Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The authors presented algorithm based on Riemannian geometric tools to align densities on Hyperbolic space. Experimental results have been demonstrated on synthetic and breast cancer data. The results have shown efficiency of the proposed method in terms of lower discrepancy and less computational time requirement. 
main_review: Post rebuttal and discussion, I want to recommend acceptance.



The overall paper is an easy read. although my main concern is
(a) The existence and uniqueness of FM should be addressed to ensure validity of Algorithm 1.
(b) Most of the propositions are true due to the choice of metric and property of PT. So why Algorithm 1 does not valid for other Riemannian manifolds? The authors should point out what is so special about Hyperbolic spaces? 
(c) The experimental results are limited only to a synthetic dataset and the batch effect removal on the breast cancer dataset. Also the comparison is subpar.

Detailed Comments:
(a) line 83, the authors should cite many other older papers like [Groisser Riemannian Newton .., 2003], [Chakraborty et al., Exact PGA ..., 2014].
(b) As mentioned earlier, most of the propositions like Proposition 5, 8 follow directly from the property of PT.
(c) The steps for Riemannian wrapped rotation should be presented as bulleted lists or algorithm for better clarity.
(d) The main comparison with [18] is not enough, few obvious baselines including doing aligning in the ambient space, using Log Euclidean framework should be presented.
limitations_and_societal_impact: The authors mentioned they did not have any potential negative societal impact which I agree.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: This paper studies the unsupervised alignment of hierarchical datasets embedded in hyperbolic spaces, by proposing a hyperbolic procrustes analysis  (HPA) method. HPA uses new hyperbolic translations, rotations and scaling methods to align datasets. Experiments on synthetic and real datasets show that HPA is able to remove noise from data collection to align datasets. 

main_review: Strengths:
The paper is well-written. The theorems and propositions in this paper are precisely formulated and clearly written. The authors clearly explained why each theorem is important to the paper.
Experiments are convincing, and the downstream tasks in 4.2 are good experiments to show the quality and utility of alignment.

Suggestions/Comments:
Line 167 ("Second, rotation applied directly to the tangent space [...] does not guarantee that the rotated points remain on the same tangent space"). I don't understand this sentence. The tangent space of L^d at \bar{h}^{(1)}, just like the tangent space of any Riemannian manifold at any point, is isometric to the Euclidean space R^d. Thus, we can identify it with R^d using an isometry then perform Euclidean rotations there (then identify things back to the tangent space). If rotations in Euclidean space do not throw points out of the space, why would that happen for rotations in the tangent space T_{\bar{h}^{(1)}} L^d ? 

If, instead of using the wrapped rotations, we (1) map points to the tangent space T_{\bar{h}^{(1)}} L^d using the log map, (2) perform Euclidean rotations on this tangent space around the center \bar{h}^{(1)}, and (3) map the points back to L^d using the exponential map, then the resulting map have the advantage of being an isometry of L^d: In fact, it will just be a rotation of L^d around the point \bar{h}^{(1)}. In particular, it preserves the mean \bar{h}^{(1)}.

Note that the authors' comment in line 165 (about rotation maps in L^d not necessarily preserving the mean) does not really apply here: Rotations in L^d can totally preserve the mean, as long as we rotate around the mean itself, as suggested above. While the rotation used in [46] indeed might not preserve the mean \bar{h}, that is only because that rotation is based at the "bottom" of the hyperboloid and not the mean \bar{h}. I don't see why we have to stick with that particular rotation, given that the hyperbolic space is homogenous so there is nothing intrinsically special about the "bottom" point.

To summarize: I feel that the rotation of L^d around \bar{h} has a few advantages over the wrapped rotations suggested in 3.3: It is (1) an isometry, (2) quite natural, and (3) intrinsic to the geometry of the hyperbolic space (i.e. not depends on any specific model or coordinate systems). Meanwhile, it still preserves the mean \bar{h} and has a closed-form expression, as the authors desire. Therefore, I do not understand why the remarks in line 165-168 are valid reasons to discard this rotation. Perhaps the authors have tried this and found that it performed worse empirically? If so, it would be nice to see that analysis.

PS. Note that if we replace the wrapped operations by the usual rotation above, then the whole alignment method is essentially just (1) map points to tangent spaces using logarithm maps, (2) translate them to the same tangent space using parallel transport, (3) perform Euclidean alignment on the tangent space, and (4) map points back to L^d using exponential map. This is a simple baseline that can be efficiently computed and can leverage whatever Eucllidean alignment techniques we want.

limitations_and_societal_impact: Yes.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: This paper proposes Hyperbolic Procrustes Analysis (HPA) for aligning point sets that live in hyperbolic spaces. The proposed algorithm uses the tools from Riemannian geometry to correctly and efficiently generalize the (generalized) Procrustes alignment to Riemannian manifolds. The paper deploys this algorithm to perform unsupervised, label-free, and hierarchical data alignment. The experiments demonstrate that HPA can yield higher accuracy when compared to optimal transport based alignment.
main_review: PROs

- I think generalizing Procrustes alignment to Riemannian manifolds carries fundamental importance. The approach presented in this paper is intuitive, principled and makes sense. 
- The paper is clearly written explaining different stages of the algorithm. It is easy to follow and understand for readers with certain background.
- The paper conducts an appropriate evaluation on hierarchical datasets and problems where such an alignment is superior to the existing optimal transport based ones.

CONs

1. Procrustes alignment only considers translation, scale and rotation to align two datasets. For a high dimensional problem, I seem to lose the motivation why (and in which cases) this would be superior to say a hierarchical OT? Can we have a general recipe in how to choose which algorithm to use? In other words, intuitively, why would this perform better than OT for a given dataset? It would be good to clarify this point. 
2. It seems like a large portion of the material addresses a general Riemannian manifold. However, when we dig to the specifics of the proofs the Lorenz model comes along. Is it possible to consider a similar alignment procedure not for hyperbolic spaces but for general manifolds? 
3. A lot of ablation studies seem to be missing here. First, as mention in Sec. 3.3, there could be multiple ways to solve the rotation problem. The paper argues for the *wrapped rotation*. I would agree, but this choice is not experimentally validated. Can we see a study of different rotational alignment strategies?
4. Prop. 10 seems to assume that the dataset is generated by an inverse Procrustes. Why should this be the case?
5. For the alignment of multiple point clouds (the generalized problem) : would it not make sense to fix one of the point clouds and align the rest? Yes or no, it would be good to discuss why.
6. Can we see visualizations from other methods as well? This might help in grasping certain problems with the state of the art.
limitations_and_societal_impact: As this work is a theoretical one, the authors avoid including a broader impact statement. However, in general, the paper really fails to discuss the limitations from an empirical perspective. Besides, I also do not think that Section 3 does a good job of discussing thoroughly the theoretical limitations. What are the theoretical cases to break the algorithm? Can we speak of a convergence basin, at least for the hyperbolic geometry?
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The need for alignment often arises since data are acquired in multiple domains, under different environmental conditions, using various acquisition equipment, and at different sites. This paper focuses on the problem of unsupervised alignment of data embedded in hyperbolic spaces. 
main_review: This paper presents the Lorentz model for unsupervised label-free hierarchical data alignment.
They also provide theoretical analysis and justification of the alignment method. The theoretical analysis provides further insight and highlights properties that may be useful. Their experiments also indicate that HPA is stable under noise and distortions.

The most interesting thing is that the authors also experiment on two datasets that arguably do not have a distinct hierarchical structure but are highly effective.


limitations_and_societal_impact: The authors has adequately addressed the limitations and potential negative societal impact of their work.
review_title: null

Meta-review: The paper looks at aligning different hyperbolic datasets (which consists of elements in the hyperbolic space). In particular, it proposes to align the means, scale the dispersions, and rotate to align the elements. The proposed approach shows good performance on different datasets. It also rigorously compares with different baselines. A minor comment. How does the analysis extent to cases when the datasets are in different dimensions? 


Decision: Accept (Poster)