Venue: NeurIPS 2022 Conference

Reviews:

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 studies the problem of modeling dynamical systems using low-rank RNNs. The paper motivates a number of domains in which low-rank RNNs have given insight into the computations underlying a particular task. The paper proposes directly fitting low-rank RNNs (meaning the connectivity matrix in the RNN is restricted to be low-rank). The paper demonstrates how this restriction allows one to learn the correct effective connectivity for a number of synthetic tasks, and also applies the method to recordings from macaque prefontal cortex while the animals performed a context-dependent decision making task. This method reveals that a rank one network is sufficient to capture the neural activity.
strengths_and_weaknesses: Strengths:
1. The paper is well written, and does a good job of motivating the problem and summarizing relevant work.
2. The paper's focus on low-rank structure as an important inductive bias for neuroscience is not particularly new, but is nonetheless important. For example, Fig 2f. is a powerful reminder of the utility of the appropriate inductive biases for systems neuroscience, where we are (generally) only able to sample a tiny fraction of the relevant neurons participating in a given task.
3. The experiments on synthetic tasks as well as the application to neural data are clear and high quality.

Weaknesses:
1. The main contribution of the paper is essentially advocating for a type of regularization (low-rank structure), one that has been discussed plenty in the literature (for example by Mastroguiseppe and Ostojic, which is already cited by the paper). While important, the overall significance and novelty does not seem that high.
2. It is not clear how one generalizes the method to RNNs beyond the ones described by equation (1). For example, what about RNNs with multiple weight matrices (such as gated architectures: GRUs or LSTMs)? These are commonly used in machine learning and neuroscience, I am curious if or how the authors would apply their method to those networks.
3. The paper studies tasks which can readily be solved by a low-rank vanilla RNN. But the space of tasks that we are interested in understanding is much greater! What happens if you apply this method to a dynamical system that is solving a more complicated task? For example, one that does not admit a low-rank solution? It would be good to get a sense of what to expect from the method in these scenarios.

Minor suggestions for improvement:
- Should explain the "effective connectivity" in the main text. It is used throughout, and thus seems important enough to put in the main text (currently the explanation is in appendix A.4).
- The entire PDF is low resolution (for exmample text) which makes it hard to read.
- This is a nitpick, but replace $J^{eff}$ with $J^{\text{eff}}$. That will prevent the "eff" from being italicized (which should only be done for mathematical variables).

Overall, I am not so worried about point (1) under weaknesses, but look forward to see if the authors can address (2) and (3) in the rebuttal.

questions: - Why use an explicit (hard) rank constraint, rather than an implicit (soft) constraint, for example by penalizing the nuclear norm of the connectivity matrix? My understanding is that the latter is admits an easier optimization problem, as the nuclear norm is the convex relaxation of a hard rank constraint.
- There are multiple gray and red curves in Fig 2. What do they correspond to? Different random seeds?
- Why is there no equivalent of Fig 2f. for the DMS task?
limitations: Yes
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 describes results from a method for estimating low rank RNNs from observed trajectories.  The method itself is just gradient descent on the set of weights that specify a low-rank RNN to match observed trajectories.  The paper demonstrates that the method can recover the effective connectivity of known low-rank RNNs, does a good job describing the behavior of general RNNs trained on a set of four tasks, and does a reasonable job recovering neural trajectories from NHPs performing one of those tasks.
strengths_and_weaknesses: Strengths

Given the widespread interest in RNNs in machine learning and neuroscience, this method may find broad applications in both of those fields.  The low-rank description is more readily interpretable.

Weaknesses

I think the paper seems to take as a given that low rank RNNs can provide a veridical description of brain dynamics and learn many tasks.  This is not at all well established.  The tasks chosen here are well-described by low-dimensional RNNs, but many tasks and natural networks appear not to have that property.  For instance, Cueva et al., (2020, PNAS) estimate the dimensionality of neural populations during the delay period of a variety of experiments (see especially their figure 7).  They find a number on the order of 5-10 for the dimensionality over a few seconds.  Leaving aside technical questions about whether a particular number is meaningful, one can read their result as decelerating but growing without bound (e.g., dimensionality $\propto \log t$), or as saturating at some time $T$.  The first interpretation argues that a low-rank description is not appropriate---or at least that rank grows to be much larger than 10.  The other possibility---that dimensionality saturates at some time T---suggests it would be impossible for the network to distinguish times $> T$.  But that seems inconsistent with the data (e.g., Lewis & Miall, 2009, Trans Royal Soc B). 

Kind of a minor issue, but submitting a set of images instead of a searchable PDF made the job of reviewing this paper more difficult than it needed to be.


questions: It would be very informative to include a situation where the dynamics are in fact relatively high-dimensional.  Note that a full-rank RNN trained on the tasks used here does not necessarily exhibit the growth in dimensionality.  One possibility is to compare to an RNN trained to distinguish a large number of temporal intervals.  
limitations: 
The paper should do a better job at pointing to the computational limitations of low-rank RNNs and their possible limitations in describing brain data and cognition.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: In this paper, the authors use low-rank RNNs for modelling neural data. The connectivity matrix is approximated using a low-rank representation, which is then inferred by minimising the loss between the predicted neural outputs and the target ones. In task-optimised RNN, relevant parameters to are trained based on task targets. The method is then applied to various cognitive tasks to validate and extract insights.  

strengths_and_weaknesses: Strengths: 
The models used are rather simple, which benefits from good interpretability. 
The experiments considered are interesting and elucidate different aspects of the model.

Weaknesses:
Novelty. The novelty of the paper in terms of the model is very limited. The paper shows that low-rank RNNs can help analyse neural data, but the contributions of the work beyond these experimental evaluations and applications of the low-rank RNNs are unclear.

Minor weaknesses:
Presentation: From the main text, it is unclear how \kappa and \v enter the analysis.
Similarly, the role of section A.2 is rather unclear. Is it to justify the effective connectivity metric?

questions: - The assumption is 'm' vectors are orthogonal to each other (and similarly for 'n'). How is this constraint enforced during training? 

- 3.2 assumes that m,n, I, and w are all zero-centred. Please expand on why this is a valid assumption.

- One can first obtain a full-ranked connectivity matrix and then approximate it using the low-ranked method suggested here (similar to what is done for SVD in the paper) rather than learning a low-ranked approximation directly. How is this alternative compared to what is suggested in the paper? On a similar note, one can train a truncated SVD directly (rather than making the approximation post-training); how is this method's performance compared with the proposed method?
limitations: N/A
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 introduces low-rank RNNs, parameterized by the left and right eigenvectors of the connectivity matrix. The authors first show on simulated full-rank vanilla RNNs trained to perform some cognitive tasks that low-rank RNNs can accurately extract the essential dynamical and task structure learned by the full-rank systems. They thereby offer a computationally interpretable low dimensional representation of task activity. The authors then test their approach on neural recordings from two monkeys performing a context-dependent decision making task. Again the trained low-rank RNNs seem to extract the behaviorally relevant low-dimensional dynamics, at the same time producing synaptic weight distributions reminiscent of those observed in the brain.
strengths_and_weaknesses: Strengths:
Generally speaking, I think this is a powerful approach that may have important implications in neuroscience. Building a low-rank structure right away into the models used for inference may profoundly ease detection of computational mechanisms from data.

Weaknesses:
See detailed questions below. My major point probably is that I don't think it's clear whether the model really extracts the neural dynamics and how an animal is truly solving the task, or whether it merely finds a parsimonious solution to the task.
questions: 1) One may perhaps still ask what is really novel about this approach, given that low-rank RNN-type models have been employed in computational neuroscience for a number of years now to obtain insight into neural dynamics and task performance. I assume the authors are the first to actually train such a model directly on simulated and real neural data and to systematically explore its abilities in identifying the dynamical structure?

2) Directly inferring a lrRNN seems to be more powerful than training a full-rank RNN and using SVD posthoc. But why actually? Can the authors offer some theoretical insight why this is more efficient? Does this simply have to do with the much lower number of parameters that need to be inferred on lrRNNs, easing the training process?

3) The authors call the CDM and DMS tasks ‘complex’, but compared to many real world tasks and those in use elsewhere in the ML community I find them almost simplistic. Could the low-rank structure be an ‘artifact’ or result of the very simple nature of these tasks? For instance, say you were to construct a very high-dimensional space where dimensions are somewhat modulated by task features like inputs, would a lrRNN always infer this same structure cos it’s really a property of the task, not of the recorded neural activity?

I think this is a relevant question since I was surprised that, despite the very simple nature of the tasks, hundreds of neurons were used for the full-rank RNNs trained on the tasks. More importantly, the neural recording data sets appear to consist likewise of hundreds of neurons which were however drawn *independently* from different sessions and trials? This appears to imply that the neurons are *not* truly dynamically connected, and do not have any correlations beyond that induced by the task structure. Thus, the inferred lrRNNs may simply reflect this task structure but not really the processing and (missing) dynamics that goes on in the neural population during a task.

4) A related question: As I assume hundreds of RNN units are not really necessary for learning these tasks, couldn’t we just train a much lower-dimensional system to begin with and thereby obtain a much more parsimonious representation of dynamics? I would guess for these tasks 5 units may actually already do the job?

5) Is the phenomenon of mixed selectivity in cortical networks at odds with the low-rank structures identified?

6) I would like to see more examples of original and fitted neural firing rates and neural trajectories. For instance, in sect. 3.1 the claim is made that essential connectivity features and neural trajectories were well reproduced by the lrRNN, but Table 1 shows only a single summary statistic w/o confidence limits. Same goes for the real neural recordings, more unit examples that illustrate neural behavior on different trials of the task would be great.

7) Since Fig. 3-5 appear to show just examples of a single training run, I would like to see more stats on how robust the training process is, i.e. how likely the same structures are to be identified again when starting from many different initial conditions for parameters?

I think point #3 is really my main issue. In any case it would have been nice to see the application of this method to at least one other data set with a very different task. But given the third point, in my mind it’s essential to see an application on a higher-dimensional set of truly simultaneously recorded neurons. 

Minor points:

- Line 129: Universal approximation properties in my mind don’t apply here: To assure this, you would need to allow for a an arbitrarily long basis expansion, commonly implying that J would need to be of full rank and, in addition, you may need many more latent dim. than observed variables.

- My understanding is that for any real physiological data set one would need to re-train the lrRNN for a number of different R’s?

- Line 146-147: says that at least rank of 9 is necessary, but Fig. 2 only shows up to rank 5.

- Would be good to have connectivity matrices from the Suppl. in main text. I think they add a lot to the understanding of the inner workings of the trained lrRNNs.

- I’m somewhat puzzled why so many units are used for lrRNNs to begin with, and then kept after training? The results on the CDM, for instance, seem to imply one just needs 5 units?

- Does the lrRNN trained on the monkey data have as many dimensions as units? Isn’t this a constraint that could also prevent detection of latent structure in the data, since there are no latent variables in the lrRNN that could account for unobserved processes?

limitations: Partly, see questions above.
review_title: null

Meta-review: metareview: Building on recent theoretical work on the dynamics of low-rank recurrent neural networks, the authors present a method called LINT for learning low-rank network models directly from data. As the reviewers point out, from a purely technical perspective, the idea is straightforward: simply optimize a low-rank parameterization of an RNN. Similar ideas have been considered under the heading of _tensorized_ neural networks [e.g. 1]. See also related works cited therein, which considers low-rank parameterizations of weight matrices [e.g. 2]. 

Though the technical innovation may be limited, the work makes up for it with connections to recent research in theoretical neuroscience and interesting experiments. The reviewers raise many important caveats and limitations (e.g. are these tasks really reflective of the complexity of "real world" tasks in ML and experimental neuroscience?). Overall, though, the reviewers and I think this paper offers valuable contributions. I encourage the authors to revise their manuscript in light of these thorough and constructive reviews. 

[1] Novikov, A., Podoprikhin, D., Osokin, A. and Vetrov, D.P., 2015. Tensorizing neural networks. Advances in neural information processing systems, 28.

[2] Denil, M., Shakibi, B., Dinh, L., Ranzato, M.A. and De Freitas, N., 2013. Predicting parameters in deep learning. Advances in neural information processing systems, 26.
confidence: Certain
Decision: Accept