  The goal of this paper is to analyze the geometric properties of deep neural
network classifiers in the input space. We specifically study the topology of
classification regions created by deep networks, as well as their associated
decision boundary. Through a systematic empirical investigation, we show that
state-of-the-art deep nets learn connected classification regions, and that the
decision boundary in the vicinity of datapoints is flat along most directions.
We further draw an essential connection between two seemingly unrelated
properties of deep networks: their sensitivity to additive perturbations in the
inputs, and the curvature of their decision boundary. The directions where the
decision boundary is curved in fact remarkably characterize the directions to
which the classifier is the most vulnerable. We finally leverage a fundamental
asymmetry in the curvature of the decision boundary of deep nets, and propose a
method to discriminate between original images, and images perturbed with small
adversarial examples. We show the effectiveness of this purely geometric
approach for detecting small adversarial perturbations in images, and for
recovering the labels of perturbed images.
