# A Lifted Bregman Formulation for the Inversion of Deep Neural Networks

arXiv: https://arxiv.org/abs/2303.01965 · HTML: https://mlsanity.com/paper/2303.01965/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 63 references that are in the map.
- First posted on arXiv 1 Mar 2023.

## Builds on (most cited first, 24 of 24)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [An Image is Worth One Word: Personalizing Text-to-Image Generation using Textual Inversion](https://mlsanity.com/paper/2208.01618/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks](https://mlsanity.com/paper/1703.10593/) (2017)
- [Interpretable Explanations of Black Boxes by Meaningful Perturbation](https://mlsanity.com/paper/1704.03296/) (2017)
- [Understanding Deep Image Representations by Inverting Them](https://mlsanity.com/paper/1412.0035/) (2014)
- [GAN Inversion: A Survey](https://mlsanity.com/paper/2101.05278/) (2021)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Understanding Deep Networks via Extremal Perturbations and Smooth Masks](https://mlsanity.com/paper/1910.08485/) (2019)
- [Coordinate Descent Algorithms](https://mlsanity.com/paper/1502.04759/) (2015)
- [Explaining Image Classifiers by Counterfactual Generation](https://mlsanity.com/paper/1807.08024/) (2018)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [Understanding and Mitigating Exploding Inverses in Invertible Neural Networks](https://mlsanity.com/paper/2006.09347/) (2020)
- [Learned convex regularizers for inverse problems](https://mlsanity.com/paper/2008.02839/) (2020)
- [Lifted Bregman Training of Neural Networks](https://mlsanity.com/paper/2208.08772/) (2022)
- [Convergent Data-driven Regularizations for CT Reconstruction](https://mlsanity.com/paper/2212.07786/) (2022)
- [Generalised Perceptron Learning](https://mlsanity.com/paper/2012.03642/) (2020)

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

- [Learning a Sparse Representation of Barron Functions with the Inverse Scale Space Flow](https://mlsanity.com/paper/2312.02671/) (2023)
- [A lifted Bregman strategy for training unfolded proximal neural network Gaussian denoisers](https://mlsanity.com/paper/2408.08742/) (2024)

## Abstract (arXiv)

We propose a novel framework for the regularised inversion of deep neural networks. The framework is based on the authors' recent work on training feed-forward neural networks without the differentiation of activation functions. The framework lifts the parameter space into a higher dimensional space by introducing auxiliary variables, and penalises these variables with tailored Bregman distances. We propose a family of variational regularisations based on these Bregman distances, present theoretical results and support their practical application with numerical examples. In particular, we present the first convergence result (to the best of our knowledge) for the regularised inversion of a single-layer perceptron that only assumes that the solution of the inverse problem is in the range of the regularisation operator, and that shows that the regularised inverse provably converges to the true inverse if measurement errors converge to zero.