Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 10: Top 5% of accepted NeurIPS papers, seminal paper
review_content: summary: The paper derives a model of grid cells from first principles, starting with the basic assumption that position is represented as a high-D vector and that it is updated as a function of direction and speed.  The authors stipulate an isotropic scaling condition in order to make the representation robust to neural noise.  The elements of the resulting high-D embedding resemble grid cells, and importantly, the ability for error correction during path integration.

main_review: I find this to be a super interesting and exciting paper.  There are by now many papers that claim to show how grid cells emerge as a result of some kind of learning rule, but in my view this is the most insightful analysis I've seen to date.  It is a sensible and mathematically grounded approach.  Although they show how the grid cells emerge from learning, the resulting solution corroborates the theory - and so we understand why this is happening.  An important part of the story is robustness to noise, which is clearly linked to the isotropic condition and hence hexagonal grids.  This is high impact work that I believe will of interest to both the NeurIPS and computational neuroscience communities.



limitations_and_societal_impact: yes
review_title: null

final_score: rating: 5: Marginally below the acceptance threshold
review_content: summary: This paper focuses on the find that how the grid cells perform path integration. Speciafically, this paper introduces the isotropic scaling condition to produce the hexagon firing pattern for grid cells. The optimization experiments show that the proposed formulation of place and grid cells demonstrated this.
main_review: Although I appreciated that the authors provide an interesting formulation for the relationship of place and grid cells, I have the following concerns:

1.	The authors claim that they don’t make any orthogonality assumption and the isotropic scaling guarantees the l_2 error in L122. However, in the proof of Theorem 3 of this error bound, explicit orthogonal assumptions are made. I expect the authors to provide, in the rebuttal, a more rigorous proof for this error bound without any orthogonal assumptions.

2.	Many literatures have shown that cos/sin functions can produce hexagon patterns, e.g., [a, b]. Can you provide more connections to those results, because I highly suspect that the three conditions introduced in this paper will result in the cosine functions (Maybe the formulation (e.g., the isotropic scaling) in this paper is mathematically equivalent to the cosine functions?), especially after seeing the Fig 7 in Appendix. If this conjecture is correct, then I see no contribution for this paper, since we can simply use the predefined functions as in [a] to do both path integration and generate hexagon. Then what’s the purpose for the optimization?
3.	The authors didn’t show that the linear model with isotropic condition has a hexagon grid pattern in the proof, thus the tittle of Sec 3.3 is misleading, “hexagon grid pattern” in the section title should be removed. 
4.	Theoretic guarantee for generating the hexagon pattern by the introduced formulation will strengthen this paper a lot.
5.	It’s not clear how to derive the third item from the second item in Eq (9).

[a] Burgess N, Barry C, O'keefe J. An oscillatory interference model of grid cell firing. Hippocampus. 2007 Sep;17(9):801-12.

[b] Fuhs MC, Touretzky DS. A spin glass model of path integration in rat medial entorhinal cortex. Journal of Neuroscience. 2006 Apr 19;26(16):4266-76.


limitations_and_societal_impact: It seems that the only contribution is to introduce the isotropic scaling to generate the hexagon pattern during path integration. However, why should we do this? What ‘s the necessity to connect hexagon pattern and path integration via optimization, if predefined cosine functions can generate hexagon pattern very well?
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper introduces a general framework for modeling 2D navigation via grid cells. The main idea is to embed a 2D position x into a higher-dimensional positional encoding v(x) and model 2D translations via group transformations in this embedding space. A specific case are linear transformations via rotation matrices that act on the embedding v(x). Under this assumption, it can be shown that hexagonal patterns which are characteristic for neural grid cells naturally emerge as an optimal encoding. 
main_review: 1. Presentation: According to the submission history, the ICML AC had concerns about the exposition and presentation of the paper. The presentation of the paper in its current form is very clear. I find the structure of the methods part adequate (Sec. 2 - Sec. 5) and easy to follow. 

2. Quality of theoretical contribution/mathematical validity: Overall, the theory sections are mathematically sound and the stated assumptions are reasonable (l. 82, l. 107), save for a number of minor flaws. It should be straightforward to address these issues in a minor revision:
- The defined set of transformations F are intimately related to the standard mathematic concept of a "flow". Curiously, the authors do not mention this connection. This would also provide a stronger justification for Condition 1 (l. 82).
- Why does the angle Phi not depend on positional differential dx. In Eq. (5), the expansion of dv would envolve both dr and dPhi. If this is an implicit requirement, it should be stated as an explicit assumption.
- What is the motivation for restricting the approach to 2D environments? One could easily conceive of a formulation for more general Euclidean domains, like R^3. Most of the theory should be analogous. 

3. Results:
- It is fascinating that the hexagonal patterns emerge from the optimization framework. It's also a strong justification for the validity of the proposed method and assumptions. Furthermore, this effect is validated quantitatively in Table 1.
- It is a little disappointing that all evaluations focus on linear transformation models. One of the motivations of the proposed framework is that it allows for more general group transformations. This should be shown empirically as a proof of concept.
- The path integration experiments lack baseline comparisons which makes it hard to put these results into context. I would like to see comparisons to recurrent [3] and PCA based [33] models in Figure 6.
- The presented ablation study (l. 289) is useful to assess under which circumstances the hexagonal patterns emerge.

Minor remarks:
- I believe that the condition of F(v, 0, Phi) = v (l. 101) is central to the proposed framework and should be stated as a part of "Condition 1" (l. 82).
- The embeddings v(x) lie on the Lie group manifold characterized by rotation matrices. This means that they are contained in an elliptic space with positive curvature. This seems counterintuitive, since the goal is to model 2D trajectory prediction in Euclidean space. E.g. there might be group transformations, such that 3 consecutive steps with 90 degree turns between them yield a closed loop. Is this a phenomenon you encountered or did you find that this effect is relevant in practice?
limitations_and_societal_impact: There is a small discussion on limitations in Sec. 8 but no dedicated section or paragraph.
review_title: null

Meta-review: The reviewers are overall positive. The presented approach is mathematically sound and shows that grid cells can emerge from an optimization framework with minimal assumptions. The main criticism raised is whether the approach is useful beyond already existing models. The authors should clarify potential applications more explicitly in the revision, as well as include the additional experiments they cited in the rebuttal.

Decision: Accept (Poster)