# Bort: Towards Explainable Neural Networks with Bounded Orthogonal Constraint

arXiv: https://arxiv.org/abs/2212.09062 · HTML: https://mlsanity.com/paper/2212.09062/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 59 references that are in the map.
- First posted on arXiv 18 Dec 2022.

## Builds on (most cited first, 30 of 46)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [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)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [CutMix: Regularization Strategy to Train Strong Classifiers with Localizable Features](https://mlsanity.com/paper/1905.04899/) (2019)
- [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)
- [RandAugment: Practical automated data augmentation with a reduced search space](https://mlsanity.com/paper/1909.13719/) (2019)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [SmoothGrad: removing noise by adding noise](https://mlsanity.com/paper/1706.03825/) (2017)
- [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)
- [RISE: Randomized Input Sampling for Explanation of Black-box Models](https://mlsanity.com/paper/1806.07421/) (2018)
- [This Looks Like That: Deep Learning for Interpretable Image Recognition](https://mlsanity.com/paper/1806.10574/) (2018)
- [Large Batch Optimization for Deep Learning: Training BERT in 76 minutes](https://mlsanity.com/paper/1904.00962/) (2019)
- [Interpretable Explanations of Black Boxes by Meaningful Perturbation](https://mlsanity.com/paper/1704.03296/) (2017)
- [Explaining Explanations: An Overview of Interpretability of Machine Learning](https://mlsanity.com/paper/1806.00069/) (2018)
- [Methods for Interpreting and Understanding Deep Neural Networks](https://mlsanity.com/paper/1706.07979/) (2017)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [Deep Learning for Case-Based Reasoning through Prototypes: A Neural Network that Explains Its Predictions](https://mlsanity.com/paper/1710.04806/) (2017)

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

- [Diversifying the Mixture-of-Experts Representation for Language Models with Orthogonal Optimizer](https://mlsanity.com/paper/2310.09762/) (2023)
- [SWEA: Updating Factual Knowledge in Large Language Models via Subject Word Embedding Altering](https://mlsanity.com/paper/2401.17809/) (2024)
- [Harnessing Orthogonality to Train Low-Rank Neural Networks](https://mlsanity.com/paper/2401.08505/) (2024)
- [Self-Explaining Hypergraph Neural Networks for Diagnosis Prediction](https://mlsanity.com/paper/2502.10689/) (2025)
- [On the Fast Adaptation of Delayed Clients in Decentralized Federated Learning: A Centroid-Aligned Distillation Approach](https://mlsanity.com/paper/2508.02993/) (2025)

## Abstract (arXiv)

Deep learning has revolutionized human society, yet the black-box nature of deep neural networks hinders further application to reliability-demanded industries. In the attempt to unpack them, many works observe or impact internal variables to improve the comprehensibility and invertibility of the black-box models. However, existing methods rely on intuitive assumptions and lack mathematical guarantees. To bridge this gap, we introduce Bort, an optimizer for improving model explainability with boundedness and orthogonality constraints on model parameters, derived from the sufficient conditions of model comprehensibility and invertibility. We perform reconstruction and backtracking on the model representations optimized by Bort and observe a clear improvement in model explainability. Based on Bort, we are able to synthesize explainable adversarial samples without additional parameters and training. Surprisingly, we find Bort constantly improves the classification accuracy of various architectures including ResNet and DeiT on MNIST, CIFAR-10, and ImageNet. Code: https://github.com/zbr17/Bort.