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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

discussion (0)

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Reference graph

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