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arxiv: 2606.12161 · v1 · pith:7MKAKAFBnew · submitted 2026-06-10 · ❄️ cond-mat.stat-mech

Path convergence in diffusion models

Roi Holtzman , Roman Beauvallet , Werner Krauth This is my paper

Pith reviewed 2026-06-27 08:04 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords diffusion modelspath convergencebackward pathsdensity estimationextrapolationone-dimensional test casegenerative modelsfinite patterns
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The pith

In one dimension, backward diffusion paths with identical noise converge to the infinite-pattern limit on a 1/sqrt(p) scale, enabling extrapolation to sample the target distribution.

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

The paper studies interpolating paths in diffusion models that connect a target distribution, known only through a finite number p of patterns, to a reference distribution that can be sampled directly. It focuses on backward paths constructed with the same diffusion noise for every pattern. In a one-dimensional test case the paths approach the p to infinity limit at a rate set by 1 over the square root of p, even though the mean-square deviation remains infinite. The observed convergence is shown to support a simple extrapolation procedure that approximates the infinite-p path, which itself samples the target distribution exactly. The authors present a proof-of-concept algorithm and suggest the same convergence-plus-extrapolation route as a possible method for density estimation and generalization.

Core claim

For backward paths with identical diffusion noise, the deviation from the p-equals-infinity path scales as 1/sqrt(p) in a one-dimensional test case, despite an infinite mean-square deviation; this scaling permits an extrapolation algorithm that approximates the infinite-pattern path and thereby samples the target distribution.

What carries the argument

Path convergence of backward diffusion trajectories under identical noise, whose scaling with the number of patterns p enables extrapolation to the infinite-p limit.

If this is right

  • The p-equals-infinity path exactly samples the target distribution.
  • Extrapolation from finite-p paths yields an approximation to the target-sampling path.
  • The method supplies a concrete algorithm for density estimation from a modest number of patterns.
  • The same extrapolation step offers a route to generalization beyond the supplied patterns.

Where Pith is reading between the lines

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

  • If the scaling persists in higher dimensions, the same extrapolation could reduce the number of patterns needed for accurate sampling in practical generative models.
  • The infinite mean-square deviation suggests that convergence is weak in an L2 sense yet still sufficient for pointwise or distributional extrapolation.
  • The approach may connect to other finite-sample corrections used in statistical mechanics sampling methods.

Load-bearing premise

The 1/sqrt(p) convergence and extrapolation observed in the one-dimensional test case can be used as a general strategy for density estimation.

What would settle it

A direct numerical check in the one-dimensional case of whether the extrapolated path at large but finite p recovers the known target distribution to within the expected 1/sqrt(p) error.

Figures

Figures reproduced from arXiv: 2606.12161 by Roi Holtzman, Roman Beauvallet, Werner Krauth.

Figure 1
Figure 1. Figure 1: Backward path construction. (a): At finite ∆ [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Path convergence, velocity fields in our test case [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence of paths for our test case with Alg. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Extrapolation for sets {p} and {q} of p patterns joint into a set {p + q} (test case, τ = 3). (a): Scatter plot for direction for x {∞} τ with respect to x {p+q} τ (orange dots: x {∞} τ towards x {q} τ ; blue dots: x {∞} τ towards x {p} τ ). (b): Change α of difference with x {∞} τ as a function of Υ with linear approx￾imation indicated (see eq. (26)). the smoothings of patterns that are a staple in kernel… view at source ↗
read the original abstract

We discuss diffusion-model paths interpolating between a target distribution known only through p patterns and a reference distribution that can be sampled. These interpolating paths can be constructed symmetrically or else in forward direction (often referred to as a "noising") from the target patterns to the reference distribution or in backward direction (as a "denoising") from the reference distribution to the patterns. For backward paths with identical diffusion noise, we consider the path convergence in number of patterns p towards the path for infinitely many patterns. In a one-dimensional test case, we show that this convergence is on a scale 1/sqrt(p), but with infinite mean square deviation. We demonstrate that the path convergence allows for extrapolation towards the p=infinity path which samples the target distribution. We provide a proof-of-concept extrapolation algorithm and propose the convergence and extrapolation of paths as a possible strategy for density estimation and generalization. We illustrate all our algorithms through pseudo-codes and provide Python implementations.

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

1 major / 0 minor

Summary. The paper analyzes diffusion-model paths interpolating between a target distribution known only through p patterns and a reference distribution. For backward paths with identical diffusion noise, it claims that in a one-dimensional test case the paths converge to the p=∞ path on a scale 1/sqrt(p) despite infinite mean square deviation. It demonstrates that this convergence permits extrapolation to the infinite-p path (which samples the target distribution), proposes the approach as a strategy for density estimation and generalization, and supplies pseudo-codes together with Python implementations.

Significance. If the 1D convergence result can be placed on a rigorous footing with an explicit mode of convergence and the extrapolation shown to be stable, the work would supply a new perspective on using path limits for density estimation in diffusion models. The provision of reproducible code is a clear strength.

major comments (1)
  1. [Abstract] Abstract: the claim that convergence occurs 'on a scale 1/sqrt(p)' while the mean square deviation is infinite does not identify the norm or topology in which the scaling holds. Infinite MSD precludes L2 convergence and leaves the well-posedness of the proposed extrapolation operator unclear; this ambiguity is load-bearing for the central claim that the observed convergence enables a usable density-estimation strategy.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and valuable feedback on our work. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that convergence occurs 'on a scale 1/sqrt(p)' while the mean square deviation is infinite does not identify the norm or topology in which the scaling holds. Infinite MSD precludes L2 convergence and leaves the well-posedness of the proposed extrapolation operator unclear; this ambiguity is load-bearing for the central claim that the observed convergence enables a usable density-estimation strategy.

    Authors: We agree that the abstract does not explicitly identify the norm or topology. In the one-dimensional test case presented in the manuscript, the 1/sqrt(p) scaling is demonstrated through direct computation of path differences and numerical simulations, indicating that the typical deviation between finite-p and infinite-p paths behaves as 1/sqrt(p). The infinite mean square deviation is due to the presence of heavy-tailed fluctuations, but the convergence holds in a weaker sense, such as in probability. The extrapolation algorithm is constructed based on this scaling and is shown to be effective in our proof-of-concept examples for density estimation. We will revise the abstract to clarify the observed mode of convergence and add discussion on the well-posedness of the extrapolation based on the numerical evidence provided. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected.

full rationale

The paper's central result is a direct numerical/analytical demonstration of path convergence scaling in a one-dimensional test case, together with an extrapolation algorithm. No equations or claims reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the 1/sqrt(p) observation and infinite-MSD statement are presented as outcomes of the test case rather than tautological renamings or imported uniqueness theorems. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the setup assumes the target distribution is accessible only via finite p patterns and that paths use identical diffusion noise in the backward direction. No free parameters, axioms beyond standard diffusion assumptions, or invented entities are mentioned.

axioms (2)
  • domain assumption Target distribution known only through p patterns; reference distribution can be sampled.
    Core setup stated in the abstract for constructing interpolating paths.
  • domain assumption Paths constructed symmetrically or in forward/backward directions with identical diffusion noise for backward case.
    Stated as the framework for studying convergence.

pith-pipeline@v0.9.1-grok · 5691 in / 1463 out tokens · 33481 ms · 2026-06-27T08:04:04.566579+00:00 · methodology

discussion (0)

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Works this paper leans on

24 extracted references · 3 canonical work pages

  1. [1]

    W., Rosenbluth M

    Metropolis N., Rosenbluth A. W., Rosenbluth M. N., Teller A. H. Teller E. J. Chem. Phys. 21 1953 1087

    1953

  2. [2]

    Binder K

    Landau D. Binder K. A guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press) 2013. ://books.google.de/books?id=hrIhAwAAQBAJ

    2013

  3. [3]

    A., Peres Y

    Levin D. A., Peres Y. Wilmer E. L. Markov Chains and Mixing Times (American Mathematical Society) 2008

    2008

  4. [4]

    Statistical Mechanics: Algorithms and Computations (Oxford University Press) 2006

    Krauth W. Statistical Mechanics: Algorithms and Computations (Oxford University Press) 2006

    2006

  5. [5]

    All of Statistics (Springer, New York) 2004

    Wasserman L. All of Statistics (Springer, New York) 2004. ://doi.org/10.1007/978-0-387-21736-9

    work page doi:10.1007/978-0-387-21736-9 2004

  6. [6]

    All of Nonparametric Statistics (Springer, New York) 2006

    Wasserman L. All of Nonparametric Statistics (Springer, New York) 2006

    2006

  7. [7]

    Ganguli S

    Sohl-Dickstein J., Weiss E., Maheswaranathan N. Ganguli S. Deep Unsupervised Learning using Nonequilibrium Thermodynamics in proc. of Proceedings of the 32nd International Conference on Machine Learning , edited by Bach F. Blei D. Vol. 37 of Proceedings of Machine Learning Research (PMLR, Lille, France) 2015 pp. 2256--2265

    2015

  8. [8]

    Abbeel P

    Ho J., Jain A. Abbeel P. presented at Advances in Neural Information Processing Systems

  9. [9]

    P., Kumar A., Ermon S

    Song Y., Sohl-Dickstein J., Kingma D. P., Kumar A., Ermon S. Poole B. Score-based generative modeling through stochastic differential equations presented at International Conference on Learning Representations 2021

    2021

  10. [10]

    Song Y. Ermon S. Generative modeling by estimating gradients of the data distribution presented at Advances in Neural Information Processing Systems Vol. 32 2019

    2019

  11. [11]

    Feynman R. P. Statistical mechanics: a set of lectures Frontiers in physics (W. A. Benjamin, Reading, Massachusetts.) 1972

    1972

  12. [12]

    Ceperley D. M. Rev. Mod. Phys. 67 1995 279–355 . ://dx.doi.org/10.1103/RevModPhys.67.279

    work page doi:10.1103/revmodphys.67.279 1995

  13. [13]

    Krauth W

    Holtzman R., Beauvallet R. Krauth W. PathConvergence software package https://github.com/jellyfysh/PathConvergence.git (2026)

    2026

  14. [14]

    L \'e vy P. Compos. Math. 7 1939 283

    1939

  15. [15]

    Hyv \"a rinen A. Dayan P. J. Mach. Learn. Res. 6 2005

    2005

  16. [16]

    Neural Comput

    Vincent P. Neural Comput. 23 2011 1661

    2011

  17. [17]

    Propp J. G. Wilson D. B. Random Structures & Algorithms 9 1996 223

    1996

  18. [18]

    Krauth W

    Holtzman R., Beauvallet R. Krauth W. Manuscript in preparation (2026)

    2026

  19. [19]

    Albergo M., Boffi N. M. Vanden-Eijnden E. J. Mach. Learn. Res. 26 2025 1

    2025

  20. [20]

    Zhang K., Yin H., Liang F. Liu J. Minimax optimality of score-based diffusion models: Beyond the density lower bound assumptions in proc. of Proceedings of the 41st International Conference on Machine Learning (PMLR) 2024 pp. 60134--60178

    2024

  21. [21]

    Chen S., Chewi S., Li J., Li Y., Salim A. Zhang A. Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions presented at NeurIPS 2022 Workshop on Score-Based Methods 2022

    2022

  22. [22]

    Suzuki T

    Oko K., Akiyama S. Suzuki T. Diffusion models are minimax optimal distribution estimators in proc. of Proceedings of the 40th International Conference on Machine Learning Vol. 202 of Proceedings of Machine Learning Research (PMLR) 2023 pp. 26517--26582

    2023

  23. [23]

    M \'e zard M

    Biroli G. M \'e zard M. SIAM Journal on Mathematics of Data Science 8 2026 46

    2026

  24. [24]

    Scarvelis C., Borde H. S. d. O. Solomon J. arXiv preprint arXiv:2310.12395 2023

    work page arXiv 2023