Optimizing the Network Topology of a Linear Reservoir Computer

Francesco Sorrentino; Nicholas A. Mecholsky; Sahand Tangerami

arxiv: 2509.23391 · v3 · submitted 2025-09-27 · 📡 eess.SY · cs.LG· cs.SY· nlin.CD

Optimizing the Network Topology of a Linear Reservoir Computer

Sahand Tangerami , Nicholas A. Mecholsky , Francesco Sorrentino This is my paper

Pith reviewed 2026-05-18 12:10 UTC · model grok-4.3

classification 📡 eess.SY cs.LGcs.SYnlin.CD

keywords reservoir computingnetwork topologylinear systemseigenvalue optimizationtemporal predictionmachine learningconnectivity design

0 comments

The pith

Optimizing the eigenvalues of a linear reservoir computer's adjacency matrix improves its performance on temporal tasks over random designs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a linear reservoir computer can be made more effective by replacing random network connections with a deliberate choice of eigenvalues in the adjacency matrix. It decouples the system's overall behavior into independent modes and tunes each mode separately to suit the task. A sympathetic reader would care because the resulting networks train and test more accurately than random linear reservoirs and often outperform nonlinear reservoirs of similar size. The method also increases interpretability, since each mode's contribution can be adjusted and understood in isolation rather than treated as a black box.

Core claim

The central claim is that the dynamics of a linear reservoir computer can be expressed as a sum of independent modal contributions, each governed by an eigenvalue of the adjacency matrix. Selecting a set of eigenvalues that optimizes performance for a given task, then realizing the corresponding connectivity, produces a reservoir that significantly outperforms randomly constructed reservoirs in both training and testing phases and frequently surpasses nonlinear reservoirs of comparable size.

What carries the argument

Decoupling the reservoir dynamics into independent modes via the eigenvalues of the adjacency matrix, allowing each mode to be optimized separately before constructing the overall network.

If this is right

Optimized linear reservoirs achieve higher accuracy than random ones of the same size in both training and testing.
They can match or exceed the performance of nonlinear reservoirs of comparable size.
The approach supplies analytical guidelines for constructing task-specific reservoir architectures.
The performance gains hold across networks of different sizes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The modal decoupling might extend to other linear dynamical systems used for signal processing or control.
If the independence assumption holds under noise, the method could simplify hyperparameter search in practical reservoir applications.
Testing the optimized networks on tasks with changing input statistics would reveal whether the eigenvalue choices remain robust.

Load-bearing premise

The reservoir dynamics can be decoupled into independent modes such that optimizing each mode separately yields the best overall connectivity.

What would settle it

A direct comparison in which a randomly connected linear reservoir achieves higher test accuracy than the eigenvalue-optimized version on the same temporal prediction task.

Figures

Figures reproduced from arXiv: 2509.23391 by Francesco Sorrentino, Nicholas A. Mecholsky, Sahand Tangerami.

**Figure 2.** Figure 2: Performance of the 10-node reservoir. Red dashed lines represent the input signal, while blue lines correspond to the reservoir computer [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Performance of the 100-node reservoir. Red dashed lines represent the input signal, while blue lines correspond to the reservoir computer [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Average normalized root mean squared error (NRMSE) of reservoir computers with their corresponding standard deviations. The yellow [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: a) Effect of optimization terms’ weights on the optimization error. b) Sensitivity of the Reservoir Training Error to Optimal Eigenvalues. Appendix A. Proof of Theorem I A coupled and a decoupled linear reservoir are described by Eqs. (3) and (7), respectively. We now consider the two regression problems, one for coupled modes and the other for decoupled modes with fit signals, hr(t) = X j κjrj(t) + κN+1, … view at source ↗

read the original abstract

Machine learning has become a fundamental approach for modeling, prediction, and control, enabling systems to learn from data and perform complex tasks. Reservoir computing is a machine learning tool that leverages high-dimensional dynamical systems to efficiently process temporal data for prediction and observation tasks. Traditionally, the connectivity of the network that underlies a reservoir computer (RC) is generated randomly, lacking a principled design. Here, we focus on optimizing the connectivity of a linear RC to improve its performance and interpretability, which we achieve by decoupling the RC dynamics into a number of independent modes. We then proceed to optimize each one of these modes to perform a given task, which corresponds to selecting an optimal RC connectivity in terms of a given set of eigenvalues of the RC adjacency matrix. Simulations on networks of varying sizes show that the optimized RC significantly outperforms randomly constructed reservoirs in both training and testing phases and often surpasses nonlinear reservoirs of comparable size. This approach provides both practical performance advantages and theoretical guidelines for designing efficient, task-specific, and analytically transparent RC architectures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a mode-decoupling procedure to pick eigenvalues for linear reservoir computers and reports simulation gains over random wiring, but the independence assumption leaves open whether joint optimization over the full adjacency matrix would close the gap.

read the letter

The main takeaway is that they replace random connectivity in linear reservoir computers with a procedure that diagonalizes the adjacency matrix, decouples the dynamics into modes, and tunes each eigenvalue separately for the task. Simulations across network sizes show the resulting reservoirs beat random ones on both training and test error and sometimes beat nonlinear reservoirs of similar size. This is a clear, practical alternative to the usual random design rule in the linear case. The work is new in spelling out the per-mode eigenvalue selection as an explicit construction rather than leaving it to chance or broad spectral tuning. It earns credit for focusing on interpretability through the modal view and for running the comparison on varying sizes. The soft spot is the decoupling step itself. The transfer function of a linear RC is a sum of terms c_i b_i / (z - lambda_i), so the residues mix the fixed input vector b with the eigenvectors. Optimizing only the lambdas while holding the modal matrix or b fixed can leave coupling that a joint search over the full W would capture. The reported gains are against random W but do not test whether varying eigenvectors at fixed eigenvalues erases the advantage. The abstract also gives no details on the optimizer, the exact tasks, or any statistical tests, so the empirical claim rests on modest evidence. This is for reservoir-computing researchers who want a more deliberate linear design rule for temporal prediction. A reader working on dynamical-system ML would get usable ideas even if they later tighten the optimality argument. Send it to peer review; the core construction is worth checking with fuller methods and ablations.

Referee Report

2 major / 2 minor

Summary. The paper proposes optimizing the topology of linear reservoir computers by decoupling the dynamics x(t+1)=W x(t)+u(t) into independent modes via the eigenbasis of the adjacency matrix W and then selecting optimal eigenvalues for a given task. Simulations on networks of varying sizes are reported to show that the resulting optimized linear RC significantly outperforms randomly constructed reservoirs in both training and testing phases and often surpasses nonlinear reservoirs of comparable size, while also providing theoretical guidelines for task-specific RC design.

Significance. If the central empirical claims hold after addressing the decoupling assumptions, the work would be moderately significant for the reservoir computing and systems identification communities. It attempts to replace random connectivity with an eigenvalue-based design procedure grounded in linear systems theory, which could improve interpretability and performance for temporal tasks. The simulation evidence of outperformance over random and some nonlinear baselines is a positive contribution, though the absence of ablations on eigenvector variation limits the strength of the optimality claim.

major comments (2)

[Method section on decoupling into independent modes] The decoupling approach (described in the method section on independent modes) assumes that separately optimizing each eigenvalue of W yields a globally optimal connectivity. However, the effective transfer function of the linear RC is sum_i (c_i b_i / (z - lambda_i)), so the residues depend on both the chosen eigenvalues and the projections of the fixed input vector b and readout c onto the eigenvectors in V. The manuscript does not demonstrate that the reported performance gains survive an ablation in which eigenvectors are varied while holding eigenvalues fixed, leaving open the possibility that residual input/readout coupling undermines the independent-mode optimality claim.
[Results section on simulation comparisons] The central simulation results (reported in the results section on networks of varying sizes) claim statistically significant outperformance, but the manuscript provides no details on the optimization algorithm for eigenvalue selection, the precise task definitions and loss functions, the number of independent trials or statistical tests performed, or the procedure for choosing the comparison baselines (random W and nonlinear reservoirs). These omissions make the empirical support for the main claim difficult to evaluate or reproduce.

minor comments (2)

[Introduction and method] Notation for the input vector and readout should be introduced consistently with the state equation x(t+1)=W x(t)+u(t) to avoid ambiguity when discussing projections onto modes.
[Results figures] Figure captions for the performance plots should explicitly state the number of random realizations, error bars, and whether the plotted curves represent means or best cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback on manuscript arXiv:2509.23391. We address each major comment below and outline the revisions we will make to improve methodological transparency and strengthen the supporting evidence.

read point-by-point responses

Referee: [Method section on decoupling into independent modes] The decoupling approach (described in the method section on independent modes) assumes that separately optimizing each eigenvalue of W yields a globally optimal connectivity. However, the effective transfer function of the linear RC is sum_i (c_i b_i / (z - lambda_i)), so the residues depend on both the chosen eigenvalues and the projections of the fixed input vector b and readout c onto the eigenvectors in V. The manuscript does not demonstrate that the reported performance gains survive an ablation in which eigenvectors are varied while holding eigenvalues fixed, leaving open the possibility that residual input/readout coupling undermines the independent-mode optimality claim.

Authors: We appreciate the referee highlighting the role of eigenvector projections in the residues of the transfer function. Our decoupling procedure transforms the system into the eigenbasis of W, after which eigenvalues are optimized independently for the task; the input vector b and readout c are projected accordingly, and we select eigenvalues to maximize task performance under this fixed basis. To directly test robustness, the revised manuscript will include a new ablation in which eigenvectors are varied (via random orthogonal matrices or alternative bases) while holding the optimized eigenvalues fixed, with performance comparisons reported to show that gains persist and are driven primarily by eigenvalue choice rather than specific eigenvector alignments. revision: yes
Referee: [Results section on simulation comparisons] The central simulation results (reported in the results section on networks of varying sizes) claim statistically significant outperformance, but the manuscript provides no details on the optimization algorithm for eigenvalue selection, the precise task definitions and loss functions, the number of independent trials or statistical tests performed, or the procedure for choosing the comparison baselines (random W and nonlinear reservoirs). These omissions make the empirical support for the main claim difficult to evaluate or reproduce.

Authors: We agree that these details are required for reproducibility and proper evaluation. In the revised manuscript we will expand the Methods and Results sections to specify: the eigenvalue optimization procedure (a gradient-based constrained optimizer with explicit bounds on spectral radius), the exact task definitions and loss functions (including input signals, prediction horizons, and mean-squared error), the number of independent trials (50 runs with distinct random seeds), the statistical tests performed (paired t-tests with p-values and confidence intervals), and the construction of baselines (random W drawn from the same spectral-radius distribution; nonlinear reservoirs implemented as standard echo-state networks with tanh activation and identical reservoir size). Full code and hyperparameters will be referenced in a supplementary repository link. revision: yes

Circularity Check

0 steps flagged

No circularity; modal decoupling is an explicit optimization method with independent simulation validation

full rationale

The paper presents an optimization procedure for linear reservoir computers that begins with an explicit modeling choice to decouple the linear dynamics into independent modes via the eigenbasis of the adjacency matrix and then tunes eigenvalues for a task. This choice is not derived from or fitted to the target performance metric by construction; instead, the resulting networks are constructed and then evaluated in separate training and testing simulations against random reservoirs and nonlinear baselines. No load-bearing step reduces to a self-citation, a renamed empirical pattern, or a parameter fit that is relabeled as a prediction. The derivation chain therefore remains self-contained and externally falsifiable through the reported performance comparisons.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the ability to decouple linear dynamics into independent modes and on treating the eigenvalues as tunable parameters chosen for each task; no new physical entities are introduced.

free parameters (1)

eigenvalues of the adjacency matrix
Selected per mode to optimize task performance; these are the explicit design variables that replace random connectivity.

axioms (1)

domain assumption Linear reservoir dynamics can be decoupled into independent modes via the eigenvalues of the adjacency matrix
Invoked when the paper states that optimization of each mode corresponds to selecting optimal connectivity.

pith-pipeline@v0.9.0 · 5718 in / 1194 out tokens · 39670 ms · 2026-05-18T12:10:22.270419+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we employ modal decomposition to transform the reservoir dynamics into independent modes... A = V Λ V^T ... ˙q_i(t) = γ(λ_i − 1)q_i(t) + γ c_i u(t)
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

each mode acts as a low-pass filter... optimization problem to determine the optimal set of eigenvalues

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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work page doi:10.1016/bs.hna.2021.12.016 2021

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work page doi:10.1063/1.4979665

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doi: 10.1109/OJCSYS.2024.3451889. Claus Metzner, Achim Schilling, Andreas Maier, and Patrick Krauss. Nonlinear neural dynamics and classification accuracy in reservoir computing.Neural Computation, 37(8):1469–1504, 07

work page doi:10.1109/ojcsys.2024.3451889 2024

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doi: 10.1162/ neco_a_01770

ISSN 0899-7667. doi: 10.1162/ neco_a_01770. URLhttps://doi.org/10.1162/neco_a_01770. Jaideep Pathak, Brian Hunt, Michelle Girvan, Zhixin Lu, and Edward Ott. Model-free prediction of large spatiotem- porally chaotic systems from data: A reservoir computing approach.Phys. Rev. Lett., 120:024102, Jan

work page doi:10.1162/neco_a_01770

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Model-free prediction of large spatiotem- porally chaotic systems from data: A reservoir computing approach.Physical Review Letters, 120:024102, 2018

doi: 10.1103/PhysRevLett.120.024102. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.120.024102. Junyi Shen, Tetsuro Miyazaki, and Kenji Kawashima. Control pneumatic soft bending actuator with feedforward hysteresis compensation by pneumatic physical reservoir computing.IEEE Robotics and Automation Letters, 10 (2):1664–1671,

work page doi:10.1103/physrevlett.120.024102

[10] [10]

Shuhei Sugiura, Ryo Ariizumi, Toru Asai, and Shun-Ichi Azuma

doi: 10.1109/LRA.2024.3523229. Shuhei Sugiura, Ryo Ariizumi, Toru Asai, and Shun-Ichi Azuma. Nonessentiality of reservoir’s fading memory for universality of reservoir computing.IEEE Transactions on Neural Networks and Learning Systems, 35(11): 16801–16815,

work page doi:10.1109/lra.2024.3523229 2024

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Francis Wyffels, Benjamin Schrauwen, and Dirk Stroobandt.Stable Output Feedback in Reservoir Computing Using Ridge Regression

doi: 10.1109/TNNLS.2023.3298013. Francis Wyffels, Benjamin Schrauwen, and Dirk Stroobandt.Stable Output Feedback in Reservoir Computing Using Ridge Regression. Springer Berlin Heidelberg, Berlin, Heidelberg,

work page doi:10.1109/tnnls.2023.3298013 2023

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Min Yan, Can Huang, Peter Bienstman, Peter Tino, Wei Lin, and Jie Sun

ISBN 978-3-540-87536-9. Min Yan, Can Huang, Peter Bienstman, Peter Tino, Wei Lin, and Jie Sun. Emerging opportunities and challenges for the future of reservoir computing.Nature Communications, 15(1):2056,

work page 2056

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doi: 10.1038/ s41467-024-45187-1

ISSN 2041-1723. doi: 10.1038/ s41467-024-45187-1. URLhttps://doi.org/10.1038/s41467-024-45187-1. Xiaoning Zhao, Yougang Sun, Yanmin Li, Ning Jia, and Junqi Xu. Applications of machine learning in real-time con- trol systems: a review.Measurement Science and Technology, 36(1):012003, nov

work page doi:10.1038/s41467-024-45187-1 2041

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Artibani, et

doi: 10.1088/1361-6501/ ad8947. URLhttps://dx.doi.org/10.1088/1361-6501/ad8947. 13

work page doi:10.1088/1361-6501/