# SmoothHess: ReLU Network Feature Interactions via Stein's Lemma

arXiv: https://arxiv.org/abs/2311.00858 · HTML: https://mlsanity.com/paper/2311.00858/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 94 references that are in the map.
- First posted on arXiv 1 Nov 2023.

## Builds on (most cited first, 30 of 51)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [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)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [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)
- [SmoothGrad: removing noise by adding noise](https://mlsanity.com/paper/1706.03825/) (2017)
- [Skin Lesion Analysis Toward Melanoma Detection 2018: A Challenge Hosted by the International Skin Imaging Collaboration (ISIC)](https://mlsanity.com/paper/1902.03368/) (2019)
- [Interpretation of Neural Networks is Fragile](https://mlsanity.com/paper/1710.10547/) (2017)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Why ReLU networks yield high-confidence predictions far away from the training data and how to mitigate the problem](https://mlsanity.com/paper/1812.05720/) (2018)
- [On the Robustness of Interpretability Methods](https://mlsanity.com/paper/1806.08049/) (2018)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Explanations can be manipulated and geometry is to blame](https://mlsanity.com/paper/1906.07983/) (2019)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [The Shapley Taylor Interaction Index](https://mlsanity.com/paper/1902.05622/) (2019)
- [On the Properties of the Softmax Function with Application in Game Theory and Reinforcement Learning](https://mlsanity.com/paper/1704.00805/) (2017)
- [Procedural Content Generation via Machine Learning (PCGML)](https://mlsanity.com/paper/1702.00539/) (2017)
- [Reliable Post hoc Explanations: Modeling Uncertainty in Explainability](https://mlsanity.com/paper/2008.05030/) (2020)
- [Smooth Grad-CAM++: An Enhanced Inference Level Visualization Technique for Deep Convolutional Neural Network Models](https://mlsanity.com/paper/1908.01224/) (2019)
- [Explaining Explanations: Axiomatic Feature Interactions for Deep Networks](https://mlsanity.com/paper/2002.04138/) (2020)
- [CXPlain: Causal Explanations for Model Interpretation under Uncertainty](https://mlsanity.com/paper/1910.12336/) (2019)
- [Detecting Statistical Interactions from Neural Network Weights](https://mlsanity.com/paper/1705.04977/) (2017)
- [How does this interaction affect me? Interpretable attribution for feature interactions](https://mlsanity.com/paper/2006.10965/) (2020)

## Built on by (most cited first, 5 of 5)

- [Axiomatic Explainer Globalness via Optimal Transport](https://mlsanity.com/paper/2411.01126/) (2024)
- [Leveraging Machine Unlearning for Cost-Efficient Preference Alignment](https://mlsanity.com/paper/2504.06659/) (2025)
- [OrdShap: Feature Position Importance for Sequential Black-Box Models](https://mlsanity.com/paper/2507.11855/) (2025)
- [Feature-Function Curvature Analysis: A Geometric Framework for Explaining Differentiable Models](https://mlsanity.com/paper/2510.27207/) (2025)
- [Local Stability and Gaussian Smoothing of Quantized Neural Networks](https://mlsanity.com/paper/2607.20153/) (2026)

## Abstract (arXiv)

Several recent methods for interpretability model feature interactions by looking at the Hessian of a neural network. This poses a challenge for ReLU networks, which are piecewise-linear and thus have a zero Hessian almost everywhere. We propose SmoothHess, a method of estimating second-order interactions through Stein's Lemma. In particular, we estimate the Hessian of the network convolved with a Gaussian through an efficient sampling algorithm, requiring only network gradient calls. SmoothHess is applied post-hoc, requires no modifications to the ReLU network architecture, and the extent of smoothing can be controlled explicitly. We provide a non-asymptotic bound on the sample complexity of our estimation procedure. We validate the superior ability of SmoothHess to capture interactions on benchmark datasets and a real-world medical spirometry dataset.