlsanity.com/paper/2006.01620/) (2020)

## Built on by (most cited first, 1 of 1)

- [Learning sparsity-promoting regularizers for linear inverse problems](https://mlsanity.com/paper/2412.16031/) (2024)

## Abstract (arXiv)

Learning-based and data-driven techniques have recently become a subject of primary interest in the field of reconstruction and regularization of inverse problems. Besides the development of novel methods, yielding excellent results in several applications, their theoretical investigation has attracted growing interest, e.g., on the topics of reliability, stability, and interpretability. In this work, a general framework is described, allowing us to interpret many of these techniques in the context of statistical learning. This is not intended to provide a complete survey of existing methods, but rather to put them in a working perspective, which naturally allows their theoretical treatment. The main goal of this dissertation is thereby to address the generalization properties of learned reconstruction methods, and specifically to perform their sample error analysis. This task, well-developed in statistical learning, consists in estimating the dependence of the learned operators with respect to the data employed for their training. A rather general strategy is proposed, whose assumptions are met for a large class of inverse problems and learned methods, as depicted via a selection of examples.