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 authors present two online biologically plausible algorithms for the irreducible representation learning method proposed by Cohen and Welling (2014). 
The first algorithm uses SVD to extract the subspace of maximal average rotation. As part of this algorithm, the authors’ present a novel algorithm for online SVD: cross-power iteration.
The second is based on PCA of time difference vectors and uses local learning rules to preserve biological plausibility.
The authors propose that these models may serve as hypotheses to guide future connectomics and neurophysiology research

Results. 
The authors find that both the SVD and PCA methods are able to recover the irreducible representations of SO(2) and S1 on synthetic datasets
These synthetic datasets are sets of random images or vectors sampled from the standard normal distribution, followed by the application of the transformations of interest (translation or rotation).

strengths_and_weaknesses: Strengths:
- The paper contributes two novel and interesting methods for learning the irreducible representations of toroidal groups with biologically plausible mechanisms. This is a valuable contribution for both computational neuroscience (as a hypothesis for neural circuitry) and machine learning (as a novel algorithm for learning and modeling transformation structure from data). 

Weaknesses:
- It would be nice to see a quantitative analysis on the learned filters, explicitly comparing to the standard translational and rotational Fourier bases, and reporting how well the learned weights approximate these bases.
- It would be useful to see how well the model trains on natural datasets.

General comments:
- the paper is framed as a neurobiological model, but it really seems more "neuromorphic" as the emphasis is on how to learn all these things with local learning rules in a neural circuit, as opposed to specific neurobiological substrates in the visual system.  There are issues such as how to compute arctan, and many others, that would need to be discussed or elaborated more to think of this in neurobiological terms.
- it will be interesting to see how these results could be extended to real-world applications, such as estimating optical flow from natural videos.


questions: see above
limitations: see above
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 manuscript proposes biologically plausible algorithms for learning group transformations from sequences of observations. Two algorithms are proposed, one based on SVD and the other on PCA. The performance of these algorithms is evaluated in synthetic experiments.

strengths_and_weaknesses: Strengths:
- the authors address a very interesting and challenging question: biological plausibility of the learning of transformation representations,

Weaknesses:
- the setting is quite restrictive, reducing the problem to previously proposed biologically plausible algorithms for PCA and SVD.
- experiments are very toy-like, with very simplified assumptions whose influence is not tested,
- relation to previous work on biologically inspired PCA and other such algorithms is elusive. It is difficult to figure out what is the originality of the proposed approaches, and why previous approaches could not directly be applied. 
questions: The assumption of line 89-90 seems fundamental to be able to the whole paper, but are not discussed. What do they entail?
One particular concern is the independence between $A_t$ and $x_t$, as it is likely that in many practical settings, transformations are performed depending on the previous state.
Moreover, assuming the the pixel intensity vector is white is also very unusual and unlikely in practice.

In addition, the experiment seem to assume that rotations in each subspace are also chosen independently, which is not necessarily the case in practice, and may play a key role in identifying the subspaces.
limitations: Limitations are not discussed.
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: The present paper studied the neural implementation of learning irreducible representation of commuting group transformations. Specifically, using the 2d rotation group as an example, the paper studied how the neural network learns the rotation groups by implementing different algorithms, including singular value decomposition and principal component analysis, and different algorithms eventually lead to different network architecture.
strengths_and_weaknesses: It is a fundamental question that how neural circuits learn the commutative group transformations. This study provides two novel solutions which rely on single neuron mechanism (implementing SVD) and network mechanism (implementing PCA) respectively to learn the group transformations. The network model implementing PCA (similarity matching specifically) to learn transformations is based on a recent study (ref. 1) so it is not clear how much difference of this part compared with the mechanism presented in ref. 1. The whole paper is structure-wise but some presentation is not clear and the comparison with earlier studies was not done systematically (see details below).

### Writing
- Eq. 3: the index t should be t-1 if I understood correctly.

- Line 141: should the $\sigma$ be the g in Lemma 1? If so, please use consistent notations across the manuscript. 

- It is not clear the derivation in Eq. 18 based on Eq. 16 and some motivations are missing. Also, it is not clear the meaning of matrices W and M in Eq. 18 and I need to guess whether they correspond to feedforward and recurrent weight matrices respectively.

- Fig. 5a: not clear which algorithm leads to the presented filters. That is, are those filters learnt from the SVD (representing the u and v), or from the network implementation of PCA (W and M)?

questions: - The Introduction (lines 44-45) mentions that an advantage of learning irreducible representation solves the need for multiplicative units and only a linear number of synaptic connections will be needed per neuron. However later in the Result section, I don’t see a detailed explanation of how this problem is solved in the proposed algorithms. Some elaborations are needed.

- Both models (Fig. 2 and Fig. 3) explicitly output the rotation angles $\theta$. In contrast, there are solid neuroscience studies that the angle is represented in a neural population code and can be read out by a linear decoder such as population vector. I am wondering how much the proposed network model and learning rules will be changed if we use a neural population code to represent the rotation angle.

limitations: - Both models considered in this paper only passively estimate the rotation angle. It would be more interesting to synthesize the transformed images, as in an autoencoder framework.
review_title: null

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: The authors depict two novel biologically plausible algorithms for online estimation of transformations using irreducible representations. By building on prior work, existing algorithms making use of commutative Lie groups for decomposing transformations are altered for biological plausibility. Possible neural implementations are depicted which can be searched for in connectomics. The results are evaluated experimentally on two small examples.
strengths_and_weaknesses: Strengths:
- Originality: The general idea of representing Rotations by toroidal groups is not new but nonetheless interesting and coming up with actual biologically plausible implementations is invaluable.
- The text is accessible: starting with a general introduction and good explanation of the problem at hand. 
- All Proofs including proof of convergence for "cross-power iteration" are readily available in the appendix


Weaknesses:
- Not clear which parts are contributions, i.e. labeled as "Theorem"
- Sadly no code provided
- Quantitative comparison of the experimental results with "biologically implausible" counterparts (i.e. Cohen and Welling) would've been interesting

Suspected errors:
- Line 73: canonical (ly)
- Line 227: canter
questions: How do the non-biologically plausible SVD-based algorithm and the altered version compare in terms of accuracy?
limitations: Limitations, such as biological implausibility of the original SVD based algorithm, are stated clearly.
review_title: null

Meta-review: metareview: This manuscript presents novel biologically plausible algorithms for learning representations for Lie groups. The derivation of the algorithms and the networks are based on previously studied biologically plausible networks. Although there are some limitations, the reviewers agree that this work is sound, clearly presented, and represents a valuable contribution to both computational/theoretical neuroscience and machine learning.
confidence: Certain
Decision: Accept