  Recently, there has been an abundance of works on designing Deep Neural
Networks (DNNs) that are robust to adversarial examples. In particular, a
central question is which features of DNNs influence adversarial robustness
and, therefore, can be to used to design robust DNNs. In this work, this
problem is studied through the lens of compression which is captured by the
low-rank structure of weight matrices. It is first shown that adversarial
training tends to promote simultaneously low-rank and sparse structure in the
weight matrices of neural networks. This is measured through the notions of
effective rank and effective sparsity. In the reverse direction, when the low
rank structure is promoted by nuclear norm regularization and combined with
sparsity inducing regularizations, neural networks show significantly improved
adversarial robustness. The effect of nuclear norm regularization on
adversarial robustness is paramount when it is applied to convolutional neural
networks. Although still not competing with adversarial training, this result
contributes to understanding the key properties of robust classifiers.
