Paper:
Distilled Wasserstein Learning for Word Embedding and Topic Modeling
We propose a novel Wasserstein method with a distillation mechanism, yielding joint learning of word embeddings and topics. The proposed method is based on the fact that the Euclidean distance between word embeddings may be employed as the underlying distance in the Wasserstein topic model. The word distributions of topics, their optimal transports to the word distributions of documents, and the embeddings of words are learned in a unified framework. When learning the topic model, we leverage a distilled underlying distance matrix to update the topic distributions and smoothly calculate the corresponding optimal transports. Such a strategy provides the updating of word embeddings with robust guidance, improving the algorithmic convergence. As an application, we focus on patient admission records, in which the proposed method embeds the codes of diseases and procedures and learns the topics of admissions, obtaining superior performance on clinically-meaningful disease network construction, mortality prediction as a function of admission codes, and procedure recommendation.

Review:
Summary ======= The authors present a Distilled Wasserstein Learning (DWL) method for simultaneously learning a topic model alongside a word embedding using a model/approach based on the Wasserstein distance applied to elements of finite simplices. This is claimed as the first such method to simultaneously fit topics alongside embeddings. In particular, their embeddings only exploit on document co-occurence rather than nearby co-occurence within a sequence (i.e. using word order information) such as with word2vec. The authors demonstrate the superiority of their embeddings against a variety of benchmarks on three tasks: mortality prediction, admissions-type prediction, and procedure recommendation, using a single corpus of  patient admission records where words are the international classification of diseases (ICD) ids of procedures and diseases.  There are a number of apparently novel features to their approach which they outline in their paper, namely:  * It is a topic model where observed word frequencies within a document are approximated as the *barycentres* (centre of mass) of a weighted sum over a low rank basis of topics (where these barycentres are with respect to some Wasserstein distance). This is a compelling idea, as the Wasserstein distance is a very natural one for probability distributions. * Development of an objective function, allowing alternating optimisation. One phase learning the documents as distributions of topics (the weights \Lambda) and the topics as distributions of words (the basis B). The other phase refining the word embeddings. * A distilling approach to avoid vanishing gradients when learning the weights \Lambda and basis B. This looks like a regularised version of the objective, which takes advantage of Sinkhorn distance and a fractional powers of the underlying embedding distance. * A smoothness regulariser on the parameters, \theta, of the embedding function, meaning it is not far from the previous iteration's value. * It is unclear whether this method will scale.   The paper is well written, and the method appears reproducible. Further, it is both dense in mathematical justification, and clearly references related work and foundational results. To my knowledge it is novel and represents a significant contribution which would be of interest to the field. I have a few concerns about the following aspects of the paper:  * There is no independent verification/clear justification that word order in the patient record data is less consistent than in natural language corpora. We rely on the superiority of this model over those that utilise word order information for their embeddings. * Sometimes the intuitive explanations fall a little short. * There is a heavy reliance throughout on foundational results from an arXiv paper [36] (hence not peer reviewed). * For the regularisation/distilling approaches, there don't appear to be any theoretical results assessing their influence on the convergence/optimality of the overall model. Moreover, they involve choosing hyperparameters whose influence it is difficult to assess other than empirically, * The evaluation is done on just one data-set (with three related tasks), and while the results show that the proposed method consistently outperforms other existing approaches the improvements are modest and there are no errorbars/significance results.    Nonetheless, I am recommending the paper for acceptance, as I feel the positive aspects outweight these negatives.   Detail ======  ### p1, lines 31-33 > ICD codes for a given patient admission is less relevant, as there is often a diversity in the observed codes for a given admission, and code order may hold less meaning  This could be more convincing. There is diversity in the words that appear around other words in natural language. Could you verify this on data?  ### p2, figure 1 I think it is possibly trying to do a little too much in one figure, or it might be the choice of perspective. Having read the paper from beginning to end, it is clearer, but it would have helped having the associated intuition earlier.  ### p3, lines 92-93 > Π(u, v) is the set of all probability measures on Ω × Ω.  I think there should be an addional requirement that the left and right marginals are u and v respectively, e.g. given some $$\gamma \in \Pi(u,v)$$ then $$\gamma(\Omega,A) = v(A) \quad \text{for all } A \subseteq \Omega$$  ### p3-p4, use of the word transfer > ...the transfer between different topics (and different documents) is achieved by the optimal transport built on the embedding-based underlying distance. For arbitrary two word embeddings, the more similar they are, the smaller ground distance we have, and more easily we can achieve transfer between them.  This is a little ambiguous and could be made clearer.   ### p7, table 1 Although the performance is consistently better the results only represent a seemingly modest improvement (especially considering the increased computational complexity).  ### p7, line 263 The variables P_m and R_m are not defined (although at a guess they are the summands of the precision and recall respectively.  ### p8, line 270, clarity > dispersed mainly accordingly to their connections with matched diseases.  Not sure what is meant by mainly (or how that is determined).  How is this to be evaluated. It is unclear how anyone other than a medical expert would be able to evaluate how meaningful this clustering is, without some additional information (even given the additional information in the supplementary materials).