Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper studies the gyro-structure of several matrix manifolds. In the process, the authors show that the affine-invariant (AI) and log-Euclidean (LE) geometries of SPD manifolds are closely related to hyperbolic geometries, and Grassmann manifolds share similar properties with gyrovector spaces. Several interesting applications based on these geometries are presented and compared against other models.
strengths_and_weaknesses: The paper is well-structured and is written in a highly readable and professional style. It builds on several recent works on the topic effectively.  The applications and associated experimental results are interesting and appear convincing.


questions: I have developed some uncertainty about the magnitude of significance of such works. The gyrovector operations seem to be basically standard geometric operations that have been understood for quite a while. The packaging of such standard operations in the gyrovector formalism seems reasonably straightforward, albeit necessary in certain contexts. There does not appear to be much particularly profound new analysis or geometry which makes this translation into the gyro-structure language possible. This may be a misunderstanding on my part which may only be clarified by a thorough reading of the supplementary materials. Unfortunately, given the volume of such supplementary material and the reviewing demands of the conference, it is often not possible to perform such a detailed reading in the available time.

If the authors could provide a response to dispel my impression noted above, I would be grateful. They may want to emphasise it in the main body of the paper as well.
limitations: These need to be discussed clearly in the main paper and not just as supplementary material.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper investigates the space of SPD matrices and Grassmann manifolds as gyro-vector spaces. The authors showed the basic definitions of these algebraic structures and proved that the SPD manifold satisfies the axioms in two different ways, i.e., the so-called AI and LE geometries.  The authors also showed that Grassmann manifolds satisfy similar properties.

As applications, the authors rebuilt SPD networks using the proposed operations of SPD matrices and showed performance improvements over SPDNets on actions recognization. The authors also use Grassmann manifolds for word embedding, showing marginal improvements over SPD models.
strengths_and_weaknesses: Pro:
- new geometric tools for deep learning

Con:
- the writing lacks motivation and clarity. (see the questions section below)
- some implementation details are missing in section 4
questions: I am convinced that the authors imported new geometric tools into deep learning, which can be potentially useful. However, it is not clear why they are needed, and how they are used in the applications.

Section 3 introduced the gyro-vector space structures of SPD and Grassmann manifolds. The author gives a list of definitions and properties. It should be better emphasized which of these definitions are new. If these are existing definitions, proper citations are needed in the body of the definitions. Similar to the formal statements: does section 3 merely review existing properties of such spaces or there are new discoveries?

The author vaguely mentioned that "We note that the expressions of the binary operation, scalar multiplication ... have already been found. However, none of them pointed out the relations in Eqs. (1) and (2)". It is not clear what "relations" this sentence refers to.

More importantly, why the gyrovector spaces are important to redefine the structure of such manifolds? What are the motivations from the application perspective? What limitations of existing techniques call for these gyrovector spaces?

Section 4 Applications. There should be details in implementing the modified SPD networks, which are missing. I suggest the authors explain in detail and formally the background problem (classification), the original SPD net, and how the gyro-vector space is plugged into this framework.
limitations: This work is mostly thoretical. In the applications, the authors proposed variations of existing deep learning techniques for the applications of word embedding and action recognition. The authors' experiments do not advance the state-of-the-art performance scores from the application perspective. The negative societal impact is therefore limited by these existing techniques.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper studied potential gyrovector space structures for SPD manifolds with AI and LE geometries and the Grassman manifolds. The authors defined gyroautomorphisms for these geometries and then verified that SPD manifolds can form gyrovector spaces with AI and LE geometries. For Grassman manifold, the gyrogroup is not gyroreductive, but it is gyrocommunitative. From these defined gyrovector spaces, the authors derived operations that can be used for neural networks. They empirically validated that such manifold networks can be applied to applications such as multi-time-series data analysis and word embeddings. 
strengths_and_weaknesses: Strengths:

1. The paper contains some original ideas. 
2. The theoretical analyses seem to be well supported by the empirical results provided.

Weaknesses:

- This paper is difficult to read, partly because of the prerequisite concepts, and partly because of the authors' writing style. The authors can help the reader by making sure that new concepts are properly defined and motivated. The information in this paper can be better organised so that the key messages are clear and consistent.


questions: - Contributions (2) and (3), Why AI geometry is left out?
- Definition 3.1, How is r useful?
limitations: N/A
review_title: null

Meta-review: metareview: The paper studies the gyrovector space structure of a few matrix manifolds. This is of broader interest to practitioners who may be interested in using geometric tools in applications. As the reviewers mention that although interesting, the analysis is per se straightforward for many manifolds. Having said that, this AC believes it is a worthwhile effort to compile such results in one place and show the practical benefits. This paper is a good attempt in this direction. In the final version, please include all the suggestions both in explanations and as well as the new results presented in the discussion period. 

confidence: Certain
Decision: Accept