Reducing Diffusion Model Memorization with Higher Order Langevin Dynamics

Benjamin Sterling; M\'onica F. Bugallo; Tom Tirer

arxiv: 2605.19170 · v2 · pith:C2HXVZ5Enew · submitted 2026-05-18 · 📊 stat.ML · cs.LG

Reducing Diffusion Model Memorization with Higher Order Langevin Dynamics

Benjamin Sterling , M\'onica F. Bugallo , Tom Tirer This is my paper

Pith reviewed 2026-05-20 07:07 UTC · model grok-4.3

classification 📊 stat.ML cs.LG

keywords diffusion modelsmemorizationhigher-order Langevin dynamicsscore functionregularizationgenerative modelsprivacy

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The pith

Higher-order Langevin dynamics govern diffusion data trajectories with low-pass filtered scores, reducing memorization.

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

The paper seeks to establish that higher-order Langevin dynamics reduce the tendency of diffusion models to reproduce exact training samples. It does this by showing that the data variable evolves under a low-pass-filtered version of the learned score function, where the amount of smoothing grows with the model order. A sympathetic reader would care because exact memorization raises privacy and copyright risks when models generate new content. The analysis also covers the optimal empirical score and distribution collapse, with real-data experiments supporting the theory.

Core claim

Higher-order Langevin dynamics introduce auxiliary variables that can be viewed as velocity and acceleration when the data variable is treated as position. These variables impose extra dynamical constraints, so the data variable's updates are driven by a low-pass-filtered version of the score function whose smoothness increases with order. This regularization prevents the model from collapsing onto individual training points and thereby mitigates memorization.

What carries the argument

Higher-order Langevin dynamics whose auxiliary variables produce a low-pass-filtered score function whose cutoff sharpens with order.

If this is right

Higher model order produces smoother score-driven trajectories and therefore less exact memorization.
The optimal empirical score under HOLD does not collapse to training points.
Real-world experiments confirm that HOLD exhibits lower memorization than standard first-order diffusion.
The approach supplies a practical advantage for generating content without reproducing protected training instances.

Where Pith is reading between the lines

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

The same auxiliary-variable construction could be tested in other score-based samplers to control overfitting.
Frequency-domain analysis of the filtered score might reveal further connections to signal-processing ideas for generative models.
Extending the order to very high values could be checked on large-scale datasets to see whether the memorization benefit saturates.

Load-bearing premise

The auxiliary variables impose dynamical constraints that actually translate into low-pass filtering of the empirical score on the data variable.

What would settle it

An experiment in which raising the order of HOLD leaves the rate of exact training-sample reproduction unchanged would falsify the claimed link between filtering and reduced memorization.

Figures

Figures reproduced from arXiv: 2605.19170 by Benjamin Sterling, M\'onica F. Bugallo, Tom Tirer.

**Figure 1.** Figure 1: Magnitudes of the Fourier Transforms |H(iω)| for different HOLD diffusion model orders n, and the Ornstein–Uhlenbeck filter with ξ = 1. One may observe that the HOLD filters are better at attenuating higher frequencies than OU, while still allowing a wide band of lower frequencies. “smoothing” is manifested by h (n) t acting on the score function as a low-pass filter with stronger attenuation of high frequ… view at source ↗

**Figure 2.** Figure 2: Determinant of the inverse covariance matrix, assuming [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: CelebA FIDs and fraction memorized (Fmem) percentages by the number of training [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Nearest training neighbors for different models at [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: CIFAR-10 FIDs and fraction memorized (Fmem) percentages by the number of training [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: Initial auxiliary variable comparisons for [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

**Figure 7.** Figure 7: Nearest training neighbors for different models at [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗

**Figure 8.** Figure 8: Training losses for ntrain = 256 for the VPSDE and HOLD n = 2, 3 SDEs. C Additional CelebA images with FIDs and Fmem This appendix section presents more generated samples from the CelebA experiment for different number of training images ntrain. D Miscellaneous details For all experiments, a learning rate of 1 × 10−4 , the Adam optimizer, 32 base channels, a dropout rate of 0.1, and 1000 diffusion model st… view at source ↗

**Figure 9.** Figure 9: Celeba comparison for ntrain = 256 training samples at 1,000,000 training iterations. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗

**Figure 10.** Figure 10: Celeba comparison for ntrain = 512 training samples at 1,000,000 training iterations. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗

**Figure 11.** Figure 11: Celeba comparison for ntrain = 1024 training samples at 1,000,000 training iterations. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗

**Figure 12.** Figure 12: Celeba comparison for ntrain = 2048 training samples at 1,000,000 training iterations. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗

read the original abstract

Diffusion/score-based models have emerged as powerful generative models, capable of generating high-quality samples that mimic the training data distribution. However, it has been observed that they are prone to reproducing training samples-known as "memorization"-potentially violating copyright and privacy. In this paper, we study the effect of Higher-Order Langevin Dynamics (HOLD) on this phenomenon. HOLD diffusion processes introduce auxiliary variables; if the data variable is interpreted as "position," then the auxiliary variables can be interpreted as "velocity" and "acceleration," depending on the chosen order of the model. They were originally proposed based on the intuition that they regularize the trajectories of the data variable by implicitly imposing additional dynamical constraints. Our work provides, to our knowledge, the first theoretical characterization of the regularization effect of HOLD. Specifically, we show that in HOLD, the dynamics of the data variable are governed by a low-pass-filtered version of the learned score function, with smoothness increasing with the order of HOLD. We then analyze the optimal empirical score and the possibility of distribution collapse. Together, our results explain the mitigation of memorization as the model order increases. Finally, we present an empirical study on real-world data that supports our theory and highlights this distinct advantage of HOLD over standard diffusion in practice.

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

HOLD gives a low-pass filter view of how higher-order dynamics curb memorization in diffusion models, but the step from learned score to optimal empirical score still looks like the main gap.

read the letter

The main point is that the authors derive a low-pass filtering effect on the learned score under Higher-Order Langevin Dynamics, where the data variable's motion gets smoother as the order rises because of the auxiliary velocity and acceleration terms. They then connect this to reduced memorization by analyzing how the optimal empirical score can drive distribution collapse, and they show an empirical edge on real data over ordinary diffusion sampling. That filtering characterization is the clearest new piece; prior Langevin work did not spell out this regularization as an explicit low-pass operation that strengthens with order. The empirical comparison is also useful because it moves the claim beyond pure theory. The soft spot is the transfer step the stress-test note flags. The filtering result applies to the learned score, yet the collapse argument rests on properties of the optimal empirical score. The abstract presents these as sequential but does not make the alignment between their high-frequency modes explicit, so the explanatory chain from filtering to memorization mitigation stays partly assumptive. If the full derivations close that gap with a direct argument, the concern shrinks; otherwise it remains the place where a referee would press hardest. This paper is for people working on sampling methods or privacy constraints in score-based generative models. A reader already thinking about alternatives to standard Langevin dynamics would find the theory and the practical comparison worth their time. It deserves a serious referee because the central claim is specific enough to check and the work includes both a derivation and real-data support. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that Higher-Order Langevin Dynamics (HOLD) reduces memorization in diffusion/score-based models. It provides the first theoretical characterization showing that the data variable dynamics are governed by a low-pass-filtered version of the learned score function, with smoothness increasing with HOLD order. It then analyzes the optimal empirical score and distribution collapse to explain the mitigation of memorization as model order increases, supported by an empirical study on real-world data.

Significance. If the explanatory chain holds, the work offers a principled, training-free approach to controlling memorization via sampling dynamics order, which is significant for privacy and copyright issues in generative models. The combination of a new theoretical regularization result with analysis of collapse modes and empirical validation is a strength.

major comments (1)

[Abstract and theoretical characterization sections] The low-pass filtering result is derived for the learned score function, but the memorization mitigation and distribution-collapse argument rely on properties of the optimal empirical score. The manuscript does not explicitly bridge how the filtered learned score prevents the high-frequency collapse modes identified for the optimal case (see the paragraph on intuition for HOLD and the subsequent analysis of the optimal empirical score). This link is load-bearing for the central claim.

minor comments (2)

[Empirical study section] The empirical study description would benefit from explicit reporting of quantitative metrics, baselines, and controls to allow direct verification of the claimed advantage over standard diffusion.
[HOLD process definition] Notation for auxiliary variables (velocity/acceleration) and the precise definition of the low-pass filter could be introduced with a short diagram or expanded equation for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback and positive assessment of the significance of our work on Higher-Order Langevin Dynamics for reducing memorization. We address the major comment below and outline the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract and theoretical characterization sections] The low-pass filtering result is derived for the learned score function, but the memorization mitigation and distribution-collapse argument rely on properties of the optimal empirical score. The manuscript does not explicitly bridge how the filtered learned score prevents the high-frequency collapse modes identified for the optimal case (see the paragraph on intuition for HOLD and the subsequent analysis of the optimal empirical score). This link is load-bearing for the central claim.

Authors: We agree that the connection between the low-pass filtering result (which holds for the learned score) and the collapse analysis (which characterizes the optimal empirical score) should be made more explicit to support the central claim. The low-pass filtering theorem shows that HOLD dynamics are governed by a smoothed version of whatever score is provided, including the learned score. The optimal-score analysis identifies that high-frequency components in the score induce collapse and memorization. Because the learned score approximates the optimal empirical score in high-density regions (where generation occurs), the same smoothing attenuates those high-frequency modes for the learned score as well. We acknowledge that this approximation-based link is currently implicit. In the revised manuscript we will add a short bridging paragraph immediately after the optimal-score collapse analysis, explicitly stating that the low-pass operator applied to the learned score suppresses the identified collapse modes via the approximation property, thereby explaining the observed mitigation of memorization with increasing HOLD order. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation of HOLD low-pass filtering and memorization mitigation

full rationale

The paper derives the low-pass-filtered dynamics on the learned score directly from the structure of the higher-order auxiliary variables in HOLD, then performs a separate analysis of the optimal empirical score and distribution collapse to connect increasing order to reduced memorization. No equations or steps reduce by construction to fitted parameters, self-referential definitions, or load-bearing self-citations; the central claims rest on independent theoretical characterization of the regularization effect rather than renaming known results or smuggling ansatzes via prior work. The derivation chain is self-contained against external dynamical analysis and does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review is based solely on the abstract; no explicit free parameters, axioms, or invented entities are stated. The core intuition that auxiliary variables impose dynamical constraints is treated as a modeling choice rather than a derived result.

pith-pipeline@v0.9.0 · 5760 in / 1234 out tokens · 27140 ms · 2026-05-20T07:07:46.328226+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1. ... xt = h^{(n)}_t * s_θ(ut,t) + x_natural_t where h^{(n)}_t = −γ̄ n √(2n−3) L^{-1} t^{n−1} exp(−t √(2n−3))
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 2. lim t→0+ DM(u^{(k)}_0, p_emp,HOLD_t) ≫ 0 for n=2,3 while DM=0 for OU

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.