Pith. sign in

REVIEW 2 major objections 4 minor 63 references

A tensor network approach for chaotic time series prediction

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A tensor-network representation of the truncated Volterra series predicts chaotic time series more accurately and trains at least ten times faster than a conventional echo state network on a 70-task benchmark.

desk verdict Solid, reproducible application of a known tensor-network Volterra model to chaotic forecasting; the result is plausible, with one moderate numerical caveat. read the letter →

arxiv 2505.17740 v2 pith:R4HPI6B4 submitted 2025-05-23 cs.LG cs.NEphysics.comp-ph

Rodrigo Martínez-Peña , Román Orús This is my paper · ORCID
classification cs.LGcs.NEphysics.comp-ph
keywords tensornetworkschaotictimeseriespredictionVolterraechostatereservoircomputingmatrixproductoperatorKhatri-Raonext-generation
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

The paper tries to establish that a tensor-network compression of the truncated Volterra series is a practical forecasting model for low-dimensional chaotic systems, one that can replace a tuned echo state network. Using the symmetry of the Volterra kernel, the design matrix is represented exactly as a matrix product operator built from repeated row-wise Khatri-Rao products, avoiding the exponential parameter growth of the full series. On 70 standardized chaotic systems with polynomial nonlinearities, the tensor-network model is reported to beat the leaky echo state network on most tasks for both short-term metrics (sMAPE and valid prediction time) and the long-term spectral Wasserstein distance, while training at least an order of magnitude faster. If correct, this gives chaos forecasting a low-hyperparameter alternative whose nonlinearity is an explicit, inspectable tensor rather than a random reservoir.

What carries the argument

The central object is the matrix product operator (MPO) decomposition of the Volterra design matrix $U$ and the coefficient matrix $H$, constructed from the repeated row-wise Khatri-Rao product $U = \tilde{U}^{\odot D}$, where $\tilde{U}$ is the $N \times (PM+1)$ matrix of delayed extended inputs. This exact representation avoids materializing the exponentially large $U$. Kernel symmetry bounds the rank of $U$ by $R = \binom{PM+D}{PM}$, and the least-squares solution is obtained through an SVD-based pseudoinverse computed in MPO form, which is what makes training feasible with only two hyperparameters, the maximum delay $M$ and maximum monomial degree $D$.

What would settle it

Take one chaotic task where the TN model underperforms the ESN, compute the symmetry error $S$ defined in the paper's appendix for the fitted kernel, then re-symmetrize the kernel and retest; if the gap closes, the central claim depends on SVD thresholding for correlated inputs rather than on the Volterra representation itself.

Watch

Extended reading notes

Core claim

The paper's central claim is that the tensor-network Volterra model outperforms a conventional echo state network for chaotic time-series prediction, in accuracy and in training cost. The argument is that the truncated Volterra series, which next-generation reservoir computing uses as its state space, has a symmetric kernel that can be stored and solved as a matrix product operator: each row of the design matrix is the vectorization of a symmetric tensor, so the full matrix is a repeated row-wise Khatri-Rao product and never needs to be materialized. The minimum-norm least-squares solution is symmetric in exact arithmetic regardless of the rank of the design matrix, which the paper proves in an appendix; numerically, however, highly correlated inputs can make the pseudoinverse break that symmetry, and the paper reports that some chaotic tasks show exactly this rank-deficiency behavior. On the 70-task benchmark the model still wins most short-term and long-term forecasting comparisons against the tuned ESN, with training times at least one order of magnitude smaller.

Load-bearing premise

The load-bearing premise is numerical: the pseudoinverse solution must keep the Volterra kernel symmetric in floating-point arithmetic for the correlated inputs that real chaotic systems produce, because the MPO compression is justified by that symmetry.

Editorial extensions

If this is right

  • On the paper's 70-task benchmark, the tensor-network model improves over the tuned ESN on almost all tasks for short-term prediction (sMAPE and valid prediction time) and on most tasks for the long-term climate metric.
  • Training with the tensor-network model is at least an order of magnitude faster: the full TN optimization runs in about an hour on two threads, while the ESN grid search takes over 60 hours.
  • Hyperparameter optimization reduces to two integers, $M$ and $D$, eliminating the reservoir-size, spectral-radius, input-strength, leak-rate, and regularization knobs of the ESN.
  • Because echo state networks can be approximated by truncated Volterra series, the paper argues any ESN performance should be matchable by a TN model with much simpler tuning.

Reading between the lines

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

  • The appendix's numerical symmetry loss suggests a cheap, testable fix: re-symmetrize the fitted kernel after the pseudoinverse and check whether accuracy on rank-deficient chaotic tasks improves; the paper does not do this.
  • The same MPO construction should carry over to spatiotemporal systems if the SVD step is swapped for a randomized or low-rank SVD, an extension the paper lists as future work rather than a result.
  • If the benchmark claims hold, tensor-network Volterra models become a credible default baseline for low-dimensional chaos forecasting, with the additional benefit that the learned MPO cores are inspectable Volterra kernels rather than an opaque reservoir.
  • A direct comparison with sparse system identification and polynomial kernel methods on the same 70 systems would reveal whether the advantage comes from the Volterra representation itself or from the MPO compression; the paper only lists this as future work.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proposes a tensor-network approach, specifically a matrix-product-operator (MPO) representation of a truncated Volterra series, for predicting chaotic time series. The model is benchmarked against a leaky echo state network (ESN) on 70 low-dimensional chaotic systems from the dysts database. The authors report that the tensor-network model achieves better performance on most tasks according to short-term metrics (sMAPE and valid prediction time) and a climate metric (Wasserstein distance between spectra), while requiring at least one order of magnitude less training time. The model depends on only two hyperparameters (maximum delay M and maximum monomial degree D). An appendix establishes theoretically that the minimal-norm least-squares solution is symmetric regardless of the rank of the design matrix, but also presents numerical examples showing that correlated inputs can break this symmetry in practice.

Significance. If the central claim is robust, the paper offers a practical, low-hyperparameter alternative to echo state networks for low-dimensional chaotic forecasting, with a clear speed advantage. The availability of source code and the use of a standardized benchmark (dysts) are strengths. However, the numerical symmetry issue documented in the appendix directly affects the model actually evaluated, and the ESN baseline is optimized under a restricted search budget; both issues must be resolved before the accuracy claim can be taken at face value.

major comments (2)
  1. [Appendix, Table 1 and closing paragraph] The appendix shows that for correlated inputs, the SVD-based pseudoinverse breaks the symmetry of the Volterra kernel, with the symmetry metric S (Eq. 31) reaching 0.13 even when rank(U)=R (Table 1, N=100 row). The manuscript then states that 'some of the chaotic tasks fulfill rank(U) < min(N, R)' and that larger rank discrepancies correlate with higher S, but it never reports S for the 70 benchmark tasks or demonstrates that the symmetry loss does not affect the prediction metrics. Since the MPO construction in 'MPOs enforcing symmetry' is justified by the exact symmetry of H, the model actually evaluated on those tasks may not be the symmetric Volterra kernel described in the theory. The central empirical claim could therefore be sensitive to SVD truncation thresholds rather than to the tensor-network structure. Please quantify S or an equivalent symmetry-loss measure for all 70 tasks, and either show that the reported results are insensitive to the truncation threshold or provide a mitigation (e.g., projecting H onto the symmetric subspace).
  2. [Validation section] The ESN baseline is optimized with a fixed reservoir size Nr=500 and a coarse grid over rho, g, epsilon, and lambda, using a single validation trajectory. The paper acknowledges the limitation and argues that a more exhaustive search would not change the conclusion, but no supporting evidence is provided. Since the abstract claims the TN model is 'better' than a 'conventional' ESN, the restricted ESN tuning makes the accuracy comparison potentially unfair. The training-time advantage (at least one order of magnitude) is independent of this issue, but the accuracy claim needs either a more extensive ESN hyperparameter search, a sensitivity analysis showing robustness to ESN tuning, or a rephrasing that restricts the claim to the explored ESN configuration.
minor comments (4)
  1. [Discussion] The introduction motivates the TN model as a remedy for NGRC's exponential parameter growth, but the benchmark only compares with ESN; a comparison with NGRC or a standard truncated Volterra model would clarify the practical benefit and is currently missing.
  2. [Validation section] The sentence 'we expect that since ESN models can be approximated by truncated Volterra series, we could always find a TN model that performs as well as the ESN' is presented as a justification, but it is only an expectation. Consider deriving it or softening the claim to avoid giving the impression of an established result.
  3. [Appendix, Eq. (31)] The claimed correlation between the rank discrepancy and the symmetry metric S is not quantified; reporting a correlation coefficient or a scatter plot would strengthen the appendix's argument.
  4. [Acknowledgements] There is a typo in the acknowledgements: 'manuscrip' should be 'manuscript'.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the TN model is trained on a dedicated set and tested on held-out trajectories, the symmetry guarantee is an external mathematical result (Batselier et al.) rather than a self-citation, and the Appendix's symmetry-loss caveat is a numerical limitation, not a step that reduces the prediction to its inputs.

full rationale

The derivation chain is: define the Volterra feature matrix U (Eq. 13), solve the least-squares problem H = U^+Y (Eq. 18), and implement H via MPOs from refs. 34/35. The claim that columns of H are symmetric follows from the row-space containment R(U) subset of S_D (Appendix Eqs. 28-29) and was proved in ref. 33 under a persistently-exciting assumption; the paper extends it in the Appendix without rank assumptions. This is independent linear algebra, not a definition of the target. The numerical results compare TN against ESN on held-out test trajectories using sMAPE, VPT, and Wasserstein metrics, so the 'predictions' are not fitted to the test data. The only self-citation is Orus's own tensor-network review (ref. 39), which is background and not load-bearing. The Appendix's Table 1 shows that for correlated inputs the numerical pseudoinverse can break symmetry (S up to 0.13), and the paper admits some chaotic tasks have rank(U) < min(N,R); this is an honest numerical limitation affecting the exact equivalence between the implemented model and the symmetric Volterra kernel, but it does not make any prediction equivalent to its inputs by construction. The statement that ESNs can be approximated by truncated Volterra series is an expectation motivating the comparison, not an input to the TN training. Hence score 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim is an empirical benchmark, so the main load-bearing inputs are the truncated-Volterra and MPO machinery from prior work, the symmetry-preservation property of the pseudoinverse (proved in exact arithmetic but numerically fragile), and the specific ESN baseline. No new particles, forces, or entities are postulated.

free parameters (3)
  • Maximum delay M = grid {1,2,3,4}, per-task selection not reported
    Chosen via grid search on a single validation trajectory; controls the memory length of the Volterra model.
  • Maximum monomial degree D = grid {2,3,4}, per-task selection not reported
    Chosen via grid search; controls the polynomial degree and hence the rank bound R.
  • ESN hyperparameters (rho, g, epsilon, lambda, Nr=500) = grid ranges: rho in [0.8,1.2], g in [0.1,1], epsilon in [0.1,1], lambda in [1e-13,1e-1]; per-task optima not reported
    Benchmark baseline depends on these choices; the claim of TN superiority is relative to this specific ESN configuration.
assumptions (5)
  • standard math Truncated Volterra series have universal approximation properties for fading-memory systems (Boyd and Chua; Grigoryeva and Ortega).
    Invoked in the Introduction to justify using truncated Volterra series as the state-space model for chaotic time series.
  • standard math The MPO construction of reference 35 exactly represents the Khatri-Rao product matrix U without truncation.
    The paper relies on Batselier's exact representation for the training matrix; the technical details are not re-derived here and the reader is referred to ref 35.
  • standard math The minimal-norm least-squares solution produces symmetric Volterra kernels even for rank-deficient U.
    Proved in the Appendix for exact arithmetic; the paper depends on this to justify enforcing symmetry in the MPO solution.
  • domain assumption ESN models can be approximated by truncated Volterra series, so a TN model can always be found that performs as well as the ESN.
    Stated in the Validation section as an expectation; not proved and not used in the experiments, but it underpins the claimed significance of the comparison.
  • domain assumption Numerical SVD truncation preserves the symmetry of the Volterra kernel for chaotic correlated inputs.
    The Appendix shows this assumption is violated for correlated inputs (S up to 0.13), yet the algorithm is still applied to chaotic tasks; the paper flags a correlation between rank discrepancies and symmetry loss.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A tensor network approach for chaotic time series prediction." pith.science (2026). https://pith.science/paper/R4HPI6B4

@misc{pith2026250517740,
  author       = {Pith},
  title        = {Pith review of: A tensor network approach for chaotic time series prediction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R4HPI6B4}},
  note         = {Machine review of arXiv:2505.17740}
}
read the original abstract

Making accurate predictions of chaotic time series is a complex challenge. Reservoir computing, a neuromorphic-inspired approach, has emerged as a powerful tool for this task. It exploits the memory and nonlinearity of dynamical systems without requiring extensive parameter tuning. However, selecting and optimizing reservoir architectures remains an open problem. Next-generation reservoir computing simplifies this problem by employing nonlinear vector autoregression based on truncated Volterra series, thereby reducing hyperparameter complexity. Nevertheless, the latter suffers from exponential parameter growth in terms of the maximum monomial degree. Tensor networks offer a promising solution to this issue by decomposing multidimensional arrays into low-dimensional structures, thus mitigating the curse of dimensionality. This paper explores the application of a previously proposed tensor network model for predicting chaotic time series, demonstrating its advantages in terms of accuracy and computational efficiency compared to conventional echo state networks. Using a state-of-the-art tensor network approach enables us to bridge the gap between the tensor network and reservoir computing communities, fostering advances in both fields.

Figures

Figures reproduced from arXiv: 2505.17740 by the authors.

Figure 1
Figure 1. Comparison of the VPT and sMAPE and Wasserstein metrics for the TN and ESN models. The blue and yellow columns correspond to the TN and ESN models. While the ESN models have a very narrow distribution, the TN models occupy a wider range of times due to the scaling of the TN algorithm with D and M. However, we find that the TN models can outperform the ESN models with at least one order of magnitude less training tim… view at source ↗
Figure 2
Figure 2. Histogram of CPU training times in seconds for the optimized hyperparameters. Discussion Chaotic time series prediction is one of the most popular applications of RC architectures, but the selection and optimization of RC models is an open problem. In this work, we explore a TN model introduced by Batselier34, 35 that provides a low-rank approximation of the truncated Volterra series. Our goal is to provide a proof-… view at source ↗
Figure 3
Figure 3. Singular values of U for N = 50,100 for P = 1, M = 4 and D = 4 when given random uncorrelated inputs (blue) and correlated data (red) from the example of [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

63 extracted references · 61 canonical work pages

  1. [1]

    & Palmer, T

    Slingo, J. & Palmer, T. Uncertainty in weather and climate prediction. Philos. Transactions Royal Soc. A: Math. Phys. Eng. Sci. 369, 4751–4767 (2011)

    work page 2011

  2. [2]

    & McLean, A

    May, R. & McLean, A. R. Theoretical ecology: principles and applications (Oxford University Press, 2007)

    work page 2007

  3. [3]

    Argyris, A. et al. Chaos-based communications at high bit rates using commercial fibre-optic links. Nature 438, 343–346 (2005)

    work page 2005

  4. [4]

    Reservoir kernels and Volterra series

    Gonon, L., Grigoryeva, L. & Ortega, J.-P. Reservoir kernels and volterra series. arXiv preprint arXiv:2212.14641 (2022)

    work page Pith review arXiv 2022

  5. [5]

    Grigoryeva, L., Ting, H. L. J. & Ortega, J.-P. Infinite-dimensional next-generation reservoir computing. arXiv preprint arXiv:2412.09800 (2024)

    work page Pith review arXiv 2024

  6. [6]

    R., Byeon, W., Wan, Z

    Vlachas, P. R., Byeon, W., Wan, Z. Y ., Sapsis, T. P. & Koumoutsakos, P. Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks. Proc. Royal Soc. A: Math. Phys. Eng. Sci. 474, 20170844 (2018)

    work page 2018

  7. [7]

    Model scale versus domain knowledge in statistical forecasting of chaotic systems

    Gilpin, W. Model scale versus domain knowledge in statistical forecasting of chaotic systems. Phys. Rev. Res. 5, 043252 (2023)

    work page 2023

  8. [8]

    & Ott, E

    Pathak, J., Hunt, B., Girvan, M., Lu, Z. & Ott, E. Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach. Phys. review letters 120, 024102 (2018)

    work page 2018

Show all 63 references
  1. [9]

    & Gauthier, D

    Griffith, A., Pomerance, A. & Gauthier, D. J. Forecasting chaotic systems with very low connectivity reservoir computers. Chaos: An Interdiscip. J. Nonlinear Sci. 29 (2019)

    2019

  2. [10]

    Vlachas, P.-R. et al. Backpropagation algorithms and reservoir computing in recurrent neural networks for the forecasting of complex spatiotemporal dynamics. Neural Networks 126, 191–217 (2020)

    2020

  3. [11]

    & Ortega, J.-P

    Grigoryeva, L., Hart, A. & Ortega, J.-P. Chaos on compact manifolds: Differentiable synchronizations beyond the takens theorem. Phys. Rev. E 103, 062204 (2021)

    2021

  4. [12]

    & Ortega, J.-P

    Grigoryeva, L., Hart, A. & Ortega, J.-P. Learning strange attractors with reservoir systems. Nonlinearity 36, 4674 (2023)

    2023

  5. [13]

    & Ortega, J.-P

    Grigoryeva, L. & Ortega, J.-P. Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems. J. Mach. Learn. Res. 19, 1–40 (2018)

    2018

  6. [14]

    & Ortega, J.-P

    Grigoryeva, L. & Ortega, J.-P. Echo state networks are universal. Neural Networks 108, 495–508 (2018)

    2018

  7. [15]

    & Ortega, J.-P

    Gonon, L. & Ortega, J.-P. Reservoir computing universality with stochastic inputs. IEEE transactions on neural networks learning systems 31, 100–112 (2019)

    2019

  8. [16]

    & Ortega, J.-P

    Gonon, L. & Ortega, J.-P. Fading memory echo state networks are universal. Neural Networks 138, 10–13 (2021)

    2021

  9. [17]

    Tutorial on training recurrent neural networks, covering BPPT, RTRL, EKF and the echo state network approach, vol

    Jaeger, H. Tutorial on training recurrent neural networks, covering BPPT, RTRL, EKF and the echo state network approach, vol. 5 (Citeseer, 2002)

    2002

  10. [18]

    & Jaeger, H

    Lukoˇseviˇcius, M. & Jaeger, H. Reservoir computing approaches to recurrent neural network training. Comput. science review 3, 127–149 (2009)

    2009

  11. [19]

    Sun, C. et al. A systematic review of echo state networks from design to application. IEEE Transactions on Artif. Intell. 5, 23–37 (2022)

    2022

  12. [20]

    A., Penny, S

    Platt, J. A., Penny, S. G., Smith, T. A., Chen, T.-C. & Abarbanel, H. D. A systematic exploration of reservoir computing for forecasting complex spatiotemporal dynamics. Neural Networks 153, 530–552 (2022)

    2022

  13. [21]

    & Ortega, J.-P

    Gonon, L., Grigoryeva, L. & Ortega, J.-P. Approximation bounds for random neural networks and reservoir systems. The Annals Appl. Probab. 33, 28–69 (2023)

    2023

  14. [22]

    Tanaka, G. et al. Recent advances in physical reservoir computing: A review. Neural Networks 115, 100–123 (2019)

    2019

  15. [23]

    Physical reservoir computing—an introductory perspective

    Nakajima, K. Physical reservoir computing—an introductory perspective. Jpn. J. Appl. Phys. 59, 060501 (2020)

    2020

  16. [24]

    & Fischer, I

    Nakajima, K. & Fischer, I. Reservoir computing (Springer, 2021)

    2021

  17. [25]

    Yan, M. et al. Emerging opportunities and challenges for the future of reservoir computing. Nat. Commun. 15, 2056 (2024)

    2024

  18. [26]

    J., Bollt, E., Griffith, A

    Gauthier, D. J., Bollt, E., Griffith, A. & Barbosa, W. A. Next generation reservoir computing. Nat. communications 12, 1–8 (2021)

    2021

  19. [27]

    Barbosa, W. A. & Gauthier, D. J. Learning spatiotemporal chaos using next-generation reservoir computing. Chaos: An Interdiscip. J. Nonlinear Sci. 32 (2022). 10/12

    2022

  20. [28]

    J., Fischer, I

    Gauthier, D. J., Fischer, I. & R¨ohm, A. Learning unseen coexisting attractors. Chaos: An Interdiscip. J. Nonlinear Sci. 32 (2022)

    2022

  21. [29]

    M., Barbosa, W

    Kent, R. M., Barbosa, W. A. & Gauthier, D. J. Controlling chaotic maps using next-generation reservoir computing.Chaos: An Interdiscip. J. Nonlinear Sci. 34 (2024)

    2024

  22. [30]

    M., Barbosa, W

    Kent, R. M., Barbosa, W. A. & Gauthier, D. J. Controlling chaos using edge computing hardware. Nat. Commun. 15, 3886 (2024)

    2024

  23. [31]

    & Chua, L

    Boyd, S. & Chua, L. Fading memory and the problem of approximating nonlinear operators with volterra series. IEEE Transactions on circuits systems 32, 1150–1161 (1985)

    1985

  24. [32]

    & Ortega, J.-P

    Grigoryeva, L. & Ortega, J.-P. Differentiable reservoir computing. J. Mach. Learn. Res. 20, 1–62 (2019)

    2019

  25. [33]

    & Wong, N

    Batselier, K., Chen, Z. & Wong, N. Tensor network alternating linear scheme for mimo volterra system identification. Automatica 84, 26–35 (2017)

    2017

  26. [34]

    Enforcing symmetry in tensor network mimo volterra identification

    Batselier, K. Enforcing symmetry in tensor network mimo volterra identification. IF AC-PapersOnLine54, 469–474 (2021)

    2021

  27. [35]

    A khatri-rao product tensor network for efficient symmetric mimo volterra identification

    Batselier, K. A khatri-rao product tensor network for efficient symmetric mimo volterra identification. IF AC-PapersOnLine 56, 7282–7287 (2023)

    2023

  28. [36]

    & Favier, G

    Khouaja, A. & Favier, G. Identification of parafac-volterra cubic models using an alternating recursive least squares algorithm. In 2004 12th European Signal Processing Conference, 1903–1906 (IEEE, 2004)

    2004

  29. [37]

    & Bouilloc, T

    Favier, G. & Bouilloc, T. Parametric complexity reduction of volterra models using tensor decompositions. In 2009 17th European Signal Processing Conference, 2288–2292 (IEEE, 2009)

    2009

  30. [38]

    & Kibangou, A

    Favier, G. & Kibangou, A. Tensor-based approaches for nonlinear and multilinear systems modeling and identification. Algorithms 16, 443 (2023)

    2023

  31. [39]

    Tensor networks for complex quantum systems

    Or´us, R. Tensor networks for complex quantum systems. Nat. Rev. Phys. 1, 538–550 (2019)

    2019

  32. [40]

    Ba˜nuls, M. C. Tensor network algorithms: A route map. Annu. Rev. Condens. Matter Phys. 14, 173–191 (2023)

    2023

  33. [41]

    & Schwab, D

    Stoudenmire, E. & Schwab, D. J. Supervised learning with tensor networks. Adv. neural information processing systems 29 (2016)

    2016

  34. [42]

    & Zhang, P

    Cheng, S., Wang, L., Xiang, T. & Zhang, P. Tree tensor networks for generative modeling.Phys. Rev. B 99, 155131 (2019)

    2019

  35. [43]

    Tomut, A. et al. Compactifai: extreme compression of large language models using quantum-inspired tensor networks. arXiv preprint arXiv:2401.14109 (2024)

    2024 arXiv

  36. [44]

    Low-rank tensor decompositions for nonlinear system identification: A tutorial with examples

    Batselier, K. Low-rank tensor decompositions for nonlinear system identification: A tutorial with examples. IEEE Control. Syst. Mag. 42, 54–74 (2022)

    2022

  37. [45]

    Chaos as an interpretable benchmark for forecasting and data-driven modelling.arXiv preprint arXiv:2110.05266 (2021)

    Gilpin, W. Chaos as an interpretable benchmark for forecasting and data-driven modelling.arXiv preprint arXiv:2110.05266 (2021)

    2021 arXiv

  38. [46]

    Horn, R. A. & Johnson, C. R. Matrix analysis (Cambridge university press, 2012)

    2012

  39. [47]

    A singularly valuable decomposition: the svd of a matrix

    Kalman, D. A singularly valuable decomposition: the svd of a matrix. The college mathematics journal 27, 2–23 (1996)

    1996

  40. [48]

    Oseledets, I. V . & Tyrtyshnikov, E. E. Breaking the curse of dimensionality, or how to use svd in many dimensions.SIAM J. on Sci. Comput. 31, 3744–3759 (2009)

    2009

  41. [49]

    Kolda, T. G. & Bader, B. W. Tensor decompositions and applications. SIAM review 51, 455–500 (2009)

    2009

  42. [50]

    Oseledets, I. V . Tensor-train decomposition.SIAM J. on Sci. Comput. 33, 2295–2317 (2011)

    2011

  43. [51]

    Verstraete, F., Garcia-Ripoll, J. J. & Cirac, J. I. Matrix product density operators: Simulation of finite-temperature and dissipative systems. Phys. review letters 93, 207204 (2004)

    2004

  44. [52]

    & Vidal, G

    Zwolak, M. & Vidal, G. Mixed-state dynamics in one-dimensional quantum lattice systems: A time-dependent superopera- tor renormalization algorithm. Phys. review letters 93, 207205 (2004)

    2004

  45. [53]

    Oseledets, I. V . Approximation of 2ˆd\times2ˆd matrices using tensor decomposition. SIAM J. on Matrix Analysis Appl. 31, 2130–2145 (2010)

    2010

  46. [54]

    & Wong, N

    Batselier, K., Ko, C.-Y . & Wong, N. Tensor network subspace identification of polynomial state space models.Automatica 95, 187–196 (2018)

    2018

  47. [55]

    Favier, G., Kibangou, A. Y . & Bouilloc, T. Nonlinear system modeling and identification using volterra-parafac models. Int. J. Adapt. Control. Signal Process. 26, 30–53 (2012). 11/12

    2012

  48. [56]

    & Favier, G

    Bouilloc, T. & Favier, G. Nonlinear channel modeling and identification using baseband volterra–parafac models. Signal Process. 92, 1492–1498 (2012)

    2012

  49. [57]

    & Batselier, K

    Memmel, E., Menzen, C. & Batselier, K. Bayesian framework for a mimo volterra tensor network. IF AC-PapersOnLine 56, 7294–7299 (2023)

    2023

  50. [58]

    echo state

    Jaeger, H. The “echo state” approach to analysing and training recurrent neural networks-with an erratum note. Bonn, Ger. Ger. Natl. Res. Cent. for Inf. Technol. GMD Tech. Rep. 148, 13 (2001)

    2001

  51. [59]

    A., Zhang, L., Nicolaou, Z

    Kaptanoglu, A. A., Zhang, L., Nicolaou, Z. G., Fasel, U. & Brunton, S. L. Benchmarking sparse system identification with low-dimensional chaos. Nonlinear Dyn. 111, 13143–13164 (2023)

    2023

  52. [60]

    & Chitre, M

    Bagge Carlson, F. & Chitre, M. New metrics between rational spectra and their connection to optimal transport. arXiv e-prints arXiv–2004 (2020)

    2020

  53. [61]

    Wikner, A. et al. Stabilizing machine learning prediction of dynamics: Novel noise-inspired regularization tested with reservoir computing. Neural Networks 170, 94–110 (2024)

    2024

  54. [62]

    & Wong, N

    Batselier, K., Yu, W., Daniel, L. & Wong, N. Computing low-rank approximations of large-scale matrices with the tensor network randomized svd. SIAM J. on Matrix Analysis Appl. 39, 1221–1244 (2018)

    2018

  55. [63]

    L., Proctor, J

    Brunton, S. L., Proctor, J. L. & Kutz, J. N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. national academy sciences 113, 3932–3937 (2016). Acknowledgements We thank Hannah Lim Jing Ting, Miguel C. Soriano, and Juan-Pa...

    2016

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.