Embeddings of tree-like graphs in hyperbolic space were recently shown to surpass their Euclidean counterparts in performance by a large margin. Inspired by these results, we present an algorithm for learning word embeddings in hyperbolic space from free text. An objective function based on the hyperbolic distance is derived and included in the skip-gram negative-sampling architecture from word2vec. The hyperbolic word embeddings are then evaluated on word similarity and analogy benchmarks. The results demonstrate the potential of hyperbolic word embeddings, particularly in low dimensions, though without clear superiority over their Euclidean counterparts. We further discuss subtleties in the formulation of the analogy task in curved spaces.
This paper presents a technique for embedding words in hyperbolic space, which extends previous non-euclidean methods to non-structured data like free text. The authors provide a new gradient based method for creating the embeddings and then evaluate them on standard word embedding benchmarks. Overall the paper is very well written and well executed. They find that in the low dimensions the approach outperforms standard Euclidean space methods while in higher dimensions this advantage disappears.
The results do not try to claim state of the art on all benchmarks, which I find refreshing and I appreciate the authors candor in giving an honest presentation of their results. Overall, I enjoyed this paper and am eager to see how the authors develop the approach further. 
However, along these same lines it would be great to have the authors provide more discussion about the next steps and potential applications for this approach. Is the interest here purely methodological? Are there potential use cases where they believe this approach might be superior to Euclidean approaches? More detail in the discussion and intro about the trajectory of this work would help the reader understand the methodological and application-specific implications. 
Pros:
- Clearly written and results are presented in a straightforward manner. 
- Extension of analogy reasoning to non-euclidean spaces.
Cons:
- Lack of clear motivation and compelling use case. 
- It would be nice to have a visualization of the approach in 2-dimensions. While Figure 3 is instructive for how analogies work in this space, it would be great to visualize an entire dataset. I'm sure that the proposed embeddings would result in  a very different space than euclidean embeddings (as the Poincare embedding paper showed), so it would be great to have at least one visualization of an embedded dataset. Presumably this would play to the strengths of the approach as it excels in lower dimensions. 
-  The largest of embedding dimension tested was 100, and it is common to use much larger embeddings of 500-d. Do the trends they observe continue to larger dimensions, e.g. is the performance gap even larger in higher dimensions?