# Tropical Decision Boundaries for Neural Networks Are Robust Against Adversarial Attacks

arXiv: https://arxiv.org/abs/2402.00576 · HTML: https://mlsanity.com/paper/2402.00576/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 45 references that are in the map.
- First posted on arXiv 1 Feb 2024.

## Builds on (most cited first, 22 of 22)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Adversarial Attacks on Neural Network Policies](https://mlsanity.com/paper/1702.02284/) (2017)
- [Adversarial Risk and the Dangers of Evaluating Against Weak Attacks](https://mlsanity.com/paper/1802.05666/) (2018)
- [Maxout Networks](https://mlsanity.com/paper/1302.4389/) (2013)
- [Adversarial Training and Robustness for Multiple Perturbations](https://mlsanity.com/paper/1904.13000/) (2019)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Robustness via curvature regularization, and vice versa](https://mlsanity.com/paper/1811.09716/) (2018)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [ME-Net: Towards Effective Adversarial Robustness with Matrix Estimation](https://mlsanity.com/paper/1905.11971/) (2019)
- [Provable Robustness of ReLU networks via Maximization of Linear Regions](https://mlsanity.com/paper/1810.07481/) (2018)
- [A Survey of Robust Adversarial Training in Pattern Recognition: Fundamental, Theory, and Methodologies](https://mlsanity.com/paper/2203.14046/) (2022)
- [Provable robustness against all adversarial lp-perturbations for p≥ 1](https://mlsanity.com/paper/1905.11213/) (2019)
- [Theoretical Analysis of Adversarial Learning: A Minimax Approach](https://mlsanity.com/paper/1811.05232/) (2018)
- [A Tropical Approach to Neural Networks with Piecewise Linear Activations](https://mlsanity.com/paper/1805.08749/) (2018)
- [Text Analysis Using Deep Neural Networks in Digital Humanities and Information Science](https://mlsanity.com/paper/2307.16217/) (2023)
- [Transformed Low-Rank Parameterization Can Help Robust Generalization for Tensor Neural Networks](https://mlsanity.com/paper/2303.00196/) (2023)
- [Tropical Support Vector Machines: Evaluations and Extension to Function Spaces](https://mlsanity.com/paper/2101.11531/) (2021)

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

- [Compound and Parallel Modes of Tropical Convolutional Neural Networks](https://mlsanity.com/paper/2504.06881/) (2025)

## Abstract (arXiv)

We introduce a simple, easy to implement, and computationally efficient tropical convolutional neural network architecture that is robust against adversarial attacks. We exploit the tropical nature of piece-wise linear neural networks by embedding the data in the tropical projective torus in a single hidden layer which can be added to any model. We study the geometry of its decision boundary theoretically and show its robustness against adversarial attacks on image datasets using computational experiments.