Paper:
We propose a simple change to existing neural network structures for better defending against gradient-based adversarial attacks. Instead of using popular activation functions (such as ReLU), we advocate the use of k-Winners-Take-All (k-WTA) activation, a C 0 discontinuous function that purposely invalidates the neural network model's gradient at densely distributed input data points. The proposed k-WTA activation can be readily used in nearly all existing networks and training methods with no significant overhead. Our proposal is theoretically rationalized. We analyze why the discontinuities in k-WTA networks can largely prevent gradient-based search of adversarial examples and why they at the same time remain innocuous to the network training. This understanding is also empirically backed. We test k-WTA activation on various network structures optimized by a training method, be it adversarial training or not. In all cases, the robustness of k-WTA networks outperforms that of traditional networks under white-box attacks.

Review:
This paper addresses the important question of improving the robustness of deep neural networks against adversarial attacks. The authors propose a surprisingly simple measure to improve adversarial robustness, namely replacing typical activation functions such as ReLU with a k-winners-take-all (k-WTA) functions, whereby the k largest values of the input vector are copied, and all other elements of the output vector are set to zero. Since the size of input and output maps varies drastically within networks, the authors instead use a sparsity ratio \gamma that calculates k as a function of input size. k-WTA networks can be trained without special treatment, but for low \gamma values the authors propose a training schedule, whereby \gamma is slowly reduced, then re-training takes place, until the desired value of \gamma is reached. The presented effect is backed up by extensive theoretical investigations that relate the increased robustness to the dense introduction of discontinuities, which makes gradient-based adversarial attacks harder. A small change in an input signal can change the identity of the "winning" inputs, and thus in a sub-sequent matrix multiplication make use of other rows or columns, thus allowing arbitrarily large effects due to small input variations. Empirical evaluations in CIFAR and SVHN for a variety of attacks and defense mechanisms demonstrate the desired effects, and illustrate the loss landscapes due to using k-WTA.
I think the paper is a valuable and novel contribution to an important topic, and is definitely suitable for publication at ICLR. In principle there is just one novel idea, namely using k-WTA activations to improve adversarial robustness, but this claim is investigated thoroughly, in theory, and demonstrated convincingly in experiments. The paper is well written and tries to address all potential questions one might have surrounding the basic idea. There is code available, and the idea should be simple to implement in practice, so I would expect this paper to have large impact on the study of adversarial robustness.
I appreciate the thorough proofs of the claims in section C of the appendix, but I did not review all proofs in  detail. 
Potential weaknesses that should be addressed:
1. Section 4: I would propose to fully define A_rob here or at least provide a reference.
2. From Table 1 it seems that using k-WTA leads to a quite noticeable drop in standard accuracy, especially for sparse \gamma, which leads to the best adversarial robustness. Can you please comment on whether the full ReLU accuracy in the natural case can always be recovered by k-WTA networks, e.g. with larger \gamma?
3. Since small changes have a big effect in k-WTA, it should be investigated how robust the k-WTA networks are with respect to more natural perturbations, e.g. noisy input, blurring, translations, rotations, occlusions, etc. as introduced in (Hendrycks and Dietterich, 2019). It would be critical if such perturbations have a stronger effect on k-WTA.
4. Is there any intuition about whether k-WTA should be used everywhere in a deep network, or whether it makes sense to mix k-WTA and ReLU functions? 
Minor comments:
5. WTA networks are very popular in computational neuroscience and are even hypothesized to represent canonical microcircuit functions (see e.g. Douglas, Martin, Whitteridge, 1989, Neural Computation, and many follow-up articles). You cite the work of Maass, 2000a,b already, it could be interesting to link your work to other papers in that field who motivate WTA from a biological perspective.