Disentangling Safe and Unsafe Image Corruptions via Anisotropy and Locality

State-of-the-art machine learning systems are vulnerable to small perturbations to their input, where _small_ is defined according to a threat model that assigns a positive threat to each perturbation. Most prior works define a task-agnostic, isotropic, and global threat, like the $\ell_p$ norm, where the magnitude of the perturbation fully determines the degree of the threat and neither the direction of the attack nor its position in space matter. However, common corruptions in computer vision, such as blur, compression, or occlusions, are not well captured by such treat models. This paper proposes a novel threat model called $\texttt{Projected Displacement}$ (PD) to study robustness beyond existing isotropic and global threat models. The proposed threat model measures the threat of a perturbation via its alignment with _unsafe directions_, defined as directions in the input space along which a perturbation of sufficient magnitude changes the ground truth class label. Unsafe directions are identified locally for each input based on observed training data. In this way, the PD-threat model exhibits anisotropy and locality. The PD-threat model is computationally efficient and can be easily integrated into existing robustness pipelines. Experiments on Imagenet-1k data indicate that, for any input, the set of perturbations with small PD threat includes _safe_ perturbations of large $\ell_p$ norm that preserve the true label, such as noise, blur and compression, while simultaneously excluding _unsafe_ perturbations that alter the true label. Unlike perceptual threat models based on embeddings of large-vision models, the PD-threat model can be readily computed for arbitrary classification tasks without pre-training or finetuning. Further additional task information such as sensitivity to image regions or concept hierarchies can be easily integrated into the assessment of threat and thus the PD threat model presents practitioners a flexible, task-driven threat specification that alleviates the limitations of $\ell_p$-threat models.