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arxiv: 2509.14225 · v3 · submitted 2025-09-17 · 💻 cs.LG · stat.ML

Defending Diffusion Models Against Membership Inference Attacks via Higher-Order Langevin Dynamics

Benjamin Sterling , Yousef El-Laham , M\'onica F. Bugallo This is my paper

Pith reviewed 2026-05-18 15:51 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords diffusion modelsmembership inference attacksLangevin dynamicsdata privacygenerative modelsauxiliary variableshigher-order dynamics
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The pith

Critically-damped higher-order Langevin dynamics defends diffusion models against membership inference attacks by mixing external randomness through auxiliary variables.

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

This paper establishes that diffusion models can be hardened against membership inference attacks by replacing standard dynamics with critically-damped higher-order Langevin dynamics. The key step is the introduction of auxiliary variables whose joint diffusion process mixes external randomness into the data trajectory at an earlier stage. Theoretical analysis shows this corrupts sensitive training points before they can be reliably recovered by an attacker. Experiments on a toy dataset and a speech dataset confirm lower attack success rates via AUROC while preserving sample quality via FID scores.

Core claim

The presence of auxiliary variables in the critically-damped higher-order Langevin dynamics mixes external randomness that helps to corrupt sensitive input data earlier on in the diffusion process, thereby defending diffusion models against membership inference attacks. The approach is theoretically investigated and validated on a toy dataset and a speech dataset using the Area Under the Receiver Operating Characteristic (AUROC) curves and the FID metric.

What carries the argument

Critically-damped higher-order Langevin dynamics that introduces auxiliary variables and performs a joint diffusion process along those variables to mix external randomness.

If this is right

  • The defense integrates directly into existing diffusion training loops without altering the learned model architecture.
  • Attack success rates drop as measured by AUROC on both toy and speech data while FID scores stay comparable.
  • The method supplies an early-stage corruption mechanism that standard first-order Langevin dynamics lacks.
  • Privacy gains are achieved by the mixing property of the auxiliary variables rather than by post-training noise injection.

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 might be tested on other generative families such as normalizing flows or GANs.
  • Adjusting the number or damping of auxiliary variables could offer a controllable privacy-utility trade-off not explored in the paper.
  • The approach may interact with existing differential-privacy mechanisms in diffusion training, though that interaction remains untested.

Load-bearing premise

The joint diffusion process over the auxiliary variables integrates into standard diffusion training without introducing new attack surfaces or forcing changes to the core model architecture that would reduce the privacy gain.

What would settle it

A direct test in which membership inference AUROC remains as high with the auxiliary-variable dynamics as with ordinary diffusion, or in which the added variables create a new distinguishable signal that an adversary can exploit.

read the original abstract

Recent advances in generative artificial intelligence applications have raised new data security concerns. This paper focuses on defending diffusion models against membership inference attacks. This type of attack occurs when the attacker can determine if a certain data point was used to train the model. Although diffusion models are intrinsically more resistant to membership inference attacks than other generative models, they are still susceptible. The defense proposed here utilizes critically-damped higher-order Langevin dynamics, which introduces several auxiliary variables and a joint diffusion process along these variables. The idea is that the presence of auxiliary variables mixes external randomness that helps to corrupt sensitive input data earlier on in the diffusion process. This concept is theoretically investigated and validated on a toy dataset and a speech dataset using the Area Under the Receiver Operating Characteristic (AUROC) curves and the FID metric.

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

3 major / 2 minor

Summary. The paper proposes a defense for diffusion models against membership inference attacks (MIA) that employs critically-damped higher-order Langevin dynamics. This introduces auxiliary variables together with a joint diffusion process over those variables; the external randomness is argued to corrupt sensitive training inputs earlier in the forward process. The approach is described as theoretically investigated and is validated on a toy dataset and a speech dataset using AUROC for attack success and FID for generation quality.

Significance. If the central claim holds, the method would supply a lightweight, architecture-preserving way to reduce MIA leakage in diffusion models by injecting controlled external randomness via higher-order dynamics. The idea of mixing randomness through auxiliary trajectories is conceptually appealing and could extend to other score-based generative models, but the current evidence base (toy + speech data, AUROC/FID only) is narrow and the theoretical analysis is not yet detailed enough to establish broad applicability.

major comments (3)
  1. Abstract: the claim that auxiliary variables 'mix external randomness that helps to corrupt sensitive input data earlier' is load-bearing for the defense, yet the abstract supplies neither the SDE for the joint process nor the initialization distribution of the auxiliary variables; without these the mechanism cannot be verified or reproduced.
  2. Experiments section (toy and speech results): AUROC and FID are reported without error bars, without the number of attack trials, and without any ablation on auxiliary initialization or joint-sampling procedure; this prevents assessment of whether the reported privacy gain is statistically reliable or sensitive to implementation choices.
  3. Method / threat model: the central privacy claim assumes the joint diffusion over auxiliaries does not open new leakage channels (e.g., attacks that observe or exploit the auxiliary trajectory). No analysis or experiment rules out such attacks, leaving the weakest assumption untested and the defense's net privacy benefit uncertain.
minor comments (2)
  1. Abstract: replace the vague phrase 'toy dataset' with the concrete dataset name and dimensionality; likewise specify the speech corpus and its preprocessing.
  2. Notation: the auxiliary variables and the joint process should be given explicit symbols and an equation reference in the main text so that later sections can refer to them unambiguously.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below, indicating where revisions will be made to improve clarity, statistical rigor, and completeness of the threat model analysis.

read point-by-point responses

  1. Referee: Abstract: the claim that auxiliary variables 'mix external randomness that helps to corrupt sensitive input data earlier' is load-bearing for the defense, yet the abstract supplies neither the SDE for the joint process nor the initialization distribution of the auxiliary variables; without these the mechanism cannot be verified or reproduced.

    Authors: We agree that the abstract should enable verification of the core mechanism. In the revised manuscript we will expand the abstract to include a concise statement of the joint SDE and the auxiliary initialization distribution, while respecting length constraints. The full mathematical derivation remains in Section 3. revision: yes

  2. Referee: Experiments section (toy and speech results): AUROC and FID are reported without error bars, without the number of attack trials, and without any ablation on auxiliary initialization or joint-sampling procedure; this prevents assessment of whether the reported privacy gain is statistically reliable or sensitive to implementation choices.

    Authors: We acknowledge the need for greater statistical transparency. We will add error bars from repeated independent runs, explicitly state the number of attack trials, and include ablations on auxiliary initialization and joint-sampling choices to demonstrate that the privacy improvements are robust. revision: yes

  3. Referee: Method / threat model: the central privacy claim assumes the joint diffusion over auxiliaries does not open new leakage channels (e.g., attacks that observe or exploit the auxiliary trajectory). No analysis or experiment rules out such attacks, leaving the weakest assumption untested and the defense's net privacy benefit uncertain.

    Authors: We appreciate this observation on the threat model. Our current formulation assumes a standard black-box attacker without access to internal trajectories. In revision we will add a dedicated paragraph in the threat-model section providing a theoretical argument that auxiliary trajectories do not introduce additional membership leakage, together with a new experiment that simulates an attacker given auxiliary information. revision: yes

Circularity Check

0 steps flagged

No circularity: defense mechanism introduces independent auxiliary variables and joint diffusion

full rationale

The paper's core proposal introduces auxiliary variables into critically-damped higher-order Langevin dynamics to mix external randomness and corrupt sensitive inputs earlier in the diffusion process. This is framed as a new theoretical construct that is then validated empirically on toy and speech datasets via AUROC and FID. No load-bearing step reduces the claimed privacy gain to a fitted parameter defined by attack success, a self-citation chain, or a renaming of known results; the joint diffusion process over auxiliaries is presented as an additive modification whose effect is measured separately from the original diffusion training. The derivation therefore remains self-contained against external benchmarks rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the unstated premise that the higher-order dynamics can be stably coupled to the standard diffusion forward process without altering the learned score function or introducing new privacy leaks; no free parameters, axioms, or invented entities are explicitly listed in the abstract.

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Forward citations

Cited by 1 Pith paper

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

  1. Reducing Diffusion Model Memorization with Higher Order Langevin Dynamics

    stat.ML 2026-05 unverdicted novelty 7.0

    Higher-order Langevin dynamics reduce memorization in diffusion models by making the data dynamics follow a low-pass-filtered score whose smoothness grows with model order.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    However, recent work has shown that they are still vulnerable to Backdoor Attacks, Membership Inference Attacks (MIA), and Adversarial Attacks [4]

    INTRODUCTION Diffusion models [1, 2] have been shown to be fundamentally less susceptible to data security issues than other generative models such as GANs [3]. However, recent work has shown that they are still vulnerable to Backdoor Attacks, Membership Inference Attacks (MIA), and Adversarial Attacks [4]. Defense against MIA is de- sirable, especially i...

    work page

  2. [2]

    Defending Diffusion Models Against Membership Inference Attacks via Higher-Order Langevin Dynamics

    BACKGROUND Here we will briefly review how traditional continuous diffusion models [7] apply to PIA; PIA will be used as a representative of such membership inference attacks. Diffusion models are a method of generating samples from an unknown intractable data distribu- tion. They possess a forward process that transforms training data into noise, for the...

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  3. [3]

    It is argued here that HOLD++ is better at defending against PIA than traditional diffusion mod- els because of its structure

    PROBLEM FORMULA TION This section will review HOLD++ and how to apply PIA to this specific diffusion method. It is argued here that HOLD++ is better at defending against PIA than traditional diffusion mod- els because of its structure. Following [15] and the previous section, we define the forward SDE of HOLD++ as:dx t = Fx tdt+Gdw, wherewis a standard Br...

    work page

  4. [4]

    METHODOLOGY This section rigorously proves that HOLD++ is R´enyi Differentially Private and that this bound only depends onϵ num, a variance addi- tion to the data that ensures numerical stability. The same modifica- tion works to achieve differential privacy on traditional continuous diffusion models, but at the end of the section we demonstrate that thi...

    work page

  5. [5]

    This section seeks to validate this claim on the Swiss Roll and LJ Speech datasets

    EXPERIMENTS AND RESULTS The theoretical section claims that PIA can be defended against using higher model ordersnand higher starting variancesβL −1. This section seeks to validate this claim on the Swiss Roll and LJ Speech datasets. The validation metric that this paper primar- ily uses is the Area Under the ROC curve (AUROC) that comes from running PIA....

    work page

  6. [6]

    This work provides a way to implicitly regularize using the diffusion process itself, without requiring direct data augmentation

    CONCLUSION It is well known that regularization helps to prevent membership inference attacks in generative models. This work provides a way to implicitly regularize using the diffusion process itself, without requiring direct data augmentation. This method works addition- ally well because existing membership inference attacks on diffu- sion models rely ...

    work page

  7. [7]

    Deep unsupervised learning using nonequilibrium thermodynamics,

    J. Sohl-Dickstein, E. A. Weiss, N. Maheswaranathan, and S. Ganguli, “Deep unsupervised learning using nonequilibrium thermodynamics,” 2015

    work page 2015

  8. [8]

    Denoising diffusion probabilis- tic models,

    J. Ho, A. Jain, and P. Abbeel, “Denoising diffusion probabilis- tic models,”Advances in Neural Information Processing Sys- tems, vol. 33, pp. 6840–6851, 2020

    work page 2020

  9. [9]

    Membership inference attacks against diffusion models,

    T. Matsumoto, T. Miura, and N. Yanai, “Membership inference attacks against diffusion models,” in2023 IEEE Security and Privacy Workshops (SPW), 2023, pp. 77–83

    work page 2023

  10. [10]

    Attacks and defenses for generative diffusion models: A comprehensive survey,

    V . T. Truong, L. B. Dang, and L. B. Le, “Attacks and defenses for generative diffusion models: A comprehensive survey,” ACM Comput. Surv., vol. 57, no. 8, Apr. 2025. [Online]. Available: https://doi.org/10.1145/3721479

    work page doi:10.1145/3721479 2025

  11. [11]

    Differentially private diffusion models,

    T. Dockhorn, T. Cao, A. Vahdat, and K. Kreis, “Differentially private diffusion models,”Transactions on Machine Learning Research, 2023. [Online]. Available: https://openreview.net/ forum?id=ZPpQk7FJXF

    work page 2023

  12. [12]

    Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang

    M. Abadi, A. Chu, I. Goodfellow, H. B. McMahan, I. Mironov, K. Talwar, and L. Zhang, “Deep learning with differential privacy,” inProceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, ser. CCS ’16. New York, NY , USA: Association for Computing Machinery, 2016, p. 308–318. [Online]. Available: https://doi.org/10.1145/297674...

    work page doi:10.1145/2976749.2978318 2016

  13. [13]

    Score-Based Generative Modeling through Stochastic Differential Equations

    Y . Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole, “Score-based generative model- ing through stochastic differential equations,”arXiv preprint arXiv:2011.13456, 2020

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  14. [14]

    Score-based generative modeling with critically-damped Langevin diffusion,

    T. Dockhorn, A. Vahdat, and K. Kreis, “Score-based generative modeling with critically-damped Langevin diffusion,”arXiv preprint arXiv:2112.07068, 2021

    work page arXiv 2021

  15. [15]

    Generative modelling with higher-order Langevin dynamics,

    Z. Shi and R. Liu, “Generative modelling with higher-order Langevin dynamics,”arXiv preprint arXiv:2404.12814, 2024

    work page arXiv 2024

  16. [16]

    Langwave: Realistic voice generation based on high- order Langevin dynamics,

    ——, “Langwave: Realistic voice generation based on high- order Langevin dynamics,” inICASSP 2024-2024 IEEE Inter- national Conference on Acoustics, Speech and Signal Process- ing (ICASSP). IEEE, 2024, pp. 10 661–10 665

    work page 2024

  17. [17]

    Noisy image restoration based on conditional accelera- tion score approximation,

    ——, “Noisy image restoration based on conditional accelera- tion score approximation,” inICASSP 2024-2024 IEEE Inter- national Conference on Acoustics, Speech and Signal Process- ing (ICASSP). IEEE, 2024, pp. 4000–4004

    work page 2024

  18. [18]

    Critically-damped third-order Langevin dynamics,

    B. Sterling and M. F. Bugallo, “Critically-damped third-order Langevin dynamics,” inICASSP 2025 - 2025 IEEE Interna- tional Conference on Acoustics, Speech and Signal Processing (ICASSP), 2025, pp. 1–5

    work page 2025

  19. [19]

    Are diffusion models vulnerable to membership inference attacks?

    J. Duan, F. Kong, S. Wang, X. Shi, and K. Xu, “Are diffusion models vulnerable to membership inference attacks?” inProceedings of the 40th International Conference on Machine Learning, ser. Proceedings of Machine Learning Research, A. Krause, E. Brunskill, K. Cho, B. Engelhardt, S. Sabato, and J. Scarlett, Eds., vol. 202. PMLR, 23– 29 Jul 2023, pp. 8717–8...

    work page 2023

  20. [20]

    An efficient membership inference attack for the diffusion model by proximal initialization,

    F. Kong, J. Duan, R. Ma, H. T. Shen, X. Shi, X. Zhu, and K. Xu, “An efficient membership inference attack for the diffusion model by proximal initialization,” inThe Twelfth International Conference on Learning Representations, 2024. [Online]. Available: https://openreview.net/forum?id=rpH9FcCEV6

    work page 2024

  21. [21]

    Bugallo , year=

    B. Sterling, C. Gueli, and M. F. Bugallo, “Critically-damped higher-order Langevin dynamics,” 2025. [Online]. Available: https://arxiv.org/abs/2506.21741

    work page arXiv 2025

  22. [22]

    R ´enyi differential privacy,

    I. Mironov, “R ´enyi differential privacy,” in2017 IEEE 30th Computer Security Foundations Symposium (CSF), 2017, pp. 263–275

    work page 2017

  23. [23]

    Grad-tts: A diffusion probabilistic model for text-to-speech,

    V . Popov, I. V ovk, V . Gogoryan, T. Sadekova, and M. Kudinov, “Grad-tts: A diffusion probabilistic model for text-to-speech,” inInternational Conference on Machine Learning, 2021. [Online]. Available: https://api.semanticscholar.org/CorpusID: 234483016

    work page 2021