# Robust and Provably Monotonic Networks

arXiv: https://arxiv.org/abs/2112.00038 · HTML: https://mlsanity.com/paper/2112.00038/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 25 references that are in the map.
- First posted on arXiv 30 Nov 2021.

## Builds on (most cited first, 10 of 10)

- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Deep Lattice Networks and Partial Monotonic Functions](https://mlsanity.com/paper/1709.06680/) (2017)
- [Certified Monotonic Neural Networks](https://mlsanity.com/paper/2011.10219/) (2020)
- [Limitations of the Lipschitz constant as a defense against adversarial examples](https://mlsanity.com/paper/1807.09705/) (2018)
- [How to Initialize your Network? Robust Initialization for WeightNorm & ResNets](https://mlsanity.com/paper/1906.02341/) (2019)
- [Pay attention to your loss: understanding misconceptions about 1-Lipschitz neural networks](https://mlsanity.com/paper/2104.05097/) (2021)
- [A better method to enforce monotonic constraints in regression and classification trees](https://mlsanity.com/paper/2011.00986/) (2020)

## Built on by (most cited first, 6 of 6)

- [Expressive Monotonic Neural Networks](https://mlsanity.com/paper/2307.07512/) (2023)
- [Finding NEEMo: Geometric Fitting using Neural Estimation of the Energy Mover's Distance](https://mlsanity.com/paper/2209.15624/) (2022)
- [Differentiable Earth Mover's Distance for Data Compression at the High-Luminosity LHC](https://mlsanity.com/paper/2306.04712/) (2023)
- [A mathematical certification for positivity conditions in Neural Networks with applications to partial monotonicity and Trustworthy AI](https://mlsanity.com/paper/2406.08525/) (2024)
- [Moment Unfolding](https://mlsanity.com/paper/2407.11284/) (2024)
- [Fast Gaussian Processes under Monotonicity Constraints](https://mlsanity.com/paper/2507.06677/) (2025)

## Abstract (arXiv)

The Lipschitz constant of the map between the input and output space represented by a neural network is a natural metric for assessing the robustness of the model. We present a new method to constrain the Lipschitz constant of dense deep learning models that can also be generalized to other architectures. The method relies on a simple weight normalization scheme during training that ensures the Lipschitz constant of every layer is below an upper limit specified by the analyst. A simple monotonic residual connection can then be used to make the model monotonic in any subset of its inputs, which is useful in scenarios where domain knowledge dictates such dependence. Examples can be found in algorithmic fairness requirements or, as presented here, in the classification of the decays of subatomic particles produced at the CERN Large Hadron Collider. Our normalization is minimally constraining and allows the underlying architecture to maintain higher expressiveness compared to other techniques which aim to either control the Lipschitz constant of the model or ensure its monotonicity. We show how the algorithm was used to train a powerful, robust, and interpretable discriminator for heavy-flavor-quark decays, which has been adopted for use as the primary data-selection algorithm in the LHCb real-time data-processing system in the current LHC data-taking period known as Run 3. In addition, our algorithm has also achieved state-of-the-art performance on benchmarks in medicine, finance, and other applications.