__ Summary and Contributions__:
This paper proposes a document-topic modeling formulation
based on a Optimal Transport distances, showing improvements
in topic coherence and a downstream document classification
task.

__ Strengths__:
The modeling approach, while somewhat intricate, seems
internally coherent. The framing and theoretical connections
are interesting, and the experimental results seem strong.

__ Weaknesses__:
The coherence results in Table 2 seem consistently and
significantly better than strong baselines, but I have no idea
why that should be the case. What about this formulation leads
to topics better semantic coherence, or utility in downstream
classification tasks in Table 3? Without some at least
hypothesized mechanism here I am kind of suspicious of the
results.
Overall I find the entire approach to lack clear motivations
beyond the aesthetic appeal of the geometric interpretation
and OT distance machinery, which admittedly seems nice. I was
left with open questions around which kinds of problems we
expect this approach to be superior on or what kinds of
extensions could be more naturally expressed in this
framework. Why use this somewhat involved machinery instead
of some other approach?

__ Correctness__:
L269: tuning the regularizer parameter lambda for the proposed
method without also doing so for the other approaches does not
seem correct.
I don't find the qualitative results very compelling.
How does this approach outperform on coherence versus baselines that leverage the pretrained semantic embeddings? This seems surprising and I would be interested to see this explained more thoroughly.

__ Clarity__:
I found L137-182 to be fairly dense, especially regarding the
"nested" OT problem being solved. The language is clear and
precise, and the figure is helpful, but some higher-level
setup or motivation might make the flows easier to
follow. Given all this, the optimization procedure on the
subsequent two pages was clear enough.

__ Relation to Prior Work__:
L157-168: what is the benefit or advantage to these two
differences vs WDL? I understand these are differences, but
the associated practical or theoretical gains were not clear
to me.
Other than the unfortunate omission of DWL, the experimental
baselines seem strong.

__ Reproducibility__: Yes

__ Additional Feedback__:
The paper could be strengthened with some more explicit
motivation or justification for why the particular OT
formulation can be expected to yield improved results, what additional benefits or flexibility we should expect from this formulation, or why this formulation could have a significant research impact.
L103: it was not immediately obvious to me whether Big Omega was the set of
topic vertices only, or their convex combinations as well.
L184: what is the norm here?
Figure 1 was very helpful.
It seems like one could use different gamma entropic
regularization parameters for the "inner" (topic-document) and
"outer" (document-topic) OT problems, was this explored?
Table 3: caption has a typo 2NG -> 20NG
Regarding results quality and convergence efficiency, it would
have very helpful to see experimental comparisons against
WDL. There is nothing that these authors can do about it, but
it is unfortunate that prior NeurIPS publication "Distilled
Wasserstein Learning" lacks available source code for
comparison. That said, was there any reason not to compare
against other Wasserstein-based approaches?
Are learning times/complexity for this model dramatically better (worse?) or comparable to the baselines?
I have read and considered the author feedback.

__ Summary and Contributions__: The paper proposes a method for encoding geometric information such as from word embeddings into topic model training, using an optimal transport formulation. Although there has been recent work with related ideas, the approach here respects more of the semantic constraints of models like LDA (e.g. normalized distributions) and is found to have better performance and faster training.

__ Strengths__: Optimal transport approaches to topic modeling are a relatively new and promising direction. The paper improves over the state of the art in this space.
Incorporating word embedding information into topic modelings has long been an important goal, and although substantial progress has been made on this before, using embeddings to encode semantics into the geometry of the learning problem is an elegant and interesting strategy that could lead to follow-on research.
The model and learning algorithm are reasonable and sound. Local convergence is established for the learning algorithm.
The experimental setups are mostly reasonable and the results are positive.

__ Weaknesses__: Although optimal transport approaches to topic modeling are relatively new, this is not the first work to attempt this, which reduces the novelty somewhat. The claimed advantages over e.g. [26] are mainly encoding normalization constraints and word count information, and a more efficient optimization algorithm. These are somewhat incremental improvements, especially by the standards of the NeurIPS conference, although they do lead to better performance.
Only 20-100 topics were used in the experiments, which is rather low by today's standards.
The classification results to evaluate downstream extrinsic performance, while appreciated, are rather limited. Only one dataset is used in the experiment. The results for all of the compared methods appear very poor, and only 3 of the models were considered. It is important to compare to baseline methods like tf-idf features.

__ Correctness__: The claims are mainly supported. The technical approach seems correct, although I do not have the expertise to check all of the details.
One concern is that in the experiments, default or best-reported hyper-parameter values were used for the baselines, while the proposed method used a (small) grid search on its regularization parameter lambda. This gives the proposed method a slight advantage. It was unclear whether the best lambda value was chosen based on performance on a validation set or on the test set.

__ Clarity__: The paper is mainly well written and well argued. However, I found the notation very confusing in Section 3 which hampered understanding the work. Sometimes symbols are used before they are properly defined. There is a general lack of textual explanation to provide intuitive understanding for the mathematical notation. The dimensionality of vectors and matrices are generally not specified.
Most notational symbols are non-standard for topic models such as LDA. It would help to include a notation table for the reader to refer back to.
Along the same lines, P and Q are each referred to as a distribution, but they seem to encode a collection of distributions. The semantics of these variables and their notation need to be clarified.

__ Relation to Prior Work__: Section 2.2 discusses previous approaches based on word embedding semantics and deep neural networks, but does not clearly explain the limitations of those approaches that are addressed by the present work.
The paper includes a discussion of its relation to related optimal transport approaches in Section 3.1, but it could make it clearer in the abstract and introduction that it is not the first paper to develop such an approach, and what its precise novelty is relative to those papers.

__ Reproducibility__: Yes

__ Additional Feedback__: Some minor suggestions and typos:
-Line 21, missing an "and"
-Line 33, "while other developed" - "while other authors developed"
-Line 50, and elsewhere in the paper, it is stated that LDA/PLSI use a squared Euclidean loss/distance. This is untrue - both models use likelihood based inference with a multinomial model, and/or Bayesian inference. The older LSI model uses a squared loss, but even the PLSI paper argued that this is insufficient (the implicit Gaussian assumption from squared errors does not hold with small counts as in text data), which motivates the probabilistic modeling approach in PLSI and LDA.
-Line 81, I believe that one of the sums over v should be a sum over u.
-Line 87, punctuation is missing before the mathematical notation in this sentence.
-Line 104, "Authors" - "The authors"
-Line 110, reference [14] is to a 2014 paper by Mikolov. The other papers by Mikolov by 2013 are more fundamental references which are better here, especially:
Mikolov, T., Sutskever, I., Chen, K., Corrado, G. S., & Dean, J. (2013). Distributed representations of words and phrases and their compositionality. In Advances in Neural Information Processing Systems (pp. 3111-3119).
-Line 123, "which neglects to word semantics"
-Line 128, "with jointly learning cost function" - "by jointly learning a cost function"
-Line 136, "bag-of-word"
-Line 189, "number of topic"
-Line 193, "alternative optimization" - should this be "alternating optimization"? This is repeated elsewhere in the paper.
-Line 202, capitalize "euclidean"
-Algorithm 1, "regularized" - "regularization"
-Line 293, "depicts topics ETM"
-Line 308, "classification accuracy downstream tasks"
---
Comments after reading the rebuttal: My concern about hyper-parameter selection was addressed more or less adequately in the rebuttal. The method may still get an unfair advantage since it was tuned on the target dataset rather than another dataset from an earlier paper, but since only 4 values were considered the advantage would be slight. The other reviews didn't raise any other red flags. I'm inclined to leave my score as-is.

__ Summary and Contributions__: The paper presents a topic model based on optimal transport and word-movers' distance.

__ Strengths__: Topic modeling continues to provide a useful test ground for inference methods, and this is no exception. I'm not particularly interested in the quantitative performance of topic evaluation metrics one way or another. For me the value is in demonstrating a potentially useful algorithmic approach and identifying where it is different from existing methods.

__ Weaknesses__: 8M tokens is not large-scale. None of these collections is competitive in modern terms, particularly in the vocabulary -- I would want to see at least 50k distinct terms. This makes me very concerned about efficiency. I'd like to know wallclock times and scaling with corpus size and vocabulary dimension.
The introduction mentions embeddings, but I actually can't tell whether the proposed method uses pre-trained embeddings or not. p4/180 is another cryptic remark: it could use word2vec, but does it? I don't see any mention in the rest of section 3. If it doesn't, this is a huge advantage over ETM and should be emphasized. If it does, the impact of the quality of the embedding space should be evaluated.
The difference between count and proportion vector for WNMF doesn't seem that important, but the difference between being unable to train on an 8M token corpus (WNMF) and getting pretty good performance (OTLDA) seems massive. What accounts for this difference?

__ Correctness__: Using a Gutenberg vocabulary for Wikipedia is a strange choice. Why would we expect out-of-copyright books to be a good source of vocab for a modern encyclopedia? The vocabulary size (4k) is extremely small for this size collection as a result.
Gensim's default LDA implementation is known to work poorly or collections that are not extremely large.
The qualitative results in table 4 are not particularly compelling. The goal in this type of evaluation is to complement the quantitative results to show that -.355 vs -.388 is something you would actually notice. I want to see a range of topic examples. I'm a bit concerned about distinctiveness between topics. (If you think "tax" and "money" is not related to government, I have bad news for you.)
Usually the UCI metric is a PMI, which can be positive or negative, but the UMass metric is a log of a ratio, and can't be positive. I'd actually prefer using the same function, but just varying the reference corpus (internal vs. external) if that's what's happening, but it should be mentioned.

__ Clarity__: Yes.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: