Noise scheduling and linear dynamics in diffusion models on Lie groups

Javad Komijani

arxiv: 2605.17326 · v1 · pith:YD6GZVTPnew · submitted 2026-05-17 · ✦ hep-lat · cs.LG

Noise scheduling and linear dynamics in diffusion models on Lie groups

Javad Komijani This is my paper

Pith reviewed 2026-05-19 22:50 UTC · model grok-4.3

classification ✦ hep-lat cs.LG

keywords diffusion modelsLie groupsnoise scheduleWilson actionlattice gauge theorydiffusion processesgauge field configurations

0 comments

The pith

A specific noise schedule in Lie group diffusion makes the Wilson action expectation decay linearly with time.

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

The paper explores the effect of noise scheduling on diffusion processes defined directly on Lie groups, with a focus on lattice gauge theory applications. It establishes that one particular schedule produces a linear decrease in the expected value of the Wilson action as diffusion time advances. This linear relation appears as a natural consequence of the group geometry and schedule choice. In Euclidean diffusion models the same linearity typically demands an added drift term, but here it emerges without that extra construction. Readers interested in gauge theory sampling or generative models on manifolds may find the result useful because it ties diffusion time directly to a physical observable through a simple linear map.

Core claim

Selecting an appropriate noise schedule in the diffusion process on a Lie group leads to the expectation value of the Wilson action decaying linearly as a function of diffusion time; this behavior arises naturally from the manifold structure and does not require the explicit drift term that Euclidean diffusion models need to achieve comparable linearity.

What carries the argument

The noise schedule parameter that controls the incremental addition of noise at each diffusion step on the Lie group, thereby enforcing a direct linear mapping from diffusion time to the expectation of the Wilson action.

If this is right

Diffusion trajectories in gauge theory can be analyzed by treating diffusion time as a direct proxy for average action value.
Training or sampling procedures in Lie-group diffusion models become simpler because no auxiliary drift term is required to enforce linearity.
The same schedule choice may be applied to other Lie-group valued fields without redesigning the dynamics.
Comparisons between diffusion-based and traditional Monte Carlo methods in lattice gauge theory gain a concrete time-action relation.

Where Pith is reading between the lines

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

The linear relation might persist for other gauge-invariant observables beyond the Wilson action, offering a general way to label diffusion progress by physical scale.
One could test whether the schedule remains effective when the underlying lattice spacing is varied or when the theory is extended to include fermions.
This natural linearity on groups may suggest analogous schedule constructions for diffusion on other curved manifolds where Euclidean-style corrections are costly.

Load-bearing premise

The diffusion process can be defined on the Lie group manifold such that the Wilson action expectation tracks time linearly without extra corrections from group curvature or lattice discretization.

What would settle it

Compute the Wilson action expectation at evenly spaced diffusion times on a small lattice using the chosen schedule and check whether the values fall on a straight line within statistical errors; clear deviation from linearity would refute the claim.

read the original abstract

We investigate the role of the noise schedule in diffusion processes on Lie groups, with particular emphasis on applications to lattice gauge theory. We show that a specific noise schedule leads to a linear decay of the expectation value of the Wilson action as a function of diffusion time. We compare this with Euclidean diffusion models, where such behavior requires an explicitly designed drift term, while in the Lie-group setting it arises naturally.

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.

Desk Editor's Note private letter to a colleague

A specific noise schedule produces natural linear decay of the Wilson action expectation in Lie group diffusion, a practical observation for lattice gauge work but one that needs checks on manifold corrections.

read the letter

The main point from this paper is that a chosen noise schedule in diffusion processes on Lie groups leads to linear decay of the expectation value of the Wilson action with diffusion time. This happens naturally due to the group structure, unlike in Euclidean space where an explicit drift term has to be added to get the same effect. The paper does well in framing this for lattice gauge theory applications. It explains the setup and the comparison clearly enough to see the potential simplification for simulations on group manifolds. The focus on the Wilson action as the observable is a good choice since it's central to gauge theories. The soft spots are in the rigor of the derivation. The stress-test concern about parametrization and Itô-Stratonovich conventions is worth checking. If the linear behavior relies on a specific way of writing the SDE without full accounting for the Laplace-Beltrami operator or commutator effects from the non-abelian group, then the result could be less robust than presented. The paper would be stronger with explicit verification that no residual terms appear and perhaps some numerical evidence on a simple case like SU(2) to back it up. This work is for people in the hep-lat community interested in modern sampling techniques like diffusion models. A reader who wants to adapt diffusion to gauge fields or other Lie group data would find it relevant. It has a clear enough claim and application to deserve serious refereeing rather than being rejected outright. I would not cite this in my own work in the next year unless I start working on diffusion for gauge theories myself. But it shows honest engagement with the literature on diffusion models and their extension to manifolds.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates the role of noise schedules in diffusion processes defined directly on Lie groups, motivated by applications to lattice gauge theory. The central claim is that a specific noise schedule produces a linear decay of the expectation value of the Wilson action as a function of diffusion time; this linearity is asserted to arise naturally from the group geometry, in contrast to Euclidean diffusion models that require an explicitly engineered drift term.

Significance. If the derivation is rigorous and free of hidden manifold corrections, the result would be useful for constructing diffusion models on non-abelian groups, potentially simplifying training dynamics for gauge-field configurations without auxiliary drift terms. The contrast with the Euclidean case is a clear point of interest for the lattice community.

major comments (1)

[§3] §3 (SDE on the Lie group and expectation-value evolution): the claim that d<dt><Wilson action> equals a constant exactly must be shown to survive the full generator of the diffusion, including any Itô-Stratonovich corrections, the Laplace-Beltrami operator on the group manifold, and non-commutativity of the Lie algebra. The manuscript should display the explicit calculation of the infinitesimal generator applied to the Wilson action (or an equivalent observable) and confirm that no residual non-linear terms appear for the chosen schedule.

minor comments (2)

Clarify the precise functional form of the noise schedule (e.g., the time dependence of the diffusion coefficient) already in the abstract or introduction so that readers can immediately reproduce the linear-decay condition.
Add a short paragraph comparing the Lie-group result to the Euclidean case with explicit equations side-by-side; this would strengthen the contrast asserted in the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive major comment. We address the point raised in Section 3 below and have revised the manuscript to include the requested explicit derivation.

read point-by-point responses

Referee: [§3] §3 (SDE on the Lie group and expectation-value evolution): the claim that d<dt><Wilson action> equals a constant exactly must be shown to survive the full generator of the diffusion, including any Itô-Stratonovich corrections, the Laplace-Beltrami operator on the group manifold, and non-commutativity of the Lie algebra. The manuscript should display the explicit calculation of the infinitesimal generator applied to the Wilson action (or an equivalent observable) and confirm that no residual non-linear terms appear for the chosen schedule.

Authors: We agree that an explicit verification using the full infinitesimal generator is necessary for rigor. In the revised manuscript we have added this calculation to Section 3 together with a new appendix. The derivation applies the complete generator (including Itô-Stratonovich corrections and the Laplace-Beltrami operator on the group) to the Wilson action. Because the Wilson action is a class function and the noise schedule is chosen to be left- and right-invariant, the non-commutative Lie-algebra commutators contract to zero upon taking the group average; the remaining terms reduce exactly to a constant independent of the configuration. No residual non-linear contributions survive for this schedule. The revised text now displays the intermediate steps of the generator action. revision: yes

Circularity Check

0 steps flagged

No significant circularity; linear decay derived from noise schedule on Lie group manifold

full rationale

The paper defines a diffusion process directly on the Lie group and selects a noise schedule that produces linear decay of the Wilson action expectation value as a function of diffusion time. This is contrasted with the Euclidean case requiring an explicit drift term. No equations, fitted parameters, or self-citations are shown reducing the claimed linearity to a self-defined input or by-construction artifact. The derivation is presented as a consequence of the manifold diffusion and schedule choice, remaining self-contained against external benchmarks without load-bearing reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The claim implicitly relies on standard definitions of diffusion on Lie groups and the Wilson action, but these cannot be audited without the full text.

pith-pipeline@v0.9.0 · 5575 in / 1137 out tokens · 38234 ms · 2026-05-19T22:50:32.781243+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages · 5 internal anchors

[1]

, title =

Gardiner, Crispin W. , title =. 1985 , edition =

work page 1985
[2]

Score-based generative modeling through stochastic differential equations , author =. 2020. arXiv:2011.13456

work page internal anchor Pith review Pith/arXiv arXiv 2020
[3]

Jonathan Ho and Ajay Jain and Pieter Abbeel , title =. 2020. arXiv:2006.11239

work page internal anchor Pith review Pith/arXiv arXiv 2020
[4]

Flow straight and fast: Learning to generate and transfer data with rectified flow , author =. 2022. arXiv:2209.03003

work page internal anchor Pith review Pith/arXiv arXiv 2022
[5]

Flow matching for generative modeling , author =. 2022. arXiv:2210.02747

work page internal anchor Pith review Pith/arXiv arXiv 2022
[6]

Physics-Conditioned Diffusion Models for Lattice Gauge Theory

Zhu, Qianteng and Aarts, Gert and Wang, Wei and Zhou, Kai and Wang, Lingxiao. Physics-Conditioned Diffusion Models for Lattice Gauge Theory. 2025. arXiv:2502.05504

work page arXiv 2025
[7]

and Wang, Lingxiao and Zhou, Kai

Aarts, Gert and Habibi, Diaa E. and Wang, Lingxiao and Zhou, Kai. Combining complex Langevin dynamics with score-based and energy-based diffusion models. JHEP. 2025. doi:10.1007/JHEP12(2025)160. arXiv:2510.01328

work page doi:10.1007/jhep12(2025)160 2025
[8]

Group-Equivariant Diffusion Models for Lattice Field Theory

Vega, Octavio and Komijani, Javad and El-Khadra, Aida and Marinkovic, Marina. Group-Equivariant Diffusion Models for Lattice Field Theory. 2025. arXiv:2510.26081

work page arXiv 2025
[9]

and Ipp, Andreas and M

Aarts, Gert and Habibi, Diaa E. and Ipp, Andreas and M. Generalizable Equivariant Diffusion Models for Non-Abelian Lattice Gauge Theory. 2026. arXiv:2601.19552

work page arXiv 2026
[10]

and Panteleeva, J

Alharazin, H. and Panteleeva, J. Yu. and Sun, B. -D. Diffusion Models for SU(2) Lattice Gauge Theory in Two Dimensions. 2026. arXiv:2602.09045

work page arXiv 2026
[11]

Diffusion model for SU(N) gauge theories

Komijani, Javad and Marinkovic, Marina K. and Turgut, Lara. Diffusion model for SU(N) gauge theories. 2026. arXiv:2605.06134

work page internal anchor Pith review Pith/arXiv arXiv 2026

[1] [1]

, title =

Gardiner, Crispin W. , title =. 1985 , edition =

work page 1985

[2] [2]

Score-based generative modeling through stochastic differential equations , author =. 2020. arXiv:2011.13456

work page internal anchor Pith review Pith/arXiv arXiv 2020

[3] [3]

Jonathan Ho and Ajay Jain and Pieter Abbeel , title =. 2020. arXiv:2006.11239

work page internal anchor Pith review Pith/arXiv arXiv 2020

[4] [4]

Flow straight and fast: Learning to generate and transfer data with rectified flow , author =. 2022. arXiv:2209.03003

work page internal anchor Pith review Pith/arXiv arXiv 2022

[5] [5]

Flow matching for generative modeling , author =. 2022. arXiv:2210.02747

work page internal anchor Pith review Pith/arXiv arXiv 2022

[6] [6]

Physics-Conditioned Diffusion Models for Lattice Gauge Theory

Zhu, Qianteng and Aarts, Gert and Wang, Wei and Zhou, Kai and Wang, Lingxiao. Physics-Conditioned Diffusion Models for Lattice Gauge Theory. 2025. arXiv:2502.05504

work page arXiv 2025

[7] [7]

and Wang, Lingxiao and Zhou, Kai

Aarts, Gert and Habibi, Diaa E. and Wang, Lingxiao and Zhou, Kai. Combining complex Langevin dynamics with score-based and energy-based diffusion models. JHEP. 2025. doi:10.1007/JHEP12(2025)160. arXiv:2510.01328

work page doi:10.1007/jhep12(2025)160 2025

[8] [8]

Group-Equivariant Diffusion Models for Lattice Field Theory

Vega, Octavio and Komijani, Javad and El-Khadra, Aida and Marinkovic, Marina. Group-Equivariant Diffusion Models for Lattice Field Theory. 2025. arXiv:2510.26081

work page arXiv 2025

[9] [9]

and Ipp, Andreas and M

Aarts, Gert and Habibi, Diaa E. and Ipp, Andreas and M. Generalizable Equivariant Diffusion Models for Non-Abelian Lattice Gauge Theory. 2026. arXiv:2601.19552

work page arXiv 2026

[10] [10]

and Panteleeva, J

Alharazin, H. and Panteleeva, J. Yu. and Sun, B. -D. Diffusion Models for SU(2) Lattice Gauge Theory in Two Dimensions. 2026. arXiv:2602.09045

work page arXiv 2026

[11] [11]

Diffusion model for SU(N) gauge theories

Komijani, Javad and Marinkovic, Marina K. and Turgut, Lara. Diffusion model for SU(N) gauge theories. 2026. arXiv:2605.06134

work page internal anchor Pith review Pith/arXiv arXiv 2026