# Fast Geometric Projections for Local Robustness Certification

arXiv: https://arxiv.org/abs/2002.04742 · HTML: https://mlsanity.com/paper/2002.04742/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Feb 2020.

## Built on by (most cited first, 21 of 21)

- [Globally-Robust Neural Networks](https://mlsanity.com/paper/2102.08452/) (2021)
- [Consistent Counterfactuals for Deep Models](https://mlsanity.com/paper/2110.03109/) (2021)
- [Efficient Neural Network Analysis with Sum-of-Infeasibilities](https://mlsanity.com/paper/2203.11201/) (2022)
- [Robust Models Are More Interpretable Because Attributions Look Normal](https://mlsanity.com/paper/2103.11257/) (2021)
- [Scalable Verification of GNN-based Job Schedulers](https://mlsanity.com/paper/2203.03153/) (2022)
- [Convex Bounds on the Softmax Function with Applications to Robustness Verification](https://mlsanity.com/paper/2303.01713/) (2023)
- [Closed-loop Analysis of Vision-based Autonomous Systems: A Case Study](https://mlsanity.com/paper/2302.04634/) (2023)
- [Incremental Verification of Neural Networks](https://mlsanity.com/paper/2304.01874/) (2023)
- [Unlocking Deterministic Robustness Certification on ImageNet](https://mlsanity.com/paper/2301.12549/) (2023)
- [Recent Advances in Large Margin Learning](https://mlsanity.com/paper/2103.13598/) (2021)
- [Relaxing Local Robustness](https://mlsanity.com/paper/2106.06624/) (2021)
- [Relational DNN Verification With Cross Executional Bound Refinement](https://mlsanity.com/paper/2405.10143/) (2024)
- [Limitations of Piecewise Linearity for Efficient Robustness Certification](https://mlsanity.com/paper/2301.08842/) (2023)
- [Interpreting Robustness Proofs of Deep Neural Networks](https://mlsanity.com/paper/2301.13845/) (2023)
- [Faithful Explanations for Deep Graph Models](https://mlsanity.com/paper/2205.11850/) (2022)
- [BEAVER: An Efficient Deterministic LLM Verifier](https://mlsanity.com/paper/2512.05439/) (2025)
- [On the Perils of Cascading Robust Classifiers](https://mlsanity.com/paper/2206.00278/) (2022)
- [Verifying Attention Robustness of Deep Neural Networks against Semantic Perturbations](https://mlsanity.com/paper/2207.05902/) (2022)
- [A Formally Verified Robustness Certifier for Neural Networks (Extended Version)](https://mlsanity.com/paper/2505.06958/) (2025)
- [VeriCompress: A Tool to Streamline the Synthesis of Verified Robust Compressed Neural Networks from Scratch](https://mlsanity.com/paper/2211.09945/) (2022)
- [Extending Neural Network Verification to a Larger Family of Piece-wise Linear Activation Functions](https://mlsanity.com/paper/2311.10780/) (2023)

## Abstract (arXiv)

Local robustness ensures that a model classifies all inputs within an ℓ2-ball consistently, which precludes various forms of adversarial inputs. In this paper, we present a fast procedure for checking local robustness in feed-forward neural networks with piecewise-linear activation functions. Such networks partition the input space into a set of convex polyhedral regions in which the network's behavior is linear; hence, a systematic search for decision boundaries within the regions around a given input is sufficient for assessing robustness. Crucially, we show how the regions around a point can be analyzed using simple geometric projections, thus admitting an efficient, highly-parallel GPU implementation that excels particularly for the ℓ2 norm, where previous work has been less effective. Empirically we find this approach to be far more precise than many approximate verification approaches, while at the same time performing multiple orders of magnitude faster than complete verifiers, and scaling to much deeper networks.