Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: The paper proposes Hyperbolic Busemann Learning for hyperbolic prototypes. Specifically, the prototypes are positioned on the ideal boundary of the Poincare ball model and the penalised Busemann loss is used to learn the projections of the input data to the Poincare ball model. The paper also provides a theoretical link between the proposed hyperbolic prototype approach and logistic regression.
main_review: The proposed hyperbolic prototype network is novel and well motivated. The usage of Busemann function to compute the distance between the ideal boundary point and the point in the Poincare ball model is practical. The empirical evaluation demonstrates the effectiveness of the proposed approach.

Some questions:
1. The derivation of the Busemann function in the Poincare ball model Eq. (7) is missing. How to obtain Eq. (7) from Eqs. (3) and (6) is not trivial. The derivation process or the corresponding reference should be given.
2. I have confusion on the penalty term. As mentioned in Section 3.2, the penalty term is the same for all prototypes, thus it is equivalent with minimizing the Busemann function of the prototypes directly. Then why does the penalty term affect the test accuracy in Section 4.1? 
3. Another confusion is in Line 145-146. It said that the Busemann function is a decreasing function of the cosine similarity, but there is no such concept of 'cosine similarity' in the hyperbolic space.

Some revision suggestions (which do not influence the overall evaluation of the paper):
1. Although hyperbolic space has shown many improvements, it still has some limitations, the 'arbitrary data' in the abstract may be inappropriate and too absolute.
2. In Line 167, there is an extra comma.
limitations_and_societal_impact: Yes.
review_title: null

final_score: rating: 8: Top 50% of accepted NeurIPS papers, clear accept
review_content: summary: This paper introduces a novel model which uses prototypes placed "at infinity" in hyperbolic space for classification. Specifically, ideal prototypes are placed on the surface of the sphere, and the distance from a point inside the sphere the this idealized point is calculated using the Busemann function, which defines a distance to points at infinity using a limit. Thus, the embedded points representing instances can be interpreted as points in hyperbolic space, and prototypes as points at infinity. The authors perform a number of rigorous evaluations and ablation analysis, demonstrating their model performs well in a number of classification settings.
main_review: ### Overall

**Originality:** This paper is reasonably novel, leveraging the Busemann loss function to provide a hyperbolic analog for prototype classification methods.

**Quality:** The submission is technically sound. The approach is reasonable, claims made are proven directly in the paper and appear to be correct.

**Clarity:** The paper is very clearly written and well organized.

**Significance:** The results are significant, extending tools and methods researchers are interested in to the hyperbolic setting.

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### Specific Comments / Questions

- (Comment) I was initially prepared to discard the connection to logistic regression in one dimension, as I assumed this would simply be an uninteresting degenerate case, however the connection the authors drew to higher-dimensional models with prototypes being an alternative to multinomial logistic regression did provide interesting motivation.
- The ablation for the penalized Busemann loss suggest that a larger number of classes will be much more sensitive to the weight of the penalty term. This seems as though it may be inherent to the nature of hyperbolic space, for which volumes of spheres increase exponentially with respect to the radius and thus (in some sense) the penalty term is severely limiting the capacity of the model. Some theoretical exploration or additional experiments on datasets with larger number of labels would provide more confidence that this is not an issue.
- Much of the claimed benefit for hyperbolic embeddings comes from their inductive bias toward modeling hierarchical data. Placing the prototypes for hierarchical data at the boundary would seemingly discard this essential property. In short, it seems as though you would not *want* to use idealized prototypes, placed on the boundary of the sphere, in the case where the prototypes themselves conform to a hierarchy. I would appreciate it if the authors could clarify this concern.
limitations_and_societal_impact: Some limitations are touched on. While the paper does not oversell it's methods, it is also not particularly self-critical, and some of the questions I raise above do suggest various limitations of the model which might be explored further.

The work is of a theoretical nature, and does not present any direct negative societal impact.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: Paper proposes an hyperbolic learning method based on ideal prototypes. The originality of the paper lies in the fact that there is no need to include prior information on the hierarchy to position the prototypes. There are located at the boundary of the Poincaré ball, and their position is computed by using algorithm defined in ref [28]. As points located close to the boundary have an infinite distance to the origin, authors propose to use a penalized busemann loss that can be defined in the hyperbolic space. The penalty term allows defining how far the embeddings can be from the origin. Paper also shows that, in 1d, it corresponds to logistic regression.
Experiments are provided, considering CIFAR10 and 100 datasets, on classification tasks. Results show that the proposed method provides better results than the baselines when the dimension is low. When no information on the structure of the prototypes is given, the mAP indicator is better than when the prototypes are inferred from the data, while hierarchical metrics such as s-mAP and c-mAP show that the learnt prototypes perform better when privileged information is provided.
main_review: The paper is very clear and well written. The introduction of the busemann loss is smart and allows defining an elegant loss, with a link with logistic regression in 1d.
My main concern is about the significance of the paper: the algorithm is shown to be effective in low dimensional embedding spaces (as often noticed in the litterature) and the performances come close to the spherical version of [28] when the dimension increases. The paper should emphasize some use cases on which it is important to have such low dimensional spaces (when the classification performance is sought, why considering low dimensional spaces that degrades the performances?). In Table 4, one can notice that ref [23] gives comparable performances when the dimension reaches 50 and one may wonder how the proposed method behaves in larger spaces (one can see that the performances seem to degrade when the dimension increases?).
In addition, some results in Table 2 seem surprising. While s-map and c-map perform better when hierarchical prototypes rather than separation prototypes are used, it is not the case for the mAP metrics. It would be interesting to give an explanation. 

Minor comments:
- Code was not made available to re-run the experiments
- Table 4: ref [23] instead of [28]
- it would strengthen the conclusion if an other dataset could be considered (not only CIFAR).
- results are only given for lower dimensional spaces (e.g. 50 instead of 99 or 100) in table 3: why not considering the same dimension?

limitations_and_societal_impact: yes
review_title: null

Meta-review: The paper proposes a new approach for classification in hyperbolic space by combining ideal prototypes at the boundary of hyperbolic space and a penalized Busemann loss. The paper is well written, clear, and relevant to the NeurIPS community. All reviewers and the AC support acceptance, especially due to the novel and interesting ideas that advance research on hyperbolic neural networks and their applications, as well as the promising empirical results of the method. Reviewers highlighted also that the approach is technically sound, claims are supported in the paper, and the good empirical evaluation. Furthermore, questions and concerns related to the derivation of the approach, motivation, and significance of results where resolved after rebuttal. When preparing the camera ready version, please incorporate the overall feedback and suggestions of reviewers into the new revision.
Decision: Accept (Poster)