  Deep neural networks have been shown to suffer from a surprising weakness:
their classification outputs can be changed by small, non-random perturbations
of their inputs. This adversarial example phenomenon has been explained as
originating from deep networks being "too linear" (Goodfellow et al., 2014). We
show here that the linear explanation of adversarial examples presents a number
of limitations: the formal argument is not convincing, linear classifiers do
not always suffer from the phenomenon, and when they do their adversarial
examples are different from the ones affecting deep networks.
  We propose a new perspective on the phenomenon. We argue that adversarial
examples exist when the classification boundary lies close to the submanifold
of sampled data, and present a mathematical analysis of this new perspective in
the linear case. We define the notion of adversarial strength and show that it
can be reduced to the deviation angle between the classifier considered and the
nearest centroid classifier. Then, we show that the adversarial strength can be
made arbitrarily high independently of the classification performance due to a
mechanism that we call boundary tilting. This result leads us to defining a new
taxonomy of adversarial examples. Finally, we show that the adversarial
strength observed in practice is directly dependent on the level of
regularisation used and the strongest adversarial examples, symptomatic of
overfitting, can be avoided by using a proper level of regularisation.
