# Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training

arXiv: https://arxiv.org/abs/2505.17638 · HTML: https://mlsanity.com/paper/2505.17638/

## Key facts

- Cited by 81 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 48 of its 68 references that are in the map.
- First posted on arXiv 23 May 2025.

## Placed in

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

## Builds on (most cited first, 30 of 48)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (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)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [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)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Extracting Training Data from Diffusion Models](https://mlsanity.com/paper/2301.13188/) (2023)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Sora: A Review on Background, Technology, Limitations, and Opportunities of Large Vision Models](https://mlsanity.com/paper/2402.17177/) (2024)
- [Diffusion Art or Digital Forgery? Investigating Data Replication in Diffusion Models](https://mlsanity.com/paper/2212.03860/) (2022)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Understanding and Mitigating Copying in Diffusion Models](https://mlsanity.com/paper/2305.20086/) (2023)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [Generalisation error in learning with random features and the hidden manifold model](https://mlsanity.com/paper/2002.09339/) (2020)
- [On Memorization in Diffusion Models](https://mlsanity.com/paper/2310.02664/) (2023)
- [The Gaussian equivalence of generative models for learning with shallow neural networks](https://mlsanity.com/paper/2006.14709/) (2020)

## Built on by (most cited first, 30 of 81)

- [On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity](https://mlsanity.com/paper/2506.03719/) (2025)
- [Provable Separations between Memorization and Generalization in Diffusion Models](https://mlsanity.com/paper/2511.03202/) (2025)
- [The Information Dynamics of Generative Diffusion](https://mlsanity.com/paper/2508.19897/) (2025)
- [Making Reconstruction FID Predictive of Diffusion Generation FID](https://mlsanity.com/paper/2603.05630/) (2026)
- [Delving into Latent Spectral Biasing of Video VAEs for Superior Diffusability](https://mlsanity.com/paper/2512.05394/) (2025)
- [Generalization of Diffusion Models Arises with a Balanced Representation Space](https://mlsanity.com/paper/2512.20963/) (2025)
- [Generalization Dynamics of Linear Diffusion Models](https://mlsanity.com/paper/2505.24769/) (2025)
- [From Navigation to Refinement: Revealing the Two-Stage Nature of Flow-based Diffusion Models through Oracle Velocity](https://mlsanity.com/paper/2512.02826/) (2025)
- [The Generation Phases of Flow Matching: a Denoising Perspective](https://mlsanity.com/paper/2510.24830/) (2025)
- [Revisiting Diffusion Model Predictions Through Dimensionality](https://mlsanity.com/paper/2601.21419/) (2026)
- [Spectrum Matching: a Unified Perspective for Superior Diffusability in Latent Diffusion](https://mlsanity.com/paper/2603.14645/) (2026)
- [A Random Matrix Theory Perspective on the Consistency of Diffusion Models](https://mlsanity.com/paper/2602.02908/) (2026)
- [Diffusion Model's Generalization Can Be Characterized by Inductive Biases toward a Data-Dependent Ridge Manifold](https://mlsanity.com/paper/2602.06021/) (2026)
- [A theory of learning data statistics in diffusion models, from easy to hard](https://mlsanity.com/paper/2603.12901/) (2026)
- [Local Coverage Governs Memorization in Diffusion Models](https://mlsanity.com/paper/2606.14390/) (2026)
- [Biased Generalization in Diffusion Models](https://mlsanity.com/paper/2603.03469/) (2026)
- [Distinct mechanisms underlying in-context learning in transformers](https://mlsanity.com/paper/2604.12151/) (2026)
- [Can AI Weather Models Predict Beyond Two Weeks? A Quantitative Benchmark and Analysis of Long Rollouts](https://mlsanity.com/paper/2605.30184/) (2026)
- [A Kinetic Energy Perspective of Flow Matching](https://mlsanity.com/paper/2602.07928/) (2026)
- [Steering Dynamical Regimes of Diffusion Models by Breaking Detailed Balance](https://mlsanity.com/paper/2602.15914/) (2026)
- [Two Calm Ends and the Wild Middle: A Geometric Picture of Memorization in Diffusion Models](https://mlsanity.com/paper/2602.17846/) (2026)
- [Asymptotic Learning Curves for Diffusion Models with Random Features Score and Manifold Data](https://mlsanity.com/paper/2603.22962/) (2026)
- [Benign Overfitting Does Not Occur in Diffusion Models](https://mlsanity.com/paper/2607.02671/) (2026)
- [An exact information theory of generalization phase transitions in Bayesian diffusion models](https://mlsanity.com/paper/2607.08041/) (2026)
- [Understanding Generalization in Diffusion Distillation via Probability Flow Distance](https://mlsanity.com/paper/2505.20123/) (2025)
- [Pseudo-likelihood produces associative memories able to generalize, even for asymmetric couplings](https://mlsanity.com/paper/2507.05147/) (2025)
- [Diffusion Models Memorize in Training -- and Generalize in Inference](https://mlsanity.com/paper/2603.13419/) (2026)
- [Manifold Generalization Provably Proceeds Memorization in Diffusion Models](https://mlsanity.com/paper/2603.23792/) (2026)
- [Understanding diffusion models requires rethinking (again) generalization](https://mlsanity.com/paper/2605.06077/) (2026)
- [Memorization Control in Diffusion Models from Denoising-centric Perspective](https://mlsanity.com/paper/2601.21348/) (2026)

## Abstract (arXiv)

Diffusion models have achieved remarkable success across a wide range of generative tasks. A key challenge is understanding the mechanisms that prevent their memorization of training data and allow generalization. In this work, we investigate the role of the training dynamics in the transition from generalization to memorization. Through extensive experiments and theoretical analysis, we identify two distinct timescales: an early time τgen at which models begin to generate high-quality samples, and a later time τmem beyond which memorization emerges. Crucially, we find that τmem increases linearly with the training set size n, while τgen remains constant. This creates a growing window of training times with n where models generalize effectively, despite showing strong memorization if training continues beyond it. It is only when n becomes larger than a model-dependent threshold that overfitting disappears at infinite training times. These findings reveal a form of implicit dynamical regularization in the training dynamics, which allow to avoid memorization even in highly overparameterized settings. Our results are supported by numerical experiments with standard U-Net architectures on realistic and synthetic datasets, and by a theoretical analysis using a tractable random features model studied in the high-dimensional limit.