  Adversarial examples have recently drawn considerable attention in the field
of machine learning due to the fact that small perturbations in the data can
result in major performance degradation. This phenomenon is usually modeled by
a malicious adversary that can apply perturbations to the data in a constrained
fashion, such as being bounded in a certain norm. In this paper, we study this
problem when the adversary is constrained by the $\ell_0$ norm; i.e., it can
perturb a certain number of coordinates in the input, but has no limit on how
much it can perturb those coordinates. Due to the combinatorial nature of this
setting, we need to go beyond the standard techniques in robust machine
learning to address this problem. We consider a binary classification scenario
where $d$ noisy data samples of the true label are provided to us after
adversarial perturbations. We introduce a classification method which employs a
nonlinear component called truncation, and show in an asymptotic scenario, as
long as the adversary is restricted to perturb no more than $\sqrt{d}$ data
samples, we can almost achieve the optimal classification error in the absence
of the adversary, i.e. we can completely neutralize adversary's effect.
Surprisingly, we observe a phase transition in the sense that using a converse
argument, we show that if the adversary can perturb more than $\sqrt{d}$
coordinates, no classifier can do better than a random guess.
