# Reducing Diffusion Model Memorization with Higher Order Langevin Dynamics

arXiv: https://arxiv.org/abs/2605.19170 · HTML: https://mlsanity.com/paper/2605.19170/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 24 of its 32 references that are in the map.
- First posted on arXiv 18 May 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 24 of 24)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Extracting Training Data from Diffusion Models](https://mlsanity.com/paper/2301.13188/) (2023)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Diffusion Art or Digital Forgery? Investigating Data Replication in Diffusion Models](https://mlsanity.com/paper/2212.03860/) (2022)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [On Memorization in Diffusion Models](https://mlsanity.com/paper/2310.02664/) (2023)
- [Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training](https://mlsanity.com/paper/2505.17638/) (2025)
- [Image Restoration by Denoising Diffusion Models with Iteratively Preconditioned Guidance](https://mlsanity.com/paper/2312.16519/) (2023)
- [On the Generalization of Diffusion Model](https://mlsanity.com/paper/2305.14712/) (2023)
- [ADIR: Adaptive Diffusion for Image Reconstruction](https://mlsanity.com/paper/2212.03221/) (2022)
- [Denoising Score Matching with Random Features: Insights on Diffusion Models from Precise Learning Curves](https://mlsanity.com/paper/2502.00336/) (2025)
- [Zero-Shot Image Restoration Using Few-Step Guidance of Consistency Models (and Beyond)](https://mlsanity.com/paper/2412.20596/) (2024)
- [Generative Modelling with High-Order Langevin Dynamics](https://mlsanity.com/paper/2404.12814/) (2024)
- [Critically Damped Third-Order Langevin Dynamics](https://mlsanity.com/paper/2409.07697/) (2024)
- [Critically-Damped Higher-Order Langevin Dynamics for Generative Modeling](https://mlsanity.com/paper/2506.21741/) (2025)
- [Defending Diffusion Models Against Membership Inference Attacks via Higher-Order Langevin Dynamics](https://mlsanity.com/paper/2509.14225/) (2025)

## Abstract (arXiv)

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.