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arxiv: 2606.31588 · v1 · pith:C2BNPIZ2new · submitted 2026-06-30 · 💻 cs.ET

Power law scaling for classification accuracy in physical neural networks

Andrei V. Ermolaev , Mathilde Hary , Anas Skalli , Go\"ery Genty , Marcin Gebski , Tomasz Czyszanowski , Stephan Reitzenstein , James A. Lott
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John M. Dudley Daniel Brunner
This is my paper

Pith reviewed 2026-07-01 02:17 UTC · model grok-4.3

classification 💻 cs.ET
keywords physical neural networksHotelling Trace Criterionpower law scalingclassification accuracyoptical neural networksscaling lawsneural computation
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The pith

Classification loss in physical neural networks follows a power law in the Hotelling Trace Criterion, with data from different substrates collapsing on task-specific curves.

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

The paper establishes that a separability measure called the Hotelling Trace Criterion predicts how well physical neural networks will classify data. This prediction works through a power-law relationship between the criterion and classification loss. The relationship holds with high correlation for standard image tasks and across physically different systems such as optical fibers and lasers. Because the criterion can be computed without training, it offers an efficient way to forecast performance after calibrating the exponent on a few examples. It also shows that training spreads capacity unevenly through the layers.

Core claim

Classification loss follows a power law in HTC, with Pearson correlation coefficients exceeding 0.99 for MNIST and ≈0.97 for Fashion-MNIST. Experimental and simulated data from physically distinct systems collapse onto a single scaling curve determined by the task rather than the substrate. Applying HTC layer-by-layer during training reveals that gradient-based optimisation distributes representational capacity unevenly across PNN layers.

What carries the argument

The Hotelling Trace Criterion (HTC), a task-conditioned measure of PNN-state separability evaluated without training.

If this is right

  • Performance predictions for new systems require only the HTC measurement once the task-specific exponent is calibrated.
  • HTC can diagnose training efficiency by showing uneven capacity distribution across layers.
  • HTC acts as a substrate-agnostic figure of merit for comparing different physical neural network implementations.

Where Pith is reading between the lines

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

  • This approach may allow hardware designers to optimize physical parameters directly for higher HTC values on target tasks.
  • The task-determined scaling suggests similar laws could apply to other computational tasks like regression if the separability measure is adapted.
  • Universal collapse across substrates points to a deeper connection between physical dynamics and information processing independent of specific implementation details.

Load-bearing premise

The power-law relationship between HTC and classification loss generalizes across arbitrary physical substrates and tasks once a task-specific exponent has been calibrated on a small set of trained systems.

What would settle it

Measuring HTC and classification loss on a new physical system for MNIST and finding that the points do not lie on the previously established power-law curve would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.31588 by Anas Skalli, Andrei V. Ermolaev, Daniel Brunner, Go\"ery Genty, James A. Lott, John M. Dudley, Marcin Gebski, Mathilde Hary, Stephan Reitzenstein, Tomasz Czyszanowski.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a). Peak accuracy is obtained at an intermedi￾ate value N ∼ 3, beyond which performance degrades markedly with increasing input power. A qualitatively similar optimum is found in the LA-VCSEL experiment, where the input power is held constant while the detun￾ing between the injection laser and the LA-VCSEL res￾onance is varied, see [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the classification loss as a function of the HTC metric on a double logarithmic scale, revealing a clear power law scaling across both the MNIST digit and fashion MNIST benchmarks. For MNIST, the dataset in￾cludes experimental measurements from the HNLF and the LA-VCSEL, complemented by numerical simulations of the HNLF and the CNON. The fashion MNIST results, being a more demanding benchmark, are cu… view at source ↗
Figure 5
Figure 5. Figure 5: (a) shows the evolution of HTC across the three hidden layers for 20 training epochs after which train￾ing has converged. All three layers exhibit an initial in￾crease in HTC, reflecting the early reorganisation of in￾ternal representations. However, the trajectories quickly diverge: the first hidden layer’s HTC saturates within a few epochs and subsequently undergoes a steady decline, while the second and… view at source ↗
read the original abstract

Physical neural networks (PNNs) harness the intrinsic complexity of physical systems to perform neural computation, potentially at speeds and energy efficiencies inaccessible to conventional digital hardware. Yet, a principled framework for quantifying and predicting their computing accuracy across diverse substrates has remained elusive. Here we introduce the Hotelling Trace Criterion (HTC), a task-conditioned measure of PNN- state separability that can be evaluated without training. We demonstrate that it predicts PNN classification performance with high fidelity across highly nonlinear optical fibres, vertical-cavity surface-emitting lasers, and coupled nonlinear oscillator networks, for benchmark tasks of different difficulty. Classification loss follows a power law in HTC, with Pearson correlation coefficients exceeding 0.99 for MNIST and $\approx$0.97 for Fashion-MNIST, noteworthy experimental and simulated data from physically distinct systems collapse onto a single scaling curve determined by the task rather than the substrate. Applying HTC layer-by-layer during training further reveals that gradient-based optimisation distributes representational capacity unevenly across PNN layers, providing a quantitative diagnostic of training and architecture efficiency invisible to standard loss monitoring. Crucially, once the scaling exponent is established from a small number of trained calibration systems, all further performance predictions require no training since performance can be derived from the much more efficient HTC measurement. These results establish HTC as a substrate-agnostic figure of merit for comparing and scaling PNNs, advancing the field further towards a complete theory connecting fundamental hardware parameters to task performance through universal scaling laws.

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

2 major / 0 minor

Summary. The manuscript introduces the Hotelling Trace Criterion (HTC) as a task-conditioned, training-free measure of state separability in physical neural networks (PNNs). It claims that classification loss follows a power-law relationship with HTC across physically distinct systems (nonlinear fibres, VCSELs, coupled oscillators), with Pearson correlations >0.99 on MNIST and ≈0.97 on Fashion-MNIST; data from these systems collapse onto task-specific curves, so that a task-specific exponent calibrated on a small set of trained systems enables subsequent training-free performance prediction from HTC alone. The work also reports that layer-by-layer HTC during training reveals uneven distribution of representational capacity.

Significance. If the power-law relationship and cross-substrate collapse are robust, the result would be significant: it supplies a substrate-agnostic, low-cost figure of merit that connects hardware parameters to task performance via a simple scaling law and supplies a diagnostic for training efficiency invisible to standard loss curves. The training-free prediction aspect, once the exponent is fixed, would be a practical advance for the field.

major comments (2)
  1. [Abstract] Abstract: the reported Pearson correlations (>0.99 and ≈0.97) are presented without error bars, the number of data points entering each fit, or the precise data-selection criteria; these omissions are load-bearing for the central claim that HTC predicts performance “with high fidelity.”
  2. [Abstract] Abstract: the scaling exponent is obtained by fitting a small number of trained calibration systems and is then used to predict performance from HTC alone; the manuscript must quantify how sensitive the claimed training-free regime is to the choice and number of calibration systems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight important aspects of statistical rigor and robustness in our presentation. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the reported Pearson correlations (>0.99 and ≈0.97) are presented without error bars, the number of data points entering each fit, or the precise data-selection criteria; these omissions are load-bearing for the central claim that HTC predicts performance “with high fidelity.”

    Authors: We agree that these details should be explicit. In the revised manuscript we will add to the abstract (or a concise parenthetical) the number of data points underlying each reported correlation, the precise selection criteria used to include configurations, and error bars or bootstrap-derived uncertainties on the Pearson coefficients. The underlying counts and criteria are already documented in the methods and supplementary figures; the revision will simply surface them at the abstract level without altering the reported values. revision: yes

  2. Referee: [Abstract] Abstract: the scaling exponent is obtained by fitting a small number of trained calibration systems and is then used to predict performance from HTC alone; the manuscript must quantify how sensitive the claimed training-free regime is to the choice and number of calibration systems.

    Authors: We acknowledge the value of quantifying calibration sensitivity. In the revised version we will include an additional analysis (main text or supplementary) that tests the stability of the fitted exponent and downstream predictions under leave-one-out cross-validation and under systematic variation in the number of calibration systems. This will demonstrate the minimal number of systems required for reliable training-free extrapolation and the variability introduced by different calibration subsets. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical scaling law validated externally

full rationale

HTC is defined independently as a training-free separability metric. The power-law relation (loss ~ HTC^alpha) is reported as an empirical fit with high Pearson correlations across distinct physical substrates (fibres, VCSELs, oscillators) and tasks; data collapse onto task-specific curves is presented as experimental evidence, not a definitional identity. Calibration of the task-specific exponent on a small set of systems followed by HTC-based prediction on new systems is a standard empirical scaling procedure, not a reduction of the output to the fitted inputs by construction. No self-citations, uniqueness theorems, or ansatzes are invoked to force the result. The chain is self-contained against the reported benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the empirical observation that HTC correlates with trained classification loss via a power law whose exponent is task-dependent; the abstract supplies no independent derivation or external benchmark for this relationship.

free parameters (1)
  • task-specific scaling exponent
    Determined from a small number of trained calibration systems; used to convert HTC values into predicted loss for new systems.
axioms (1)
  • domain assumption HTC computed on untrained PNN states is a faithful proxy for the separability achieved after gradient-based training
    Invoked when the authors state that HTC predicts performance without training.

pith-pipeline@v0.9.1-grok · 5833 in / 1270 out tokens · 52622 ms · 2026-07-01T02:17:35.176269+00:00 · methodology

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Reference graph

Works this paper leans on

83 extracted references · 51 canonical work pages · 6 internal anchors

  1. [1]

    and Hary, Mathilde and Leybov, Lev and Ryczkowski, Piotr and Skalli, Anas and Brunner, Daniel and Genty, Goëry and Dudley, John M

    Ermolaev, Andrei V. and Hary, Mathilde and Leybov, Lev and Ryczkowski, Piotr and Skalli, Anas and Brunner, Daniel and Genty, Goëry and Dudley, John M. , year =. Limits of nonlinear and dispersive fiber propagation for an optical fiber-based extreme learning machine , volume =. Optics Letters , publisher =. doi:10.1364/ol.562186 , number =

    work page doi:10.1364/ol.562186

  2. [2]

    Nature communications , volume=

    Toward a formal theory for computing machines made out of whatever physics offers , author=. Nature communications , volume=. 2023 , publisher=

    2023

  3. [3]

    arXiv preprint arXiv:2601.15340 , year=

    Learning Nonlinear Heterogeneity in Physical Kolmogorov-Arnold Networks , author=. arXiv preprint arXiv:2601.15340 , year=

    work page arXiv

  4. [4]

    Dudley, J. M. and Genty, G. and Heidt, A. and Sylvestre, T. and Travers, J. C. and Taylor, J. R. , year =. Fibre supercontinuum generation: Progress and perspectives , volume =. Europhysics Letters EPL , publisher =

  5. [5]

    , publisher =

    Agrawal, Govind P. , publisher =. Nonlinear Fiber Optics , year =

  6. [6]

    Dudley and Goëry Genty , journal =

    Mikko Närhi and Lauri Salmela and Juha Toivonen and Cyril Billet and John M. Dudley and Goëry Genty , journal =. Machine learning analysis of extreme events in optical fibre modulation instability , year =. doi:10.1038/s41467-018-07355-y , publisher =

    work page doi:10.1038/s41467-018-07355-y

  7. [7]

    Dudley, J. M. and Genty, G. and Coen, S. , title =. Reviews of Modern Physics , year =. doi:10.1103/RevModPhys.78.1135 , issue =

    work page doi:10.1103/revmodphys.78.1135

  8. [8]

    and Feehan, James S

    Heidt, Alexander M. and Feehan, James S. and Price, Jonathan H. V. and Feurer, Thomas , journal =. Limits of coherent supercontinuum generation in normal dispersion fibers , year =

  9. [9]

    On the implementation of nonlinearities in optical neural networks: opinion , year =

    Niyazi Ulas Dinc and Ilker Oguz and Mustafa Yildirim and Christophe Moser and Demetri Psaltis , journal =. On the implementation of nonlinearities in optical neural networks: opinion , year =

  10. [10]

    Dudley and Goëry Genty and Arnaud Mussot and Amin Chabchoub and Fr

    John M. Dudley and Goëry Genty and Arnaud Mussot and Amin Chabchoub and Fr. Nature Reviews Physics , title =. 2019 , number =. doi:10.1038/s42254-019-0100-0 , publisher =

    work page doi:10.1038/s42254-019-0100-0 2019

  11. [11]

    Optics Express , title =

    Fran. Optics Express , title =. 2012 , number =. doi:10.1364/OE.20.022783 , keywords =

    work page doi:10.1364/oe.20.022783 2012

  12. [12]

    and Rimoldi, Cristina and Falk, Tiago H

    Fischer, Bennet and Chemnitz, Mario and Zhu, Yi and Perron, Nicolas and Roztocki, Piotr and Maclellan, Benjamin and Di Lauro, Luigi and Aadhi, A. and Rimoldi, Cristina and Falk, Tiago H. and Morandotti, Roberto , journal =. Neuromorphic Computing via Fission-based Broadband Frequency Generation , year =. doi:10.1002/advs.202303835 , earlyaccessdate =

    work page doi:10.1002/advs.202303835

  13. [13]

    Prucnal , journal =

    Aashu Jha and Chaoran Huang and Paul R. Prucnal , journal =. Reconfigurable all-optical nonlinear activation functions for neuromorphic photonics , year =. doi:10.1364/OL.398234 , keywords =

    work page doi:10.1364/ol.398234

  14. [14]

    Larger and M

    L. Larger and M. C. Soriano and D. Brunner and L. Appeltant and J. M. Gutierrez and L. Pesquera and C. R. Mirasso and I. Fischer , journal =. Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing , year =. doi:10.1364/OE.20.003241 , keywords =

    work page doi:10.1364/oe.20.003241

  15. [15]

    and Fermann, Martin E

    Lee, Kevin F. and Fermann, Martin E. , journal =. Supercontinuum neural network and analog computing evaluation , year =. doi:10.1103/PhysRevA.109.033521 , eissn =

    work page doi:10.1103/physreva.109.033521

  16. [16]

    Nanophotonics , doi =

    Optical neuromorphic computing via temporal up-sampling and trainable encoding on a telecom device platform , author =. Nanophotonics , doi =

  17. [17]

    Research progress in optical neural networks: theory, applications and developments , year =

    Liu, Jia and Wu, Qiuhao and Sui, Xiubao and Chen, Qian and Gu, Guohua and Wang, Liping and Li, Shengcai , journal =. Research progress in optical neural networks: theory, applications and developments , year =. doi:10.1186/s43074-021-00026-0 , publisher =

    work page doi:10.1186/s43074-021-00026-0

  18. [18]

    Theory of Neuromorphic Computing by Waves: Machine Learning by Rogue Waves, Dispersive Shocks, and Solitons , year =

    Marcucci, Giulia and Pierangeli, Davide and Conti, Claudio , journal =. Theory of Neuromorphic Computing by Waves: Machine Learning by Rogue Waves, Dispersive Shocks, and Solitons , year =. doi:10.1103/PhysRevLett.125.093901 , issue =

    work page doi:10.1103/physrevlett.125.093901

  19. [19]

    Journal of Physics: Photonics , volume =

    A complete, parallel and autonomous photonic neural network in a semiconductor multimode laser , author =. Journal of Physics: Photonics , volume =. 2021 , publisher =

    2021

  20. [20]

    Nature communications , volume=

    Parallel photonic information processing at gigabyte per second data rates using transient states , author=. Nature communications , volume=. 2013 , publisher=

    2013

  21. [21]

    Optical Materials Express , volume=

    Computational metrics and parameters of an injection-locked large area semiconductor laser for neural network computing , author=. Optical Materials Express , volume=. 2022 , publisher=

    2022

  22. [22]

    Azzam and Jonathan George and Alexander V

    Mario Miscuglio and Armin Mehrabian and Zibo Hu and Shaimaa I. Azzam and Jonathan George and Alexander V. Kildishev and Matthew Pelton and Volker J. Sorger , journal =. All-optical nonlinear activation function for photonic neural networks. [. 2018 , number =. doi:10.1364/OME.8.003851 , keywords =

    work page doi:10.1364/ome.8.003851 2018

  23. [23]

    Optical computing with supercontinuum generation in photonic crystal fibers , volume =

    Azka Maula Iskandar Muda and Uğur Teğin , journal =. Optical computing with supercontinuum generation in photonic crystal fibers , volume =. 2024 , abstract =

    2024

  24. [24]

    Advanced Photonics , title =

    Oguz, Ilker and Hsieh, Jih-Liang and Dinc, Niyazi Ulas and Te. Advanced Photonics , title =. 2024 , issn =. doi:10.1117/1.ap.6.1.016002 , publisher =

    work page doi:10.1117/1.ap.6.1.016002 2024

  25. [25]

    Nature Computational Science , title =

    Te. Nature Computational Science , title =. 2021 , issn =. doi:10.1038/s43588-021-00112-0 , publisher =

    work page doi:10.1038/s43588-021-00112-0 2021

  26. [26]

    Nonlinear processing with linear optics , year =

    Yildirim, Mustafa and Dinc, Niyazi Ulas and Oguz, Ilker and Psaltis, Demetri and Moser, Christophe , journal =. Nonlinear processing with linear optics , year =. doi:10.1038/s41566-024-01494-z , earlyaccessdate =

    work page doi:10.1038/s41566-024-01494-z

  27. [27]

    Nonlinear optical feature generator for machine learning , year =

    Yildirim, Mustafa and Oguz, Ilker and Kaufmann, Fabian and Escale, Marc Reig and Grange, Rachel and Psaltis, Demetri and Moser, Christophe , journal =. Nonlinear optical feature generator for machine learning , year =. doi:10.1063/5.0158611 , orcid-numbers =

    work page doi:10.1063/5.0158611

  28. [28]

    All-optical neural network with nonlinear activation functions , year =

    Ying Zuo and Bohan Li and Yujun Zhao and Yue Jiang and You-Chiuan Chen and Peng Chen and Gyu-Boong Jo and Junwei Liu and Shengwang Du , journal =. All-optical neural network with nonlinear activation functions , year =. doi:10.1364/OPTICA.6.001132 , keywords =

    work page doi:10.1364/optica.6.001132

  29. [29]

    Holography in artificial neural networks , volume =

    Psaltis, Demetri and Brady, David and Gu, Xiang-Guang and Lin, Steven , year =. Holography in artificial neural networks , volume =. Nature , publisher =. doi:10.1038/343325a0 , number =

    work page doi:10.1038/343325a0

  30. [30]

    Wetzstein, Gordon and Ozcan, Aydogan and Gigan, Sylvain and Fan, Shanhui and Englund, Dirk and Soljačić, Marin and Denz, Cornelia and Miller, David A. B. and Psaltis, Demetri , year =. Inference in artificial intelligence with deep optics and photonics , volume =. Nature , publisher =. doi:10.1038/s41586-020-2973-6 , number =

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

  31. [31]

    Stumpf, M. C. and Pekarek, S. and Oehler, A. E. H. and S\". Self-referencable frequency comb from a 170-fs, 1.5- m solid-state laser oscillator , volume =. Applied Physics B , publisher =. 2010 , pages =. doi:10.1007/s00340-009-3854-8 , number =

    work page doi:10.1007/s00340-009-3854-8 2010

  32. [32]

    Corwin, K. L. and Newbury, N. R. and Dudley, J. M. and Coen, S. and Diddams, S. A. and Weber, K. and Windeler, R. S. , year =. Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber , volume =. Physical Review Letters , publisher =

  33. [33]

    and Amans, D

    Brainis, E. and Amans, D. and Massar, S. , year =. Scalar and vector modulation instabilities induced by vacuum fluctuations in fibers: Numerical study , volume =. Physical Review A , pages =. doi:10.1103/physreva.71.023808 , number =

    work page doi:10.1103/physreva.71.023808

  34. [34]

    Nanophotonics , volume =

    Nonlinear inference capacity of fiber-optical extreme learning machines , author =. Nanophotonics , volume =

  35. [35]

    Perspective on the next generation of ultra-low noise fiber supercontinuum sources and their emerging applications in spectroscopy, imaging, and ultrafast photonics , volume =

    Rampur, Anupamaa and Spangenberg, Dirk-Mathys and Sierro, Benoît and H\". Perspective on the next generation of ultra-low noise fiber supercontinuum sources and their emerging applications in spectroscopy, imaging, and ultrafast photonics , volume =. Applied Physics Letters , publisher =. doi:10.1063/5.0053436 , number =

    work page doi:10.1063/5.0053436

  36. [36]

    Optical computing with supercontinuum generation in photonic crystal fibers , volume =

    Iskandar Muda and Azka Maula and Te. Optical computing with supercontinuum generation in photonic crystal fibers , volume =. Optics Express , publisher =. 2025 , pages =. doi:10.1364/oe.539374 , number =

    work page doi:10.1364/oe.539374 2025

  37. [37]

    A Unified Framework for Reservoir Computing and Extreme Learning Machines based on a Single Time-delayed Neuron , volume =

    Ort\'. A Unified Framework for Reservoir Computing and Extreme Learning Machines based on a Single Time-delayed Neuron , volume =. Scientific Reports , publisher =. doi:10.1038/srep14945 , number =

    work page doi:10.1038/srep14945

  38. [38]

    , year =

    Zhu, Zhaoming and Brown, Thomas G. , year =. Polarization properties of supercontinuum spectra generated in birefringent photonic crystal fibers , volume =. Journal of the Optical Society of America B , publisher =

  39. [39]

    and Knight, Jonathan C

    Coen, Stéphane and Chau, Alvin Hing Lun and Leonhardt, Rainer and Harvey, John D. and Knight, Jonathan C. and Wadsworth, William J. and Russell, Philip St. J. , year =. Supercontinuum generation by stimulated Raman scattering and parametric four-wave mixing in photonic crystal fibers , volume =. Journal of the Optical Society of America B , publisher =

  40. [40]

    Extreme learning machine: Theory and applications , year =

    Huang, Guang-Bin and Zhu, Qin-Yu and Siew, Chee-Kheong , journal =. Extreme learning machine: Theory and applications , year =. doi:10.1016/j.neucom.2005.12.126 , publisher =

    work page doi:10.1016/j.neucom.2005.12.126 2005

  41. [41]

    and Miscuglio, Mario and Al-Qadasi, Mohammed and Mukherjee, Avilash and Lampe, Lutz and Nichols, Mitchell and Tait, Alexander N

    Huang, Chaoran and Sorger, Volker J. and Miscuglio, Mario and Al-Qadasi, Mohammed and Mukherjee, Avilash and Lampe, Lutz and Nichols, Mitchell and Tait, Alexander N. and Ferreira de Lima, Thomas and Marquez, Bicky A. and Wang, Jiahui and Chrostowski, Lukas and Fok, Mable P. and Brunner, Daniel and Fan, Shanhui and Shekhar, Sudip and Prucnal, Paul R. and S...

    work page doi:10.1080/23746149.2021.1981155 2021

  42. [42]

    and Pandit, Tej and Merkel, Cory and Kubendran, Rajkumar and Aimone, James B

    Kudithipudi, Dhireesha and Schuman, Catherine and Vineyard, Craig M. and Pandit, Tej and Merkel, Cory and Kubendran, Rajkumar and Aimone, James B. and Orchard, Garrick and Mayr, Christian and Benosman, Ryad and Hays, Joe and Young, Cliff and Bartolozzi, Chiara and Majumdar, Amitava and Cardwell, Suma George and Payvand, Melika and Buckley, Sonia and Kulka...

    work page doi:10.1038/s41586-024-08253-8

  43. [43]

    Optically accelerated extreme learning machine using hot atomic vapors , year =

    Azam, Pierre and Kaiser, Robin , journal =. Optically accelerated extreme learning machine using hot atomic vapors , year =. doi:10.1103/physrevapplied.22.034041 , publisher =

    work page doi:10.1103/physrevapplied.22.034041

  44. [44]

    Deep linear discriminant analysis on fisher networks: A hybrid architecture for person re-identification , year =

    Wu, Lin and Chunhua Shen and Hengel, Anton van den , journal =. Deep linear discriminant analysis on fisher networks: A hybrid architecture for person re-identification , year =. doi:10.1016/j.patcog.2016.12.022 , publisher =

    work page doi:10.1016/j.patcog.2016.12.022 2016

  45. [45]

    Generalized Fisher Score for Feature Selection

    Gu, Quanquan and Li, Zhenhui and Han, Jiawei , title =. 2012 , copyright =. doi:10.48550/ARXIV.1202.3725 , keywords =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1202.3725 2012

  46. [46]

    Carroll, T. L. , journal =. Do reservoir computers work best at the edge of chaos? , year =. doi:10.1063/5.0038163 , publisher =

    work page doi:10.1063/5.0038163

  47. [47]

    and McMahon, Peter L

    Momeni, Ali and Rahmani, Babak and Scellier, Benjamin and Wright, Logan G. and McMahon, Peter L. and Wanjura, Clara C. and Li, Yuhang and Skalli, Anas and Berloff, Natalia G. and Onodera, Tatsuhiro and Oguz, Ilker and Morichetti, Francesco and del Hougne, Philipp and Le Gallo, Manuel and Sebastian, Abu and Mirhoseini, Azalia and Zhang, Cheng and Marković,...

    work page doi:10.1038/s41586-025-09384-2

  48. [48]

    and Czyszanowski, Tomasz and Brunner, Daniel , title =

    Skalli, Anas and Sunada, Satoshi and Goldmann, Mirko and Gebski, Marcin and Reitzenstein, Stephan and Lott, James A. and Czyszanowski, Tomasz and Brunner, Daniel , title =. 2025 , copyright =. doi:10.48550/ARXIV.2503.16943 , keywords =

    work page doi:10.48550/arxiv.2503.16943 2025

  49. [49]

    Scalable Photonic Neural Networks via Surrogate Scattering-Matrix Inverse Design

    Muda, Azka Maula Iskandar and Teğin, Uğur , title =. 2026 , copyright =. doi:10.48550/ARXIV.2604.21301 , keywords =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.21301 2026

  50. [50]

    Linear discriminant analysis: A detailed tutorial , year =

    Tharwat, Alaa and Gaber, Tarek and Ibrahim, Abdelhameed and Hassanien, Aboul Ella , journal =. Linear discriminant analysis: A detailed tutorial , year =. doi:10.3233/aic-170729 , month = May, publisher =

    work page doi:10.3233/aic-170729

  51. [51]

    The Generalization of Student’s Ratio , year =

    Hotelling, Harold , journal =. The Generalization of Student’s Ratio , year =. doi:10.1214/aoms/1177732979 , publisher =

    work page doi:10.1214/aoms/1177732979

  52. [52]

    Berkeley Symp

    Harold Hotelling , title =. Berkeley Symp. on Math. Statist. and Prob. , pages =. 1951 , volume =

    1951

  53. [53]

    Hotelling

    Hotelling, H. , journal =. Analysis of a complex of statistical variables into principal components. , year =. doi:10.1037/h0071325 , publisher =

    work page doi:10.1037/h0071325

  54. [54]

    Discriminative clustering via extreme learning machine , year =

    Huang, Gao and Liu, Tianchi and Yang, Yan and Lin, Zhiping and Song, Shiji and Wu, Cheng , journal =. Discriminative clustering via extreme learning machine , year =. doi:10.1016/j.neunet.2015.06.002 , publisher =

    work page doi:10.1016/j.neunet.2015.06.002 2015

  55. [55]

    and Minhas, R

    Mohammed, A.A. and Minhas, R. and Jonathan Wu, Q.M. and Sid-Ahmed, M.A. , journal =. Human face recognition based on multidimensional PCA and extreme learning machine , year =. doi:10.1016/j.patcog.2011.03.013 , publisher =

    work page doi:10.1016/j.patcog.2011.03.013 2011

  56. [56]

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

    Ramachandran, Rahul Uma and Massar, Serge , title =. 2026 , copyright =. doi:10.48550/ARXIV.2605.19152 , keywords =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.19152 2026

  57. [57]

    1972 , url =

    Keinosuke Fukunaga , title =. 1972 , url =

    1972

  58. [58]

    Rao, M. M. , journal =. Discriminant analysis , year =. doi:10.1007/bf02865898 , publisher =

    work page doi:10.1007/bf02865898

  59. [59]

    Minimum Class Variance Extreme Learning Machine for Human Action Recognition , year =

    Iosifidis, Alexandros and Tefas, Anastasios and Pitas, Ioannis , journal =. Minimum Class Variance Extreme Learning Machine for Human Action Recognition , year =. doi:10.1109/tcsvt.2013.2269774 , publisher =

    work page doi:10.1109/tcsvt.2013.2269774 2013

  60. [60]

    Fiete, R. D. and Barrett, H. H. and Smith, W. E. and Myers, K. J. , journal =. Hotelling trace criterion and its correlation with human-observer performance , year =. doi:10.1364/josaa.4.000945 , publisher =

    work page doi:10.1364/josaa.4.000945

  61. [61]

    Lawley, D. N. , journal =. A Generalization of Fisher’s z Test , year =. doi:10.2307/2332232 , publisher =

    work page doi:10.2307/2332232

  62. [62]

    , publisher =

    Bishop, Christopher M. , publisher =. Pattern Recognition and Machine Learning , year =

  63. [63]

    Photonic neuromorphic computing using vertical cavity semiconductor lasers , year =

    Skalli, Anas and Robertson, Joshua and Owen-Newns, Dafydd and Hejda, Matej and Porte, Xavier and Reitzenstein, Stephan and Hurtado, Antonio and Brunner, Daniel , journal =. Photonic neuromorphic computing using vertical cavity semiconductor lasers , year =. doi:10.1364/ome.450926 , publisher =

    work page doi:10.1364/ome.450926

  64. [64]

    Multilayer Fisher extreme learning machine for classification , year =

    Lai, Jie and Wang, Xiaodan and Xiang, Qian and Wang, Jian and Lei, Lei , journal =. Multilayer Fisher extreme learning machine for classification , year =. doi:10.1007/s40747-022-00867-7 , publisher =

    work page doi:10.1007/s40747-022-00867-7

  65. [65]

    and Barrett, Harrison H

    Smith, Warren E. and Barrett, Harrison H. , journal =. Hotelling trace criterion as a figure of merit for the optimization of imaging systems , year =. doi:10.1364/josaa.3.000717 , publisher =

    work page doi:10.1364/josaa.3.000717

  66. [66]

    Nature communications , volume=

    Information processing using a single dynamical node as complex system , author=. Nature communications , volume=. 2011 , publisher=

    2011

  67. [67]

    and Farhat, N

    Psaltis, D. and Farhat, N. H. , title =. Opt.\ Lett. , volume =. 1985 , doi =

    1985

  68. [68]

    and Soriano, M

    Larger, L. and Soriano, M. C. and Brunner, D. and Appeltant, L. and Gutierrez, J. M. and Pesquera, L. and Mirasso, C. R. and Fischer, I. , title =. Opt.\ Express , volume =. 2012 , doi =

    2012

  69. [69]

    and Schneider, B

    Duport, F. and Schneider, B. and Smerieri, A. and Haelterman, M. and Massar, S. , title =. Opt.\ Express , volume =. 2012 , doi =

    2012

  70. [70]

    and Riou, M

    Torrejon, J. and Riou, M. and Araujo, F. A. and Tsunegi, S. and Khalsa, G. and Querlioz, D. and Bortolotti, P. and Cros, V. and Yakushiji, K. and Fukushima, A. and Kubota, H. and Yuasa, S. and Stiles, M. D. and Grollier, J. , title =. Nature , volume =. 2017 , doi =

    2017

  71. [71]

    and Talatchian, P

    Romera, M. and Talatchian, P. and Tsunegi, S. and Araujo, F. A. and Cros, V. and Bortolotti, P. and Yakushiji, K. and Fukushima, A. and Kubota, H. and Yuasa, S. and Ernoult, M. and Vodenicarevic, D. and Hirtzlin, T. and Locatelli, N. and Querlioz, D. and Grollier, J. , title =. Nature , volume =. 2018 , doi =

    2018

  72. [72]

    and Mejaouri, S

    Dion, G. and Mejaouri, S. and Sylvestre, J. , title =. J.\ Appl.\ Phys. , volume =. 2018 , doi =

    2018

  73. [73]

    and Brunner, D

    Hary, M. and Brunner, D. and Leybov, L. and Ryczkowski, P. and Dudley, J. M. and Genty, G. , title =. Nanophotonics , volume =. 2025 , doi =

    2025

  74. [74]

    and Verstraeten, D

    Dambre, J. and Verstraeten, D. and Schrauwen, B. and Massar, S. , title =. Sci.\ Rep. , volume =. 2012 , doi =

    2012

  75. [75]

    Nature , volume=

    Deep physical neural networks trained with backpropagation , author=. Nature , volume=. 2022 , publisher=

    2022

  76. [76]

    Journal of Lightwave Technology , volume=

    40 Gbps with electrically parallel triple and septuple 980 nm VCSEL arrays , author=. Journal of Lightwave Technology , volume=. 2020 , publisher=

    2020

  77. [77]

    Nature Photonics , volume=

    Photonics for artificial intelligence and neuromorphic computing , author=. Nature Photonics , volume=. 2021 , publisher=

    2021

  78. [78]

    and Angelatos, G

    Hu, F. and Angelatos, G. and Khan, S. A. and Vives, M. and T\". Tackling Sampling Noise in Physical Systems for Machine Learning Applications: Fundamental Limits and Eigentasks , journal =. 2023 , doi =

    2023

  79. [79]

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

    Information Processing Capacity of Stationary Physical Systems: Theory, Data-efficient Estimation Methods, and Photonic Demonstration , author=. arXiv preprint arXiv:2605.19152 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  80. [80]

    Nature Reviews Physics , volume=

    Physics for neuromorphic computing , author=. Nature Reviews Physics , volume=. 2020 , publisher=

    2020

Showing first 80 references.