# On Error Propagation of Diffusion Models

arXiv: https://arxiv.org/abs/2308.05021 · HTML: https://mlsanity.com/paper/2308.05021/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 21 of its 35 references that are in the map.
- First posted on arXiv 9 Aug 2023.

## Placed in

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

## Builds on (most cited first, 21 of 21)

- [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)
- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Diffusion-LM Improves Controllable Text Generation](https://mlsanity.com/paper/2205.14217/) (2022)
- [Sequence Level Training with Recurrent Neural Networks](https://mlsanity.com/paper/1511.06732/) (2015)
- [Deep Exploration via Bootstrapped DQN](https://mlsanity.com/paper/1602.04621/) (2016)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Training generative neural networks via Maximum Mean Discrepancy optimization](https://mlsanity.com/paper/1505.03906/) (2015)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [Bridging the Gap between Training and Inference for Neural Machine Translation](https://mlsanity.com/paper/1906.02448/) (2019)
- [Input Perturbation Reduces Exposure Bias in Diffusion Models](https://mlsanity.com/paper/2301.11706/) (2023)
- [Alleviating Exposure Bias in Diffusion Models through Sampling with Shifted Time Steps](https://mlsanity.com/paper/2305.15583/) (2023)
- [Consistent Diffusion Models: Mitigating Sampling Drift by Learning to be Consistent](https://mlsanity.com/paper/2302.09057/) (2023)
- [On the Equivalence of Consistency-Type Models: Consistency Models, Consistent Diffusion Models, and Fokker-Planck Regularization](https://mlsanity.com/paper/2306.00367/) (2023)

## Built on by (most cited first, 30 of 31)

- [Elucidating the Exposure Bias in Diffusion Models](https://mlsanity.com/paper/2308.15321/) (2023)
- [Improving the Diffusability of Autoencoders](https://mlsanity.com/paper/2502.14831/) (2025)
- [A Fourier Space Perspective on Diffusion Models](https://mlsanity.com/paper/2505.11278/) (2025)
- [Deep MMD Gradient Flow without adversarial training](https://mlsanity.com/paper/2405.06780/) (2024)
- [3D-Adapter: Geometry-Consistent Multi-View Diffusion for High-Quality 3D Generation](https://mlsanity.com/paper/2410.18974/) (2024)
- [Adding Conditional Control to Diffusion Models with Reinforcement Learning](https://mlsanity.com/paper/2406.12120/) (2024)
- [Timestep-Aware Correction for Quantized Diffusion Models](https://mlsanity.com/paper/2407.03917/) (2024)
- [FlexiDiT: Your Diffusion Transformer Can Easily Generate High-Quality Samples with Less Compute](https://mlsanity.com/paper/2502.20126/) (2025)
- [XGeM: A Multi-Prompt Foundation Model for Multimodal Medical Data Generation](https://mlsanity.com/paper/2501.04614/) (2025)
- [Q-Drift: Quantization-Aware Drift Correction for Diffusion Model Sampling](https://mlsanity.com/paper/2603.18095/) (2026)
- [Exposure Bias Can Alleviate Itself via Directional and Frequency Rectification in Flow Matching](https://mlsanity.com/paper/2606.28226/) (2026)
- [Improved Diffusion-based Generative Model with Better Adversarial Robustness](https://mlsanity.com/paper/2502.17099/) (2025)
- [On Efficiency-Effectiveness Trade-off of Diffusion-based Recommenders](https://mlsanity.com/paper/2510.17245/) (2025)
- [Autoregressive Visual Decoding from EEG Signals](https://mlsanity.com/paper/2602.22555/) (2026)
- [Elucidating the SNR-t Bias of Diffusion Probabilistic Models](https://mlsanity.com/paper/2604.16044/) (2026)
- [Consistent Diffusion: Denoising Diffusion Model with Data-Consistent Training for Image Restoration](https://mlsanity.com/paper/2412.12550/) (2024)
- [Flow-Matching Based Refiner for Molecular Conformer Generation](https://mlsanity.com/paper/2510.04878/) (2025)
- [MixFlow Training: Alleviating Exposure Bias with Slowed Interpolation Mixture](https://mlsanity.com/paper/2512.19311/) (2025)
- [Bias mitigation in graph diffusion models](https://mlsanity.com/paper/2604.01709/) (2026)
- [The Reward Was in Your Data All Along: Correcting Flow Matching with Discriminator-Guided RL](https://mlsanity.com/paper/2606.19162/) (2026)
- [Spectral Prior for Reducing Exposure Bias in Diffusion Models](https://mlsanity.com/paper/2607.22091/) (2026)
- [Q&C: When Quantization Meets Cache in Efficient Image Generation](https://mlsanity.com/paper/2503.02508/) (2025)
- [A Start To End Machine Learning Approach To Maximize Scientific Throughput From The LCLS-II-HE](https://mlsanity.com/paper/2505.23858/) (2025)
- [DidSee: Diffusion-Based Depth Completion for Material-Agnostic Robotic Perception and Manipulation](https://mlsanity.com/paper/2506.21034/) (2025)
- [Cloud Model Characteristic Function Auto-Encoder: Integrating Cloud Model Theory with MMD Regularization for Enhanced Generative Modeling](https://mlsanity.com/paper/2508.04447/) (2025)
- [ReflexFlow: Rethinking Learning Objective for Exposure Bias Alleviation in Flow Matching](https://mlsanity.com/paper/2512.04904/) (2025)
- [Reconstruction-Anchored Diffusion Model for Text-to-Motion Generation](https://mlsanity.com/paper/2601.14788/) (2026)
- [Public Diffusion Models, Private Images: Key-Controlled Inversion for Conditional Reconstruction](https://mlsanity.com/paper/2606.22988/) (2026)
- [Post-training Quantization for Hybrid Iterative Generative Models](https://mlsanity.com/paper/2608.13932/) (2026)
- [Let It Go or Learn to Self-Correct: Continuous Diffusion for Constrained Discrete Tasks](https://mlsanity.com/paper/2609.09009/) (2026)

## Abstract (arXiv)

Although diffusion models (DMs) have shown promising performances in a number of tasks (e.g., speech synthesis and image generation), they might suffer from error propagation because of their sequential structure. However, this is not certain because some sequential models, such as Conditional Random Field (CRF), are free from this problem. To address this issue, we develop a theoretical framework to mathematically formulate error propagation in the architecture of DMs, The framework contains three elements, including modular error, cumulative error, and propagation equation. The modular and cumulative errors are related by the equation, which interprets that DMs are indeed affected by error propagation. Our theoretical study also suggests that the cumulative error is closely related to the generation quality of DMs. Based on this finding, we apply the cumulative error as a regularization term to reduce error propagation. Because the term is computationally intractable, we derive its upper bound and design a bootstrap algorithm to efficiently estimate the bound for optimization. We have conducted extensive experiments on multiple image datasets, showing that our proposed regularization reduces error propagation, significantly improves vanilla DMs, and outperforms previous baselines.