text: This paper continues a trend of incremental improvements to Wasserstein GANs (WGAN),  where the latter were proposed in order to alleviate the difficulties encountered in training GANs.  Originally, Arjovsky et al.  [1] argued that the Wasserstein distance was superior to many others typically used for GANs.  An important feature of WGANs is the requirement for the discriminator to be 1-Lipschitz,  which [1] achieved simply by clipping the network weights.  Recently, Gulrajani et al. [2] proposed a gradient penalty "encouraging" the discriminator to be 1-Lipschitz.  However, their approach estimated continuity on points between the generated and the real samples,  and thus could fail to guarantee Lipschitz-ness at the early training stages.  The paper under review overcomes this drawback by estimating the continuity on perturbations of the real samples.  Together with various technical improvements, this leads to state-of-the-art practical performance both in terms of generated images and in semi-supervised learning.   In terms of novelty, the paper provides one core conceptual idea followed by several tweaks aimed at improving the practical performance of GANs.  The key conceptual idea is to perturb each data point twice and use a Lipschitz constant to bound the difference in the discriminator’s response on the perturbed points.   The proposed method is used in eq. (6) together with the gradient penalty from [2].  The authors found that directly perturbing the data with Gaussian noise led to inferior results  and therefore propose to perturb the hidden layers using dropout.  For supervised learning they demonstrate less overfitting for both MNIST and CIFAR 10.   They also extend their framework to the semi-supervised setting of Salismans et al 2016 and report improved image generation.  The authors do an excellent comparative job in presenting their experiments.  They compare numerous techniques (e.g., Gaussian noise, dropout) and demonstrates the applicability of the approach for a wide range of tasks.  They use several criteria to evaluate their performance (images, inception score, semi-supervised learning, overfitting, weight histogram) and compare against a wide range of competing papers.  Where the paper could perhaps be slightly improved is writing clarity.  In particular, the discussion of M and M' is vital to the point of the paper,  but could be written in a more transparent manner.  The same goes for the semi-supervised experiment details and the CIFAR-10 augmentation process.  Finally, the title seems uninformative.  Almost all progress is incremental,  and the authors modestly give credit to both [1] and [2],  but the title is neither memorable nor useful in expressing the novel idea.  [1] Martin Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein gan. [2] Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron Courville. Improved training of wasserstein gans.