Image classifiers can not be made robust to small perturbations

The sensitivity of image classifiers to small perturbations in the input is often viewed as a defect of their construction. We demonstrate that this sensitivity is a fundamental property of classifiers. For any arbitrary classifier over the set of $n$-by-$n$ images, we show that for all but one class it is possible to change the classification of all but a tiny fraction of the images in that class with a tiny modification compared to the diameter of the image space when measured in any $p$-norm, including the hamming distance. We then examine how this phenomenon manifests in human visual perception and discuss its implications for the design considerations of computer vision systems.