# Learning Gradients of Convex Functions with Monotone Gradient Networks

arXiv: https://arxiv.org/abs/2301.10862 · HTML: https://mlsanity.com/paper/2301.10862/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 20 references that are in the map.
- First posted on arXiv 25 Jan 2023.

## Builds on (most cited first, 11 of 11)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Map-Guided Curriculum Domain Adaptation and Uncertainty-Aware Evaluation for Semantic Nighttime Image Segmentation](https://mlsanity.com/paper/2005.14553/) (2020)
- [Optimal Transport for Multi-source Domain Adaptation under Target Shift](https://mlsanity.com/paper/1803.04899/) (2018)
- [On approximating nabla f with neural networks](https://mlsanity.com/paper/1910.12744/) (2019)
- [Input Convex Gradient Networks](https://mlsanity.com/paper/2111.12187/) (2021)

## Built on by (most cited first, 7 of 7)

- [Structure-Preserving Digital Twins via Conditional Neural Whitney Forms](https://mlsanity.com/paper/2508.06981/) (2025)
- [Convexity in ReLU Neural Networks: beyond ICNNs?](https://mlsanity.com/paper/2501.03017/) (2025)
- [Learning truly monotone operators with applications to nonlinear inverse problems](https://mlsanity.com/paper/2404.00390/) (2024)
- [Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systems](https://mlsanity.com/paper/2407.08868/) (2024)
- [Time-changed normalizing flows for accurate SDE modeling](https://mlsanity.com/paper/2312.14698/) (2023)
- [Gradient Networks](https://mlsanity.com/paper/2404.07361/) (2024)
- [GradNetOT: Learning Optimal Transport Maps with GradNets](https://mlsanity.com/paper/2507.13191/) (2025)

## Abstract (arXiv)

While much effort has been devoted to deriving and analyzing effective convex formulations of signal processing problems, the gradients of convex functions also have critical applications ranging from gradient-based optimization to optimal transport. Recent works have explored data-driven methods for learning convex objective functions, but learning their monotone gradients is seldom studied. In this work, we propose C-MGN and M-MGN, two monotone gradient neural network architectures for directly learning the gradients of convex functions. We show that, compared to state of the art methods, our networks are easier to train, learn monotone gradient fields more accurately, and use significantly fewer parameters. We further demonstrate their ability to learn optimal transport mappings to augment driving image data.