Deep neural networks (DNNs) have had many successes, but they suffer from two major issues: (1) a vulnerability to adversarial examples and (2) a tendency to elude human interpretation. Interestingly, recent empirical and theoretical evidence suggests that these two seemingly disparate issues are actually connected. In particular, robust models tend to provide more interpretable gradients than non-robust models. However, whether this relationship works in the opposite direction remains obscure. With this paper, we seek empirical answers to the following question: can models acquire adversarial robustness when they are trained to have interpretable gradients? We introduce a theoretically inspired technique called Interpretation Regularization (IR), which encourages a model's gradients to (1) match the direction of interpretable target salience maps and (2) have small magnitude. To assess model performance and tease apart factors that contribute to adversarial robustness, we conduct extensive experiments on MNIST and CIFAR-10 with both 2 and ∞ attacks. We demonstrate that training the networks to have interpretable gradients improves their robustness to adversarial perturbations. Applying the network interpretation technique SmoothGrad [59] yields additional performance gains, especially in cross-norm attacks and under heavy perturbations. The results indicate that the interpretability of the model gradients is a crucial factor for adversarial robustness. Code for the experiments can be found at https ://githu b.com/a1noa ck/inter p_regul ariza tion.

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Killing vectors are generators of symmetries in a spacetime. This article defines certain generalizations of Killing vectors, called affine symmetry tensors, or simply affine tensors. While the affine vectors of the Minkowski spacetime are well known, and partial results for valence n = 2 have been discussed, affine tensors of valence n > 2 have never been exhibited. In this article, we discuss a computational algorithm to compute affine tensors in Minkowski spacetime, and discuss the results for affine tensors of valence 2 ≤ n ≤ 7. After comparison with analogous results concerning Killing tensors, we make several conjectures about the spaces of affine tensors in Minkowski spacetime. KEYWORDS: Affine Symmetry Tensors; Affine Vectors; Killing Tensors; Killing Vectors; Minkowski Spacetime; Dimension; Maple CAS; Lie Derivative; Generalized Killing Tensor