Paper:
Generative Adversarial Networks (GANs), when trained on large datasets with diverse modes, are known to produce conflated images which do not distinctly belong to any of the modes. We hypothesize that this problem occurs due to the interaction between two facts: (1) For datasets with large variety, it is likely that the modes lie on separate manifolds. (2) The generator (G) is formulated as a continuous function, and the input noise is derived from a connected set, due to which G's output is a connected set. If G covers all modes, then there must be some portion of G's output which connects them. This corresponds to undesirable, conflated images. We develop theoretical arguments to support these intuitions. We propose a novel method to break the second assumption via learnable discontinuities in the latent noise space. Equivalently, it can be viewed as training several generators, thus creating discontinuities in the G function. We also augment the GAN formulation with a classifier C that predicts which noise partition/generator produced the output images, encouraging diversity between each partition/generator. We experiment on MNIST, celebA, STL-10, and a difficult dataset with clearly distinct modes, and show that the noise partitions correspond to different modes of the data distribution, and produce images of superior quality.

Review:
Summary:
The paper studies the problem of learning distributions with disconnected support. The paper is very well written, and the analysis is mostly correct, with some important exceptions. However, there are a number of claims that are unverified, and very important baselines are missing. I suggest improving the paper taking into account the following remarks and I will strongly consider improving the score.
Detailed comments:
- The paper is very well written, which is a big plus.
- There are a number of claims in the paper that are not supported by experiments, citations, or a theorem.
- Sections 3.1 - 3.3 can be summarized to "Connected prior + continuous generator => connected support". Thus, to allow for disconnected support, the authors propose to have a discontinuous generator. However to me it seems that a trivial and important attack to this problem is to allow a simple disconnected prior, such as a mixture between uniforms, or at least an approximately disconnected (given the superexponential decay of the gaussian pdf) of a mixture of gaussians, which is very common. The authors fail to mention this obvious alternative, or explore it further, which I think weakens the paper.
- Another standard approach to attacking diverse datasets such as imagenet is adding noise in the intermediate layers of the generator (this was done by EBGAN and the Improved GAN paper by Salimans et al.). It seems to me that this baseline is missing.
- Section 3.4, paragraph 3, "the outputs corresponding to vectors linearly interpolated from z_1 to z_2 show a smooth". Actually, this is known to not perform very well often, indeed the interpolations are done through great circles in z_1 and z_2. See https://www.youtube.com/watch?v=myGAju4L7O8 for example.
- Lemma 1 is correct, but the analysis on the paragraph following is flat out wrong. The fact that a certain z has high density doesn't imply that the sample g_\theta(z) has high density! You're missing the Jacobian term appearing in the change of variables. Indeed, it's common to see neural nets spreading appart regions of high probability to the extent that each individual output point has low density (this is due in its totallity to the fact that ||\nabla_x g_\theta(z)|| can be big.
- Borrowing from the previous comment, the evidence to support result 5 is insufficient. I think the authors have the right intuition, but no evidence or citation is presented to motivate result 5. Indeed, DCGANs are known to have extremely sharp interpolations, suggesting that small jumps in z lead to large jumps in images, thus having the potential to assign low probability to tunnels.
- A citation, experiment or a theorem is missing showing that the K of a generator is small enough in an experiment with separated manifolds. Until that evidence is presented, section 3.5 is anecdotal.
- The second paragraph of section 3.6 is a very astute observation, but again it is necessary to show some evidence to verify this intuition.
- The authors then propose to partition the prior space by training separate first layers for the generator in a maximally discriminative way, and then at inference time just sampling which layer to use uniformly. It's important to note that this has a problem when the underlying separated manifolds in the data are not equiprobable. For example, if we use N = 2 in CelebRoom but we use 30% faces and 70% bedrooms, I would still expect tunneling due to the fact that one of the linear layers has to cover both faces and bedrooms.
- MNIST is known to be a very poor benchmark for image generation, and it should be avoided.
- I fail to see an improvement in quality in CelebA. It's nice to see some minor form on clustering when using generator's prediction, but this has been seen in many other algorithms (e.g. ALI) with much better results long before. I have to say also the official baseline for 64x64 images in wgangp (that I've used several times) gives much better results than the ones presented in this paper https://github.com/igul222/improved_wgan_training/blob/master/gan_64x64.py .
- The experiments in celebRoom are quite nice, and a good result, but we are still missing a detailed analysis for most of the assumptions and improvements claimed in the paper. It's very hard to make very precise claims about the improvements of this algorithm in such a complex setting without having even studied the standard baselines (e.g. noise at every layer of the generator, which has very public and well established code https://github.com/openai/improved-gan/blob/master/imagenet/generator.py).
- I would like to point a lot of tunneling issues can be seen and studied in toy datasets. The authors may want to consider doing targeted experiments to evaluate their assumptions.
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After the rebuttal I've increased my score. The authors did a great job at addressing some of the concerns. I still think there is more room to be done as to justifying the approach, dealing properly with tunneling when we're not in the somewhat artificial case of equiprobable partitions, and primarily at understanding the extent to which tunneling is a problem in current methods. The revision is a step forward in this direction, but still a lot remains to be done. I would like to see simple targeted experiments aimed at testing how much and in what way tunneling is a problem in current methods before I see high dimensional non quantitative experiments.
In the case where the paper gets rejected I would highly recommend the acceptance at the workshop due to the paper raising interesting questions and hinting to a partial solution, even though the paper may not be at a state to be published at a conference venue like ICLR.