Geometry-aware Distance Measure for Diverse Hierarchical Structures in Hyperbolic Spaces

• The motivation of some operations is questionable:
  • Why is learning projection matrix M_ij necessary when from Fig. 1, the main difference between (c) and (d) is the curvature? Could you provide a more detailed justification?
  • How to compare with other methods with curvature c as a single learnable parameter? How to compare with other methods with curvature c as a tunable hyperparameter? I suggest authors include these comparisons in the experimental section (or feel free to point these out if I missed them) and provide discussions for the comparison of these methodologies.
  • What is the motivation of using M^res? Why will M^res mitigate instability of training? Can you provide any intuition?
  • The motivation of using different curvatures might be due to the artifact that classes are placed at the boundary of the space as pictured in Fig. 1. In Fig. 1 (b) why can’t we shift the dog-wolf pair up, closer to the origin of the space? Why not rescale the weight of the subtrees (for example, scale up the dog-wolf pair) so that you can embed the full tree? I’m asking because there exists combinatorial algorithms [1] to embed trees into Poincare disk with arbitrarily low distortion.

[1] Rik Sarkar. Low Distortion Delaunay Embedding of Trees in Hyperbolic Plane. In Proceedings of the 19th International Conference on Graph Drawing, GD’11, pages 355–366, Berlin, Heidelberg, 2012. Springer-Verlag.

• Writing of the proof of Prop. 4.1:
  • Since this is the core motivation of using adaptive curvature, please consider adding some proof sketch/intuition in the main body, and provide more explanation for terminologies for the proof in the Appendix.
  • line 212: Could you define “better conform to” formally?
  • Please add the citation for Theorem B.1 in its statement in line 956-957, like the format you use for Theorem B.4.
  • It would be better to explain terminologies in Sec B.2 for readers without extensive background in hyperbolic geometry. Readers might be familiar with operations in line 162-176, but not something like “deformed hyperbolic surface”, “genus”, “homotopy classes”, “fundamental group” etc., from line 972-986.

I would love to raise my score if you can address all my concerns and questions below.

1.There are several inaccuracies in the mathematical expressions throughout the article that require careful review and correction by the authors. 2.How was the hierarchical structure of the dataset used in the experiments generated? 3.In Table 3, the absence of the latest comparison algorithms from this year diminishes the persuasive power of the experiments.

Overall, despite the interesting starting motivations, the current manuscript fails to include the necessary definitions for readers to understand the Authors' idea. Each of the following points is, regrettably, fatal and also prevents us from judging the validity of other parts, such as experiment parts since, in general, experiments are conducted to verify whether the Authors idea is correct or not.

1. The specific forms of the matrix generator $g\_{t}$ and curvature generator $g\_{c}$ are not given in the paper. One might say it is a general form, but leaving these as a general form is unacceptable. If we do not restrict the function class of $g\_{t}$ and $g\_{c}$, we can learn any distance structure for the feature pairs as $\\boldsymbol{M}\_{ij}$ can depend on $ij$. This means the proposed method can overfit any data infinitely, and we can no longer say the model considers specific representation space, e.g., hyperbolic space.
2. Many mathematical terms are used in Theorem 4.2 without definitions. For example,

• systematic error
• relatively continuously
• absolute value calculation to define $|\\cdot|$ (for a vector, like $\\boldsymbol{M \otimes x} - \\boldsymbol{M' \otimes x}$)

are not defined. Also, I could not see "the mean of error" of what in Line 264. To be honest, I am unsure whether this theorem is correct since I do not see any trick in Theorem's setting that makes the low-rank matrix $\\boldsymbol{M}'$ approach $\\boldsymbol{M}$ or the feature vectors bound in the subspace of the kernel (linear-algebraic sense) of $\\boldsymbol{M}' - \\boldsymbol{M}$. In any case, we cannot verify it without specific definitions of those terms given by the Authors.

Weaknesses

1. The authors mention that their work is inspired by the Gu et. al, 2019 paper in section 2.3 (L141-147) - Learning mixed curvature representations in Products of Model Spaces - the nature of the work is similar in terms of learning adaptive curvatures however the authors do not provide a comparison with this method in their experiments

2. There are several details about the experimental setup and comparisons which are unclear from the main paper and the Appendix. The classification accuracy measurement is mentioned as “similarities between samples and prototypes” (L1685), which seems to be different from the standard classification setups used in literature where the accuracies are measured by the softmax logits of the outputs from the classifier layer or k-NN accuracy for classification. Since the features are projected to the hyperbolic space, is the similarity computed as inverse hyperbolic distances to the prototypes or using a different method? What are the backbones and loss function used for experiments in Table 1 and did the authors experiment with Hyperbolic backbones? Additionally, how do the results for the experiments in Table 1 compare with euclidean/non-hyperbolic methods with equivalent setups?

3. While the authors mention the complexity analysis for low rank decomposition, what are the training times for the different settings and how much of an increase does the proposed methodology lead to in terms of actual training time for the experiments?