mlsanityData to 14 Oct 2026

On the Asymptotic Mean Square Error Optimality of Diffusion Models

arXiv 2403.02957 ↗First posted 5 Mar 2024cs.LG stat.MLcited by 10

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion models (DMs) as generative priors have recently shown great potential for denoising tasks but lack theoretical understanding with respect to their mean square error (MSE) optimality. This paper proposes a novel denoising strategy inspired by the structure of the MSE-optimal conditional mean estimator (CME). The resulting DM-based denoiser can be conveniently employed using a pre-trained DM, being particularly fast by truncating reverse diffusion steps and not requiring stochastic re-sampling. We present a comprehensive (non-)asymptotic optimality analysis of the proposed diffusion-based denoiser, demonstrating polynomial-time convergence to the CME under mild conditions. Our analysis also derives a novel Lipschitz constant that depends solely on the DM's hyperparameters. Further, we offer a new perspective on DMs, showing that they inherently combine an asymptotically optimal denoiser with a powerful generator, modifiable by switching re-sampling in the reverse process on or off. The theoretical findings are thoroughly validated with experiments based on various benchmark datasets

Key facts. Cited by 10 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 62 of its 108 references that are in the map. First posted on arXiv 5 Mar 2024.

Builds on

62 of 108 references.

Closest ancestors · 28

Same direction, newest first

  1. 2024Broadening Target Distributions for Accelerated Diffusion Models via a Novel Analysis Approach21 Feb
  2. 2023Fast ODE-based Sampling for Diffusion Models in Around 5 Steps30 Nov
  3. 2023Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion1 Oct
  4. 2023KL Convergence Guarantees for Score diffusion models under minimal data assumptions23 Aug
  5. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization7 Aug
  6. 2023Learning Mixtures of Gaussians Using the DDPM Objective3 Jul
  7. 2023Lipschitz Singularities in Diffusion Models20 Jun
  8. 2023Improved Convergence of Score-Based Diffusion Models via Prediction-Correction23 May
  9. 2023The probability flow ODE is provably fast19 May
  10. 2023Iterative α-(de)Blending: a Minimalist Deterministic Diffusion Model5 May
  11. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers6 Mar
  12. 2023Consistency Models2 Mar
16 more
  1. 2023Diffusion Model for Generative Image Denoising5 Feb
  2. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  3. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  4. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  5. 2022Diffusion Models: A Comprehensive Survey of Methods and Applications2 Sep
  6. 2022Understanding Diffusion Models: A Unified Perspective25 Aug
  7. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  8. 2022DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps2 Jun
  9. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  10. 2021Variational Diffusion Models1 Jul
  11. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  12. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  13. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  14. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  15. 2020Denoising Diffusion Implicit Models6 Oct
  16. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 34

  1. 2017Attention Is All You Needcited by 60,361
  2. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,713
  3. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  4. 2022Flow Matching for Generative Modelingcited by 5,702
  5. 2017Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithmscited by 4,349
  6. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  7. 2018Neural Ordinary Differential Equationscited by 3,779
  8. 2016Wide Residual Networkscited by 3,070
  9. 2021Image Super-Resolution via Iterative Refinementcited by 1,511
  10. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,354
  11. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,323
  12. 2014Image Super-Resolution Using Deep Convolutional Networkscited by 1,224
22 more
  1. 2017The Perception-Distortion Tradeoffcited by 587
  2. 2023Denoising Diffusion Models for Plug-and-Play Image Restorationcited by 340
  3. 2016The Little Engine that Could: Regularization by Denoising (RED)cited by 307
  4. 2021Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contractioncited by 307
  5. 2020Deep Learning Techniques for Inverse Problems in Imagingcited by 291
  6. 2021Deblurring via Stochastic Refinementcited by 249
  7. 2023Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restorationcited by 165
  8. 2022(Certified!!) Adversarial Robustness for Free!cited by 132
  9. 2020Generative Modeling with Denoising Auto-Encoders and Langevin Samplingcited by 119
  10. 2021Noise2Score: Tweedie's Approach to Self-Supervised Image Denoising without Clean Imagescited by 113
  11. 2022Speech Enhancement with Score-Based Generative Models in the Complex STFT Domaincited by 102
  12. 2019Robust and interpretable blind image denoising via bias-free convolutional neural networkscited by 90
  13. 2023Image Denoising: The Deep Learning Revolution and Beyond -- A Survey Paper --cited by 71
  14. 2022Diffusion Model Based Posterior Sampling for Noisy Linear Inverse Problemscited by 60
  15. 2022MIMO Channel Estimation using Score-Based Generative Modelscited by 50
  16. 2023WaveDM: Wavelet-Based Diffusion Models for Image Restorationcited by 47
  17. 2023DiracDiffusion: Denoising and Incremental Reconstruction with Assured Data-Consistencycited by 15
  18. 2021An Asymptotically MSE-Optimal Estimator based on Gaussian Mixture Modelscited by 14
  19. 2015Information Extraction Under Privacy Constraintscited by 9
  20. 2021A General Derivative Identity for the Conditional Mean Estimator in Gaussian Noise and Some Applicationscited by 9
  21. 2022On the Mean Square Error Optimal Estimator in One-Bit Quantized Systemscited by 6
  22. 2023Denoising Diffusion Medical Modelscited by 6

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 3; 2025: 3; 2026: 4.

10 papers.

  1. 2024Generative Diffusion Models for High Dimensional Channel Estimationcited by 26
  2. 2024Solving Prior Distribution Mismatch in Diffusion Models via Optimal Transportcited by 2
  3. 2025Enhancements in Score-based Channel Estimation for Real-Time Wireless Systemscited by 2
  4. 2025Robust MIMO Channel Estimation Using Energy-Based Generative Diffusion Modelscited by 1
  5. 2024Sparse Bayesian Generative Modeling for Compressive Sensingcited by 0
  6. 2025CSI Prediction Using Diffusion Modelscited by 0
  7. 2026Near-Field Wideband Channel Estimation for XL-MIMO Systems via Denoising Diffusion Modelcited by 0
  8. 2026Mixture-of-Experts Diffusion Models for Adaptive Massive MIMO Channel Estimation via Variational Bayesian Inferencecited by 0
  9. 2026A Perception vs. Distortion Perspective on Score-Based Generative Channel Estimationcited by 0
  10. 2026Diffusion Models for Radio Map Estimation: Theoretical Performance Analysis and Sampling Rate Guidelinecited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2023The Missing U for Efficient Diffusion Models
  2. 2026Transformers Learn the Optimal DDPM Denoiser for Multi-Token GMMs
  3. 2025Fewer Denoising Steps or Cheaper Per-Step Inference: Towards Compute-Optimal Diffusion Model Deployment
  4. 2025Diffusion Models: A Mathematical Introduction
  5. 2025Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Bounds
  6. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions
  7. 2024Statistical guarantees for denoising reflected diffusion models
  8. 2023Interpreting and Improving Diffusion Models from an Optimization Perspective