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The susceptibility of modern machine learning classifiers to adversarialexamples has motivated theoretical results suggesting that these might beunavoidable. However, these results can be too general to be applicable tonatural data distributions. Indeed, humans are quite robust for tasks involvingvision. This apparent conflict motivates a deeper dive into the question: Areadversarial examples truly unavoidable? In this work, we theoreticallydemonstrate that a key property of the data distribution -- concentration onsmall-volume subsets of the input space -- determines whether a robustclassifier exists. We further demonstrate that, for a data distributionconcentrated on a union of low-dimensional linear subspaces, exploiting datastructure naturally leads to classifiers that enjoy good robustness guarantees,improving upon methods for provable certification in certain regimes.

Adversarial Examples Might be Avoidable: The Role of Data Concentration in Adversarial Robustness