Paper: How connectivity structure shapes rich and lazy learning in neural circuits (ICLR-2024)

Reviewer comment (Evaluation):
The metrics introduced in Section 2.2 are restricted to measuring changes in weights or kernels over the course of training. Particularly given the fact that the stated goal is to investigate how task structure affects feature learning, it would be useful to also track a measure of task-kernel alignment, for instance the kernel-target alignment $y^{\top} K y / |y|^2 tr K$ or the task norm $y^{\top} K^{-1} y$ throughout training.

Author response:
**Tracking task-kernel alignment during training**: We thank the reviewer for the insightful comment. In line with the reviewer’s suggestion of tracking impact on not just the final trained network but also during the learning process (and related comments by other reviewers), we have now added Appendix Figure 14 tracking the tangent kernel alignment as well as the alignment of the kernel and task throughout training. Figure 14A examines the kernel alignment (alignment of the current kernel to the initial one) in the idealized linear setting with 2D input in Fig. 2 in Atanasov et al. across different initialization schemes: random rank-1 initialization, random full-rank initialization and initially aligned low-rank initialization. We see that low-rank random initialization lead to greater movements, i.e. growing to be more dissimilar to the initial kernel (lower alignment value). Figure 14B looks at the task kernel alignment, $y^T K y / |y|^2 TrK$. Regardless of the initializations, the network kernel moves to be more aligned to the task over training. Low-rank initialization, without being specifically tuned to the task, is the latest to achieve alignment. 

Figure 14C examines the kernel alignment (alignment of the current kernel to the initial one) in a non-idealized setting: sMNIST task. Again, random low-rank initialization leads to more movement over training, i.e. more dissimilar to the initial kernel. For Figure 14D, with multiple (10) output targets, we track the centered kernel alignment (CKA) of the NTK and class label, e.g. used in Baratin et al. (2021). We again saw that all initializations grow to be more aligned with the class label through training, but higher rank initialization gets to a higher alignment value faster and with less kernel rotation. We have already trained for nearly 20000 SGD iterations and we are not sure if training longer would eventually lead to similar final alignment values across initializations. We note that although achieving higher CKA over training can indicate greater degree of feature learning, a hallmark of the standard rich regime, this does not contradict our findings centered on effective learning regime that is defined based on the amount of change post-training (see Introduction). To incorporate these new findings in the main text, we have added sentences in Simulation Results to bring these Appendix Figures to the attention of an interested reader.

Reviewer comment (Evaluation):
On the whole, I found the Discussion to be a weak point of the manuscript. Section 4.1 does not answer the vital question of precisely what experimentally-measurable biological phenomena the present work can explain, contextualize, or predict. As it stands, this section is in some sense a restatement of Zador's ideas around the importance of task-aligned initialization in the language of lazy learning. Moreover, the discussion of implications for deep learning promised by the section title seems to be missing.

Author response:
**Improving Discussion Section**: In response to the reviewer’s crucial comment regarding more concrete experimental explanation or prediction, we have now updated the manuscript, and the paragraph reads as: “*We study the impact of effective weight rank on learning regime due to its potential implications in neuroscience. In particular, learning regimes are indicative of the amount of change undergone through learning, which bear consequences for metabolic costs and catastrophic forgetting (McCloskey & Cohen, 1989; Plaçais & Preat, 2013; Mery & Kawecki, 2005). These considerations are especially pertinent when we examine the various learning regimes evident in neural systems. For instance, during developmental phases, neural systems undergo resource-intensive, plasticity-driven transformations characterized by significant synaptic changes. These transformations are in contrast to the more subdued adjustments observed in mature neural circuits (Lohmann & Kessels, 2014). Based on this understanding, we predict that a circuit's alignment with specific tasks is likely established either through evolutionary processes or during these early developmental phases. As such, the specialization of a neural circuit (e.g., ventral vs dorsal (Bakhtiari et al., 2021)) likely stems from its engagement with tasks that share overlapping computational requirements, ensuring that each circuit is optimally configured for the tasks it is predisposed to perform. Conversely, circuits with high-rank structures, due to their inherent flexibility, are expected to be less specialized, potentially engaging in the learning of a broader range of tasks. With this in mind, our framework could be used as a tool to compare the connectivities across brain regions (and species) to predict their function and flexibility…* ” 

For deep learning, we note that while low-rank initialization is not common practice, low-rank adaptation and other update parametrizations have recently been popular in large model training schemes. See for example, Low-rank Adaptation (LoRA) in Hu et al. (2021). As such, we argue that the impact of low-rank structures on learning dynamics is quite relevant for current AI practices, and our study contributes to examining how having low-rank structures impact the effective learning regime. The methods and results we present here have the potential to be adapted and built upon to study rankedness and learning regime in several settings, and we are keen to see future work stemming from these ideas.
We also note that our study has the tangential benefit of informing the deep learning community about potential applications of their models in other fields such as neuroscience. However, due to the significantly heavier balance of Section 4.1 in discussing neuroscience implication, we have removed “Deep Learning” in the section heading. We have updated Section 4.1 reflecting these points.

Reviewer comment (Presentation):
Section 4.2 contains far too many ideas to be jammed into its single paragraph; for clarity the authors should split it into at least two paragraphs.

Author response:
Upon re-reading Section 4.2, we agree with the reviewer that Section 4.2 is jammed with too many ideas. As per the reviewer’s suggestion, we have now broken 4.2 into smaller paragraphs for readability. We also thank the reviewer for pointing out the long sentence involving neuroscience implications and Tishby’s proposal. This sentence suffers the same problem as the entire section that the reviewer pointed out: too many ideas jammed together, which compromises clarity. We, thus, have removed the mention of Tishby’s proposal since it is creating confusion and distraction from our main point in the sentence, which is asking for deeper exploration into the neuroscience implications in the future.

Reviewer comment (Theory):
The relevance to neuroscience of neural collapse and of Tishby's proposals regarding the information bottleneck needs to be justified. The sentence where they are mentioned is much too long as it is, and these are both somewhat subtle and controversial topics.

Author response:
Upon re-reading Section 4.2, we agree with the reviewer that Section 4.2 is jammed with too many ideas. As per the reviewer’s suggestion, we have now broken 4.2 into smaller paragraphs for readability. We also thank the reviewer for pointing out the long sentence involving neuroscience implications and Tishby’s proposal. This sentence suffers the same problem as the entire section that the reviewer pointed out: too many ideas jammed together, which compromises clarity. We, thus, have removed the mention of Tishby’s proposal since it is creating confusion and distraction from our main point in the sentence, which is asking for deeper exploration into the neuroscience implications in the future.

Reviewer comment (Writing):
In the block of citations in the second sentence of the introduction, the work of Canatar, Bordelon, and Pehlevan (2021) should be cited before Xie et al 2022, as the latter is based on the results developed in the former work.

Author response:
**Citing Canatar et al.**: We thank the reviewer for catching this missed citation. We worked hard to be comprehensive in our citation, and yet, we still somehow let this highly relevant reference slip through our attention. We have thus added it to exactly where the reviewer suggested.

Reviewer comment (Writing):
Is "tabula rasa" the appropriate way to describe Gaussian or Uniform random initialization with non-vanishing variance?

Author response:
**Tabula rasa**: we thank the reviewer for bringing this to our attention. "Tabula rasa" means "blank slate”. However, Gaussian or Uniform random initialization, with non-vanishing variance, is not exactly a “blank state” in the strictest sense, as they can still exhibit inherent bias. We have thus replaced “tabula rasa” with “random” in the offending sentence.

Reviewer comment (Evaluation):
Why do the eigenvalues in Figure 2 appear not to be sorted by magnitude? Also, why is $\frac{\sum\_{i} |\lambda|\_{i}}{|\lambda\_{1} N}$ the relevant notion of effective rank in this setting?

Author response:
**Sorting eigenspectrum plots and effective rank**: We thank the reviewer for the comment. The eigenvalues were not sorted because we used *numpy.linalg.eigvals()*, which does not guarantee sorted eigenvalues and we didn’t ensure that once we were able to clearly observe the faster decay trend in the eigenspectrum of the experimentally-driven connectivities was still visible. As per the reviewer’s comment, we have now sorted the eigenvalues in the eigenspectrum plots. 

As for the effective rank measure — which captures the proportion of eigenvalues on the order of the leading one — we chose it because it has been used before (e.g. Murray et al., ICLR’23) and would correspond to the area under the curve of the eigenspectrum plots scaled relative to the leading one, thereby informing us regarding the eigenspectrum decay. Low rank matrices would have fewer eigenvalues on the order of the leading one. That said, singular values are used more generally for capturing the effective rank. Hence, we have also added plots that measure effective rank using singular values (Appendix Figure 13) and a sentence in Simulation Results referring to that figure. These plots all support our main point here: these example experimentally-driven structures exhibit lower effective rank compared to null.

Reviewer comment (Presentation):
Figures 2 and 3 each occupy more than half of a page, but both contain a significant amount of whitespace. Is it possible to combine them side-by-side into a single figure?

Author response:
**Combining Figure 2 and 3**: We thank the reviewer for this suggestion and we have now combined them into one figure. This indeed gave us a lot more space to address the reviewers’ important points.

Reviewer comment (Theory):
Given that the proof of Theorem 1 is left to an Appendix, the paper would benefit from providing more explanation / intuition for the main result. I can imagine an intuition that the network "needs to" adjust its representation more to address the misalignment. But a more thorough presentation of the key proof steps in the main text, and/or illustrative examples to provide intuition for the learning dynamics, would be very helpful.

Author response:
**More explanation/intuition of the main theoretical result**: we thank the reviewer for this important comment. Our intuition is indeed in line with that of the reviewer’s. As per the reviewer’s suggestion, we have added a sentence explaining the intuition right after the Theorem statement. The sentence reads as “*The intuition of Theorem 1 result is that, when two random vectors are drawn in high-dimensional spaces, corresponding to the low-rank initial network and the task, the probability of them being nearly orthogonal is very high; this then necessitates greater movement to eventually learn the task direction*“. We have also updated the proof in the Appendix with more explanations for clarity.  

In addition, we followed the reviewer’s suggestion and updated the main text with an outline of key proof steps; roughly these steps are: 1) The kernel alignment (Eq. 9) consists of three factors: $Tr(K^{(0)} K^{(f)})$, $\| K^{(0)} \|$ and $\| K^{(f)} \|$. We first derived the expression for each of them, which then gives us an expression for the kernel alignment. 2) Since we are interested in the expected alignment over tasks, we take the expectation of the expression found in step 1 over $\beta$ (i.e. the task definition). 3) We write the resulting expression in step 2 in terms of the singular values of the weights, and we show that its optimized when the singular values $s_i$ are distributed evenly across dimensions.

Reviewer comment (Novelty):
It is not clear to me how the findings of this paper will be useful to either the ML or neuroscience communities. I do not mean to claim that they aren't useful, but rather that the paper does not present a clear case for why they are. From an ML standpoint, low-rank initializations are uncommon. From a neuroscience standpoint, the applicability of this theoretical framework seems uncertain given that the learning algorithms used in the brain are not well characterized and may differ substantially from SGD. Moreover, it is not clear to me what we gain by knowing that low-rank neural connectivity gives rise to richer learning than a counterfactual in which brains had higher-rank connectivity. Is the idea to compare between brain regions with connectivities of different ranks to make predictions about different representation learning dynamics? Or across species? More concrete and precise proposals for how these findings could be used would be helpful.

Author response:
**Better articulation of neuroscience and ML implications**: this is another important comment from the reviewer, and other reviewers have also raised this concern. We apologize for having not made these points clear in our initial submission. In response to this, we have updated Section 4.1 in the manuscript and the paragraph reads as: “*We study the impact of effective weight rank on learning regime due to its potential implications in neuroscience. In particular, learning regimes are indicative of the amount of change undergone through learning, which bear consequences for metabolic costs and catastrophic forgetting (McCloskey & Cohen, 1989; Plaçais & Preat, 2013; Mery & Kawecki, 2005). These considerations are especially pertinent when we examine the various learning regimes evident in neural systems. For instance, during developmental phases, neural systems undergo resource-intensive, plasticity-driven transformations characterized by significant synaptic changes. These transformations are in contrast to the more subdued adjustments observed in mature neural circuits (Lohmann & Kessels, 2014). Based on this understanding, we predict that a circuit's alignment with specific tasks is likely established either through evolutionary processes or during these early developmental phases. As such, the specialization of a neural circuit (e.g., ventral vs dorsal (Bakhtiari et al., 2021)) likely stems from its engagement with tasks that share overlapping computational requirements, ensuring that each circuit is optimally configured for the tasks it is predisposed to perform. Conversely, circuits with high-rank structures, due to their inherent flexibility, are expected to be less specialized, potentially engaging in the learning of a broader range of tasks. With this in mind, our framework could be used as a tool to compare the connectivities across brain regions (and species) to predict their function and flexibility…*” 

For deep learning, we note that while low-rank initialization is not common practice, low-rank adaptation and other update parametrizations have recently been popular in large model training schemes. See for example, Low-rank Adaptation (LoRA) in Hu et al. (2021). As such, we argue that the impact of low-rank structures on learning dynamics is quite relevant for current AI practices, and our study contributes to examining how having low-rank structures impact the effective learning regime. The methods and results we present here have the potential to be adapted and built upon to study rankedness and learning regime in several settings, and we are keen to see future work stemming from these ideas. We also note that our study has the tangential benefit of informing the deep learning community about potential applications of their models in other fields such as neuroscience. However, due to the significantly heavier balance of Section 4.1 in discussing neuroscience implication, we have removed “Deep Learning” in the section heading. We have updated Section 4.1 reflecting these points.

Reviewer comment (Evaluation):
The richness/laziness of learning dynamics by itself tells us very little about the representations a network learns. If the goal is to understand the impact of weight initialization on representation learning, the paper would benefit from more direct consideration of this question.

Author response:
**Impact on representation learning**: We thank the reviewer for the insightful comment. In line with the reviewer’s suggestion of tracking impact on not just the final trained network but also during the learning process (and related comments by other reviewers), we have now added Appendix Figure 14 tracking the tangent kernel alignment as well as the alignment of the kernel and task throughout training. Figure 14A examines the kernel alignment (alignment of the current kernel to the initial one) in the idealized linear setting with 2D input in Fig. 2 in Atanasov et al. across different initialization schemes: random rank-1 initialization, random full-rank initialization and initially aligned low-rank initialization. We see that low-rank random initializations lead to greater movements, i.e. growing to be more dissimilar to the initial kernel (lower alignment value). Figure 14B looks at the task kernel alignment, $y^T K y / |y|^2 TrK$. Regardless of the initializations, the network kernel moves to be more aligned to the task over training. Low-rank initialization, without being specifically tuned to the task, is the latest to achieve alignment. 

Figure 14C examines the kernel alignment (alignment of the current kernel to the initial one) in a non-idealized setting: sMNIST task. Again, random low-rank initialization leads to more movement over training, i.e. more dissimilar to the initial kernel. For Figure 14D, with multiple (10) output targets, we track the centered kernel alignment (CKA) of the NTK and class label, e.g. used in Baratin et al. (2021). We again saw that all initializations grow to be more aligned with the class label through training, but higher rank initialization gets to a higher alignment value faster and with less kernel rotation. We have already trained for nearly 20000 SGD iterations and we are not sure if training longer would eventually lead to similar final alignment values across initializations. We note that although achieving higher CKA over training can indicate greater degree of feature learning, a hallmark of the standard rich regime, this does not contradict our findings centered on effective learning regime that is defined based on the amount of change post-training (see Introduction). To incorporate these new findings in the main text, we have added sentences in Simulation Results to bring these Appendix Figures to the attention of an interested reader.

Reviewer comment (Evaluation):
Do the kernels associated with low-rank-initialized networks grow more aligned with the target function than those of high-rank-initialized networks? Or do they merely grow "as aligned" as the lazier networks with high-rank initializations?

Author response:
1) *“While I appreciate the authors' additional analysis in Figure 14, the results in Figure 14D appear to be in tension with the intuition underlying Theorem 1 would predict. Although the authors say that their findings are not contradictory (obviously they are not, assuming all are true!) it makes the findings of the paper harder to interpret for me. If the authors have a clear understanding of why high-rank initializations result in greater target alignment during learning, despite being overall in a "lazier" regime, an explanation would be very helpful.”*

We apologize for not having articulated how this new result integrates into our existing one, and **we hope our intuition below will better clarify and suggest how this new result in Figure 14 is actually in line with the main finding of Theorem 1**. The intuition is in line with the intuition that the reviewer asked us to articulate in their previous comment, *“in high-D spaces, when you draw two random vectors corresponding to the task and the low-rank initialization, they have a high chance of being nearly orthogonal to each other”*. This necessitates more movement for the low-rank initialization to be aligned to the task; consequently, this need for greater movement would require more training (as seen in Figure 14B). If training is stopped early (e.g. due to resource limitations), as likely the case in Figure 14D, they would achieve less final alignment within the training window. 

Regarding *“If the authors have a clear understanding of why high-rank initializations result in greater target alignment during learning, despite being overall in a "lazier" regime”*, we would like to remark that cannot be concluded from Figure 14D alone. First of all, these different initializations all lead to similar final alignment in the setting in Figure 14B. Second, we mentioned in our previous response to this reviewer that *“... we are not sure if training longer would eventually lead to similar final alignment values across initializations…”* Based on the upward trend at the end of Figure 14D, we predict that the alignment should continue to rise if we train longer. We note that for sMNIST, we concluded our training upon reaching 97% accuracy, a criteria informed by both published results and our computational resources. We also experimented with halving and doubling the training iterations and observed similar trends.  Whether they would all converge to similar alignment values at the end of training under a wide range of settings ties to one of the proposed future directions (see below). 

In response to the reviewer’s feedback, we have updated the caption of Figure 14 clarifying these points. Also, since this discussion is in line with our proposed future direction on deeper investigation of the link between rankedness, the learning regime and the consequent impact on representation (including kernel task alignment) and generalization properties of converged solutions under a wide range of settings, we have updated the corresponding sentence in Discussion alerting the reader to Appendix Figure 14.

Reviewer comment (Experiments):
While many controls are done, the initial dynamic regime of the network (stable, unstable or chaotic) is not controlled for (see questions below).

Author response:
**Confounding factor from dynamical regime**: We thank the reviewer for this insightful comment/question. Indeed, it is plausible that the learning regime in RNNs could be influenced by the initial dynamical regime. With stable dynamics (non chaotic) , for instance, activity typically settles into steady states in absence of inputs over time steps, potentially needing more weight adjustments during training to effectively propagate information to the last time step. The study of learning dynamics and dynamical regime of RNNs is rich and involved and there are multiple ways to control for dynamics stability. A common method is through the leading weight eigenvalue of the connectivity matrix, which directly influences the top Lyapunov exponent dictating the rate of trajectory expansion.

As such, we now add details about a new control where we compare the leading weight eigenvalue (Appendix Figure 19), and we observed a consistent trend: higher-rank initializations lead to effectively lazier learning post-training. We note that a deeper exploration of the relationship between learning regime and various notions of the dynamical regime is a promising avenue for future work but falls outside the scope of the present paper, and we express this view in the discussion. We thank the reviewer for this insightful comment/question, and in response, we have updated the manuscript that added sentences in Simulation Results to reflect these points and alert interested readers to Appendix Figure 18.

Reviewer comment (Theory):
I am not entirely convinced by the evidence given for the statement that the brain has low-rank connectivity. The paper repeatedly cites Song et al., 2005 and Mastrogiuseppe & Ostojic, 2018 as evidence for low-rank connectivity in the brain. The latter is a purely theoretical study, which I am not sure why it used as evidence for biology. The former indeed studies biological neural circuits, and found that the distribution of synaptic connection strength can be fitted by a lognormal distribution, as well as the circuits having overrepresented bidirectional connectives. While my intuition is that this implies low-rank connectivity, it can (and likely should) be made clear if this is the case. In general the relation between local connectivity patterns and (global) rank is not always obvious a priori (potentially, see Shao & Ostojic 2023).

Author response:
**Evidence that the brain uses low-rank connectivity**: we thank the reviewer for pointing out this important and subtle point. In response to the reviewer’s important point, we have toned down our claim throughout the manuscript (Abstract, Introduction and Discussion in particular): there are evidence (e.g. Song et al.) \textbf{suggestive} of effective low-rank structures in the brain, rather than hard evidence. We also removed the citation to Mastrogiuseppe & Ostojic here, as the reviewer pointed out. We originally had a sentence in between the two citations but it got lost in the editing. 

The extent to which the brain utilizes low-rank structures remains an open question, given the significant variation in neural circuit structures across regions and species. On top of that, as the reviewer pointed out, while local connectivity statistics can offer some predictive insight into the global low-rank structure, this relationship is not always immediately apparent (Shao & Ostojic, 2023). Our theoretical results contribute to this discourse by providing tools that link connectivity features, particularly effective rank, with learning (see Section 4.1). We have updated the manuscript reflecting this discussion in Related Works and Limitations & Future Directions.

Reviewer comment (Theory):
Biological constraints are only used as initialisation and not enforced during training. This decreases the relation of the simulations to biology, as neurons in the brain can (generally) not just decide to forget about e.g. Dale’s law during learning.

Author response:
**Biological constraints enforced during training**: we agree with the reviewer that enforcing constraints (e.g. Dale’s law) would be important for biological relevance. We initially didn’t do this to test our theoretical prediction, which assumes gradient-descent update without constraints. Following the reviewer’s comment, we have added a new figure (see Appendix Figure 17) with Dale’s law enforced for the EI experiments and observed similar trend as before: EI initialization led to greater changes post-training. We have also added sentences in Simulation Results alerting readers to this appendix figure.

Reviewer comment (Experiments):
The authors claim to study continuous-time networks, however, $dt$ is set equal to $\tau_m$, and tasks have around 10 time steps after discretisation. It could be questioned how well the simulations reflect the continuous-time equations.

Author response:
**Continuous RNN settings**: we thank the reviewer for this important comment. We have now added Appendix Figure 18 to show our main conclusions — that higher rank random initializations lead to effectively lazier learning — also hold for the case when $\tau_m > dt$ and with longer sequence length. Specifically, we kept $\tau_m=100$ but changed $dt$ from $100$ to $20$, which increased the sequence length by a factor of 5. We have also added sentences in Simulation Results alerting readers to this appendix figure.

Reviewer comment (Writing):
Equation 10, last line should say d instead of 3 above the equality sign.

Author response:
**Minor fixes**: we thank the reviewer for catching these and we have fixed them.

Reviewer comment (Presentation):
Figure 3B left, title is cut off.

Author response:
**Minor fixes**: we thank the reviewer for catching these and we have fixed them.

Reviewer comment (Presentation):
The 5th reference seems formatted wrong.

Author response:
**Minor fixes**: we thank the reviewer for catching these and we have fixed them.

Reviewer comment (Evaluation):
I would like the authors to comment on whether the difference in lazy versus rich learning is potentially confounded by the regime of the initial dynamics (stable, unstable or chaotic), instead of the rank of the initial matrix. Three results that lead me to hypothesise there is at least some influence of dynamic regime on the learning: (a) For Figure 1, the large full rank networks will likely be initially be chaotic, given a gain of 1.5 (transition at $\sqrt(2)$ for ReLU networks; Kadmon & Somplinsky 2015). However at least part of the truncated networks (despite rescaling by the norm) will likely be not. (b) The outlier eigenvalue in the networks of Figure 2 might lead to (unstable) dynamics initially exploding in the direction of the corresponding eigenvector, instead of following a chaotic regime. (c) Finally, in Figure 9, the differences between low-rank and full-rank networks diminishes when both are initialised in the stable regime.

Author response:
**Confounding factor from dynamical regime**: We thank the reviewer for this insightful comment/question. Indeed, it is plausible that the learning regime in RNNs could be influenced by the initial dynamical regime. With stable dynamics (non chaotic) , for instance, activity typically settles into steady states in absence of inputs over time steps, potentially needing more weight adjustments during training to effectively propagate information to the last time step. The study of learning dynamics and dynamical regime of RNNs is rich and involved and there are multiple ways to control for dynamics stability. A common method is through the leading weight eigenvalue of the connectivity matrix, which directly influences the top Lyapunov exponent dictating the rate of trajectory expansion.

As such, we now add details about a new control where we compare the leading weight eigenvalue (Appendix Figure 19), and we observed a consistent trend: higher-rank initializations lead to effectively lazier learning post-training. We note that a deeper exploration of the relationship between learning regime and various notions of the dynamical regime is a promising avenue for future work but falls outside the scope of the present paper, and we express this view in the discussion. We thank the reviewer for this insightful comment/question, and in response, we have updated the manuscript that added sentences in Simulation Results to reflect these points and alert interested readers to Appendix Figure 18.

Reviewer comment (Experiments):
Were the Dale’s law networks balanced? E.g. increasing inhibitory strength if less than 50% of neurons are inhibitory, so that the expected input to neurons stays zero, would put the Dale’s law networks in similar regimes as the controls (see Rajan & Abbott, 2006).

Author response:
**Balance Dale’s Law**: we thank the reviewer for this question and we apologize that it wasn’t clear. We used 80% excitatory and indeed balanced the initialization according to standard practices. We have updated Setup & Simulation Details reflecting this point.

Reviewer comment (Theory):
The Eigenspectrum, and as a result the initial dynamics, of the full-rank random networks are most likely approximately constant between networks, whereas the (non-zero part of the) eigenspectrum of the low-rank networks will vary more over initialisations - is this related to the observed high variance in richer learning between the low-rank models?

Author response:
**High variance due to variable eigenspectrum in low-rank network**: However, our newly added Appendix Figure 19, which maintains a relatively stable leading eigenvalue by fixing the dominant eigenvalue across comparisons, still results in high variance for low-rank initializations. This suggests that there may be other factors contributing to this high variance, and it would be interesting to examine this in the future. This point is also reflected in the sentences added to Simulation Results, as a part of responses to the reviewer's earlier comment about the dynamical regime as a possible confounding factor.

Reviewer comment (Evaluation):
The paper could provide a more finer-grained description and insight into how the initialization scheme affects the final solution. This could be done by showing how the existing metrics (from Sect 2.3) change during learning as a function of training epoch, rather than just reporting them for the final solution (especially in the rich regime), and/or by introducing additional finer-grained metrics. There may be different explanations for how the weights evolve during learning that are interesting and important; for example, is the largest rank component changing first or changing in magnitude the most during learning? Does this occur even if the task is low-rank but the low-rank initialization is not perfectly aligned with the task?

Author response:
**More fine-grained description of the impact of initialization schemes**: In line with the reviewer’s suggestion of tracking impact on not just the final trained network but also during the learning process (and related comments by other reviewers), we have now added Appendix Figure 14 tracking the tangent kernel alignment as well as the alignment of the kernel and task throughout training. Figure 14A examines the kernel alignment (alignment of the current kernel to the initial one) in the idealized linear setting with 2D input in Fig. 2 in Atanasov et al. across different initialization schemes: random rank-1 initialization, random full-rank initialization and initially aligned low-rank initialization. We see that low-rank random initializations lead to greater movements, i.e. growing to be more dissimilar to the initial kernel (lower alignment value). Figure 14B looks at the task kernel alignment, $y^T K y / |y|^2 TrK$. Regardless of the initializations, the network kernel moves to be more aligned to the task over training. Low-rank initialization, without being specifically tuned to the task, is the latest to achieve alignment. 

Figure 14C examines the kernel alignment (alignment of the current kernel to the initial one) in a non-idealized setting: sMNIST task. Again, random low-rank initialization leads to more movement over training, i.e. more dissimilar to the initial kernel. For Figure 14D, with multiple (10) output targets, we track the centered kernel alignment (CKA) of the NTK and class label, e.g. used in Baratin et al. (2021). We again saw that all initializations grow to be more aligned with the class label through training, but higher rank initialization gets to a higher alignment value faster and with less kernel rotation. We have already trained for nearly 20000 SGD iterations and we are not sure if training longer would eventually lead to similar final alignment values across initializations. We note that although achieving higher CKA over training can indicate greater degree of feature learning, a hallmark of the standard rich regime, this does not contradict our findings centered on effective learning regime that is defined based on the amount of change post-training (see Introduction).

In addition, to examine how the leading component changes relative to the rest, we track the evolution of the kernel effective rank, inspired by Baratin et al. (2021) in Appendix Figure 15. Here, the effective rank is the ratio of the trace to the dominant one, thereby capturing how much the leading eigenvalue evolves relative to the rest. We found that the kernel effective rank is getting closer to the target/task dimension over training. 

Please note that we focused the additional analyses on the network kernel rather than individual layer weights, as the former is more reflective of the network’s function as a whole. To incorporate these new findings in the main text, we have added sentences in Simulation Results to bring these Appendix Figures to the attention of an interested reader.

Reviewer comment (Theory):
The question of initialization effects could be addressed both experimentally and in the linear setting where the authors make analytical statements. For example, Proposition 1 appears to consider a low-rank 1 initialization completely aligned with the task (initialized with $\beta$), but it would be helpful to understand in general how a low-rank initialization compares with a high-rank initialization for learning a fixed low-rank task, rather than considering an expectation over all tasks. More specifically: in general, would a low-rank initialization be better for a low-rank task, and why or why not? Also, how are weights and singular components evolving to facilitate the learning?

Author response:
**Helpful to understand low vs high rank initialization for learning some fixed (low rank) task, rather than an expectation over all tasks? Would a low-rank initialization be better for a low rank task and why/why not?** We thank the reviewer for these questions and we apologize for any confusions we caused. Before directly answering the reviewer’s question, we would first like to clarify that in our RNN experiments (e.g. 2AF), we examined a fixed task (although there is some degree of stochasticity in data presentation due to SGD). Our theoretical result — focusing on the expectation — primarily aims to explain the randomness/uncertainty arising from random initialization that has no knowledge of the specific/fixed task prior to training. Importantly, looking at the expectation across tasks also gives some insights into how flexible the circuit is when learning across a broad range of tasks. 

Regarding the question of whether low-rank initialization would help for a low-rank task, we again remark that all the neuroscience tasks studied here are known to be low-rank. Our idealized linear feedforward task, accompanying Theorem 1, is also low-rank. As such, this discussion would connect back to the gist of our results: without having prior knowledge about the task, i.e. without explicitly aligning the circuit, random low-rank initialization would require greater adjustments (or harder time) to learn these tasks (even if these tasks are low-rank). These low-rank structures would mainly be helpful (requiring less changes) if they were aligned to the task statistics prior to training (the aligned initialization results). We also remark that more nuanced interplay of learning dynamics across different initialization rank and task dimension is left for future work, as mentioned in Discussion. We have also updated the Discussion section in response to the reviewer’s great question.

Reviewer comment (Theory):
The paper could be made stronger if the theoretical case studied were closer to the experiments, though it is understandable that it is easier to analyze the linear feedforward setting and that the theory still provides insight for the RNN experiments.

Author response:
**Feedforward theory and RNN experiments**: We agree with the reviewer’s comment. Note that the new addition of feedforward experiments in non-idealized setting (in response to Reviewer rK41) would be a step closer to what the reviewer suggests about closer theory and experiment setting.

Reviewer comment (Writing):
The presentation and writing, while generally clear, could be improved. At times, the writing felt too verbose and could be made clearer. For example, the sentences “A nexus between deep learning and neuroscience has expanded the applications of deep learning theoretical frameworks to study the learning dynamics of biological neural network” and “Our work builds on these studies by further exploring the precursors of these regime” could be clearer.

Author response:
**Improving presentation**: We thank the reviewer for this suggestion. We fixed several more verbose sentences in addition to the one the reviewer pointed out, including the very first sentence in Introduction. We have also fixed the typo that the reviewer pointed out for Figure 3B (now in Figure 2) in the original submission.

Reviewer comment (Presentation):
Minor presentation issue: In Fig. 3B, the left title is not visible.

Author response:
**Improving presentation**: We thank the reviewer for this suggestion. We fixed several more verbose sentences in addition to the one the reviewer pointed out, including the very first sentence in Introduction. We have also fixed the typo that the reviewer pointed out for Figure 3B (now in Figure 2) in the original submission.

Reviewer comment (Theory):
For the proof in the Appendix of Theorem 1, why is it necessary for the scale $\sigma = \||W_i(0)||_F$ to be small? Also, small relative to what?

Author response:
**Small \sigma relative to what**: we thank the reviewer for this question. Here, small $\sigma$ means $\sigma<<1$. This small initial weight variance assumption allows us to use  t the analytical expression ofr the tangent kernel after training derived by Atanasov et al., which is valid up to terms that are at least quadratic in  $\sigma$. We have updated the manuscript clarifying this point.

Reviewer comment (Theory):
Is there evidence that the initial connectivity structures in biology are low-rank?

Author response:
**Evidence for low-rank structure in neural circuits**: We appreciate the reviewer's great question. In response to this reviewer and reviewer Cn5Y's comments, we have added an overview of the implications of local connectivity structures (in both Related Works as well as Limitations and Future Work), as evidenced in works like Song et al. (2005), and their potential influence on global low-rankedness, for example, as discussed in Shao & Ostojic (2023). While local connectivity statistics can indeed provide some predictive insight into global low-rank structures, the relationship is not always immediately obvious. Additionally, the degree to which the brain employs low-rank structures is still an open question, considering the vast variability in neural circuit structures across different regions and species. Consequently, we have moderated our claims regarding the low-rankedness in neural circuits throughout our manuscript (Abstract, Introduction and Discussion in particular). Importantly, we believe our theoretical findings significantly contribute to this ongoing discourse by offering predictions that connect specific connectivity features, especially effective rank, with learning (see Section 4.1). We have updated the manuscript reflecting this point in Limitations and Future Work.