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arxiv: 2606.30384 · v1 · pith:C4UL7QABnew · submitted 2026-06-29 · 💻 cs.LG · cond-mat.dis-nn· nlin.CD· physics.data-an

Scalar Representations of Neural Network Training Dynamics

Pedro Jim\'enez-Gonz\'alez , Miguel C. Soriano , Lucas Lacasa This is my paper

Pith reviewed 2026-06-30 07:04 UTC · model grok-4.3

classification 💻 cs.LG cond-mat.dis-nnnlin.CDphysics.data-an
keywords neural network trainingtraining dynamicsLyapunov exponentscalar embeddingtemporal networkschaos in optimizationMNISTasymptotic states
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The pith

Scalar embeddings of neural network training trajectories preserve key dynamical features including Lyapunov exponents.

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

The paper establishes that treating neural network parameter trajectories as temporal networks and applying scalar embedding techniques produces a low-dimensional representation of the training process. This embedding preserves dynamical features observed in the full parameter space, such as the appearance of sensitivity to initial conditions at particular learning rates and an accurate match to the maximum Lyapunov exponent computed directly. The embedded trajectory is further used to extract a characteristic decorrelation time and to examine the statistical structure of asymptotic states via a spacing observable. A sympathetic reader would care because analyzing high-dimensional loss landscapes directly is intractable, so a faithful low-dimensional proxy could make the study of optimization dynamics more tractable.

Core claim

By embedding the training trajectories of a multilayer perceptron on MNIST as scalar time series using temporal network methods, the resulting low-dimensional representation preserves the main dynamical features observed in the original parameter space, including the emergence of sensitivity to initial conditions for specific learning rate regimes and an accurate reconstruction of the network's maximum Lyapunov exponent. The embedded space also yields a characteristic time analogous to a Lyapunov time that captures the typical decorrelation between initially close trajectories in the high-dimensional system, and the distributions of rescaled asymptotic spacings collapse onto a common skew lo

What carries the argument

The scalar embedding of the training trajectory viewed as a temporal network, which reduces the high-dimensional parameter dynamics to a one-dimensional time series while retaining invariants such as the maximum Lyapunov exponent.

If this is right

  • The embedded trajectory defines a characteristic time after which exponential separation saturates, corresponding to the decorrelation time between close trajectories in the original system.
  • Rescaled spacings of asymptotic training states produce distributions that collapse onto a skew lognormal form independent of initial conditions.
  • The method supplies a practical framework for visualizing and quantifying chaotic properties of optimization without operating in the full parameter space.
  • Statistical organization of long-term training states can be probed through observables defined solely in the embedded space.

Where Pith is reading between the lines

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

  • If the embedding preserves dynamics across architectures, it could enable real-time monitoring of chaotic regimes during training using only reduced-dimensional projections.
  • The collapse to a common spacing distribution hints that asymptotic optimization states may obey universal statistical laws independent of starting point or task details.
  • Applying the same embedding to recurrent or transformer models would test whether the reported Lyapunov preservation and skew-lognormal spacing are architecture-independent features.

Load-bearing premise

That scalar embedding methods developed for general temporal networks can be applied to neural-network parameter trajectories while retaining the dynamical quantities such as Lyapunov exponents and decorrelation times that are treated as ground truth.

What would settle it

Direct computation of the maximum Lyapunov exponent in the full high-dimensional parameter space for the same network and learning rates, followed by comparison to the exponent extracted from the scalar embedding; a statistically significant mismatch would falsify the preservation claim.

Figures

Figures reproduced from arXiv: 2606.30384 by Lucas Lacasa, Miguel C. Soriano, Pedro Jim\'enez-Gonz\'alez.

Figure 1
Figure 1. Figure 1: Sketch of the methodology to extract a scalar embedding of a temporal network, based on the methodology in [ [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Training loss trajectories for three representative [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The black curve shows the averaged network Maxi [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Maximum Lyapunov Exponent estimated from the [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 5
Figure 5. Figure 5: Semi-log plot of the distances d(t) between the scalar embeddings of a reference trajectory and five inde￾pendently perturbed trajectories, shown as a function of the training epoch t for η = 10. Each color corresponds to a dif￾ferent perturbation. This agreement is particularly remarkable given the drastic dimensionality reduction involved: original tra￾jectories evolving in a space of tens of thousands o… view at source ↗
Figure 8
Figure 8. Figure 8: Black markers show the embedding-based separa [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Example of the embedding-based decorrelation [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Scatter plot of τ acc dec versus τ emb dec across initial con￾ditions for η = 10. Each point corresponds to one initial condition, with both decorrelation times averaged over the five perturbations for each initial condition. Furthermore, we argue that the correlation coefficient is not higher precisely because estimation of decorrela￾tion times via accuracy trajectories has some problems –as identified e… view at source ↗
Figure 13
Figure 13. Figure 13: Asymptotic embedded states as a function of the [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Rescaled spacing distributions for η = 10. Markers corresponding to 10 independent initial conditions (CI 0–9), shown in terms of the transformed variable log10(∆′ ), with ∆′ = ∆/⟨∆⟩. Each set of markers represents the empirical histogram of a single initial condition. The solid line shows the skew-normal fit to the pooled data, yielding fitted param￾eters α = −2.78, ξ = 0.36, and ω = 0.91. The interpreta… view at source ↗
Figure 15
Figure 15. Figure 15: Percentage of positive local Lyapunov exponents, [PITH_FULL_IMAGE:figures/full_fig_p011_15.png] view at source ↗
read the original abstract

Training in artificial neural networks can be viewed as a trajectory evolving through a high-dimensional loss landscape. However, the large number of trainable parameters makes the direct analysis of these dynamics challenging. In this work, we treat such training trajectories as temporal networks and apply recently proposed strategies for the scalar embedding of temporal networks. We investigate whether such a scalar embedding provides a meaningful low-dimensional representation of neural network training dynamics. Using a multilayer perceptron trained on the MNIST classification task, we show that the embedding preserves the main dynamical features observed in the original parameter space, including the emergence of sensitivity to initial conditions for specific learning rate regimes and an accurate reconstruction of the network's maximum Lyapunov exponent. We then use the embedded scalar trajectory to define a characteristic time, analogous to a Lyapunov time, after which the exponential separation between initially close embedded trajectories saturates. This characteristic time captures the typical decorrelation time between initially close network trajectories in the original high-dimensional system. Finally, we investigate the statistical organization of asymptotic training states through a spacing observable defined in the embedded space. We find that the distributions of rescaled asymptotic spacings collapse onto a common form across initial conditions and are compatible with a skew lognormal distribution. Altogether, our results suggest that scalar low-dimensional embeddings provide a useful framework for studying and visualizing the dynamical properties of neural network optimization trajectories.

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 / 1 minor

Summary. The paper claims that treating neural network training trajectories as temporal networks and applying scalar embedding techniques yields a low-dimensional representation that preserves key dynamical features. Specifically, for an MLP on MNIST, the embedding captures sensitivity to initial conditions in certain learning rate regimes, accurately reconstructs the maximum Lyapunov exponent, defines a characteristic decorrelation time, and shows that asymptotic spacing distributions collapse to a skew lognormal form.

Significance. Should the embedding method prove faithful, it would provide a valuable tool for analyzing and visualizing the high-dimensional dynamics of neural network training, potentially uncovering universal statistical properties in the organization of training trajectories. The approach bridges concepts from temporal networks and dynamical systems with machine learning optimization.

major comments (2)
  1. [Abstract] The assertion of an 'accurate reconstruction' of the maximum Lyapunov exponent lacks any quantitative error metrics, ablation checks on embedding parameters, or description of the validation procedure against the high-dimensional trajectory. This is central to the claim that the embedding preserves dynamical features and prevents assessment of its soundness.
  2. [Results] No explicit cross-validation, such as comparing divergence rates of nearby trajectories in the original versus embedded space, is described to support the preservation of the Lyapunov exponent and decorrelation time. This leaves the central claim dependent on the untested assumption that the embedding map is dynamically faithful for parameter trajectories.
minor comments (1)
  1. [Abstract] The abstract introduces 'temporal networks' and 'scalar embedding' without a short definition or citation, which could hinder accessibility for readers outside the subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments highlighting the need for stronger quantitative validation of the embedding's dynamical faithfulness. We address each point below and have revised the manuscript accordingly to include the requested metrics, ablations, and cross-validations.

read point-by-point responses

  1. Referee: [Abstract] The assertion of an 'accurate reconstruction' of the maximum Lyapunov exponent lacks any quantitative error metrics, ablation checks on embedding parameters, or description of the validation procedure against the high-dimensional trajectory. This is central to the claim that the embedding preserves dynamical features and prevents assessment of its soundness.

    Authors: We agree that the original manuscript presented the reconstruction primarily through visual comparison without quantitative error metrics or ablations. In the revision we will add the relative error between the maximum Lyapunov exponent computed from the embedded trajectory and the value obtained directly from the high-dimensional parameter trajectory. We will also include ablation studies varying the embedding dimension and time delay, together with an explicit description of the validation procedure used to compare against the original dynamics. revision: yes

  2. Referee: [Results] No explicit cross-validation, such as comparing divergence rates of nearby trajectories in the original versus embedded space, is described to support the preservation of the Lyapunov exponent and decorrelation time. This leaves the central claim dependent on the untested assumption that the embedding map is dynamically faithful for parameter trajectories.

    Authors: We acknowledge that explicit cross-validation strengthens the central claim. The revised manuscript will report direct comparisons of the divergence rates between pairs of nearby trajectories evaluated both in the original high-dimensional parameter space and in the scalar embedded space. These comparisons will be used to confirm preservation of the Lyapunov exponent and the characteristic decorrelation time. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained empirical comparisons

full rationale

The paper applies existing scalar embedding techniques for temporal networks to NN parameter trajectories and validates preservation of dynamical features (e.g., Lyapunov exponent reconstruction, decorrelation times, spacing statistics) via direct comparison on an MLP trained on MNIST. No equations, procedures, or self-citations in the provided text reduce any reported prediction or reconstruction to a quantity defined by fitting the same data or by the embedding map itself. The validations are presented as independent checks against the original high-dimensional trajectory, and the central claims do not rely on load-bearing self-citations or ansatzes that collapse by construction. This is the normal case of an externally benchmarked application study.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that scalar embeddings developed for temporal networks remain faithful to the dynamical invariants of neural-network optimization trajectories; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Scalar embedding strategies for temporal networks preserve the main dynamical features of neural-network training trajectories
    This premise is required for the claim that the low-dimensional representation is meaningful and for the subsequent Lyapunov and spacing analyses.

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