# Bilevel approaches for learning of variational imaging models

arXiv: https://arxiv.org/abs/1505.02120 · HTML: https://mlsanity.com/paper/1505.02120/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2015.

## Built on by (most cited first, 17 of 17)

- [Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond](https://mlsanity.com/paper/2101.11517/) (2021)
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- [Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processes](https://mlsanity.com/paper/2510.05568/) (2025)
- [Dual Meta-Learning with Longitudinally Generalized Regularization for One-Shot Brain Tissue Segmentation Across the Human Lifespan](https://mlsanity.com/paper/2308.06774/) (2023)
- [An Adaptively Inexact Method for Bilevel Learning Using Primal-Dual Style Differentiation](https://mlsanity.com/paper/2412.06436/) (2024)
- [Learned Discrepancy Reconstruction and Benchmark Dataset for Magnetic Particle Imaging](https://mlsanity.com/paper/2501.05583/) (2025)
- [Generative models and Bayesian inversion using Laplace approximation](https://mlsanity.com/paper/2203.07755/) (2022)
- [Optimal Regularization Under Uncertainty: Distributional Robustness and Convexity Constraints](https://mlsanity.com/paper/2510.03464/) (2025)
- [Bilevel Imaging Learning Problems as Mathematical Programs with Complementarity Constraints: Reformulation and Theory](https://mlsanity.com/paper/2110.02273/) (2021)
- [An incremental algorithm for non-convex AI-enhanced medical image processing](https://mlsanity.com/paper/2505.08324/) (2025)
- [Deep regularization networks for inverse problems with noisy operators](https://mlsanity.com/paper/2506.07008/) (2025)
- [Learning spatially varying regularisation parameters of low regularity for image reconstruction](https://mlsanity.com/paper/2608.25127/) (2026)

## Abstract (arXiv)

We review some recent learning approaches in variational imaging, based on bilevel optimisation, and emphasize the importance of their treatment in function space. The paper covers both analytical and numerical techniques. Analytically, we include results on the existence and structure of minimisers, as well as optimality conditions for their characterisation. Based on this information, Newton type methods are studied for the solution of the problems at hand, combining them with sampling techniques in case of large databases. The computational verification of the developed techniques is extensively documented, covering instances with different type of regularisers, several noise models, spatially dependent weights and large image databases.