  Deep networks have recently been shown to be vulnerable to universal
perturbations: there exist very small image-agnostic perturbations that cause
most natural images to be misclassified by such classifiers. In this paper, we
propose the first quantitative analysis of the robustness of classifiers to
universal perturbations, and draw a formal link between the robustness to
universal perturbations, and the geometry of the decision boundary.
Specifically, we establish theoretical bounds on the robustness of classifiers
under two decision boundary models (flat and curved models). We show in
particular that the robustness of deep networks to universal perturbations is
driven by a key property of their curvature: there exists shared directions
along which the decision boundary of deep networks is systematically positively
curved. Under such conditions, we prove the existence of small universal
perturbations. Our analysis further provides a novel geometric method for
computing universal perturbations, in addition to explaining their properties.
