# Understanding Impacts of High-Order Loss Approximations and Features in Deep Learning Interpretation

arXiv: https://arxiv.org/abs/1902.00407 · HTML: https://mlsanity.com/paper/1902.00407/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Feb 2019.

## Built on by (most cited first, 22 of 22)

- [Salient ImageNet: How to discover spurious features in Deep Learning?](https://mlsanity.com/paper/2110.04301/) (2021)
- [Improving Deep Learning Interpretability by Saliency Guided Training](https://mlsanity.com/paper/2111.14338/) (2021)
- [ProtoPFormer: Concentrating on Prototypical Parts in Vision Transformers for Interpretable Image Recognition](https://mlsanity.com/paper/2208.10431/) (2022)
- [Sketching Curvature for Efficient Out-of-Distribution Detection for Deep Neural Networks](https://mlsanity.com/paper/2102.12567/) (2021)
- [SSSE: Efficiently Erasing Samples from Trained Machine Learning Models](https://mlsanity.com/paper/2107.03860/) (2021)
- [Understanding and Combating Robust Overfitting via Input Loss Landscape Analysis and Regularization](https://mlsanity.com/paper/2212.04985/) (2022)
- [Core Risk Minimization using Salient ImageNet](https://mlsanity.com/paper/2203.15566/) (2022)
- [Token-Modification Adversarial Attacks for Natural Language Processing: A Survey](https://mlsanity.com/paper/2103.00676/) (2021)
- [SMOOT: Saliency Guided Mask Optimized Online Training](https://mlsanity.com/paper/2310.00772/) (2023)
- [A Survey of Neural Trees](https://mlsanity.com/paper/2209.03415/) (2022)
- [Interpretable Artificial Intelligence through the Lens of Feature Interaction](https://mlsanity.com/paper/2103.03103/) (2021)
- [Improving Attribution Methods by Learning Submodular Functions](https://mlsanity.com/paper/2104.09073/) (2021)
- [SmoothHess: ReLU Network Feature Interactions via Stein's Lemma](https://mlsanity.com/paper/2311.00858/) (2023)
- [Data-Centric Debugging: mitigating model failures via targeted data collection](https://mlsanity.com/paper/2211.09859/) (2022)
- [On Dissipativity of Cross-Entropy Loss in Training ResNets](https://mlsanity.com/paper/2405.19013/) (2024)
- [Unraveling the Hessian: A Key to Smooth Convergence in Loss Function Landscapes](https://mlsanity.com/paper/2409.11995/) (2024)
- [H-Sets: Hessian-Guided Discovery of Set-Level Feature Interactions in Image Classifiers](https://mlsanity.com/paper/2604.22045/) (2026)
- [MaskSearch: Querying Image Masks at Scale](https://mlsanity.com/paper/2305.02375/) (2023)
- [Deep Metric Tensor Regularized Policy Gradient](https://mlsanity.com/paper/2305.11017/) (2023)
- [Reducing Adversarial Training Cost with Gradient Approximation](https://mlsanity.com/paper/2309.09464/) (2023)
- [Aerial Image Classification in Scarce and Unconstrained Environments via Conformal Prediction](https://mlsanity.com/paper/2504.17655/) (2025)
- [Improving Adversarial Robustness of Attribution via Implicit Regularization](https://mlsanity.com/paper/2605.29983/) (2026)

## Abstract (arXiv)

Current methods to interpret deep learning models by generating saliency maps generally rely on two key assumptions. First, they use first-order approximations of the loss function neglecting higher-order terms such as the loss curvatures. Second, they evaluate each feature's importance in isolation, ignoring their inter-dependencies. In this work, we study the effect of relaxing these two assumptions. First, by characterizing a closed-form formula for the Hessian matrix of a deep ReLU network, we prove that, for a classification problem with a large number of classes, if an input has a high confidence classification score, the inclusion of the Hessian term has small impacts in the final solution. We prove this result by showing that in this case the Hessian matrix is approximately of rank one and its leading eigenvector is almost parallel to the gradient of the loss function. Our empirical experiments on ImageNet samples are consistent with our theory. This result can have implications in other related problems such as adversarial examples as well. Second, we compute the importance of group-features in deep learning interpretation by introducing a sparsity regularization term. We use the L0-L1 relaxation technique along with the proximal gradient descent to have an efficient computation of group feature importance scores. Our empirical results indicate that considering group features can improve deep learning interpretation significantly.