# Hidden Monotonicity: Explaining Deep Neural Networks via their DC Decomposition

arXiv: https://arxiv.org/abs/2601.07700 · HTML: https://mlsanity.com/paper/2601.07700/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 44 of its 66 references that are in the map.
- First posted on arXiv 12 Jan 2026.

## Builds on (most cited first, 30 of 44)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Learning Deep Features for Discriminative Localization](https://mlsanity.com/paper/1512.04150/) (2015)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Sanity Checks for Saliency Maps](https://mlsanity.com/paper/1810.03292/) (2018)
- [SmoothGrad: removing noise by adding noise](https://mlsanity.com/paper/1706.03825/) (2017)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [Grad-CAM++: Improved Visual Explanations for Deep Convolutional Networks](https://mlsanity.com/paper/1710.11063/) (2017)
- [Interpretable Explanations of Black Boxes by Meaningful Perturbation](https://mlsanity.com/paper/1704.03296/) (2017)
- [Captum: A unified and generic model interpretability library for PyTorch](https://mlsanity.com/paper/2009.07896/) (2020)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Methods for Interpreting and Understanding Deep Neural Networks](https://mlsanity.com/paper/1706.07979/) (2017)
- [Top-down Neural Attention by Excitation Backprop](https://mlsanity.com/paper/1608.00507/) (2016)
- [On the (In)fidelity and Sensitivity for Explanations](https://mlsanity.com/paper/1901.09392/) (2019)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Explaining NonLinear Classification Decisions with Deep Taylor Decomposition](https://mlsanity.com/paper/1512.02479/) (2015)
- [Towards Robust Vision Transformer](https://mlsanity.com/paper/2105.07926/) (2021)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [Towards Best Practice in Explaining Neural Network Decisions with LRP](https://mlsanity.com/paper/1910.09840/) (2019)
- [Certified Monotonic Neural Networks](https://mlsanity.com/paper/2011.10219/) (2020)
- [Don't Lie to Me! Robust and Efficient Explainability with Verified Perturbation Analysis](https://mlsanity.com/paper/2202.07728/) (2022)

## Abstract (arXiv)

It has been demonstrated in various contexts that monotonicity leads to better explainability in neural networks. However, not every function can be well approximated by a monotone neural network. We demonstrate that monotonicity can still be used in two ways to boost explainability. First, we use an adaptation of the decomposition of a trained ReLU network into two monotone and convex parts, thereby overcoming numerical obstacles from an inherent blowup of the weights in this procedure. Our proposed saliency methods - SplitCAM and SplitLRP - improve on state of the art results on both VGG16 and Resnet18 networks on ImageNet-S across all Quantus saliency metric categories. Second, we exhibit that training a model as the difference between two monotone neural networks results in a system with strong self-explainability properties.