Paper:
We make the following striking observation: fully convolutional VAE models trained on 32×32 ImageNet can generalize well, not just to 64×64 but also to far larger photographs, with no changes to the model. We use this property, applying fully convolutional models to lossless compression, demonstrating a method to scale the VAE-based 'Bits-Back with ANS' algorithm for lossless compression (Townsend et al., 2019) to large color photographs, and achieving state of the art for compression of full size ImageNet images. We release Craystack, an open source library for convenient prototyping of lossless compression using probabilistic models, along with full implementations of all of our compression results 1 .
of magnitude, resulting from vectorization.
We release an open source implementation based on 'Craystack', a Python package which we have written for general prototyping of lossless compression with ANS.

Review:
Summary:
This paper focuses on lossless source compression with bits back coding for hierarchical fully convolutional VAEs. The focus/contribution is three-fold: 1. Improve the compression rate performance by adapting the discretization of latent space required for the entropy coder ANS. The newly proposed discretization scheme allows for a dependency structure that is not restricted to a Markov chain structure in the encoder model q(z|x) and in the generative part of the model p(x,z). This is in contrast with bit-swap[1], which requires a markov chain structure. The dependency structure that is allowed in the proposed method is widely known to perform better than a markov chain structure, which can explain why it improves significantly over Bit-swap [1] (another hierarchical VAE compression algorithm that uses bits back coding.) 2. Increasing compression speed by implementing a vectorized version of ANS, and heaving an ANS head in the shape of a pair of arrays matching that of the latent variable and the observed variable. The latter allows for simultaneous encoding of the latent with the prior distribution and the image with the decoder distribution. 3. Showing that a model trained on a low-resolution imagenet 32 dataset can generalize its compression capabilities to higher resolution datasets with convincing results. 
Decision: Accept.
This paper is clearly written, makes clear claims and supports these claims with convincing experiments. The contributions are of practical use and I expect future work to benefit from this paper. 
Supporting arguments for decision:
The paper is well motivated; off the shelf compression algorithms such as PNG are also not trained on every dataset separately, and cross-dataset generalization is important if this model should be used in practice for many different images from different datasets and of different resolutions.  
The paper clearly supports the main claims. It improves upon the previous bits-back coding-based hierarchical VAE [1]. The only hypothesis that is not checked is the one that hypothesizes that the lower bpd for higher resolution images is due to the lower ratio of edge pixels versus non-edge pixels, but this is not a dealbreaker from my point of view. 
I would like the authors to revise their statement of state of the art compression performance on page 7 directly below table 2. “The fact that HiLLoC achieves state of the art compression rates relative to the baselines even under a change of distribution is striking, and provides strong evidence of its efficacy as a general method for lossless compression of natural images”. This is sentence should be made more nuanced as the proposed model only improves on Bit-Swap, but is still significantly outperformed by Local bits back coding (LBB [2]), and in the case of cifar-10 also by integer discrete flows (IDF [3]). On the other hand, it would be useful to still state that LBB is trained on every dataset separately, as well as IDF. Note also that in [3], a model that is trained on Imagenet32 and evaluated on the other datasets is also reported (see table 1 in [3]). It would be beneficial for the author to include the scores of this model, as the proposed method seems to perform slightly better at generalizing to new datasets.
Because of the buffer of initial bits required by bit-back coding, the compression/decompression of several data points has to be sequential if one wants to amortize this cost over several data points. Compression methods that don’t rely on bits-back coding, such as IDF [3], do not have this issue and can compress/decompress data points in parallel. Since this influences the practical usability of the model, it would be transparent to mention this. 
My final main question is on the equivalence of evaluation methods of Bit-Swap and Hilloc on imagenet. The Bit-Swap paper states: “For MNIST, CIFAR-10 and Imagenet (32 × 32) we report the bitrates, shown in Table 5, as a result of compressing 100 datapoints in sequence (averaged over 100 experiments)...”. This means that Bit-Swap is not evaluated on the full test set of Imagenet 32 (as this contains 50000 images), as opposed to Hilloc. Do the authors think this is a problem? 
Furthermore, in the case of “full” Imagenet, Bit-swap uses a subset of 100 images for evaluation and crops them to a multiple of 32 pixels in height and width, so that bit-swap can compress patches and the result is the average of patches for on image. Hilloc appears to take 500 random images and does not state anything about cropping. Could the authors comment on this?
Additional feedback to improve paper (not part of decision assessment):
- In the introduction, first paragraph: “ the method can achieve an expected message length equal to the variational free energy, often referred to as the evidence lower bound (ELBO) of the model. “ → “ the method can achieve an expected message length equal to the variational free energy, often referred to as the negative evidence lower bound (ELBO) of the model. “
- Section 3.2, last paragraph: It is not clear if in practice the latent and image are actually encoded in parallel as the author states that this is “in theory” possible. 
- Page 4: “... we found that most of the compute time for our compression was spent in neural net inference, …” I assume you mean “inference” in any part of the encoder or decoder, and not specifically approximate inference of the encoder network. Perhaps clarify this to avoid confusion?
- Section 4: When referring to the ResnetVAE by Kingma et al, it would be appropriate to also cite [4], as this is very similar to resnetVAE’s and was released earlier.
[1] F. H. Kingma, P. Abbeel, and J. Ho. Bit-Swap: recursive bits-back coding for lossless compression with hierarchical latent variables. In International Conference on Machine Learning (ICML), 2019.
[2] Jonathan Ho, Evan Lohn, and Pieter Abbeel. Compression with Flows via Local Bits-Back Coding. arXiv e-prints, 2019.
[3] Emiel Hoogeboom, Jorn W. T. Peters, Rianne van den Berg, and Max Welling. Integer Discrete Flows and Lossless Compression. arXiv e-prints, 2019.
[4] C. K. Sønderby, T. Raiko, L. Maaløe, S. K. Sønderby, and O. Winther. Ladder variational autoencoders. In Advances in Neural Information Processing Systems (NIPS), 2016.