# A Generative Variational Model for Inverse Problems in Imaging

arXiv: https://arxiv.org/abs/2104.12630 · HTML: https://mlsanity.com/paper/2104.12630/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Apr 2021.

## Built on by (most cited first, 14 of 14)

- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [Regularising Inverse Problems with Generative Machine Learning Models](https://mlsanity.com/paper/2107.11191/) (2021)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging](https://mlsanity.com/paper/2212.12499/) (2022)
- [Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces](https://mlsanity.com/paper/2205.14627/) (2022)
- [Optimal Regularization for a Data Source](https://mlsanity.com/paper/2212.13597/) (2022)
- [Compressed Sensing MRI Reconstruction Regularized by VAEs with Structured Image Covariance](https://mlsanity.com/paper/2210.14586/) (2022)
- [Learning of Patch-Based Smooth-Plus-Sparse Models for Image Reconstruction](https://mlsanity.com/paper/2412.13070/) (2024)
- [A Note on the Regularity of Images Generated by Convolutional Neural Networks](https://mlsanity.com/paper/2204.10588/) (2022)
- [Error Analysis of Bayesian Inverse Problems with Generative Priors](https://mlsanity.com/paper/2601.17374/) (2026)
- [Learned reconstruction methods for inverse problems: sample error estimates](https://mlsanity.com/paper/2312.14078/) (2023)
- [Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overview](https://mlsanity.com/paper/2402.12072/) (2024)
- [Actively Inferring Optimal Measurement Sequences](https://mlsanity.com/paper/2502.18142/) (2025)
- [Parameter-free structure-texture image decomposition by unrolling](https://mlsanity.com/paper/2503.13354/) (2025)

## Abstract (arXiv)

This paper is concerned with the development, analysis and numerical realization of a novel variational model for the regularization of inverse problems in imaging. The proposed model is inspired by the architecture of generative convolutional neural networks; it aims to generate the unknown from variables in a latent space via multi-layer convolutions and non-linear penalties, and penalizes an associated cost. In contrast to conventional neural-network-based approaches, however, the convolution kernels are learned directly from the measured data such that no training is required. The present work provides a mathematical analysis of the proposed model in a function space setting, including proofs for regularity and existence/stability of solutions, and convergence for vanishing noise. Moreover, in a discretized setting, a numerical algorithm for solving various types of inverse problems with the proposed model is derived. Numerical results are provided for applications in inpainting, denoising, deblurring under noise, super-resolution and JPEG decompression with multiple test images.