Certified geometric robustness -- Super-DeepG

The adoption of vision neural networks in regulated industries requires formal robustness guarantees, especially in safety-critical domains such as healthcare, autonomous vehicles, and aerospace. However, current approaches are confined to incomplete statistical verification or robustness to $\ell_p$-norm and affine transforms, which cover only a narrow subset of perturbations to the image formation process. In particular, robustness to camera motion remains an open problem despite being key to deploy many vision applications. We present a formal verification approach that targets robustness against 3D motion perturbations of the capturing camera. We first establish a closed-form mapping from camera pose to pixel values. By analyzing the continuity properties of the resulting homographies, we show that recent work on Lipschitz optimization and piecewise continuity can be extended to derive tight linear bounds on perturbed pixel values. Our approach applies to scenes with predominantly planar structure, such as ground planes in augmented reality, road markings and traffic signs in autonomous driving, or planar workspaces in robotic manipulation. This enables the first formal verification of projective geometry transforms, without complex simulation, surrogate networks, or explicit image-formation models. We validate our implementation and show up to 89% speedup and 7% tighter bounds over prior work. We then evaluate our method on the VNN-COMP benchmark and reveal systematic weaknesses to projective perturbations. Finally, we demonstrate a real-world case study on a safety-critical runway classifier, highlighting practical vulnerabilities to camera motion, and addressing a key challenge in the certification of learned models. Data and code are publicly available at https://github.com/jeangud/homography-verification .

A primary challenge in AI safety is the existence of adversarial examples -- slightly distorted inputs that cause a neural network (NN) to misclassify. To mitigate this problem, recent research focuses on the computation of robustness certifications, which, for a given input, determine the largest distortion the input may receive without breaking the network's prediction. Robustness certifications can be interpreted as an axis-aligned hyper-rectangle (multi-dimensional intervals). Most existing approaches focus on maximizing the certification's volume, but recent intractability results prohibit the computation of volume-optimal certifications in reasonable time. We introduce the apothem measure and show how to compute apothem-optimal certifications in a linear number of calls to a NN verifier (oracle) w.r.t. the input domain's diameter. Moreover, we prove that we cannot have a volume-optimal, oracle-based algorithm, even if we discard the oracle costs. Also, we introduce dual certifications -- an interval including all instances of a class -- thus providing apothem-minimum upper bounds to a robustness certification. Further, we present the ParallelepipedoNN system, which we evaluate on the standard MNIST and Fashion MNIST benchmarks. A preliminary comparison with existing work on the same datasets reveals at least two-fold improvement w.r.t. the minimum edge length.

Vision models have been found to be susceptible to perturbations such as motion blur induced at runtime by a shaking camera. This impedes their deployment in critical applications since phenomena such as slightly blurred vision might lead to failures, for example an object detector missing objects. While methods such as data augmentation or Adversarial Training can improve empirical robustness, they lack formal safety guarantees, making it difficult to identify and mitigate hidden vulnerabilities. We introduce a novel Certified Training approach that leverages an efficient encoding of convolutional perturbations to train provably robust models. Our method significantly outperforms Adversarial Training, achieving, for example, over 80% robust accuracy against motion blur of reasonable intensity on CIFAR10 while maintaining comparable standard accuracy.