Dyson Brownian motion and random matrix dynamics of weight matrices during learning

Biagio Lucini (Swansea University); Chanju Park (Swansea University); Gert Aarts (Swansea University); Ouraman Hajizadeh (Graz)

REVIEW 3 major objections 5 minor 1 cited by

Dyson Brownian motion and random matrix dynamics of weight matrices during learning

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Learning weight matrices are governed by Dyson Brownian motion.

desk verdict A clean restatement of the authors' earlier DBM framework with a thin, unfinished transformer add-on; the α/|B| scaling is proven for the Gaussian RBM, but the abstract's 'generic' claim outruns the assumptions. read the letter →

arxiv 2411.13512 v1 pith:GZESF6OG submitted 2024-11-20 cond-mat.dis-nn cs.LGhep-lat

Gert Aarts (Swansea University) , Ouraman Hajizadeh (Graz) , Biagio Lucini (Swansea University) , Chanju Park (Swansea University) This is my paper · ORCID

classification cond-mat.dis-nncs.LGhep-lat MSC 15B5260J6582C31

keywords DysonBrownianmotionrandommatrixtheoryeigenvaluerepulsionCoulombgaslinearscalingrulerestrictedBoltzmannmachinetransformerMarchenko-Pasturdistribution

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that mini-batch stochastic gradient descent on weight matrices can be reduced to Dyson Brownian motion for the eigenvalues of $X = W^T W$. The resulting eigenvalue update carries a drift, a repulsive Coulomb interaction, and Gaussian noise, and the stationary distribution is a Coulomb gas whose dependence on learning rate $\alpha$ and batch size $|B|$ enters only through the ratio $\alpha/|B|$. This derives the empirically observed linear scaling rule from stochastic matrix dynamics rather than treating it as a heuristic. The authors confirm the full Coulomb-gas picture analytically in a Gaussian restricted Boltzmann machine and give empirical support in a small transformer, where the spectrum flows from a Marchenko-Pastur law at initialization to a distribution with a heavy tail.

What carries the argument

The engine of the argument is the Dyson Brownian motion eigenvalue equation for $X = W^T W$: each eigenvalue moves under a deterministic drift, a pairwise repulsive Coulomb term $\sum_{j \neq i} g_i^2/(x_i-x_j)$, and a Gaussian noise term, with explicit factors of $\alpha$ and $|B|$ shown in Eq. 2.4. Passing to the Fokker-Planck equation converts this into the stationary Coulomb gas measure (Eq. 2.5), in which the potential is a sum of single-eigenvalue terms $V_i(x_i)$ and the noise variance controls the repulsion strength. The combination $\alpha/|B|$ in Eq. 2.6 is the universal scaling variable; the model-dependent part is the ratio $\tilde V_i/\tilde g_i^2$. The Marchenko-Pastur distribution plays the role of the initial condition for this flow.

What would settle it

Train the same transformer on the same data at several pairs $(\alpha, |B|)$ with equal ratio $\alpha/|B|$ and compare the stationary eigenvalue spacing distribution after unfolding; any systematic dependence on the individual values of $\alpha$ or $|B|$ would falsify the claimed universal scaling.

Watch

Extended reading notes

Core claim

The central claim is that the learning dynamics of weight matrices can be projected onto the eigenvalues $x_i$ of $X = W^T W$, yielding the update $x_i \to x_i + \alpha \tilde K_i + \frac{\alpha^2}{|B|} \sum_{j \neq i} \tilde g_i^2/(x_i - x_j) + \frac{\alpha}{\sqrt{|B|}} \sqrt{2} \tilde g_i \eta_i$ (Eq. 2.4). The induced Coulomb repulsion among eigenvalues makes the level-spacing statistics Wigner-Dyson, and the stationary solution of the associated Fokker-Planck equation is the Coulomb gas $P_s \propto \prod_{i<j}|x_i-x_j| \exp(-\sum_i V_i(x_i)/g_i^2)$, so the only universal control parameter is $\alpha/|B|$ (Eq. 2.6). Learning is therefore described as a spectral flow: eigenvalues start from a Marchenko-Pastur density and, as training proceeds, reorganize around the minima of a model-dependent potential, with residual repulsion that vanishes only as $\alpha/|B| \to 0$. This is established in full analytic detail for the Gaussian restricted Boltzmann machine and asserted empirically for a transformer.

Load-bearing premise

The load-bearing premise is that the systematic part of the eigenvalue update is the gradient of a potential that splits into one term per eigenvalue; this is proved for the Gaussian restricted Boltzmann machine but assumed without proof for generic architectures, so if it fails the stationary Coulomb gas and the $\alpha/|B|$ scaling do not follow.

Editorial extensions

If this is right

At any finite $\alpha/|B|$, the trained spectrum is not the target spectrum: eigenvalue repulsion prevents exact recovery, and the deviation shrinks as the ratio is reduced.
Running with doubled learning rate and doubled batch size leaves the stationary spectral distribution unchanged, so hyperparameter pairs with equal $\alpha/|B|$ are dynamically equivalent.
Level-spacing statistics of trained weight matrices should follow the Wigner surmise, as a universal signature of the underlying eigenvalue repulsion.
In the transformer experiment, the bulk of the spectrum remains approximately Marchenko-Pastur while a heavy tail grows to carry 15\%--25\% of the spectral weight, so the flow away from the initial law is visible in standard spectral-density diagnostics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the $\alpha/|B|$ scaling survives in settings with adaptive optimizers, grid searches over learning rate and batch size could be replaced by a one-dimensional scan over their ratio, at least for the spectral sector of the dynamics.
The Coulomb-gas picture suggests that spectral tails in trained networks carry information about the effective loss landscape; monitoring the tail fraction during training could serve as a cheap probe of when the landscape's structure changes.
The derivation assumes Gaussian noise from the central limit theorem and a finite, time-independent batch-gradient variance; architectures with heavy-tailed gradient noise would be a natural test bed for corrections to Wigner-Dyson statistics.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take. The genuinely new content in this paper is thin: the DBM equation, the α/|B| scaling, and the Gaussian RBM verification are all carried over from the authors' earlier paper (Ref [8]). What this paper adds is a concise restatement plus an exploratory nano-GPT study. Viewed as a workshop synthesis, that is fine; but the abstract's claim that the dynamics 'can generically be described' by DBM is stronger than what is shown.

What the paper does well: the Gaussian RBM part is analytically controlled. The drift K_i and potential V_i are known explicitly, the noise strength is computed, and the numerical demonstration of α/|B| scaling—varying α and |B| independently—is convincing. The picture of eigenvalue evolution from Marchenko-Pastur to Coulomb gas with eigenvalue repulsion is clean and pedagogically useful. The transformer study shows Wigner-surmise level spacing and a non-MP tail, and the authors are honest that it is empirical and incomplete.

Now the soft spots, in proportion. The main issue is the assumption before Eq. (2.5): the drift must be a gradient of a separable potential, K_i = -dV_i/dx_i. For a generic loss and generic architecture, the projected gradient dynamics on eigenvalues does not automatically take that form, and the paper provides no argument beyond the solvable RBM. Relatedly, Eq. (2.1) replaces minibatch gradient fluctuations with entrywise independent Gaussians. In reality the same minibatch perturbs all weights, so the noise is correlated across matrix elements; the DBM framework requires independent, rotationally invariant matrix noise for the eigenvalue process to close. Neither of these is a fatal flaw in the RBM, where everything is solvable, but they undercut the 'generic' claim and the universal α/|B| conclusion for arbitrary architectures.

The transformer section is the most novel but also the least complete. No code or data are provided for it; the spacing histograms lack error bars; the MP fits use two free parameters with no uncertainty or cross-checks. The authors explicitly say work is in progress, so I read it as a case study, not as evidence for the universal claim.

Citation-wise, leaning on Ref [8] is legitimate—it is the source of the framework—and the paper cites the relevant RMT and ML literature. No citation red flags.

Bottom line: as a full journal paper, this deserves genuine refereeing rather than desk rejection. The RBM derivation is worth checking, and the scaling claim is important if true. I'd ask the authors to soften the generic claim, justify or drop the separable-potential assumption, discuss the noise-correlation issue, and either add code/data for the transformer or present it as an explicit preliminary study. For a workshop paper, it is acceptable as is. I would not cite this paper in my own work; I'd cite Ref [8] instead.

Referee Report

3 major / 5 minor

Summary. The paper claims that the stochastic gradient descent update of a weight matrix W can generically be described by Dyson Brownian motion on the eigenvalues of X = W^T W. The authors write a stochastic eigenvalue equation, Eq. (2.4), and the corresponding stationary Coulomb gas distribution, Eq. (2.5), from which they derive the universal dependence of the stationary spectrum on the ratio alpha/|B| of learning rate to batch size, thereby explaining the empirically observed linear scaling rule. The Gaussian restricted Boltzmann machine is treated analytically, and a nano-GPT transformer is studied empirically, where the authors report Wigner-surmise level statistics and a spectral evolution from a Marchenko-Pastur distribution toward a heavier-tailed distribution.

Significance. If the central claim were established, connecting stochastic gradient descent to Dyson Brownian motion and predicting alpha/|B| scaling of the stationary spectral distribution would be a valuable contribution with consequences for hyperparameter tuning and for the random-matrix description of learning. The paper is honest about the limits of the transformer analysis, and it ships code and data for the RBM part, which is a concrete strength. However, the generic claim is only proven for the Gaussian RBM; for generic architectures it rests on unproven assumptions about noise isotropy, eigenvector decoupling, and separability of the effective potential. The paper's value at present is therefore as a controlled RBM result plus an intriguing but incomplete empirical study, not as a demonstration of the generic universality claimed in the abstract.

major comments (3)

[§2, Eq. (2.3)-(2.4)] The reduction of the matrix update to the closed eigenvalue equation (2.3) is not derived for general losses. For X = W^T W, the first-order perturbation of an eigenvalue involves the instantaneous eigenvectors u_i: d x_i = u_i^T (W^T A + A^T W) u_i dt plus a noise term whose covariance is u_i^T W^T C W u_i dt, where A is the deterministic gradient term and C is the covariance of the minibatch gradient noise. Because the same minibatch perturbs all entries of W jointly, Cov(delta W_ij, delta W_kl) is generically nonzero for (i,j) != (k,l), so the noise is not entry-wise independent. Eq. (2.3) assumes that the eigenvalue process closes in the variables {x_i} alone with a scalar noise strength g_i; without an argument that the eigenvector coupling vanishes or decouples, the statement in the abstract that the dynamics 'can generically be described' by Dyson Brownian motion is not supported.
[§2, Eq. (2.5)-(2.6)] The stationary Coulomb gas distribution (2.5) and the alpha/|B| scaling conclusion (2.6) rest on the assumption, stated just before Eq. (2.5), that the drift K_i can be written as K_i = -dV_i/dx_i with a separable potential V_i(x_i). For a generic loss landscape, the projection of the gradient flow of W^T W onto its eigenvalues need not be a gradient flow on a separable potential; if the potential is non-separable, Eq. (2.5) is not the stationary distribution and the universal alpha/|B| scaling does not follow. The paper proves this assumption only in the Gaussian RBM (via Ref. [8]), and Section 3.2 explicitly states that the Coulomb gas potential is unknown for the transformer. This is a load-bearing gap, not a presentation issue.
[§3.2, Figs. 3-4] The transformer experiment does not test the central scaling claim. Agreement with the Wigner surmise after spectral unfolding is a universal fluctuation property that is compatible with many stochastic dynamics and does not identify the specific Dyson Brownian motion of Eq. (2.4). The Marchenko-Pastur fits in Fig. 4 use two free parameters (sigma^2 and area A) and leave a 15-25% tail unexplained, and the paper states that the exact spectrum or Coulomb gas potential is not known. Most importantly, no variation of alpha or |B| is reported for the transformer, so Eq. (2.6) is neither confirmed nor refuted there. The empirical support for the generic claim is therefore substantially weaker than the abstract suggests.

minor comments (5)

[§2, Eq. (2.2)] The noise term in Eq. (2.2) is written as sqrt(Var(delta X_ij)) eta_ij, but delta X = W'^T W' - W^T W is nonlinear in delta W and includes the cross term W^T delta W + delta W^T W; the variance and the covariance structure of the noise for X should be computed from this full expression rather than inherited from Eq. (2.1).
[§3.1, Eq. (3.2)] The continuous-time Dyson equation (3.2) and the discrete update (2.4) are not explicitly matched; the paper should state how the discrete SGD step with learning rate alpha and batch size |B| maps to the continuous-time parameter tau, so that the alpha/|B| dependence in Fig. 1 is unambiguous.
[§3.2, Fig. 4] The text says that the statistical uncertainty is determined by repeating training at least 50 times with bootstrap analysis, but Fig. 4 shows no error bars; please add them or state explicitly that they are smaller than the symbol size.
[§3.2] There is a typo in 'an Marchenko-Pastur distribution'; it should be 'a Marchenko-Pastur distribution'.
[§2, Eq. (2.5)] In Eq. (2.5), Z is defined as the integral of P_s, but P_s already contains 1/Z; the definition is circular as written. It would be clearer to define Z as the integral of the unnormalized density product over |x_i - x_j| exp(-sum_i V_i(x_i)/g_i^2).

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity: the α/|B| scaling follows from the CLT noise scaling in Eq. (2.1); the main caveats are the explicitly stated separable-potential assumption and reliance on same-authors Ref. [8] for the RBM verification.

full rationale

The argument chain is not circular in the prohibited sense. Eq. (2.4) is obtained by inserting the central-limit-theorem scaling of minibatch-SGD fluctuations (noise variance proportional to α²/|B|) into the Dyson Brownian motion eigenvalue equation, and Eq. (2.6) is the algebraic evaluation of the stationary Coulomb-gas exponent V_i/g_i² = (|B|/α) Vtilde_i/gtilde_i²; the α/|B| dependence is therefore a consequence of the assumed noise scaling, not a re-importation of the empirical linear scaling rule, and the paper separately cites independent empirical support (Refs. [15,16]) and alternative derivations (Refs. [17,18]). The load-bearing assumption, stated explicitly before Eq. (2.5), is that the projected drift is a gradient of a separable potential, K_i = -dV_i/dx_i; the paper concedes this is not available for the transformer ('the exact spectrum or Coulomb gas potential are not known', Sec. 3.2), so the generic claim is conditional rather than circularly derived. The only notable self-citation concern is that the Gaussian RBM drift, noise strength, and both verification figures are taken from the same-authors Ref. [8] ('We follow here closely our previous work [8]', 'Figures from Ref. [8]'), so the paper's demonstration for the RBM is inherited rather than reproduced. This is a load-bearing caveat, but Ref. [8] is an analytic derivation with released code, not an unverified assertion of the target result. Hence no prediction reduces by construction; score 2 reflects the modest self-citation and unproven-generality limitation rather than circularity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central DBM and alpha/|B| scaling result rests on a sequence of assumptions: Gaussian mini-batch noise, the standard DBM reduction to eigenvalue dynamics, a separable potential for the drift, and identification of the discrete-time update with a continuous-time equilibrium. The only place all assumptions are controlled is the Gaussian RBM, where the drift and noise are taken from the authors' own Ref [8]. The transformer part uses no derived potential and is purely empirical. No new physical entities are introduced; the Coulomb gas potential is an emergent effective object from RMT.

free parameters (3)

MP fit sigma^2 for transformer K^T K spectrum = 0.34 at iteration 0; 0.6 to 0.9 by iteration 5000 depending on layer
Fitted to the spectral density histogram of X = K^T K in Fig. 3 (bottom) and tracked in Fig. 4 (right). It characterizes the bulk of the spectrum but is not a parameter of the DBM derivation.
MP fit area A for transformer K^T K spectrum = 1.0 at iteration 0; 0.75 to 0.85 by iteration 5000
Fitted together with sigma^2. The claimed 15-25% tail spectral weight is inferred from the fitted area, so the tail fraction is a derived quantity from these fits.
RBM mean level spacing proportionality a_fit = 2.2750 +/- 0.0326
Linear fit of the mean eigenvalue spacing versus sqrt(alpha/|B| kappa^2 Omega) in Fig. 1 (right). It confirms the predicted scaling direction but is a fitted constant, not predicted from first principles.

assumptions (6)

domain assumption Mini-batch SGD gradient noise is Gaussian with variance scaling as 1/|B| after the central limit theorem (Eq. 2.1).
Used in Sec. 2 to write the stochastic weight update. Requires i.i.d. minibatch gradients with finite variance, not justified for all loss landscapes or for AdamW used in Sec. 3.2.
ad hoc to paper The drift K_i in the eigenvalue dynamics derives from a separable potential V_i(x_i) via K_i = -dV_i/dx_i.
Stated immediately before Eq. (2.5). This is required for the Coulomb gas stationary distribution and the alpha/|B| scaling. It is verified only in the Gaussian RBM (from Ref [8]); for generic architectures it is unverified, and the transformer section does not use it.
domain assumption The continuous-time Dyson Brownian motion stationary distribution (2.5) describes the discrete-time SGD update (2.1).
The paper jumps from a discrete update to a continuous-time Fokker-Planck equilibrium without stating a small-learning-rate limit or checking discrete-time corrections. This is implicit throughout Sec. 2.
standard math The standard Dyson Brownian motion reduction applies to X = W^T W, with off-diagonal noise giving Coulomb repulsion and diagonal noise giving independent Gaussian terms (Eq. 2.3).
Taken from Dyson [7] and Mehta [6]; standard in RMT, but its application to the specific gradient-noise process for X = W^T W is asserted rather than derived here.
domain assumption The Gaussian RBM drift K_i and noise strength g_i^2 in Eq. (3.2) are exact, as derived in Ref [8].
The analytical RBM verification is not re-derived in this paper; Figs. 1 and 2 are taken from Ref [8], so the soundness of this pillar depends on an unstated self-cited derivation.
standard math The initial eigenvalue distribution of X = W^T W follows the Marchenko-Pastur law (3.3) with sigma^2 = 1/3 and r = 1/4 from uniform initialization.
Standard RMT result for rectangular random matrices, used to set the initial condition in the transformer analysis.

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Cite this review

Pith. "Pith review of Dyson Brownian motion and random matrix dynamics of weight matrices during learning." pith.science (2026). https://pith.science/paper/GZESF6OG

@misc{pith2026241113512,
  author       = {Pith},
  title        = {Pith review of: Dyson Brownian motion and random matrix dynamics of weight matrices during learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GZESF6OG}},
  note         = {Machine review of arXiv:2411.13512}
}

read the original abstract

During training, weight matrices in machine learning architectures are updated using stochastic gradient descent or variations thereof. In this contribution we employ concepts of random matrix theory to analyse the resulting stochastic matrix dynamics. We first demonstrate that the dynamics can generically be described using Dyson Brownian motion, leading to e.g. eigenvalue repulsion. The level of stochasticity is shown to depend on the ratio of the learning rate and the mini-batch size, explaining the empirically observed linear scaling rule. We verify this linear scaling in the restricted Boltzmann machine. Subsequently we study weight matrix dynamics in transformers (a nano-GPT), following the evolution from a Marchenko-Pastur distribution for eigenvalues at initialisation to a combination with additional structure at the end of learning.

Figures

Figures reproduced from arXiv: 2411.13512 by the authors.

**Figure 2.** Gaussian RBM: Evolution of eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Transformer: Evolution during training of the eigenvalue distribution of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Transformer: Evolution of fit parameters area [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Random Matrix Theory for Stochastic Gradient Descent
hep-lat 2024-12 conditional novelty 4.0 of 10

SGD weight-matrix eigenvalue fluctuations follow random matrix predictions, with variance proportional to learning rate divided by batch size, the linear scaling rule.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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