__ Summary and Contributions__: This paper aims to resolve two limitations of bits-back compression: (a) transmission of the auxiliary bitstream degrades the rates, (b) bits-back coding only applies to the lossless compression.
The authors propose an index coding (iREC) which allows encoding the latent representation z with codelength close to the relative entropy. The key idea is to use importance sampling and index coding together with shared randomness, where p(z) is assumed to be shared and multiple realizations of z ~ p(z) can be shared between the encoder and decoder. For computational reasons, the authors propose using auxiliary variables to represent z and apply important sampling on the auxiliary space.

__ Strengths__: I think the biggest contribution is in the algorithm itself. The proposed idea of index coding together with importance sampling, introduction of auxiliary variables and beam search, is interesting and seems novel.
The problem considered is a single image compression, which may be a narrow application, but I think it has potential usage. For the single image compression setting, the empirical results show iREC outperforms other the variants of bits-back compression.

__ Weaknesses__: The authors articulate several benefits of iREC, but those were not fully convincing.
i) Lossy compression: While the proposed method is immediately applicable to lossy compression, the performance on lossy compression seems to be clearly worse than several other baselines.
ii) Being able to handle continuous latent space is a plus, but the gain from that ability was not clear.
Another limitation is, as mentioned by the authors, its computation cost.

__ Correctness__: I have a question regarding the statement "We present Index coding (iREC), a REC algorithm that scales to ...." (line 163) Does iREC provably achieve communication cost provably close to KLD? The authors state "We refer to algorithms that achieve communication cost provably close to KL [ q(z) || p(z) ] as REC algorithms." (line 151)

__ Clarity__: The paper is overall well written. The author provides a high-level intuition at various places (e.g., stochastic coding scheme), which I think greatly helps readers to follow the paper.

__ Relation to Prior Work__: The relation to prior and directly related work was presented clearly.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: This paper propose a method with relative entropy coding that can be applied both lossless and lossy image compression. The latent representation is coded with codelength close to the relative entropy for single images, which can outperform existing bits-back methods that requires auxiliary bits.

__ Strengths__: This paper contains complete theoretical derivation, with very detailed algorithm description with source code in supplementary materials.
It provides a new lossless image compression pipeline based on VAE. The proposed method shows much bitrate save when comparing with existing bits-back methods on single image.

__ Weaknesses__: The performance compared with no bits-back method is not that competitive. The highest resolution image used in the paper is Kodak, but lossless single image lossless compression might be more essential for images with higher resolution. Besides, the proposed method show clear performance gap with state-of-the-art methods in lossy image compression, but its provides an new perspective for lossy image compression.

__ Correctness__: Yes, except that in Tab.1 the proposed method should not be included in the bits-back methods, since it is fundamentally different from those bits-back methods.

__ Clarity__: Yes

__ Relation to Prior Work__: Yes

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The paper presents index coding based REC method (iREC) to extend the use of VAE (continuous latent variable) for both lossless and lossy compression. The proposed method can encode an image with codelength close to the negative ELBO. The paper presents experiments on benchmark datasets and demonstrate improved performance on both lossless and lossy compression task.

__ Strengths__: - The presented algorithm (iREC) is a novel approach (in terms of a coding scheme) that proposes the sampling of auxiliary variables instead of directly sampling from the learned posterior of the latent variables.
- The presented results are impressive as the method is outperforming others recently proposed method in the compression literature in case of lossless compression and in the case of lossy compression, the paper presents comparable performance to the current state-of-the-art method.

__ Weaknesses__: - One of the biggest weaknesses is the clear discussion/motivation on why other methods are problematic. This is important because the use of continuous latent space in this kind of problem has been rigorously discussed in the ML literature and without clearly differentiating with prior works, the contribution is weakened.
- The experiments for ablation is missing. For example, the effect of bias of importance sampling (or benefit of the beam search) is not demonstrated empirically.

__ Correctness__: Some of the claims are not validated empirically (missing ablation studies). Empirical methodology correct.

__ Clarity__: The paper is poorly written with a lot of typos, and unclear and vague statements.
E.g.,
- "who to our knowledge" : line 255
- "As our model we used the ..": line 257

__ Relation to Prior Work__: The relation to prior work is made vague. The statements like "We introduce REC" in line 47 contradicts with "they present a REC algorithm" (line 160).

__ Reproducibility__: Yes

__ Additional Feedback__: - Authors are encouraged to clearly validate all the claims made in the paper empirically.
- The proofreading and correcting grammatical errors could enhance the strength of the paper.

__ Summary and Contributions__: This paper proposes a method to compress the latent representation generated by the encoder of an autoencoder. The method is abased on a bits-back scheme and addresses the problem of such scheme requiring a large context to be effective, ie being inefficient to encode a single image.

__ Strengths__: The addressed problem, ie compression of the latent space variables of an autoencoder is definitely relevant and timely. The paper is extremely well written and contains a good tutorial on bits-back methods that make the paper accessible.

__ Weaknesses__: 32x32 Cifar and Imagenet are a bit unsuited to evaluate the quality of the proposed scheme. The Kodak dataset is much more appropriate, however it starts to be a bit outdated nowadays. The authors could for example consider the Class B test sequences of MPEG for HEVC or VVC. I would also like to see JPEG2000 in Fig 3a) and 3b)

__ Correctness__: Yes.

__ Clarity__: Very well written, good tutorial.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: