Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: This paper proposes a model for hyperbolic space using multi-component floating point in the Poincare upper-half space model which can be supported on GPU and with small numerical error. Experimental results show good training time performance. 
main_review: This paper offers a new model to support hyperbolic space multi-component floating-point representations. The proposed method offers good trade-off between numerical error magnitude and hardware cost. This paper is well organised for readers to comprehend the problems and the proposed solution. The authors provide enough background discussing the related models and their limitations in computational precision and existing framework support.  The proposed algorithms is clearly demonstrated with sound mathematical analysis and supported with experimental results comparing training time on several graph-related data sets. 
limitations_and_societal_impact: The author does mention that one of the limitation is that some  embeddings problems may not benefit from the proposed method.

Some questions and suggestions:
1. Can the proposed method be possibly benchmarked for its performance on model trained using it? Such as model accuracy. 
2. While the authors does provide a thorough introduction on multi-component fp representations, a figure illustrating it directly would be better.
3. I would suggest to summarise or have a table for training time also in relative improvements. As it seems the saving time in training time can be huge compared to 3% improvements in MAP, which is also presented in relative comparison. 
4. Can we also have some experiments on how does the proposed method perform on problems that it may not benefit from?
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: Solving "the NaN problem" in the Poincaré upper-half space model using multiple-component floating-point arithmetic.
main_review: In this paper, the authors propose a simple, feasible-on-GPUs, and easy-to-understand solution for numerically accurate learning on Poincaré upper-half space model using multiple-component floating-point arithmetic in the addition & subtraction computations.

The work is technically sound and well organized. 
But (1) Poincar\'e Ball model and Lorentz model are more popular in hyperbolic embedding, how about using MCF on these models? (2) Many works use SGD or Adam for optimization, how about using these optimizers?

Totally, the work is new and important to the research community.
limitations_and_societal_impact: The authors adequately addressed the limitations and potential negative societal impact of their work
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: To solve the 'NaN' problem in hyperbolic embeddings, the paper proposes to learn the hyperbolic embeddings in the Poincare upper-half space model using multi-component floating-point (MCF). Theoretically, the paper proves that numerical errors can be reduced to any degree by simply increasing the number of components of MCF. In experiments, it shows the proposed MCF can have better results while not compromising much on efficiency.
main_review: The paper focuses on the numerical issues in the hyperbolic embeddings. The task and experimental settings closely follow the previous work on tiling-based models, but the proposed method is novel since it uses the Poincare upper-half space to learn the hyperbolic embeddings and it utilizes MCF to alleviate the numerical errors. The theoretical contribution is solid and clearly presented. However, the improvements in empirical evaluation is very small. The proposed method is quite close with H-Tiling. The gap is smaller than std sometimes.

*Post-rebuttal*: I appreciate the response of the authors. I am keeping my score.
limitations_and_societal_impact: Yes, the authors addressed the limitations. The potential negative societal impact is not discussed since they thought there are none, which I agree.
review_title: null

Meta-review: The paper proposes a new approach to numerically represent hyperbolic space with high precision using multi-component floats. Insufficient precision is known to cause problems for hyperbolic representation learning and this paper proposes an interesting method to alleviate this issue. The contributions are somewhat incremental as there exists closely related prior work (also in terms of organization of the paper) on improving numerical accuracy for hyperbolic embeddings by Yu and De Sa (2019). However, an important advantage of the proposed method is that it allows for easy GPU-training what significantly expands it applicability compared to prior work. Reviewers highlighted also the sound mathematical analysis and empirical evaluation in the manuscript. After rebuttal, all reviewers supported therefore acceptance of the paper. When preparing the camera ready version of the manuscript, please take the feedback and comment from the reviewers into account to improve the paper.
Decision: Accept (Poster)