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The existence of adversarial examples in which an imperceptible change in theinput can fool well trained neural networks was experimentally discovered bySzegedy et al in 2013, who called them "Intriguing properties of neuralnetworks". Since then, this topic had become one of the hottest research areaswithin machine learning, but the ease with which we can switch between any twodecisions in targeted attacks is still far from being understood, and inparticular it is not clear which parameters determine the number of inputcoordinates we have to change in order to mislead the network. In this paper wedevelop a simple mathematical framework which enables us to think about thisbaffling phenomenon from a fresh perspective, turning it into a naturalconsequence of the geometry of $\mathbb{R}^n$ with the $L_0$ (Hamming) metric,which can be quantitatively analyzed. In particular, we explain why we shouldexpect to find targeted adversarial examples with Hamming distance of roughly$m$ in arbitrarily deep neural networks which are designed to distinguishbetween $m$ input classes.

A Simple Explanation for the Existence of Adversarial Examples with Small Hamming Distance