Tree Tensor Network Reservoir Computing: Hierarchical Ensemble with Invariant Phase Boundaries

Chih-Chieh Chen; Daiki Sasaki; Tomah Sogabe

REVIEW 2 major objections 4 minor 85 references

Tree Tensor Network Reservoir Computing: Hierarchical Ensemble with Invariant Phase Boundaries

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that a tree tensor network reservoir, subdivided into independent sub-reservoirs, has a single asymptotic stability boundary at $\sigma_T=\sqrt{2}$, and that this design matches or beats echo-state networks on…

desk verdict Solid theoretical core for tree-tensor reservoirs with a clean σ=√2 asymptotic boundary; the NARMA performance claims are currently undercut by outlier-removed means and seed reuse in tuning. read the letter →

arxiv 2607.24127 v1 pith:6DMQWJAX submitted 2026-07-27 quant-ph

Daiki Sasaki , Chih-Chieh Chen , Tomah Sogabe This is my paper · ORCID

classification quant-ph

keywords treetensornetworkreservoircomputingechostatetime-seriespredictionpropertymean-fieldtheoryphasetransitionhierarchicalensemble

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes Tree Tensor Network Reservoir Computing (TTN-RC), a quantum-inspired reservoir for time-series prediction whose internal reservoir is a random binary tree tensor network rather than a sparse random matrix. The central claim is that, in the limit where each sub-reservoir becomes large, three separate theoretical indicators—the concentration/divergence of the TTN output, the expected contraction rate $C$, and the mean-field reservoir-state variance—all single out the same critical value $\sigma_T=\sqrt{2}$ as the stability boundary. To make the topology usable, the paper introduces a hierarchical ensemble that keeps the total reservoir size fixed while partitioning it into $M$ independent trees, which prevents the exponential blow-up or collapse of outputs. On the tested NARMA benchmarks, TTN-RC is competitive with or better than echo-state networks, especially at higher task order and larger reservoir size, giving a practical design rule: initialize tensors with variance $\sigma_T^2/(d_{\alpha}d_{\beta})$ and tune $\sigma_T$ near $\sqrt{2}$ while choosing $M$ to control tree depth.

What carries the argument

The load-bearing object is the random Tree Tensor Network: a binary tree of three-leg tensors whose elements are drawn independently from $\mathcal{N}(0,\sigma_T^2/(d_{\alpha}d_{\beta}))$ and contracted from the leaves up to produce the activation potential. The hierarchical ensemble partitions a reservoir of total size $N_x$ into $M$ independent trees, each of size $\tilde N_x=N_x/M$, so the effective depth is $\log_2\tilde N_x$; this is what turns an exponentially concentrating or diverging single tree into a tunable reservoir. The argument is carried by two exact recursions, $V_g^{(l)}=\sigma_T^2(V_g^{(l-1)})^2$ for the output variance and $V_J^{(l)}=(\sigma_T^2/2)^{n_{l-1}}V_J^{(l-1)}$ for the Jacobian variance, whose solutions give the closed-form variance $V_g^{(\tilde L)}=(\sigma_T^2)^{\tilde N_x-1}/2^{\tilde N_x}$ and the expected contraction rate $C$. A mean-field treatment, treating the pre-activation potential as Gaussian with that variance, supplies the reservoir-state moments used to locate the performance-optimal region.

What would settle it

Take a single tree with $\tilde N_x=1024$, draw tensor elements from a zero-mean Gaussian with variance $\sigma_T^2/\chi^2$ as in Eq. (10), and scan $\sigma_T$ across $[0.5,3.5]$ while recording the TTN output variance, the $C=1$ contour, and the mean-field reservoir variance; the transition should sharpen around $\sigma_T=\sqrt{2}$ as $\tilde N_x$ grows. Repeating the same scan with unit-variance Gaussian draws should move the boundary to $\sigma_T=3$; observing a boundary at some other value, or no convergence of the three indicators in the large-$\tilde N_x$ limit, would falsify the central claim.

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Extended reading notes

Core claim

The paper’s central discovery is that a random tree tensor network used as a reservoir has a well-defined asymptotic stability boundary. For a tree of size $\tilde N_x$ and tensor-element standard deviation $\sigma_T$, the variance of the TTN output over the tensor randomness is $V_g^{(\tilde L)} = (\sigma_T^2)^{\tilde N_x-1}/2^{\tilde N_x}$; as $\tilde N_x \to \infty$, this variance vanishes for $\sigma_T < \sqrt{2}$ and diverges for $\sigma_T > \sqrt{2}$. The Jacobian-based expected contraction constant $C = \frac{\pi N_x}{8\sqrt{2}M}\left(\frac{\sigma_T}{\sqrt{2}}\right)^{(N_x-M)/M}$ gives the same boundary through $C=1$, and the mean-field variance of the reservoir state changes sharply at the same point when the ensemble number $M$ is small. The paper thus claims that concentration, expected contraction, and mean-field stability converge to a single critical value $\sigma_T=\sqrt{2}$ in the large-per-tree limit, and that this boundary is a design principle for tensor-network reservoir computing.

Load-bearing premise

The whole $\sqrt{2}$ boundary rests on initializing every tensor element independently from a zero-mean Gaussian with variance $\sigma_T^2/(d_{\alpha}d_{\beta})$; if that normalization is changed, for instance to unit-variance draws, the same analysis places the boundary at $\sigma_T=3$, so the claimed critical value is a property of this initialization scheme rather than of tree tensor networks in general.

Editorial extensions

If this is right

With the $\sqrt{2}$ boundary established, choosing $\sigma_T$ near $\sqrt{2}$ and adjusting $M$ to control tree depth replaces spectral-radius tuning as the main hyperparameter rule for TTN-RC.
The condition $C<1$ is a usable expected-contraction indicator: in the numerical ESP index, the region left of the $C=1$ contour exhibits $I_{\mathrm{ESP}}(100)\le 10^{-7}$, matching the predicted echo-state regime.
Larger $M$ widens the smooth crossover around the boundary, so shallow-tree ensembles are easier to tune, while deep single trees have a sharp boundary that is harder to locate.
On the tested NARMA tasks, TTN-RC matches or beats echo-state networks at equal total reservoir size and equal hyperparameter search budget, with the advantage growing for NARMA5, 7, and 10.
The optimal $\sigma_T$ for task performance falls inside the slope region of the mean-field variance $v_x$, so the mean-field statistics identify the performance peak, not just the stability boundary.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural testable extension is to check whether the crossover near $\sigma_T=\sqrt{2}$ obeys a scaling collapse in $\epsilon=(\tilde N_x-1)(2-\sigma_T^2)/2$ with a universal exponent; the paper's Appendix D already proposes $\beta=1$ for the mean-field order parameter.
The paper's comparison with unit-variance initialization suggests that the phase boundary is not intrinsic to tree topology; probing other tensor-element distributions (sparse, signed, or heavy-tailed) would reveal a family of boundaries and show which features of the reservoir are universal.
Since performance peaks near but not exactly at the stability boundary, TTN-RC can serve as a clean testbed for edge-of-chaos ideas in reservoir computing, with the mean-field variance providing a parameter-free proxy for the useful nonlinear regime.
The ensemble perspective implies a practical recipe for quantum-inspired reservoirs: prefer shallow trees with moderate bond dimension and $\sigma_T$ near $\sqrt{2}$; deep trees are only usable when $M$ is large enough to keep $\tilde N_x$ small.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The theoretical part of this paper is genuinely good. The closed-form variance and Jacobian statistics for tree tensor reservoirs (Appendix F) are clean, the derivations are transparent, and the asymptotic σ_T=√2 boundary follows from their normalization without fitted parameters. The hierarchical ensemble is a natural and sensible fix for the exponential concentration problem. The paper also does the honest thing by noting that Ref. [56]'s entanglement transition translates to σ=3 under a unit-variance draw, so the boundary is a design principle for this initialization scheme, not a universal property of all TTNs. The title's "invariant" should be read as "bond-dimension invariant under the chosen normalization," and the paper mostly says that.

The soft spots are in the empirical section. Figure 3 averages after removing outliers (1.5×IQR) from 10 reservoir realizations and reports no error bars. That can bias the mean in favor of whichever model happens to have fewer outliers, and with n=10 the apparent TTN-RC advantage at Nx=1024 is hard to distinguish from seed noise. More seriously, Appendix G shows that the validation performance used for Optuna selection is averaged over the same 10 seeds that are later used for the reported test NMSE. That makes the test numbers dependent on the tuning procedure; the paper needs either fresh seeds for testing or a nested protocol. The absence of an MPS-RC baseline is also a gap, given the program of extending it, though the paper acknowledges this.

None of this undermines the theoretical core. The derivations check out, the asymptotic analysis is sound, and the qualitative claim that performance peaks near the slope of the mean-field variance is supported by the heatmaps even if the quantitative comparison to ESN is not yet established. This is a paper that a serious referee can fix with a revision: add error bars or drop outlier removal, separate tuning and test seeds, and add the MPS-RC comparison. The theory alone justifies a referee's time.

I would bring it to a reading group with a caveat about the empirics, and I would cite the theoretical results in my own work. Recommended for peer review: yes, with expectations of a substantive empirical revision.

Referee Report

2 major / 4 minor

Summary. The manuscript proposes Tree Tensor Network Reservoir Computing (TTN-RC), a reservoir computing model whose internal state is produced by a random tree tensor network, equipped with a hierarchical ensemble that partitions the reservoir into M independent sub-trees. The authors derive closed-form expressions for the variance of the TTN output and of its Jacobian, yielding an expected contraction constant C, and a mean-field theory for the reservoir-state variance. They show that in the large per-tree limit Ñ_x→∞, the divergence/concentration transition, the condition C=1, and the mean-field transition all converge to σ_T=√2. They benchmark TTN-RC against Echo State Networks on NARMA tasks of orders 1, 3, 5, 7, and 10 and report competitive or improved NMSE, especially at larger reservoir sizes and higher task orders.

Significance. The theoretical analysis is a strength: the variance and Jacobian computations in Appendix F are explicit, self-contained, and checkable, with no fitted parameters, and the convergence of three independent indicators to σ_T=√2 is a clean, falsifiable prediction about the reservoir statistics. The hierarchical ensemble idea is a simple and practical remedy to the concentration/divergence problem. If the empirical performance claims are confirmed, the work would provide a genuinely useful design principle for tensor-network reservoirs. However, the empirical evidence as presented does not currently meet the bar for the stated claim of 'competitive or improved performance'; the statistical analysis needs strengthening.

major comments (2)

[Section 3.1, Fig. 3, Appendix G] The central empirical claim that TTN-RC is competitive or improved over ESN is not statistically substantiated. Fig. 3's caption states that outliers outside 1.5×IQR were removed before averaging, but the number of removed points per condition is not reported, and no standard deviations, confidence intervals, or per-seed values are shown; with only 10 realizations, the visible TTN-RC advantage at Nx=1024 could be seed noise. In addition, Appendix G describes a protocol in which the same Nseed=10 reservoir seeds are used both to select hyperparameters (validation RMSE) and to compute the reported test NMSE; the test numbers are therefore not independent of the selection process. The 'competitive or improved performance' bullet in Section 1 is a main contribution and needs to be supported by (i) error bars or per-seed distributions, (ii) the outlier counts, and (iii) a validation/test seed split or an equivalent nested procedure.
[Section 5 and Section 4.1] The paper explicitly acknowledges in the Conclusion that 'more performance evaluations against the MPS-RC on a variety of tasks would be desirable.' Given that the Introduction frames TTN-RC as an extension of the pioneering MPS-RC, the absence of a direct MPS-RC baseline leaves the claimed benefit of the tree topology (as opposed to the one-dimensional MPS structure) untested. Without this comparison, the paper can only claim TTN-RC is competitive with a classical ESN for the tested tasks, which is weaker than the tensor-network-specific advance suggested by the framing.

minor comments (4)

[Section 2.3 and Eq. (26)] The symbol V_g is used for two different quantities: in Eq. (26) it denotes the variance of the TTN output, while in Section 2.3 it denotes the variance of the Jacobian entries. Using a distinct symbol such as V_J for the Jacobian variance would prevent confusion, especially since the two quantities follow different recurrences (Eqs. (F.8) and (F.23)).
[Fig. 3 and Fig. 4] All reported empirical heatmaps and line plots lack error bars or dispersion measures. Even if the central performance claim is revised to be weaker, reporting per-seed medians with interquartile ranges would substantially improve the transparency of the benchmark results.
[Section 4.3 and title] The term 'invariant phase boundaries' in the title could be misread as independence of the initialization scheme. The paper itself notes in Section 4.3 that the unit-variance initialization of Ref. [56] yields a different critical value (σ_T=3). A sentence clarifying that the invariants are the convergence of the three theoretical indicators under the normalization of Eq. (10) would prevent misinterpretation.
[Eq. (27)] The expression for C uses Nx and M, while the surrounding text defines Ñ_x=Nx/M. Writing C explicitly in terms of Ñ_x first and then substituting would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the theoretical indicators are closed-form consequences of the explicitly stated random-tensor initialization, and the benchmark comparison is an observation, not a derivation from fitted inputs.

full rationale

The paper's theoretical chain is self-contained. The tensor distribution is stated in Eq. (10) as zero-mean Gaussian with variance sigma_T^2/(d_alpha d_beta); Appendix F derives the TTN output variance V_g = (sigma_T^2)^{Ntilde_x-1}/2^{Ntilde_x} (Eq. 30) and the Jacobian variance V_J = (pi^2/8)(sigma_T^2/2)^{Ntilde_x-1} (Eq. F.24), from which the expected contraction rate C in Eq. (27) and the C=1 boundary in Eq. (28) follow algebraically. The mean-field quantities q_x, mu_x, v_x in Eqs. (22)-(24) are Gaussian integrals over this same V_g, with no fitted parameters. The claimed asymptotic boundary sigma_T = sqrt(2) is the explicit threshold of (sigma_T^2/2)^{Ntilde_x-1} in the Ntilde_x -> infinity limit, and the C=1 and mean-field thresholds tend to the same value by direct formulas rather than by construction from measured data. The empirical NARMA comparison is a benchmark result, not a quantity derived from the theory, and the paper explicitly labels the GGO-based Delta_g as an interpretive indicator rather than a quantitative predictor (Sec. 4.2 and Appendix B), while also noting that the effect of M on performance is not fully explained by the theory. Citations to the authors' prior MPS-RC work [39] provide a method and a lemma for this interpretive proxy, but the main theoretical indicators are rederived in the appendices and do not rely on any unverified self-citation; the prior lemma is a parameter-free published result. No fitted parameter is renamed as a prediction, and no claimed derivation reduces to its own inputs by construction. Any concerns about outlier removal or seed reuse in the empirical section are statistical robustness issues, not circularity of the theoretical derivation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central theoretical claims rest on the model's defining distributions and standard Gaussian calculus; no free parameters are fitted in the derivations. The listed domain assumptions are all stated in the paper. The most consequential is the tensor initialization variance scaling, which sets the value of the critical boundary.

assumptions (6)

domain assumption Random tensor elements are i.i.d. zero-mean Gaussian with variance σ_T^2/(dα dβ).
Model definition in Eq. (10) and (15); the entire variance and Jacobian analysis, and the σ_T = √2 boundary, are specific to this normalization. With unit-variance Gaussians (Ref [56]) the critical value is different.
domain assumption Input encoding uses ϕ0(x)=cos(πx/2) and ϕ1(x)=sin(πx/2).
Eq. (8); the Jacobian variance V_J depends on derivatives of these functions, so the contraction rate C changes for other encodings.
domain assumption Sigmoid activation with Lipschitz constant K = 1/4.
Used in Appendix A to define C = Ñ_x K Λ; a different activation changes the contraction condition.
domain assumption Mean-field Gaussian approximation for the pre-activation, with stationary-state statistics and the input-driven term ignored.
Stated in Section 2.3 and Appendix E; used to compute q_x, v_x, and ρ, and to locate the mean-field transition.
domain assumption The asymptotic boundary is taken in the thermodynamic limit Ñ_x → ∞.
The σ_T = √2 convergence of the indicators holds only in this limit; at finite Ñ_x the boundaries shift, as the paper's own contours show.
standard math Underlying Gaussian integral identities and independence of uncorrelated jointly Gaussian variables.
Used in Appendices E and F for the mean-field and Jacobian calculations.

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Cite this review

Pith. "Pith review of Tree Tensor Network Reservoir Computing: Hierarchical Ensemble with Invariant Phase Boundaries." pith.science (2026). https://pith.science/paper/6DMQWJAX

@misc{pith2026260724127,
  author       = {Pith},
  title        = {Pith review of: Tree Tensor Network Reservoir Computing: Hierarchical Ensemble with Invariant Phase Boundaries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6DMQWJAX}},
  note         = {Machine review of arXiv:2607.24127}
}

abstract

We propose Tree Tensor Network Reservoir Computing (TTN-RC), a quantum-inspired reservoir computing framework for time-series prediction that uses the hierarchical structure of Tree Tensor Networks as a random reservoir. To control the exponential concentration or divergence of TTN outputs, we introduce a hierarchical ensemble method that partitions a fixed-size reservoir into multiple independent sub-reservoirs. In the tested NARMA benchmarks, TTN-RC achieves competitive or improved performance compared with conventional Echo State Networks, especially for tasks requiring higher-order nonlinear processing and longer contextual dependence. We also derive an expected contraction rate based on the reservoir Jacobian and develop a mean-field description of the reservoir-state statistics. These analyses identify an asymptotic stability boundary at $\sigma_{T}=\sqrt{2}$ in the large per-tree-size limit, where several theoretical indicators converge. Our results provide a design principle for tensor-network-based reservoir computing and clarify how hierarchical reservoir topology controls stability and nonlinear information processing.

Figures

Figures reproduced from arXiv: 2607.24127 by the authors.

**Figure 1.** Conceptual diagram of TTN-RC (L = 3): (a) shows the recursive structure in which the reservoir state is obtained by applying an activation function to the sum of (i) the contraction result of the random TTN function and (ii) the weighted external input, and then feeding that state back as input. (b) shows only the random TTN function component. Here, the encoded reservoir state at the current time is propagated by s… view at source ↗

**Figure 2.** The recursive structure of the ensemble TTN-RC. Each tree is contracted and recursed independently, but their outputs are concatenated to form the overall reservoir state vector. The NMSE is the mean squared error normalized by the variance of the true series. This normalization makes the metric independent of the data’s scale. It is calculated as: NMSE = PT t=1(d(t) − y(t))2 PT t=1(d(t) − ¯d) 2 (18) A lower NMSE va… view at source ↗

**Figure 3.** Comparison of the performance between TTN-RC and ESN on the NARMA1, 3, 5, 7, and 10 tasks. The horizontal axis represents the order of the NARMA task, n = 1, 3, 5, 7, 10, and the vertical axis represents the NMSE. In the left panel, the total reservoir size is fixed at Nx = 128, while in the right panel, it is fixed at Nx = 1024. For TTN-RC, the results are shown for values of M corresponding to N˜ x = 4, 8, and 16.… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Performance evaluation of the TTN-RC. (a): Heatmap of the total memory capacity, C tot STM, for the STM task. (b, c): Heatmaps of the R2 score for the NARMA5 and NARMA10 tasks, respectively. (d): Heatmap of the ESP metric, IESP(100). The light-green line indicates the …

**Figure 5.** Figure 5: Heatmaps of theoretical metrics. The horizontal axis represents σT , and the vertical axis represents the number of trees M. Nx = 256 for all calculations. (a): Heatmap of C, obtained by calculating the reservoir’s Jacobian. Values of C ≥ 10 are saturated to the same c…

**Figure 4.** Figure 4: Figure C1 shows the heatmaps of the RMSE for the training data, test data, and the generalization error. The generalization RMSE is defined as Generalization RMSE = Test RMSE − Train RMSE (C.2) Although the trends of the Train RMSE and Test RMSE appear nearly identical…

Discussion (0). Continue with ORCID to comment.

Reference graph

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Kubota T, Takahashi H and Nakajima K 2021Physical Review Research3043135

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echo state

Jaeger H 2001 The “echo state” approach to analysing and training recurrent neural networks – with an erratum note Tech. rep. GMD Report 148, German National Research Center for Information Technology

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[2]

The contour line for C = 1 also deviates from the critical value σT = √ 2 at large M

For large M, the growth is smoother, and a crossover regime emerges. The contour line for C = 1 also deviates from the critical value σT = √ 2 at large M. The crossover regime becomes wider and shifts toward σT > √ 2 for large M. The consistency with the experimental results will be demonstrated later. Fig. 5(b) provides a heatmap of the MFT reservoir-sta...
[3]

0.00.10.20.30.40.50.60.7 '2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5f) 64 32 16 8 4 2 1

Results 3.1. Comparison with ESN We compared the performance of TTN-RC and ESN on the NARMA1, 3, 5, 7, and 10 tasks using the NMSE. The lengths of the training data, validation data (for hyperparameter optimization), and test data were set to 3000, 500, and 500, respectively. A washout period of 500 steps was applied before the training data. The hyperpar...
[4]

Better performance over ESNs In the tested NARMA settings, TTN-RC showed advantages over the ESN baseline in several higher-order tasks

Discussion 4.1. Better performance over ESNs In the tested NARMA settings, TTN-RC showed advantages over the ESN baseline in several higher-order tasks. This result suggests a potential benefit of adopting a hierarchical TN reservoir topology for predicting time series with long-range correlations. Although this study focused on a comparison between TTN-R...
[5]

In MPS-RC, the only phase transition point in the thermodynamic limit was atσ T = 0. By analogy with statistical physics, this behavior can be interpreted as a finite- temperature-like transition in the TTN reservoir statistics, in contrast to the MPS-RC case where the corresponding thermodynamic-limit transition point is at σT = 0. This analogy should be...
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It would be interesting to study their potential links

This suggests a similar scaling behavior, but the exact locations of critical lines are different. It would be interesting to study their potential links
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0.00.51.01.52.02.53.03.5f) 25612864321684 #G

Conclusion In this work, we introduced Tree Tensor Network Reservoir Computing (TTN-RC). In the tested NARMA tasks, TTN-RC demonstrated competitive or improved performance compared with the standard ESN baseline, especially in higher-order settings. We established theoretical tools for model selection, including an expected contraction indicatorCfor stabi...

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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echo state

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The contour line for C = 1 also deviates from the critical value σT = √ 2 at large M

For large M, the growth is smoother, and a crossover regime emerges. The contour line for C = 1 also deviates from the critical value σT = √ 2 at large M. The crossover regime becomes wider and shifts toward σT > √ 2 for large M. The consistency with the experimental results will be demonstrated later. Fig. 5(b) provides a heatmap of the MFT reservoir-sta...

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0.00.10.20.30.40.50.60.7 '2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5f) 64 32 16 8 4 2 1

Results 3.1. Comparison with ESN We compared the performance of TTN-RC and ESN on the NARMA1, 3, 5, 7, and 10 tasks using the NMSE. The lengths of the training data, validation data (for hyperparameter optimization), and test data were set to 3000, 500, and 500, respectively. A washout period of 500 steps was applied before the training data. The hyperpar...

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Better performance over ESNs In the tested NARMA settings, TTN-RC showed advantages over the ESN baseline in several higher-order tasks

Discussion 4.1. Better performance over ESNs In the tested NARMA settings, TTN-RC showed advantages over the ESN baseline in several higher-order tasks. This result suggests a potential benefit of adopting a hierarchical TN reservoir topology for predicting time series with long-range correlations. Although this study focused on a comparison between TTN-R...

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In MPS-RC, the only phase transition point in the thermodynamic limit was atσ T = 0. By analogy with statistical physics, this behavior can be interpreted as a finite- temperature-like transition in the TTN reservoir statistics, in contrast to the MPS-RC case where the corresponding thermodynamic-limit transition point is at σT = 0. This analogy should be...

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It would be interesting to study their potential links

This suggests a similar scaling behavior, but the exact locations of critical lines are different. It would be interesting to study their potential links

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0.00.51.01.52.02.53.03.5f) 25612864321684 #G

Conclusion In this work, we introduced Tree Tensor Network Reservoir Computing (TTN-RC). In the tested NARMA tasks, TTN-RC demonstrated competitive or improved performance compared with the standard ESN baseline, especially in higher-order settings. We established theoretical tools for model selection, including an expected contraction indicatorCfor stabi...

work page arXiv