__ Summary and Contributions__: This paper establishes a link between Reservoir Computing (RC) and Recurrent Kernels (RK) by showing that Kernel Methods (KM) with Random Features (RF) can be equivalent to RC and thus RC can converge to a deterministic kernel as the number of neurons N tends to infinity. The link is clarified through RK as a limit of such approximation.
The authors also present Structured Reservoir Computing (SRC) as an alternative to RC (and thus RK) that is more memory efficient and yields faster prediction times.
Results are supported by empirical studies on random and chaotic time series.

__ Strengths__: The paper is well written, and the grounds for claims and the links between the different elements are well established in Section 2.
All claims are supported by corresponding empirical results or complexity analysis.

__ Weaknesses__: As an uninitiated person to the almost all components studied here, it is hard for me to detect flaws in the theoretical derivations shown here, which are at the heart of the paper.
I list some clarity aspects below on the parts I could follow, but I could not detect any major flaw.

__ Correctness__: The theoretical claims are proven and empirically supported.

__ Clarity__: Overall the paper is very well written and even though I could not follow all the mathematical derivations, the ideas were clearly presented, the text is easy to follow and the supporting evidence is presented clearly.
From Equation (6) and downwards, there are a lot of occurrences of expressions in the form of k(.) for which k(.) is applied on a single component. All previous uses of k(.) define the kernel operation applied on two components but it is unclear to me what this operation is when applied on a single component.

__ Relation to Prior Work__: The links with previous works seem clearly defined and put in the relation to the current work.
Note that I am unfamiliar with the prior work.

__ Reproducibility__: Yes

__ Additional Feedback__: I wish to apologize to the authors for such a poorly constructive review on the hard work presented here. I do not think I have the required expertise and theoretical background to further comment on the work.
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Given the rebuttal and other reviewers appreciation I keep my positive score. My lack of expertise in the area prevents me to further comment.

__ Summary and Contributions__: This work aims to explore the kernel limit for Reservoir Computing. The authors proved and tested its convergence theoretically and empirically. The authors further proposed methods, namely recurrent kernel and structured RC, that showed the proposed methods are more memory and computationally efficient than conventional RC.
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Thanks for the detailed response, my concerns were addressed. I'd like to increase my score

__ Strengths__: Training RNNs remains challenging, RC could be one of the approaches to simplify this and yield more efficient RNN models.

__ Weaknesses__: 1. The applicability to other real-world tasks is unknown
2. Some experimental settings/results are unclear, see my comments in "Additional feedback"

__ Correctness__: 1. I am not able to judge the correctness of convergence theorem
2. The main claim is not well supported by current empirical results

__ Clarity__: Could be improved, particularly experiment section could be clear

__ Relation to Prior Work__: Relation to prior work is well presented

__ Reproducibility__: Yes

__ Additional Feedback__: 1. In Table 2, The results are averaged on how many different runs? For measuring the time costs, each different run would lead to quite different results, a single run number is not reliable. Why does KR have the same number for all N?
.
2. Is the conclusion section missing?
3. In line 181, it says the convergence can be achieved even though the assumptions in previous theoretical study are not satisfied, then what is the point of the theoretical study?
4. I am a little bit confused by "forward" and "train", usually train stage of RNNs contains "forward" and "backward", here the "train" means the part of linear regression only?

__ Summary and Contributions__: The authors connect two important methods, Reservoir Computing (RC) and kernel methods, both theoretically and with experiments. By comparing Recurrent Kernels (RK) to structured and unstructured RC neural networks, they show that both methods converge numerically to the same limit. They attempt to derive this limit theoretically but with very restrictive assumptions compared to practice. They also show that their structured RC and RK methods can reduce computational complexity and are able to predict a chaotic time-series accurately.

__ Strengths__: Understanding how neural networks behave in the RC setting, where the network is initialized with random weights and an output is trained with linear regression, can help us understand the inner workings of Recurrent Neural Networks (RNNs) in general, how to better train them in machine learning and how they behave in models of the brain in neuroscience. Establishing a link between kernel methods, random features and RC, is an important and novel contribution in helping the community study RNNs further. The paper was well written and all theoretical claims and proofs were well-stated and explained. Demonstrating that RC nets converge to their kernel limit is an important finding, yet some of the assumptions were too restrictive. Numerical experiments further demonstrated the authors claims and they were well explained and plotted.

__ Weaknesses__: One weakness of the paper is that there is somewhat of a gap between theory and practice. The necessary components for their theory to hold is to have random network and input weights for every time-step, while keeping the network stable and exhibiting the echo-state property. While the latter might be possible by selecting certain hyperparameters, having random weights is never done in practice or in their experiments.

__ Correctness__: My two main concerns in this paper are regarding the theoretical assumptions necessary for RC convergence to its kernel limit:
1 - Random Weights: Their theory assumes that the weight matrices of the network (W_r) and input (W_i) are randomly redrawn every time-step. The authors explain that this assumption was necessary as correlations are difficult to take into account yet no further explanation was given. The main problem with this assumption is that it does not reflect the use of RC networks in practice where its weights are fixed. Ideally there should be a theoretical exploration of fixed weights and if that it is too difficult at least an explanation on why both random and fixed weights give similar convergence as demonstrated in the Supplementary Figure 3.
2. Stability Analysis: One of the theoretical assumptions of the paper is that the function f is bounded by a constant K. That means that the network must be stable and also exhibit the echo state property. The authors explain that by having a finite memory this property is achieved and they demonstrate stability for the dataset as input in Supplemental Figure 4. What is not clear to me is in the theoretical setting of random weights, how are we guaranteed stability and/or the echo-state property, that are necessary for convergence?

__ Clarity__: The paper is well written and all theoretical claims and numerical results were well presented.

__ Relation to Prior Work__: Relating kernel and random feature methods to reservoir computing is a novel approach in this paper, while discussed previously by [5], the theoretical and experimental findings of this paper is a new contribution to the field. The authors clearly discuss relations to other contributions in the introduction. I suggest considering discussing Inubushi & Yoshimura 2017, especially concerning network memory and the echo-state property.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The paper introduces structured reservoir computing as an efficient
approach to RC and prediction of chaotic time series, together with
recurrent random kernels. The paper formally connects and compares RC
and recurrent kernels with structured reservoir computing, including a
comparison of computational complexities / memory complexities for the
different approaches.

__ Strengths__: The work establishes formal connections between RC and random
features. Convergence is investigated and along with complexity
analysis is a major and important component of the paper. The claims
appear to be sound and the overall work is relevant to the NeurIPS
community. I have not previously seen these connections being
investigated, and believe the work would be interesting for
implementations on resource-constrained hardware.
The work clearly points out limitations of the different approaches.

__ Weaknesses__: The paper makes quite good use of the available space: while it would
be interesting to see additional empirical evaluation in the main part
of the paper, there is nothing in the paper that should be
replaced. The focus on only chaotic time series prediction makes sense
in light of the original ESN papers, even though it would be nice to
see additional applications.

__ Correctness__: As far as I can tell claims and methods are correct. The paper
contains only limited empirical evaluation (but I believe that no
additional empirical evaluation is necessary).

__ Clarity__: The paper is very well organised and well written.

__ Relation to Prior Work__: Important relevant prior works are discussed and cited, and the paper
manages to place itself well into their context.

__ Reproducibility__: Yes

__ Additional Feedback__: It should be possible to reproduce the major results; however reproducing the practical aspects of the work may be difficult.