# A mathematical certification for positivity conditions in Neural Networks with applications to partial monotonicity and Trustworthy AI

arXiv: https://arxiv.org/abs/2406.08525 · HTML: https://mlsanity.com/paper/2406.08525/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 61 references that are in the map.
- First posted on arXiv 12 Jun 2024.

## Builds on (most cited first, 21 of 21)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [A Survey on Explainable Artificial Intelligence (XAI): Towards Medical XAI](https://mlsanity.com/paper/1907.07374/) (2019)
- [A Survey on Neural Network Interpretability](https://mlsanity.com/paper/2012.14261/) (2020)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [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)
- [Efficiently Computing Local Lipschitz Constants of Neural Networks via Bound Propagation](https://mlsanity.com/paper/2210.07394/) (2022)
- [Counterexample-Guided Learning of Monotonic Neural Networks](https://mlsanity.com/paper/2006.08852/) (2020)
- [Constrained Monotonic Neural Networks](https://mlsanity.com/paper/2205.11775/) (2022)
- [How to Incorporate Monotonicity in Deep Networks While Preserving Flexibility?](https://mlsanity.com/paper/1909.10662/) (2019)
- [Monotonic classification: an overview on algorithms, performance measures and data sets](https://mlsanity.com/paper/1811.07155/) (2018)
- [Deontological Ethics By Monotonicity Shape Constraints](https://mlsanity.com/paper/2001.11990/) (2020)
- [The Curious Case of Convex Neural Networks](https://mlsanity.com/paper/2006.05103/) (2020)
- [NeuralSens: Sensitivity Analysis of Neural Networks](https://mlsanity.com/paper/2002.11423/) (2020)
- [Size and depth of monotone neural networks: interpolation and approximation](https://mlsanity.com/paper/2207.05275/) (2022)
- [Testing Monotonicity of Machine Learning Models](https://mlsanity.com/paper/2002.12278/) (2020)
- [Robust and Provably Monotonic Networks](https://mlsanity.com/paper/2112.00038/) (2021)
- [Monotonicity Regularization: Improved Penalties and Novel Applications to Disentangled Representation Learning and Robust Classification](https://mlsanity.com/paper/2205.08247/) (2022)
- [Monotonicity for AI ethics and society: An empirical study of the monotonic neural additive model in criminology, education, health care, and finance](https://mlsanity.com/paper/2301.07060/) (2023)
- [NN2Poly: A polynomial representation for deep feed-forward artificial neural networks](https://mlsanity.com/paper/2112.11397/) (2021)

## Built on by (most cited first, 1 of 1)

- [MonoKAN: Certified Monotonic Kolmogorov-Arnold Network](https://mlsanity.com/paper/2409.11078/) (2024)

## Abstract (arXiv)

Artificial Neural Networks (ANNs) have become a powerful tool for modeling complex relationships in large-scale datasets. However, their black-box nature poses trustworthiness challenges. In certain situations, ensuring trust in predictions might require following specific partial monotonicity constraints. However, certifying if an already-trained ANN is partially monotonic is challenging. Therefore, ANNs are often disregarded in some critical applications, such as credit scoring, where partial monotonicity is required. To address this challenge, this paper presents a novel algorithm (LipVor) that certifies if a black-box model, such as an ANN, is positive based on a finite number of evaluations. Consequently, since partial monotonicity can be expressed as a positivity condition on partial derivatives, LipVor can certify whether an ANN is partially monotonic. To do so, for every positively evaluated point, the Lipschitzianity of the black-box model is used to construct a specific neighborhood where the function remains positive. Next, based on the Voronoi diagram of the evaluated points, a sufficient condition is stated to certify if the function is positive in the domain. Unlike prior methods, our approach certifies partial monotonicity without constrained architectures or piece-wise linear activations. Therefore, LipVor could open up the possibility of using unconstrained ANN in some critical fields. Moreover, some other properties of an ANN, such as convexity, can be posed as positivity conditions, and therefore, LipVor could also be applied.