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arxiv: 2605.19152 · v2 · pith:I6XPGV2Fnew · submitted 2026-05-18 · 📊 stat.ML · cs.ET· cs.IT· cs.LG· cs.NE· math.IT· physics.optics

Information Processing Capacity of Stationary Physical Systems: Theory, Data-efficient Estimation Methods, and Photonic Demonstration

Rahul Uma Ramachandran , Serge Massar This is my paper

Pith reviewed 2026-05-22 08:58 UTC · model grok-4.3

classification 📊 stat.ML cs.ETcs.ITcs.LGcs.NEmath.ITphysics.optics
keywords information processing capacitystationary physical systemsphotonic computingnonlinear opticsmachine learning performancedata-efficient estimationKerr effecteffective dimensionality
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The pith

Stationary physical systems have total information processing capacity bounded by their readout count, which correlates with machine learning performance.

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

The paper extends the information processing capacity framework to stationary physical computing systems and derives basic bounds on what such systems can compute. Individual capacities range from zero to one and add up to no more than the number of readouts, with noise lowering the total. It also supplies data-efficient estimation techniques based on extrapolation and quasi-random sampling, then tests them in a photonic experiment where laser power and fiber length change the distribution of capacities toward higher nonlinear orders. The central result is that the summed capacity tracks how well the system performs on standard machine learning benchmarks and gives a direct reading of the system's effective computational dimensionality.

Core claim

For stationary physical computing systems the individual information processing capacities are each bounded between zero and one, their sum over a complete basis is bounded by the number of readouts, and this bound is strictly reduced by noise. Finite-sample estimators carry a known positive bias whose asymptotic form is derived, and data-efficient methods using Richardson extrapolation together with Sobol quasi-random sampling are introduced to estimate the capacities. In a photonic setup based on picosecond pulses in nonlinear optical fibre, increasing laser power and fibre length shifts the capacity distribution toward higher-order nonlinear terms through the Kerr effect. The total IPC is

What carries the argument

Information processing capacity (IPC) for stationary systems, quantified as the normalized correlation between input sequences and system readouts, which decomposes the system's computational ability into independent linear and nonlinear contributions.

If this is right

  • Physical hardware can be ranked for machine-learning use by measuring its total IPC without running the actual tasks.
  • Noise in the physical system directly reduces the usable computational capacity below the readout limit.
  • The data-efficient estimators allow reliable IPC measurement from far fewer samples than naive methods require.
  • In photonic devices the Kerr nonlinearity systematically moves capacity into higher-order terms as power or length increases.
  • Total IPC supplies a task-independent estimate of the effective dimensionality available for computation.

Where Pith is reading between the lines

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

  • Designers could tune physical parameters such as nonlinearity or noise level to target desired capacity distributions for specific applications.
  • The stationarity assumption might be relaxed in future work to cover slowly varying systems while preserving similar bounds.
  • IPC could serve as a quick screening tool when selecting among candidate hardware platforms for embedded learning.
  • Applying the same measurement protocol to other physical substrates would test whether the readout-bound and performance correlation hold more generally.

Load-bearing premise

The physical computing system under study must be stationary.

What would settle it

Observing a summed IPC that exceeds the number of readouts in a clearly stationary system, or measuring no correlation between total IPC and performance on machine-learning benchmarks, would falsify the central bounds and the claimed link to task performance.

Figures

Figures reproduced from arXiv: 2605.19152 by Rahul Uma Ramachandran, Serge Massar.

Figure 1
Figure 1. Figure 1: Illustration of capacity visualizations: (a) Capacity matrix when the input dimensionality [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Raw capacities CN (yl , X) as a function of the number of samples N for selected basis functions. The curves all start at 1 for N = K = 72, and then decrease for larger N, illustrating the asymptotic behaviour proven in Prop. 5.2. The data for this panel is obtained using the experimental system described in Section 7 using average power=-1.9 dBm and fiber length =40m. (b,c) Results for the synthetic d… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Schematic of the Experiment. Programmable Spectral Filter (PSF), Variable Optical [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Capacity matrices for 2-dimensional inputs compared at different laser powers (the mea [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Capacity bar-plots for 5-dimensional inputs compared at different laser powers (the [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Total capacities as a function of nonlinear phase [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Capacities at different laser powers and fiber lengths compared with accuracies on machine [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Estimating effective dimensionality using factor analysis and IPC. (a) Experimentally [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
read the original abstract

Physical computing systems provide a promising route toward hardware-native machine learning, but their computational capabilities remain difficult to characterize in a principled, task-independent, and data-efficient way. We extend the Information Processing Capacity (IPC) framework to stationary physical computing systems and establish several fundamental results: individual capacities are bounded between zero and one, their sum over a complete basis is bounded by the number of readouts, and noise strictly reduces this bound. We address the finite-sample estimation of IPC and derive the asymptotic form of the systematic positive bias affecting naive estimators. Building on these results, we introduce data-efficient estimation methods based on Richardson extrapolation and Sobol quasi-random sampling. We validate the framework experimentally using a photonic computing system based on picosecond laser pulses propagating through a nonlinear optical fibre. By varying the laser power and fibre length, we observe systematic shifts of the IPC distribution toward higher-order nonlinear capacities induced by the Kerr effect. Finally, we demonstrate that the total IPC strongly correlates with performance on benchmark machine-learning tasks and provides a reliable estimate of the effective dimensionality of the system. These results establish IPC as a practical bridge between the intrinsic dynamics of physical computing systems and their machine-learning performance.

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

1 major / 1 minor

Summary. The paper extends the Information Processing Capacity (IPC) framework to stationary physical computing systems. It derives fundamental bounds (individual capacities in [0,1], sum bounded by number of readouts, strict reduction by noise), obtains the asymptotic bias of naive finite-sample estimators, and introduces data-efficient methods via Richardson extrapolation and Sobol quasi-random sampling. In a photonic experiment with picosecond pulses in a nonlinear fibre, laser power and fibre length are varied to induce Kerr-driven shifts in the IPC distribution; the authors report that total IPC correlates strongly with benchmark ML task performance and serves as a reliable estimator of effective system dimensionality.

Significance. If the central claims hold, the work supplies a task-independent, theoretically grounded metric for characterizing physical computing systems, directly linking intrinsic dynamics to ML utility. The stationarity-based bounds and bias analysis are useful contributions; the photonic demonstration illustrates practical applicability in a hardware-native setting.

major comments (1)
  1. [Photonic demonstration] Photonic demonstration section: the reported correlation between total IPC and ML task performance is obtained solely by sweeping laser power and fibre length within a single Kerr-nonlinear system. Because these same parameters simultaneously control nonlinearity strength, effective degrees of freedom, and the observed IPC distribution, the experiment does not isolate IPC as an independent estimator of dimensionality; an alternative explanation that both quantities are driven by the same physical changes remains viable.
minor comments (1)
  1. [Abstract] The abstract states that noise strictly reduces the sum bound, but the precise dependence on noise variance or readout SNR is not quantified in the provided summary; a short explicit statement would clarify the result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and for recognizing the potential significance of extending the IPC framework to stationary physical systems. We address the major comment on the photonic demonstration below.

read point-by-point responses

  1. Referee: Photonic demonstration section: the reported correlation between total IPC and ML task performance is obtained solely by sweeping laser power and fibre length within a single Kerr-nonlinear system. Because these same parameters simultaneously control nonlinearity strength, effective degrees of freedom, and the observed IPC distribution, the experiment does not isolate IPC as an independent estimator of dimensionality; an alternative explanation that both quantities are driven by the same physical changes remains viable.

    Authors: We agree that the experiment varies laser power and fibre length within a single Kerr-nonlinear fibre system, so these parameters jointly influence nonlinearity, effective degrees of freedom, and the IPC distribution. This design means the observed correlation between total IPC and ML benchmark performance could be explained by the shared physical changes rather than IPC acting as a fully independent estimator. The IPC values are nevertheless obtained from the system's measured response to a complete input basis using the stationary framework, which is independent of the particular ML tasks. The tasks themselves are standard benchmarks applied to the same physical outputs. The results therefore show that IPC tracks changes in effective dimensionality and task performance across different operating regimes induced by the Kerr effect. To address this point, we have revised the discussion section to explicitly note the correlational character of the evidence and to state that experiments across distinct physical platforms would be needed to further separate the contributions. This addition clarifies the scope of the demonstration without altering the reported observations. revision: partial

Circularity Check

0 steps flagged

No significant circularity; bounds derived from stationarity assumption and definitions, empirical correlation independent of fitted inputs.

full rationale

The paper derives bounds on individual capacities [0,1] and their sum from the stationarity assumption and readout definitions, which is a standard mathematical consequence rather than a self-referential fit. Estimation methods correct for bias via extrapolation and sampling without redefining the target IPC. The photonic demonstration varies physical parameters (power, length) to shift IPC distribution and reports correlation with ML tasks; this is an empirical observation, not a reduction of the claim to its inputs by construction. No self-citation load-bearing steps or ansatz smuggling identified in the provided abstract and context. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the stationarity assumption and standard properties of linear readouts; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The physical system is stationary.
    Invoked to extend IPC framework and derive the 0-1 bounds, sum bound, and noise reduction.

pith-pipeline@v0.9.0 · 5760 in / 1245 out tokens · 34735 ms · 2026-05-22T08:58:11.214732+00:00 · methodology

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Works this paper leans on

61 extracted references · 61 canonical work pages · 1 internal anchor

  1. [1]

    Physics for neuromorphic computing,

    D. Markovi´ c, A. Mizrahi, D. Querlioz, and J. Grollier, “Physics for neuromorphic computing,” Nature Reviews Physics, vol. 2, no. 9, pp. 499–510, 2020

    work page 2020

  2. [2]

    Trends in extreme learning machines: A review,

    G. Huang, G.-B. Huang, S. Song, and K. You, “Trends in extreme learning machines: A review,”Neural Networks, vol. 61, pp. 32–48, 2015

    work page 2015

  3. [3]

    Toward a formal theory for computing machines made out of whatever physics offers,

    H. Jaeger, B. Noheda, and W. G. Van Der Wiel, “Toward a formal theory for computing machines made out of whatever physics offers,”Nature communications, vol. 14, no. 1, p. 4911, 2023

    work page 2023

  4. [4]

    Recent advances in physical reservoir computing: A review,

    G. Tanaka, T. Yamane, J. B. H´ eroux, R. Nakane, N. Kanazawa, S. Takeda, H. Numata, D. Nakano, and A. Hirose, “Recent advances in physical reservoir computing: A review,” Neural Networks, vol. 115, pp. 100–123, 2019

    work page 2019

  5. [5]

    Opportunities for neuromorphic computing algorithms and applications,

    C. D. Schuman, S. R. Kulkarni, M. Parsa, J. P. Mitchell, P. Date, and B. Kay, “Opportunities for neuromorphic computing algorithms and applications,”Nature Computational Science, vol. 2, no. 1, pp. 10–19, 2022

    work page 2022

  6. [6]

    Nonlinear inference capacity of fiber-optical extreme learning machines,

    S. Saeed, M. M¨ uft¨ uo˘ glu, G. R. Cheeran, T. Bocklitz, B. Fischer, and M. Chemnitz, “Nonlinear inference capacity of fiber-optical extreme learning machines,”Nanophotonics, vol. 14, no. 16, pp. 2749–2760, 2025

    work page 2025

  7. [7]

    Edge of chaos and prediction of computational performance for neural circuit models,

    R. Legenstein and W. Maass, “Edge of chaos and prediction of computational performance for neural circuit models,”Neural Networks, vol. 20, no. 3, pp. 323–334, 2007. Echo State Networks and Liquid State Machines. 29

    work page 2007

  8. [8]

    The NeuroBench framework for benchmarking neuro- morphic computing algorithms and systems,

    J. Yik, S. H. Ahmed, Z. Ahmed,et al., “The NeuroBench framework for benchmarking neuro- morphic computing algorithms and systems,”Nature Communications, vol. 16, p. 1586, 2025

    work page 2025

  9. [9]

    Benchmarking keyword spotting effi- ciency on neuromorphic hardware,

    P. Blouw, X. Choo, E. Hunsberger, and C. Eliasmith, “Benchmarking keyword spotting effi- ciency on neuromorphic hardware,” inProceedings of the 7th annual neuro-inspired computa- tional elements workshop, pp. 1–8, 2019

    work page 2019

  10. [10]

    Principles and metrics of extreme learning machines using a highly nonlinear fiber,

    M. Hary, D. Brunner, L. Leybov, P. Ryczkowski, J. M. Dudley, and G. Genty, “Principles and metrics of extreme learning machines using a highly nonlinear fiber,”Nanophotonics, vol. 14, no. 16, pp. 2733–2748, 2025

    work page 2025

  11. [11]

    Analysis of a complex of statistical variables into principal components,

    H. Hotelling, “Analysis of a complex of statistical variables into principal components,”Journal of Educational Psychology, vol. 24, no. 6, pp. 417–441, 1933

    work page 1933

  12. [12]

    Determination of the number of factors and the experimental error in a data matrix,

    E. R. Malinowski, “Determination of the number of factors and the experimental error in a data matrix,”Analytical Chemistry, vol. 49, no. 4, pp. 612–617, 1977

    work page 1977

  13. [13]

    Short term memory in echo state networks,

    H. Jaeger, “Short term memory in echo state networks,” Technical Report 152, Fraunhofer Institute for Autonomous Intelligent Systems, 2002

    work page 2002

  14. [14]

    Information processing capacity of dynamical systems,

    J. Dambre, D. Verstraeten, B. Schrauwen, and S. Massar, “Information processing capacity of dynamical systems,”Scientific reports, vol. 2, no. 1, pp. 1–7, 2012

    work page 2012

  15. [15]

    Memory and forecasting capacities of nonlinear recurrent networks,

    L. Gonon, L. Grigoryeva, and J.-P. Ortega, “Memory and forecasting capacities of nonlinear recurrent networks,”Physica D: Nonlinear Phenomena, vol. 414, p. 132721, 2020

    work page 2020

  16. [16]

    Unifying framework for information processing in stochastically driven dynamical systems,

    T. Kubota, H. Takahashi, and K. Nakajima, “Unifying framework for information processing in stochastically driven dynamical systems,”Physical Review Research, vol. 3, no. 4, p. 043135, 2021

    work page 2021

  17. [17]

    Asymptotic evaluation of the information processing capacity in reservoir comput- ing,

    Y. Saito, “Asymptotic evaluation of the information processing capacity in reservoir comput- ing,”Neurocomputing, vol. 665, p. 132128, 2026

    work page 2026

  18. [18]

    Short-term memory in orthogonal neural networks,

    O. L. White, D. D. Lee, and H. Sompolinsky, “Short-term memory in orthogonal neural networks,”Phys. Rev. Lett., vol. 92, p. 148102, Apr 2004

    work page 2004

  19. [19]

    Minimum complexity echo state network,

    A. Rodan and P. Tino, “Minimum complexity echo state network,”IEEE transactions on neural networks, vol. 22, no. 1, pp. 131–144, 2010

    work page 2010

  20. [20]

    Optimal nonlinear information pro- cessing capacity in delay-based reservoir computers,

    L. Grigoryeva, J. Henriques, L. Larger, and J.-P. Ortega, “Optimal nonlinear information pro- cessing capacity in delay-based reservoir computers,”Scientific reports, vol. 5, no. 1, p. 12858, 2015

    work page 2015

  21. [21]

    Delay-based reservoir computing: tackling performance degradation due to system response time,

    S. Ort´ ın and L. Pesquera, “Delay-based reservoir computing: tackling performance degradation due to system response time,”Optics Letters, vol. 45, no. 4, pp. 905–908, 2020

    work page 2020

  22. [22]

    Input-driven bifurcations and information processing capacity in spintronics reservoirs,

    N. Akashi, T. Yamaguchi, S. Tsunegi, T. Taniguchi, M. Nishida, R. Sakurai, Y. Wakao, and K. Nakajima, “Input-driven bifurcations and information processing capacity in spintronics reservoirs,”Physical Review Research, vol. 2, no. 4, p. 043303, 2020

    work page 2020

  23. [23]

    Master memory function for delay-based reservoir com- puters with single-variable dynamics,

    F. K¨ oster, S. Yanchuk, and K. L¨ udge, “Master memory function for delay-based reservoir com- puters with single-variable dynamics,”IEEE Transactions on Neural Networks and Learning Systems, vol. 35, no. 6, pp. 7712–7725, 2022. 30

    work page 2022

  24. [24]

    Deriving task specific performance from the information processing capacity of a reservoir computer,

    T. H¨ ulser, F. K¨ oster, K. L¨ udge, and L. Jaurigue, “Deriving task specific performance from the information processing capacity of a reservoir computer,”Nanophotonics, vol. 12, no. 5, pp. 937–947, 2023

    work page 2023

  25. [25]

    Limitations of the recall capabilities in delay-based reservoir computing systems,

    F. K¨ oster, D. Ehlert, and K. L¨ udge, “Limitations of the recall capabilities in delay-based reservoir computing systems,”Cognitive Computation, vol. 15, no. 5, pp. 1419–1426, 2023

    work page 2023

  26. [27]

    Utilizing rate-independent hysteresis for analog computing,

    L. Jaurigue and K. L¨ udge, “Utilizing rate-independent hysteresis for analog computing,”Neu- romorphic Computing and Engineering, vol. 5, no. 4, p. 044007, 2025

    work page 2025

  27. [28]

    Short-term Memory of Deep RNN

    C. Gallicchio, “Short-term memory of deep RNN,”arXiv preprint arXiv:1802.00748, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  28. [29]

    Gaus- sian states of continuous-variable quantum systems provide universal and versatile reservoir computing,

    J. Nokkala, R. Mart´ ınez-Pe˜ na, G. L. Giorgi, V. Parigi, M. C. Soriano, and R. Zambrini, “Gaus- sian states of continuous-variable quantum systems provide universal and versatile reservoir computing,”Communications Physics, vol. 4, no. 1, p. 53, 2021

    work page 2021

  29. [30]

    Information pro- cessing capacity of spin-based quantum reservoir computing systems,

    R. Mart´ ınez-Pe˜ na, J. Nokkala, G. L. Giorgi, R. Zambrini, and M. C. Soriano, “Information pro- cessing capacity of spin-based quantum reservoir computing systems,”Cognitive Computation, vol. 15, no. 5, pp. 1440–1451, 2023

    work page 2023

  30. [31]

    All-optical reservoir computing,

    F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,”Optics express, vol. 20, no. 20, pp. 22783–22795, 2012

    work page 2012

  31. [32]

    Information processing capacity of a single- node reservoir computer: an experimental evaluation,

    B. Vettelschoss, A. R¨ ohm, and M. C. Soriano, “Information processing capacity of a single- node reservoir computer: an experimental evaluation,”IEEE Transactions on Neural Networks and Learning Systems, vol. 33, no. 6, pp. 2714–2725, 2021

    work page 2021

  32. [33]

    Task-independent computational abilities of semicon- ductor lasers with delayed optical feedback for reservoir computing,

    K. Harkhoe and G. Van der Sande, “Task-independent computational abilities of semicon- ductor lasers with delayed optical feedback for reservoir computing,”Photonics, vol. 6, no. 4, 2019

    work page 2019

  33. [34]

    Information processing capacity of spintronic oscillator,

    S. Tsunegi, T. Kubota, A. Kamimaki, J. Grollier, V. Cros, K. Yakushiji, A. Fukushima, S. Yuasa, H. Kubota, K. Nakajima,et al., “Information processing capacity of spintronic oscillator,”Advanced Intelligent Systems, vol. 5, no. 9, p. 2300175, 2023

    work page 2023

  34. [35]

    Deep photonic reservoir computing recurrent network,

    Y.-W. Shen, R.-Q. Li, G.-T. Liu, J. Yu, X. He, L. Yi, and C. Wang, “Deep photonic reservoir computing recurrent network,”Optica, vol. 10, no. 12, pp. 1745–1751, 2023

    work page 2023

  35. [36]

    Memory of recurrent networks: Do we compute it right?,

    G. Ballarin, L. Grigoryeva, and J.-P. Ortega, “Memory of recurrent networks: Do we compute it right?,”Journal of Machine Learning Research, vol. 25, no. 243, pp. 1–38, 2024

    work page 2024

  36. [37]

    Memory capacity of nonlinear recurrent net- works: Is it informative?,

    G. Ballarin, L. Grigoryeva, and J.-P. Ortega, “Memory capacity of nonlinear recurrent net- works: Is it informative?,” inGeometric Science of Information(F. Nielsen and F. Barbaresco, eds.), (Cham), pp. 53–64, Springer Nature Switzerland, 2026

    work page 2026

  37. [38]

    Extreme learning machine: Theory and applica- tions,

    G.-B. Huang, Q.-Y. Zhu, and C.-K. Siew, “Extreme learning machine: Theory and applica- tions,”Neurocomputing, vol. 70, no. 1, pp. 489–501, 2006. Neural Networks. 31

    work page 2006

  38. [39]

    Deep learning with coherent nanophotonic circuits,

    Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund,et al., “Deep learning with coherent nanophotonic circuits,”Nature photonics, vol. 11, no. 7, pp. 441–446, 2017

    work page 2017

  39. [40]

    Fully hardware-implemented memristor convolutional neural network,

    P. Yao, H. Wu, B. Gao, J. Tang, Q. Zhang, W. Zhang, J. J. Yang, and H. Qian, “Fully hardware-implemented memristor convolutional neural network,”Nature, vol. 577, no. 7792, pp. 641–646, 2020

    work page 2020

  40. [41]

    Deep physical neural networks trained with backpropagation,

    L. G. Wright, T. Onodera, M. M. Stein, T. Wang, D. T. Schachter, Z. Hu, and P. L. McMahon, “Deep physical neural networks trained with backpropagation,”Nature, vol. 601, no. 7894, pp. 549–555, 2022

    work page 2022

  41. [42]

    Multilayer spintronic neural networks with ra- diofrequency connections,

    A. Ross, N. Leroux, A. De Riz, D. Markovi´ c, D. Sanz-Hern´ andez, J. Trastoy, P. Bortolotti, D. Querlioz, L. Martins, L. Benetti,et al., “Multilayer spintronic neural networks with ra- diofrequency connections,”Nature Nanotechnology, vol. 18, no. 11, pp. 1273–1280, 2023

    work page 2023

  42. [43]

    Hardware implementation of real-time extreme learning machine in fpga: analysis of precision, resource occupation and performance,

    J. V. Frances-Villora, A. Rosado-Mu˜ noz, J. M. Mart´ ınez-Villena, M. Bataller-Mompean, J. F. Guerrero, and M. Wegrzyn, “Hardware implementation of real-time extreme learning machine in fpga: analysis of precision, resource occupation and performance,”Computers & Electrical Engineering, vol. 51, pp. 139–156, 2016

    work page 2016

  43. [44]

    Photonic extreme learning machine by free-space optical propagation,

    D. Pierangeli, G. Marcucci, and C. Conti, “Photonic extreme learning machine by free-space optical propagation,”Photonics Research, vol. 9, no. 8, pp. 1446–1454, 2021

    work page 2021

  44. [45]

    Experimental property reconstruction in a photonic quantum extreme learning machine,

    A. Suprano, D. Zia, L. Innocenti, S. Lorenzo, V. Cimini, T. Giordani, I. Palmisano, E. Polino, N. Spagnolo, F. Sciarrino,et al., “Experimental property reconstruction in a photonic quantum extreme learning machine,”Physical Review Letters, vol. 132, no. 16, p. 160802, 2024

    work page 2024

  45. [46]

    Weak Kerr nonlinearity boosts the performance of frequency-multiplexed photonic extreme learning machines: a multifaceted approach,

    M. Zajnulina, A. Lupo, and S. Massar, “Weak Kerr nonlinearity boosts the performance of frequency-multiplexed photonic extreme learning machines: a multifaceted approach,”Opt. Express, vol. 33, pp. 7601–7619, Feb 2025

    work page 2025

  46. [47]

    Tackling sampling noise in physical systems for machine learning applications: Fundamental limits and eigentasks,

    F. Hu, G. Angelatos, S. A. Khan, M. Vives, E. T¨ ureci, L. Bello, G. E. Rowlands, G. J. Ribeill, and H. E. T¨ ureci, “Tackling sampling noise in physical systems for machine learning applications: Fundamental limits and eigentasks,”Physical Review X, vol. 13, no. 4, p. 041020, 2023

    work page 2023

  47. [48]

    On the distribution of points in a cube and the approximate evaluation of integrals,

    I. Sobol’, “On the distribution of points in a cube and the approximate evaluation of integrals,” USSR Computational Mathematics and Mathematical Physics, vol. 7, no. 4, pp. 86–112, 1967

    work page 1967

  48. [49]

    Photonic extreme learning machine based on frequency multiplexing,

    A. Lupo, L. Butschek, and S. Massar, “Photonic extreme learning machine based on frequency multiplexing,”Opt. Express, vol. 29, pp. 28257–28276, Aug 2021

    work page 2021

  49. [50]

    Programmable photonic extreme learning machines,

    J. R. Rausell-Campo, A. Hurtado, D. P´ erez-L´ opez, and J. Capmany Francoy, “Programmable photonic extreme learning machines,”Laser & Photonics Reviews, vol. 19, no. 9, p. 2400870, 2025

    work page 2025

  50. [51]

    Neuromorphic computing via fission-based broad- band frequency generation,

    B. Fischer, M. Chemnitz, Y. Zhu, N. Perron, P. Roztocki, B. MacLellan, L. Di Lauro, A. Aadhi, C. Rimoldi, T. H. Falk, and R. Morandotti, “Neuromorphic computing via fission-based broad- band frequency generation,”Advanced Science, vol. 10, no. 35, p. 2303835, 2023

    work page 2023

  51. [52]

    Gradient-based learning applied to document recognition,

    Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,”Proceedings of the IEEE, vol. 86, no. 11, pp. 2278–2324, 1998. 32

    work page 1998

  52. [53]

    Learning to tell two spirals apart,

    K. J. Lang and M. J. Witbrock, “Learning to tell two spirals apart,” inProceedings of the 1988 Connectionist Models Summer School, 1988

    work page 1988

  53. [54]

    Computational metrics and parameters of an injection-locked large area semiconductor laser for neural network computing [invited],

    A. Skalli, X. Porte, N. Haghighi, S. Reitzenstein, J. A. Lott, and D. Brunner, “Computational metrics and parameters of an injection-locked large area semiconductor laser for neural network computing [invited],”Opt. Mater. Express, vol. 12, pp. 2793–2804, Jul 2022

    work page 2022

  54. [55]

    R. A. Horn and C. R. Johnson,Matrix Analysis. Cambridge University Press, 1985

    work page 1985

  55. [56]

    III. on the expression of the product of any two legendre’s coefficients by means of a series of legendre’s coefficients,

    J. C. Adams, “III. on the expression of the product of any two legendre’s coefficients by means of a series of legendre’s coefficients,”Proceedings of the Royal Society of London, vol. 27, no. 185-189, pp. 63–71, 1878

    work page

  56. [57]

    A. R. Edmonds,Angular momentum in quantum mechanics, vol. 4. Princeton university press, 1996

    work page 1996

  57. [58]

    The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam,

    L. F. Richardson, “The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam,”Philosophical Transactions of the Royal Society of London, Series A: Containing Papers of a Mathematical or Physical Character, vol. 210, pp. 307–357, 01 1911

    work page 1911

  58. [59]

    Nonlinear Schr¨ odinger kernel for hardware acceleration of machine learning,

    T. Zhou, F. Scalzo, and B. Jalali, “Nonlinear Schr¨ odinger kernel for hardware acceleration of machine learning,”Journal of Lightwave Technology, vol. 40, no. 5, pp. 1308–1319, 2022

    work page 2022

  59. [60]

    Supercontinuum neural network and analog computing eval- uation,

    K. F. Lee and M. E. Fermann, “Supercontinuum neural network and analog computing eval- uation,”Phys. Rev. A, vol. 109, p. 033521, Mar 2024

    work page 2024

  60. [61]

    Deep time-delay reservoir computing: Dynamics and memory capacity,

    M. Goldmann, F. K¨ oster, K. L¨ udge, and S. Yanchuk, “Deep time-delay reservoir computing: Dynamics and memory capacity,”Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 30, no. 9, 2020

    work page 2020

  61. [62]

    Code for Information Processing Capacity of Stationary Physical Sys- tems

    R. U. Ramachandran, “Code for Information Processing Capacity of Stationary Physical Sys- tems.”https://doi.org/10.5281/zenodo.20229947, 2026. 33

    work page doi:10.5281/zenodo.20229947 2026