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arxiv: 2604.06732 · v1 · submitted 2026-04-08 · 💻 cs.LG

Recognition: no theorem link

Extraction of linearized models from pre-trained networks via knowledge distillation

Fumito Kimura, Jun Ohkubo

Pith reviewed 2026-05-10 17:56 UTC · model grok-4.3

classification 💻 cs.LG
keywords Koopman operatorknowledge distillationlinearized modelspre-trained neural networksclassification tasksnumerical stabilityMNISTFashion-MNIST
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0 comments X

The pith

Integrating Koopman operator theory with knowledge distillation extracts linearized models from pre-trained networks that outperform standard approximations.

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

The authors seek to create linear models from existing trained neural networks so that machine learning can run on hardware designed for linear math after a basic nonlinear step. They combine the mathematical tool of Koopman operators, which turn nonlinear evolution into linear form, with knowledge distillation to have the linear model learn the behavior of the original network. Tests on standard image datasets show this produces higher accuracy and more stable computations than the usual way of fitting Koopman operators with least squares. Readers should care if they want to deploy powerful networks on specialized linear hardware without rebuilding everything from scratch.

Core claim

The paper establishes a framework that uses Koopman operator theory together with knowledge distillation to derive a linearized model from a pre-trained neural network for classification. This framework is demonstrated to achieve higher classification accuracy and better numerical stability than the conventional least-squares-based Koopman approximation method when tested on the MNIST and Fashion-MNIST datasets.

What carries the argument

The central mechanism is the fusion of the Koopman operator, which linearizes the system's dynamics, and knowledge distillation, which allows the linearized model to replicate the outputs of the pre-trained nonlinear network.

If this is right

  • The resulting linearized models support efficient implementation on linear-operation hardware such as photonic circuits.
  • Classification performance is preserved better than with direct least-squares Koopman fitting.
  • The approach increases numerical stability for practical use of the models.
  • It demonstrates a viable path to hybrid linear-nonlinear learning systems.

Where Pith is reading between the lines

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

  • This technique might apply to other machine learning tasks beyond classification, such as regression or sequence prediction.
  • Further work could test whether the method scales to larger, more complex networks and datasets.
  • The success suggests distillation can recover nonlinear information lost in linear approximations.

Load-bearing premise

That the knowledge distillation process can transfer the necessary nonlinear classification capability into the linear Koopman model without substantial degradation.

What would settle it

If experiments on additional datasets show the proposed method no longer consistently exceeds the accuracy or stability of the least-squares Koopman baseline, the advantage would be called into question.

Figures

Figures reproduced from arXiv: 2604.06732 by Fumito Kimura, Jun Ohkubo.

Figure 1
Figure 1. Figure 1: FIG. 1. The naive method for extracting a linearized model fr [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A di [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of test accuracy between the proposed mod [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

Recent developments in hardware, such as photonic integrated circuits and optical devices, are driving demand for research on constructing machine learning architectures tailored for linear operations. Hence, it is valuable to explore methods for constructing learning machines with only linear operations after simple nonlinear preprocessing. In this study, we propose a framework to extract a linearized model from a pre-trained neural network for classification tasks by integrating Koopman operator theory with knowledge distillation. Numerical demonstrations on the MNIST and the Fashion-MNIST datasets reveal that the proposed model consistently outperforms the conventional least-squares-based Koopman approximation in both classification accuracy and numerical stability.

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.

Referee Report

0 major / 3 minor

Summary. The paper proposes a framework to extract linearized models from pre-trained neural networks for classification by integrating Koopman operator theory with knowledge distillation. Motivated by hardware constraints favoring linear operations (e.g., photonic circuits), the method is demonstrated on MNIST and Fashion-MNIST, where it reportedly outperforms the standard least-squares Koopman approximation in both classification accuracy and numerical stability.

Significance. If the empirical results hold under rigorous validation, the work offers a practical bridge between nonlinear pre-trained models and linear computational hardware, leveraging two established techniques (Koopman linearization and distillation) without introducing obvious circularity. The numerical demonstrations on standard datasets provide a concrete starting point for hardware-aware model extraction, though extension to deeper networks or regression tasks would increase impact.

minor comments (3)
  1. The abstract states outperformance 'consistently' but provides no quantitative deltas, confidence intervals, or mention of multiple random seeds; adding these would improve clarity without altering the central claim.
  2. Notation for the combined loss (Koopman reconstruction plus distillation term) should be introduced with an explicit equation in the methods section to avoid ambiguity when comparing to the baseline least-squares formulation.
  3. Figure captions for the stability and accuracy plots would benefit from stating the exact network architectures, layer counts, and training epochs used, allowing readers to reproduce the setup more easily.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; empirical integration of established methods

full rationale

The paper's derivation chain consists of applying standard Koopman operator theory for linearization combined with knowledge distillation to extract models from pre-trained networks. The central result is an empirical demonstration on MNIST and Fashion-MNIST that this combination yields higher classification accuracy and numerical stability than plain least-squares Koopman approximation. No step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the method is a novel but non-circular synthesis of two independent, externally validated techniques, with performance claims resting on direct numerical comparison rather than tautological renaming or internal fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the approach rests on standard assumptions from Koopman theory and knowledge distillation without introducing new free parameters or entities visible at this level.

axioms (2)
  • domain assumption Koopman operator theory can approximate the nonlinear dynamics of a neural network via linear operators in a lifted function space
    Invoked as the core mechanism for linearization in the proposed framework
  • domain assumption Knowledge distillation can transfer classification behavior from a pre-trained network to a linearized model
    Central to the extraction process described

pith-pipeline@v0.9.0 · 5384 in / 1229 out tokens · 26947 ms · 2026-05-10T17:56:54.849615+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

36 extracted references · 25 canonical work pages · 2 internal anchors

  1. [1]

    LeCun, Y

    Y . LeCun, Y . Bengio, and G. Hinton, “Deep learning,” Nature, vol. 521, no. 7553, pp. 436–444, May 2015. DOI:10.1038/nature14539

    work page doi:10.1038/nature14539 2015

  2. [2]

    Strubell, A

    E. Strubell, A. Ganesh, and A. McCallum. “Energy and pol- icy considerations for deep learning in NLP ,” Proc. 57th Ann . Meeting Assoc. Comp. Linguistics, pp. 3645–3650, July 2019 . DOI:10.18653/v1/P19-1355

    work page doi:10.18653/v1/p19-1355 2019

  3. [3]

    Energy and policy considerations for modern deep learning research,

    E. Strubell, A. Ganesh, and A. McCallum. “Energy and policy considerations for modern deep learning research,” Proc. A AAI Conf. Artificial Intelligence, vol. 34, no. 09, pp. 13693–13 696, February 2020. DOI:10.1609/aaai.v34i09.7123

    work page doi:10.1609/aaai.v34i09.7123 2020

  4. [4]

    Pruning and quantization for deep neural network acceleration: A su r- vey,

    T. Liang, J. Glossner, L. Wang, S. Shi, and X. Zhang, “Pruning and quantization for deep neural network acceleration: A su r- vey,” Neurocomputing, vol. 461, pp. 370–403, October 2021. DOI:10.1016/j.neucom.2021.07.045

    work page doi:10.1016/j.neucom.2021.07.045 2021

  5. [5]

    A survey on deep neural network pruning: Taxonomy, comparison, analysis, and rec- ommendations,

    H. Cheng, M. Zhang and J. Q. Shi, “A survey on deep neural network pruning: Taxonomy, comparison, analysis, and rec- ommendations,” IEEE Trans. Pattern Analysis & Machine In- telligence vol. 46, no. 12, pp. 10558–10578, December 2024. DOI:10.1109/TPAMI.2024.3447085

    work page doi:10.1109/tpami.2024.3447085 2024

  6. [6]

    Deep learning with coherent nanophotonic c ir- cuits,

    Y . Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljaˇ ci´ c, “Deep learning with coherent nanophotonic c ir- cuits,” Nature Photon., vol. 11, no. 7, pp. 441–446, June 2017. DOI:10.1038/nphoton.2017.93

    work page doi:10.1038/nphoton.2017.93 2017

  7. [7]

    Infer- ence in artificial intelligence with deep optics and photon- ics,

    G. Wetzstein, A. Ozcan, S. Gigan, S. Fan, D. Englund, M. Soljaˇ ci´ c, C. Denz, D. A. B. Miller, and D. Psaltis, “Infer- ence in artificial intelligence with deep optics and photon- ics,” Nature, vol. 588, no. 7836, pp. 39–47, December 2020. DOI:10.1038/s41586-020-2973-6

    work page doi:10.1038/s41586-020-2973-6 2020

  8. [8]

    Roadmap on Neuromorphic Photonics,

    B. J. Shastri, A. N. Tait, T. Ferreira de Lima, W. H. P . Per- nice, H. Bhaskaran, C. D. Wright, and P . R. Prucnal, “Pho- tonics for artificial intelligence and neuromorphic comput ing,” Nature Photon. , vol. 15, no. 2, pp. 102–114, January 2021. DOI:10.1038/s41566-020-00754-y

    work page doi:10.1038/s41566-020-00754-y 2021

  9. [9]

    Fully nonlinear neu- romorphic computing with linear wave scattering,

    C. C. Wanjura and F. Marquardt, “Fully nonlinear neu- romorphic computing with linear wave scattering,” Nature Phys., vol. 20, no. 9, pp. 1434–1440, September 2024. DOI:10.1038/s41567-024-02534-9

    work page doi:10.1038/s41567-024-02534-9 2024

  10. [10]

    Single-chip photonic deep neural network with forward-only training,

    S. Bandyopadhyay, A. Sludds, S. Krastanov, R. Hamerly, N. Harris, D. Bunandar, M. Streshinsky, M. Hochberg, and D. Englund, “Single-chip photonic deep neural network with forward-only training,” Nature Photon. , vol. 18, no. 12, pp. 1335–1343, December 2024. DOI:10.1038 /s41566-024-01567- z

    2024

  11. [11]

    A sur- vey on silicon photonics for deep learning,

    F. P . Sunny, E. Taheri, M. Nikdast, and S. Pasricha, “A sur- vey on silicon photonics for deep learning,” ACM J. Emerging Tech. Comp. System , vol. 17, no. 4, article no. 61, June 2021. DOI:10.1145/3459009

    work page doi:10.1145/3459009 2021

  12. [12]

    A r- tificial neural networks for photonic applications—from al go- rithms to implementation: tutorial,

    P . Freire, E. Manuylovich, J. Prilepsky, and S. Turitsyn, “A r- tificial neural networks for photonic applications—from al go- rithms to implementation: tutorial,” Adv. Opt. Photon. vol. 15, no. 3, pp. 739–834 September 2023. DOI:10.1364 /AOP .484119

    2023

  13. [13]

    Memristor-based spiking neural networks: cooperative development of neural network architecture/algorithms and memristors,

    S. Abreu, I. Boikov, M. Goldmann, T. Jonuzi, A. Lupo, S. Masaad, L. Nguyen, E. Picco, G. Pourcel, A. Skalli, L. Ta- landier, B. V ettelschoss, E. A. Vlieg, A. Argyris, P . Bienst - man, D. Brunner, J. Dambre, L. Daudet, J.D. Domenech, I. Fischer, F. Horst, S. Massar, C. R. Mirasso, B. J. O ffrein, A. Rossi, M. C. Soriano, S. Sygletos, and S. K. Turitsyn, “...

    work page doi:10.1016/j.revip.2024.100093 2024

  14. [14]

    An optical neural network using less than 1 photon per multiplication,

    T. Wang, S.-Y . Ma, L. G. Wright, T. Onodera, B. C. Richard, and P . L. McMahon, “An optical neural network using less than 1 photon per multiplication,” Nature Comm., vol. 13, no. 1, ar- ticle no. 123, January 2022. DOI:10.1038 /s41467-021-27774-8

    2022

  15. [15]

    Integrated photonic en- coder for low power and high-speed image processing,

    X. Wang, B. Redding, N. Karl, C. Long, Z. Zhu, J. Skowronek, S. Pang, D. Brady, and R. Sarma, “Integrated photonic en- coder for low power and high-speed image processing,” Na- ture Comm. , vol ˙15, no. 1, article no. 4510, May 2024. DOI:10.1038/s41467-024-48099-2

    work page doi:10.1038/s41467-024-48099-2 2024

  16. [16]

    Blending optimal control and biolog- ically plausible learning for noise-robust physical neura l net- works,

    S. Sunada, T. Niiyama, K. Kanno, R. Nogami, A. R¨ ohm, T. Awano, and A. Uchida, “Blending optimal control and biolog- ically plausible learning for noise-robust physical neura l net- works,” Phys. Rev. Lett. , vol. 134, no. 1, article no. 017301, January 2025. DOI:10.1103/PhysRevLett.134.017301

    work page doi:10.1103/physrevlett.134.017301 2025

  17. [17]

    All-optical no n- linear activation function for photonic neural networks,

    M. Miscuglio, A. Mehrabian, Z. Hu, S. I. Azzam, J. George, A. V . Kildishev, M. Pelton, and V . J. Sorger, “All-optical no n- linear activation function for photonic neural networks,” Opt. Mater . Expressvol. 8, no. 12, pp. 3851–3863, December 2018. DOI:10.1364/OME.8.003851

    work page doi:10.1364/ome.8.003851 2018

  18. [18]

    All-optical neural network with nonli n- ear activation functions,

    Y . Zuo, B. Li, Y . Zhao, Y . Jiang, Y .-C. Chen, P . Chen, G.-B. Jo, J. Liu, and S. Du, “All-optical neural network with nonli n- ear activation functions,” Optica, vol. 6, no. 9, pp. 1132–1137, September 2019. DOI:10.1364/OPTICA.6.001132

    work page doi:10.1364/optica.6.001132 2019

  19. [19]

    Experimental realization of any discrete unitary operato r,

    M. Reck, A. Zeilinger, H. J. Bernstein, and P . Philip, “Experimental realization of any discrete unitary operato r,” Phys. Rev. Lett. , vol. 73, no. 1, pp. 58–61, July 1994. DOI:10.1103/PhysRevLett.73.58

    work page doi:10.1103/physrevlett.73.58 1994

  20. [20]

    Pho- tonic matrix multiplication lights up photonic accelerato r and beyond,

    H. Zhou, J. Dong, J. Cheng, W. Dong, C. Huang, Y . Shen, Q. Zhang, M. Gu, C. Qian, H. Chen, Z. Ruan, and X. Zhang, “Pho- tonic matrix multiplication lights up photonic accelerato r and beyond,” Light Sci. Appl. vol. 11, no. 1, article no. 30, January

  21. [21]

    DOI:10.1038/s41377-022-00717-8

    work page doi:10.1038/s41377-022-00717-8

  22. [22]

    Integrated pho- tonic tensor processing unit for a matrix multiply: A review ,

    N. Peserico, B. J. Shastri and V . J. Sorger, “Integrated pho- tonic tensor processing unit for a matrix multiply: A review ,” J. Lightwave Tech., vol. 41, no. 12, pp. 3704–3716, June, 2023. DOI:10.1109/JLT.2023.3269957

    work page doi:10.1109/jlt.2023.3269957 2023

  23. [23]

    Universal photonic artificial intelligence a cceler- ation,

    S. R. Ahmed, R. Baghdadi, M. Bernadskiy, N. Bowman, R. Braid, J. Carr, C. Chen, P . Ciccarella, M. Cole, J. Cooke, K. Desai, C. Dorta, J. Elmhurst, B. Gardiner, E. Greenwald, S. Gupta, P . Husbands, B. Jones, A. Kopa, H. J. Lee, A. Mad- havan, A. Mendrela, N. Moore, L. Nair, A. Om, S. Patel, R. Patro, R. Pellowski, E. Radhakrishnani, S. Sane, N. Sarkis, J...

    work page doi:10.1038/s41586-025-08854-x 2025

  24. [24]

    Hamiltonian systems and transformation in Hilbert space,

    B. O. Koopman, “Hamiltonian systems and transformation in Hilbert space,” Proc. Natl. Acad. Sci. , vol. 17, no. 5, pp. 315– 318, May 1931. DOI: 10.1073 /pnas.17.5.315

    1931

  25. [25]

    Koopman operator, geometry, and learning of dy- namical systems,

    I. Mezi´ c, “Koopman operator, geometry, and learning of dy- namical systems,” Notices of Amer . Math. Soc. , vol. 68, no. 6, pp. 1087–1105, August 2021. DOI: 10.1090 /noti2306

    2021

  26. [26]

    Journal of Statistical Mechanics: Theory and Experiment , author =

    N. Sugishita, K. Kinjo, and J. Ohkubo, “Extraction of non- linearity in neural networks with Koopman operator,” J. Stat. Mech., vol. 2024, no. 7, article no. 073401, July 2024. DOI:10.1088/1742-5468/ad5713

    work page doi:10.1088/1742-5468/ad5713 2024

  27. [27]

    Represent- ing neural network layers as linear operations via Koop- man operator theory,

    N. S. Aswani, S. Jabari and M. Shafique, “Represent- ing neural network layers as linear operations via Koop- man operator theory,” Proc. 3rd Workshop on Unify- ing Representations in Neural Models, December, 2025. https://openreview.net/forum?id=lksyVbX4qW

    2025

  28. [28]

    Distilling the Knowledge in a Neural Network

    G. Hinton, O. Vinyals, and J. Dean, “Distilling the knowledg e in a neural network,” arXiv preprint, arXiv:1503.02531

    work page internal anchor Pith review Pith/arXiv arXiv

  29. [29]

    Modern Koopman theory for dynamical systems,

    S. L. Brunton, M. Budis ˇi´ c, E. Kaiser, and J. N. Kutz, “Modern Koopman theory for dynamical systems,” SIAM Rev., vol. 64, pp. 229–340, May 2022. DOI: 10.1137 /21M1401243

    2022

  30. [30]

    A data- driven approximation of the Koopman operator: Extending dy - namic mode decomposition,

    M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A data- driven approximation of the Koopman operator: Extending dy - namic mode decomposition,” J. Nonlinear Sci. , vol. 25, no. 6, pp. 1307–1346, June (2015). DOI: 10.1007 /s00332-015-9258-5

    2015

  31. [31]

    Dynamic mode decomposition of numerical and experimental data,

    P . J. Schmid, “Dynamic mode decomposition of numerical and experimental data,” J. Fluid Mech. , vol. 656, pp. 5–28, July

  32. [32]

    DOI: 10.1017/S0022112010001217

    work page doi:10.1017/s0022112010001217

  33. [33]

    Gradient-based learning applied to document recognition , journal =

    Y . LeCun, L. Bottou, Y . Bengio, and P . Ha ffner, “Gradient- based learning applied to document recognition,” Proc. IEEE, vol. 86, no. 11, pp. 2278–2324, November, 1998. DOI: 10.1109/5.726791

    work page doi:10.1109/5.726791 1998

  34. [34]

    MNIST handwritten digit database,

    Y . LeCun, C. Cortes, and C. J. Burges, “MNIST handwritten digit database,” A TT Labs [Online]. Available: http://yann. lecun.com/exdb/mnist, 2, 2010

    2010

  35. [35]

    Deep residual learning for image recognition

    K. He, X. Zhang, S. Ren, and J. Sun. “Deep residual learn- ing for image recognition,” Proc. IEEE Conf. Comp. Vi- sion Pattern Recognition, pp. 770–778, June 2016. DOI: 10.1109/CVPR.2016.90

    work page doi:10.1109/cvpr.2016.90 2016

  36. [36]

    Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms

    H. Xiao, K. Rasul, and R. V ollgraf. “Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithm s,” arXiv preprint, arXiv:1708.07747

    work page internal anchor Pith review arXiv