mlsanityData to 14 Oct 2026

On the Posterior Distribution in Denoising: Application to Uncertainty Quantification

arXiv 2309.13598 ↗First posted 24 Sep 2023cs.CV cs.LG stat.MLcited by 33

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Denoisers play a central role in many applications, from noise suppression in low-grade imaging sensors, to empowering score-based generative models. The latter category of methods makes use of Tweedie's formula, which links the posterior mean in Gaussian denoising (ie the minimum MSE denoiser) with the score of the data distribution. Here, we derive a fundamental relation between the higher-order central moments of the posterior distribution, and the higher-order derivatives of the posterior mean. We harness this result for uncertainty quantification of pre-trained denoisers. Particularly, we show how to efficiently compute the principal components of the posterior distribution for any desired region of an image, as well as to approximate the full marginal distribution along those (or any other) one-dimensional directions. Our method is fast and memory-efficient, as it does not explicitly compute or store the high-order moment tensors and it requires no training or fine tuning of the denoiser. Code and examples are available on the project webpage in https://hilamanor.github.io/GaussianDenoisingPosterior/ .

Key facts. Cited by 33 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 28 of its 66 references that are in the map. First posted on arXiv 24 Sep 2023.

Builds on

28 of 66 references.

Closest ancestors · 4

Same direction, newest first

  1. 2022Conffusion: Confidence Intervals for Diffusion Models17 Nov
  2. 2022GENIE: Higher-Order Denoising Diffusion Solvers11 Oct
  3. 2022Maximum Likelihood Training for Score-Based Diffusion ODEs by High-Order Denoising Score Matching16 Jun
  4. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 24

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  3. 2018A Style-Based Generator Architecture for Generative Adversarial Networkscited by 6,013
  4. 2015Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learningcited by 4,448
  5. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  6. 2016Gaussian Error Linear Units (GELUs)cited by 3,502
  7. 2021SwinIR: Image Restoration Using Swin Transformercited by 1,887
  8. 2017Enhanced Deep Residual Networks for Single Image Super-Resolutioncited by 1,552
  9. 2016Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoisingcited by 1,358
  10. 2020Training Generative Adversarial Networks with Limited Datacited by 1,186
  11. 2019MaskGAN: Towards Diverse and Interactive Facial Image Manipulationcited by 628
  12. 2022Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Modelcited by 583
12 more
  1. 2021Label-Efficient Semantic Segmentation with Diffusion Modelscited by 499
  2. 2020Plug-and-Play Image Restoration with Deep Denoiser Priorcited by 470
  3. 2017Learning Deep CNN Denoiser Prior for Image Restorationcited by 408
  4. 2018Noise2Void - Learning Denoising from Single Noisy Imagescited by 364
  5. 2016The Little Engine that Could: Regularization by Denoising (RED)cited by 307
  6. 2022Image-to-Image Regression with Distribution-Free Uncertainty Quantification and Applications in Imagingcited by 107
  7. 2018A Poisson-Gaussian Denoising Dataset with Real Fluorescence Microscopy Imagescited by 71
  8. 2021Estimating High Order Gradients of the Data Distribution by Denoisingcited by 64
  9. 2019Probabilistic Noise2Void: Unsupervised Content-Aware Denoisingcited by 51
  10. 2019High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithmcited by 36
  11. 2022Semantic uncertainty intervals for disentangled latent spacescited by 22
  12. 2020Interval Neural Networks: Uncertainty Scorescited by 14

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 2; 2024: 8; 2025: 10; 2026: 13.

33 papers.

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  1. 2024Zero-Shot Unsupervised and Text-Based Audio Editing Using DDPM Inversioncited by 60
  2. 2024Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributionscited by 53
  3. 2024Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structurecited by 38
  4. 2023Uncertainty Quantification via Neural Posterior Principal Componentscited by 16
  5. 2024Exploring Low-Dimensional Subspaces in Diffusion Models for Controllable Image Editingcited by 15
  6. 2024Free Hunch: Denoiser Covariance Estimation for Diffusion Models Without Extra Costscited by 14
  7. 2025Uncertainty Quantification for Machine Learning in Healthcare: A Surveycited by 14
  8. 2025Object-AVEdit: An Object-level Audio-Visual Editing Modelcited by 14
  9. 2025InvFusion: Bridging Supervised and Zero-shot Diffusion for Inverse Problemscited by 10
  10. 2024What Secrets Do Your Manifolds Hold? Understanding the Local Geometry of Generative Modelscited by 7
  11. 2025When Diffusion Models Memorize: Inductive Biases in Probability Flow of Minimum-Norm Shallow Neural Netscited by 6
  12. 2026Self-Supervised Learning from Noisy and Incomplete Datacited by 5
21 more
  1. 2024Hierarchical Uncertainty Exploration via Feedforward Posterior Treescited by 4
  2. 2025EigenScore: OOD Detection using Covariance in Diffusion Modelscited by 4
  3. 2023Uncertainty Visualization via Low-Dimensional Posterior Projectionscited by 3
  4. 2025PriorGuide: Test-Time Prior Adaptation for Simulation-Based Inferencecited by 3
  5. 2026Uncertainty in a Single Pass: A Closed-Form Identity for One-Step Flow Matchingcited by 3
  6. 2026Amortized Moment Matching for Visual Generationcited by 2
  7. 2024Shallow Diffuse: Robust and Invisible Watermarking through Low-Dimensional Subspaces in Diffusion Modelscited by 1
  8. 2025A Simple Combination of Diffusion Models for Better Quality Trade-Offs in Image Denoisingcited by 1
  9. 2025Heteroscedasticity of Denoising Score Matching with Generalised Smooth Noisecited by 1
  10. 2026Spatially Adaptive Noise Injectioncited by 1
  11. 2025Conditional Distribution Quantization in Machine Learningcited by 0
  12. 2025QUTCC: Quantile Uncertainty Training and Conformal Calibration for Imaging Inverse Problemscited by 0
  13. 2026The Value of Covariance Matching in Gaussian DDPMs and the Lanczos Samplercited by 0
  14. 2026PFM-HR: Pose Flow Matching for Humanoid Robotscited by 0
  15. 2026Improved denoising diffusion probabilistic models with efficient non-diagonal covariance modelingcited by 0
  16. 2026The Loss Floor of Denoising Score Matching: Fisher Geometry from Schr"odinger Bridgescited by 0
  17. 2026A Lagrangian View of Flow Matchingcited by 0
  18. 2026Zero-SNR Analyticity of the Scalar MMSE Is Equivalent to Gaussianitycited by 0
  19. 2026On the spectral properties of generative denoiser Jacobianscited by 0
  20. 2026Posterior Flow Matchingcited by 0
  21. 2026Intrinsic Geometry Matching: learning observation-equivariant process geometrycited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2025Exploiting the Exact Denoising Posterior Score in Training-Free Guidance of Diffusion Models
  2. 2022Score-based Denoising Diffusion with Non-Isotropic Gaussian Noise Models
  3. 2024Diffusion Priors for Variational Likelihood Estimation and Image Denoising
  4. 2024Listening to the Noise: Blind Denoising with Gibbs Diffusion
  5. 2023Diffusion Model for Generative Image Denoising
  6. 2025Optimal Denoising in Score-Based Generative Models: The Role of Data Regularity
  7. 2024On the Asymptotic Mean Square Error Optimality of Diffusion Models
  8. 2024Enhancing Diffusion Models for Inverse Problems with Covariance-Aware Posterior Sampling