# Total Deep Variation: A Stable Regularizer for Inverse Problems

arXiv: https://arxiv.org/abs/2006.08789 · HTML: https://mlsanity.com/paper/2006.08789/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jun 2020.

## Built on by (most cited first, 9 of 9)

- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problem](https://mlsanity.com/paper/2101.08286/) (2021)
- [Bayesian Uncertainty Estimation of Learned Variational MRI Reconstruction](https://mlsanity.com/paper/2102.06665/) (2021)
- [Optimal Regularization for a Data Source](https://mlsanity.com/paper/2212.13597/) (2022)
- [PAC-Bayesian Learning of Optimization Algorithms](https://mlsanity.com/paper/2210.11113/) (2022)
- [WARPd: A linearly convergent first-order method for inverse problems with approximate sharpness conditions](https://mlsanity.com/paper/2110.12437/) (2021)
- [Learning-Assisted Algorithm Unrolling for Online Optimization with Budget Constraints](https://mlsanity.com/paper/2212.01689/) (2022)
- [A Generalization Result for Convergence in Learning-to-Optimize](https://mlsanity.com/paper/2410.07704/) (2024)
- [Regularizing Attention Scores with Bootstrapping](https://mlsanity.com/paper/2604.01339/) (2026)

## Abstract (arXiv)

Various problems in computer vision and medical imaging can be cast as inverse problems. A frequent method for solving inverse problems is the variational approach, which amounts to minimizing an energy composed of a data fidelity term and a regularizer. Classically, handcrafted regularizers are used, which are commonly outperformed by state-of-the-art deep learning approaches. In this work, we combine the variational formulation of inverse problems with deep learning by introducing the data-driven general-purpose total deep variation regularizer. In its core, a convolutional neural network extracts local features on multiple scales and in successive blocks. This combination allows for a rigorous mathematical analysis including an optimal control formulation of the training problem in a mean-field setting and a stability analysis with respect to the initial values and the parameters of the regularizer. In addition, we experimentally verify the robustness against adversarial attacks and numerically derive upper bounds for the generalization error. Finally, we achieve state-of-the-art results for numerous imaging tasks.