# Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure

arXiv: https://arxiv.org/abs/2410.24060 · HTML: https://mlsanity.com/paper/2410.24060/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 36 of its 48 references that are in the map.
- First posted on arXiv 31 Oct 2024.

## Placed in

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

## Builds on (most cited first, 30 of 36)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [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)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Diffusion Art or Digital Forgery? Investigating Data Replication in Diffusion Models](https://mlsanity.com/paper/2212.03860/) (2022)
- [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)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [On Memorization in Diffusion Models](https://mlsanity.com/paper/2310.02664/) (2023)
- [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (2024)
- [Learning Mixtures of Gaussians Using the DDPM Objective](https://mlsanity.com/paper/2307.01178/) (2023)
- [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024)
- [Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality](https://mlsanity.com/paper/2410.18784/) (2024)
- [Analysis of learning a flow-based generative model from limited sample complexity](https://mlsanity.com/paper/2310.03575/) (2023)
- [A Good Score Does not Lead to A Good Generative Model](https://mlsanity.com/paper/2401.04856/) (2024)

## Built on by (most cited first, 30 of 38)

- [Towards a Mechanistic Explanation of Diffusion Model Generalization](https://mlsanity.com/paper/2411.19339/) (2024)
- [Low-dimensional adaptation of diffusion models: Convergence in total variation](https://mlsanity.com/paper/2501.12982/) (2025)
- [An Analytical Theory of Spectral Bias in the Learning Dynamics of Diffusion Models](https://mlsanity.com/paper/2503.03206/) (2025)
- [Bigger Isn't Always Memorizing: Early Stopping Overparameterized Diffusion Models](https://mlsanity.com/paper/2505.16959/) (2025)
- [Locality in Image Diffusion Models Emerges from Data Statistics](https://mlsanity.com/paper/2509.09672/) (2025)
- [Understanding Representation Dynamics of Diffusion Models via Low-Dimensional Modeling](https://mlsanity.com/paper/2502.05743/) (2025)
- [A Closer Look at Model Collapse: From a Generalization-to-Memorization Perspective](https://mlsanity.com/paper/2509.16499/) (2025)
- [Selective Underfitting in Diffusion Models](https://mlsanity.com/paper/2510.01378/) (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)
- [A solvable model of learning generative diffusion: theory and insights](https://mlsanity.com/paper/2501.03937/) (2025)
- [Generalization Dynamics of Linear Diffusion Models](https://mlsanity.com/paper/2505.24769/) (2025)
- [Algorithm- and Data-Dependent Generalization Bounds for Diffusion Models](https://mlsanity.com/paper/2506.03849/) (2025)
- [Diffusion models under low-noise regime](https://mlsanity.com/paper/2506.07841/) (2025)
- [A theory of learning data statistics in diffusion models, from easy to hard](https://mlsanity.com/paper/2603.12901/) (2026)
- [Deeper Diffusion Models Amplify Bias](https://mlsanity.com/paper/2505.17560/) (2025)
- [Antithetic Noise in Diffusion Models](https://mlsanity.com/paper/2506.06185/) (2025)
- [Geometric Regularity in Deterministic Sampling Dynamics of Diffusion-based Generative Models](https://mlsanity.com/paper/2506.10177/) (2025)
- [When and how can inexact generative models still sample from the data manifold?](https://mlsanity.com/paper/2508.07581/) (2025)
- [Kernel-Smoothed Scores for Denoising Diffusion: A Bias-Variance Study](https://mlsanity.com/paper/2505.22841/) (2025)
- [WaveletDiff: Multilevel Wavelet Diffusion For Time Series Generation](https://mlsanity.com/paper/2510.11839/) (2025)
- [Two Calm Ends and the Wild Middle: A Geometric Picture of Memorization in Diffusion Models](https://mlsanity.com/paper/2602.17846/) (2026)
- [Beyond One-Hot Labels: Semantic Mixing for Model Calibration](https://mlsanity.com/paper/2504.13548/) (2025)
- [Toward Theoretical Insights into Diffusion Trajectory Distillation via Operator Merging](https://mlsanity.com/paper/2505.16024/) (2025)
- [Understanding Generalization in Diffusion Distillation via Probability Flow Distance](https://mlsanity.com/paper/2505.20123/) (2025)
- [Diffusion Models Memorize in Training -- and Generalize in Inference](https://mlsanity.com/paper/2603.13419/) (2026)
- [On the Collapse Errors Induced by the Deterministic Sampler for Diffusion Models](https://mlsanity.com/paper/2508.16154/) (2025)
- [The two clocks and the innovation window: When and how generative models learn rules](https://mlsanity.com/paper/2605.10019/) (2026)
- [Evaluating the Representation Space of Diffusion Models via Self-Supervised Principles](https://mlsanity.com/paper/2606.09718/) (2026)
- [Gaussian Universality for Diffusion Models](https://mlsanity.com/paper/2501.07741/) (2025)

## Abstract (arXiv)

In this work, we study the generalizability of diffusion models by looking into the hidden properties of the learned score functions, which are essentially a series of deep denoisers trained on various noise levels. We observe that as diffusion models transition from memorization to generalization, their corresponding nonlinear diffusion denoisers exhibit increasing linearity. This discovery leads us to investigate the linear counterparts of the nonlinear diffusion models, which are a series of linear models trained to match the function mappings of the nonlinear diffusion denoisers. Surprisingly, these linear denoisers are approximately the optimal denoisers for a multivariate Gaussian distribution characterized by the empirical mean and covariance of the training dataset. This finding implies that diffusion models have the inductive bias towards capturing and utilizing the Gaussian structure (covariance information) of the training dataset for data generation. We empirically demonstrate that this inductive bias is a unique property of diffusion models in the generalization regime, which becomes increasingly evident when the model's capacity is relatively small compared to the training dataset size. In the case that the model is highly overparameterized, this inductive bias emerges during the initial training phases before the model fully memorizes its training data. Our study provides crucial insights into understanding the notable strong generalization phenomenon recently observed in real-world diffusion models.