Alhussein Fawzi, Seyed-Mohsen Moosavi-Dezfooli, Pascal Frossard. Robustness of classifiers: from adversarial to random noise. NIPS 2016: 1624-1632.

Fawzi et al. study robustness in the transition from random samples to semi-random and adversarial samples. Specifically they present bounds relating the norm of an adversarial perturbation to the norm of random perturbations – for the exact form I refer to the paper. Personally, I find the definition of semi-random noise most interesting, as it allows to get an intuition for distinguishing random noise from adversarial examples. As in related literature, adversarial examples are defined as

$r_S(x_0) = \arg\min_{x_0 \in S} \|r\|_2$ s.t. $f(x_0 + r) \neq f(x_0)$

where $f$ is the classifier to attack and $S$ the set of allowed perturbations (e.g. requiring that the perturbed samples are still images). If $S$ is mostly unconstrained regarding the direction of $r$ in high dimensional space, Fawzi et al. consider $r$ to be an adversarial examples – intuitively, and adversary can choose $r$ arbitrarily to fool the classifier. If, however, the directions considered in $S$ are constrained to an $m$-dimensional subspace, Fawzi et al. consider $r$ to be semi-random noise. In the extreme case, if $m = 1$, $r$ is random noise. In this case, we can intuitively think of $S$ as a randomly chosen one dimensional subspace – i.e. a random direction in multi-dimensional space.

Also find this summary on ShortScience.org.
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