Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper presents a tessellation based approach to mapping between continuous and discrete distributions which learns quantization boundaries in continuous space. The approach has several desirable mathematical and computational properties that give rise to more efficient scaling in dimensions than previous approaches. The approach is used in two ways--Voronoi dequantization and Voronoi mixtures--which are demonstrated on data and shown to be competitive with baselines. The authors concluded by noting some limitations and directions for extension. 
strengths_and_weaknesses: The paper is a mostly clear and interesting presentation of an interesting approach to bridging discrete and continuous distributions. Strengths include: clear mathematical and general exposition, an interesting mix of mathematical and computational considerations, and interesting empirical tests. There are not many weaknesses, though I found myself wanting to understand the advantages of the Voronoi approaches better. The experiments did what the authors proposed (show competitiveness), but it would have been nice to have stronger intuition. 
questions: Following the above discussion, I would be interested in understanding the advantage of the Voronoi approaches. 
limitations: The authors do mention a couple limitations and directions for future work, which I applaud. There is no reason to discuss social impact for this work.
review_title: null

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: A dequantization method is presented that is able to map between discrete and continuous distributions using continuous, invertible normalizing flows. Two applications are presented, namely Voronoi dequantization and Voronoi mixture models.
strengths_and_weaknesses: The results are well founded, and figures visually explain the concepts well. Though the text is often a bit convoluted, where the text explanation tends to be over-complex and hard to understand for non-experts in the field.

Although it is great how much space is given to explaining the background and catching the reader up, more results and experiments would perhaps be even more intriguing. 

Could it be that no Appendix was present with the submission, even though it was referred to?
questions: In Figure 5, the caption could directly explain what "Samples" refers to. Line 325 also mentions how the method could potentially make gradient-based optimization difficult, has this been tried in any experiments? It would be interesting to see such results, or these kinds of applications with the framework.
limitations: The limitations are clear.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: The paper proposes a new method for modelling discrete data using normalizing flows by extending previous works on dequantization based method (like argmax flows) using a "tessellation" (not introduced or defined in the paper) based approach that automatically learns the boundaries for quantization. This is explore both for discrete to continuous and continuous to discrete.
strengths_and_weaknesses: **Strengths:**
- The paper describes sound extensions to normalizing flows for modelling discrete data and explores novel ways of incorporating tessellation based approaches in this context.
- The experiments are done on a variety of datasets including language modelling etc to study the empirical performance of the method. 
- Significance: Getting good performance of generative models on discrete data is challenging and thus the problem is quite hard. This paper explores an interesting extension to previous methods to tackle this problem.

**Weakness:**
- Unfortunately, the paper is not clear on technical details. Important concepts are assumed to be known by the reader instead of properly introducing them e.g. tessellation, anchor points, Voronoi tessellation etc. The paper will greatly benefit from having a brief notations and definitions section. I also believe the paper will benefit greatly from having intuitive explanations accompany technical details to make it easier to follow.

- Similarly, several appropriate references are missing e.g. for dequantization, the paper by Uria, 2013 needs to be cited. Similarly, statements/claims as in line 46-48 need to be supported with relevant references. 

- It was not immediately clear to me why is it useful to go from continuous case to the discrete case? What applications will this be important in? Apart from testing this on simple UCI datasets, I do not see any specific experiments for this.

- I also found the experiments fairly limited in scope. There are no experiments to show the capability of the model on modeling categorical data with large number of classes. The language modeling task have small vocabularies. I'd recommended to evaluate the model on word-based language models with large vocabularies.

- Similar to RAD flows, I believe the method will suffer from exploding gradients near the boundaries. How is the training stabalised to address this behaviour?
questions: Please see in the strengths and weakness section.
limitations: Limitations have been discussed.
review_title: null

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper applies Voronoi tessellation to allow the application of normalizing flows on discrete data. The main contributions are Voronoi dequantization, which is a new mapping method between discrete values and continuous space, and a disjoint mixture model, which uses an invertible mapping between unbounded support and convex polytopes. Experimental result on a variety of datasets shows that the proposed method outperforms existing baselines. 
strengths_and_weaknesses: Strengths:
- The Voronoi tessellation idea is simple, novel, and intuitive. Additionally, it addresses an important problem of modeling discrete data with normalizing flows.
- The writing is very clear. All mathematical details are concise yet complete. The authors also provide enough context to distinguish their contribution from related works. 
- The proposed method for dequantization and disjoint mixture modeling significantly improves performance with little additional computational cost. 

Weaknesses:
There are no major flaws that I can think of. The paper can be further improved if the authors can provide a few alternatives to the current invertible mapping to convex polytopes and compare performance and computation costs. Other design choices can be explored as mentioned in the conclusion section, but the current content seems sufficient for a paper.

questions: - Since the experiments mainly compared with other flow models, it would be interesting to see how the proposed methods compare with state-of-the-art methods. Is normalizing flow outperformed by other methods even with the Voronoi dequantization and mixture modeling, or can it actually have better performance? 
limitations: The authors have discussed limitations, along with possible ways to address them, in the conclusion.
review_title: null

Meta-review: metareview: The authors develop a tesselation based approach to map between discrete and continuous spaces. They use this approach to dequantize data to port likelihood based models on continuous spaces to discrete spaces and to scale mixture models where each mixture component has disjoint support. From the view of normalizing flows, the approach is neat, and the results are supportive. The two outstanding things, I'd encourage to authors to work on are 1) the accessibility of the writing and 2) placing the work in the context of generative models outside of normalizing flows 
confidence: Certain
Decision: Accept