# Memorization and Regularization in Generative Diffusion Models

arXiv: https://arxiv.org/abs/2501.15785 · HTML: https://mlsanity.com/paper/2501.15785/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 18 of its 48 references that are in the map.
- First posted on arXiv 27 Jan 2025.

## Placed in

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

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [On Memorization in Diffusion Models](https://mlsanity.com/paper/2310.02664/) (2023)
- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [Understanding Hallucinations in Diffusion Models through Mode Interpolation](https://mlsanity.com/paper/2406.09358/) (2024)
- [Optimal score estimation via empirical Bayes smoothing](https://mlsanity.com/paper/2402.07747/) (2024)
- [Closed-Form Diffusion Models](https://mlsanity.com/paper/2310.12395/) (2023)
- [A Good Score Does not Lead to A Good Generative Model](https://mlsanity.com/paper/2401.04856/) (2024)
- [Mathematical analysis of singularities in the diffusion model under the submanifold assumption](https://mlsanity.com/paper/2301.07882/) (2023)
- [Wasserstein proximal operators describe score-based generative models and resolve memorization](https://mlsanity.com/paper/2402.06162/) (2024)
- [SIDE: Surrogate Conditional Data Extraction from Diffusion Models](https://mlsanity.com/paper/2410.02467/) (2024)
- [An Inversion-based Measure of Memorization for Diffusion Models](https://mlsanity.com/paper/2405.05846/) (2024)
- [Investigating Memorization in Video Diffusion Models](https://mlsanity.com/paper/2410.21669/) (2024)
- [Reducing Training Sample Memorization in GANs by Training with Memorization Rejection](https://mlsanity.com/paper/2210.12231/) (2022)
- [Capacity Control is an Effective Memorization Mitigation Mechanism in Text-Conditional Diffusion Models](https://mlsanity.com/paper/2410.22149/) (2024)
- [Elucidating Flow Matching ODE Dynamics with Respect to Data Geometries and Denoisers](https://mlsanity.com/paper/2412.18730/) (2024)

## Built on by (most cited first, 30 of 40)

- [Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training](https://mlsanity.com/paper/2505.17638/) (2025)
- [Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive](https://mlsanity.com/paper/2510.02305/) (2025)
- [Understanding Representation Dynamics of Diffusion Models via Low-Dimensional Modeling](https://mlsanity.com/paper/2502.05743/) (2025)
- [Taking a Big Step: Large Learning Rates in Denoising Score Matching Prevent Memorization](https://mlsanity.com/paper/2502.03435/) (2025)
- [On the Interpolation Effect of Score Smoothing in Diffusion Models](https://mlsanity.com/paper/2502.19499/) (2025)
- [Making Reconstruction FID Predictive of Diffusion Generation FID](https://mlsanity.com/paper/2603.05630/) (2026)
- [Flow Matching for Efficient and Scalable Data Assimilation](https://mlsanity.com/paper/2508.13313/) (2025)
- [Generalization of Diffusion Models Arises with a Balanced Representation Space](https://mlsanity.com/paper/2512.20963/) (2025)
- [Decoupled Diffusion Sampling for Inverse Problems on Function Spaces](https://mlsanity.com/paper/2601.23280/) (2026)
- [Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spaces](https://mlsanity.com/paper/2505.02508/) (2025)
- [PerturbDiff: Functional Diffusion for Single-Cell Perturbation Modeling](https://mlsanity.com/paper/2602.19685/) (2026)
- [Unifying and extending Diffusion Models through PDEs for solving Inverse Problems](https://mlsanity.com/paper/2504.07437/) (2025)
- [Diffusion Model's Generalization Can Be Characterized by Inductive Biases toward a Data-Dependent Ridge Manifold](https://mlsanity.com/paper/2602.06021/) (2026)
- [Iterative Flow Matching: Path Correction and Gradual Refinement for Enhanced Generative Modeling](https://mlsanity.com/paper/2502.16445/) (2025)
- [When and how can inexact generative models still sample from the data manifold?](https://mlsanity.com/paper/2508.07581/) (2025)
- [Predicting Forced Responses of Probability Distributions via the Fluctuation-Dissipation Theorem and Generative Modeling](https://mlsanity.com/paper/2504.13333/) (2025)
- [Kernel-Smoothed Scores for Denoising Diffusion: A Bias-Variance Study](https://mlsanity.com/paper/2505.22841/) (2025)
- [A Kinetic Energy Perspective of Flow Matching](https://mlsanity.com/paper/2602.07928/) (2026)
- [Two Calm Ends and the Wild Middle: A Geometric Picture of Memorization in Diffusion Models](https://mlsanity.com/paper/2602.17846/) (2026)
- [Closed-form conditional diffusion models for data assimilation](https://mlsanity.com/paper/2603.21291/) (2026)
- [Tessellations of Semi-Discrete Flow Matching](https://mlsanity.com/paper/2605.07513/) (2026)
- [Generalization, memorization, and overfitting for diffusion models trained in the lazy high-dimensional regime](https://mlsanity.com/paper/2608.23938/) (2026)
- [Regularization can make diffusion models more efficient](https://mlsanity.com/paper/2502.09151/) (2025)
- [A PDE Perspective on Generative Diffusion Models](https://mlsanity.com/paper/2511.05940/) (2025)
- [On The Hidden Biases of Flow Matching Samplers](https://mlsanity.com/paper/2512.16768/) (2025)
- [Diffusion Models Memorize in Training -- and Generalize in Inference](https://mlsanity.com/paper/2603.13419/) (2026)
- [Conditional flow matching for physics-constrained inverse problems with finite training data](https://mlsanity.com/paper/2603.14135/) (2026)
- [Towards Irreversible Machine Unlearning for Diffusion Models](https://mlsanity.com/paper/2512.03564/) (2025)
- [On the role of memorization in learned priors for geophysical inverse problems](https://mlsanity.com/paper/2603.19629/) (2026)
- [Diffusion Models Preferentially Memorize Prototypical Examples or: Why Does My Diffusion Model Love Slop?](https://mlsanity.com/paper/2605.30642/) (2026)

## Abstract (arXiv)

Diffusion models have emerged as a powerful framework for generative modeling. At the heart of the methodology is score matching: learning gradients of families of log-densities for noisy versions of the data distribution at different scales. When the loss function adopted in score matching is evaluated using empirical data, rather than the population loss, the minimizer corresponds to the score of a time-dependent Gaussian mixture. However, use of this analytically tractable minimizer leads to data memorization: in both unconditioned and conditioned settings, the generative model returns the training samples. This paper contains an analysis of the dynamical mechanism underlying memorization. The analysis highlights the need for regularization to avoid reproducing the analytically tractable minimizer; and, in so doing, lays the foundations for a principled understanding of how to regularize. Numerical experiments investigate the properties of: (i) Tikhonov regularization; (ii) regularization designed to promote asymptotic consistency; and (iii) regularizations induced by under-parameterization of a neural network or by early stopping when training a neural network. These experiments are evaluated in the context of memorization, and directions for future development of regularization are highlighted.