mlsanityData to 7 Oct 2026

Infinite-Dimensional Diffusion Models

arXiv 2302.10130 ↗First posted 20 Feb 2023stat.ML cs.LG math.PRcited by 38

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion models have had a profound impact on many application areas, including those where data are intrinsically infinite-dimensional, such as images or time series. The standard approach is first to discretize and then to apply diffusion models to the discretized data. While such approaches are practically appealing, the performance of the resulting algorithms typically deteriorates as discretization parameters are refined. In this paper, we instead directly formulate diffusion-based generative models in infinite dimensions and apply them to the generative modelling of functions. We prove that our formulations are well posed in the infinite-dimensional setting and provide dimension-independent distance bounds from the sample to the target measure. Using our theory, we also develop guidelines for the design of infinite-dimensional diffusion models. For image distributions, these guidelines are in line with current canonical choices. For other distributions, however, we can improve upon these canonical choices. We demonstrate these results both theoretically and empirically, by applying the algorithms to data distributions on manifolds and to distributions arising in Bayesian inverse problems or simulation-based inference.

Key facts. Cited by 38 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 25 of its 48 references that are in the map. First posted on arXiv 20 Feb 2023.

Builds on

25 of 48 references.

Closest ancestors · 17

Same direction, newest first

  1. 2023Score-based Diffusion Models in Function Space14 Feb
  2. 2022Your diffusion model secretly knows the dimension of the data manifold23 Dec
  3. 2022Wavelet Diffusion Models are fast and scalable Image Generators29 Nov
  4. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  5. 2022Spectral Diffusion Processes28 Sep
  6. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  7. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  8. 2022Wavelet Score-Based Generative Modeling9 Aug
  9. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  10. 2022Score-Based Generative Models Detect Manifolds2 Jun
  11. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
  12. 2021Conditional Image Generation with Score-Based Diffusion Models26 Nov
5 more
  1. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  2. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  3. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  4. 2020Denoising Diffusion Probabilistic Models19 Jun
  5. 2020Improved Techniques for Training Score-Based Generative Models16 Jun

Other references · 8

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,339
  4. 2021CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputationcited by 577
  5. 2019The frontier of simulation-based inferencecited by 490
  6. 2014Dimension-independent likelihood-informed MCMCcited by 36
  7. 2021hIPPYlib-MUQ: A Bayesian Inference Software Framework for Integration of Data with Complex Predictive Models under Uncertaintycited by 4
  8. 2020Convergence of Preconditioned Hamiltonian Monte Carlo on Hilbert Spacescited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 6; 2025: 15; 2026: 17.

38 papers.

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  1. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEscited by 33
  2. 2024Wavelet Diffusion Neural Operatorcited by 28
  3. 2024Warped Diffusion: Solving Video Inverse Problems with Image Diffusion Modelscited by 21
  4. 2025FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modelingcited by 19
  5. 2025Conditioning Diffusions Using Malliavin Calculuscited by 15
  6. 2025Stochastic Process Learning via Operator Flow Matchingcited by 14
  7. 2024Wasserstein Bounds for generative diffusion models with Gaussian tail targetscited by 10
  8. 2024Universal Functional Regression with Neural Operator Flowscited by 9
  9. 2025Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equationscited by 9
  10. 2025Flow Matching Neural Processescited by 6
  11. 2025Probability-Flow ODE in Infinite-Dimensional Function Spacescited by 5
  12. 2025Preconditioned Langevin Dynamics with Score-Based Generative Models for Infinite-Dimensional Linear Bayesian Inverse Problemscited by 5
26 more
  1. 2025Flow Straight and Fast in Hilbert Space: Functional Rectified Flowcited by 5
  2. 2025Scale-Adaptive Generative Flows for Multiscale Scientific Datacited by 4
  3. 2026Supervised Guidance Training for Infinite-Dimensional Diffusion Modelscited by 3
  4. 2026Physics-informed diffusion models in spectral spacecited by 3
  5. 2025Generative Unordered Flow for Set-Structured Data Generationcited by 2
  6. 2025A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fieldscited by 2
  7. 2025PRISMA: Improving the Accuracy-Latency Frontier of Diffusion-based PDE Solvers Using Physics-Informed Spectral Attentioncited by 2
  8. 2025Residual-augmented flow matching operators for probabilistic partial differential equationscited by 2
  9. 2026Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamicscited by 2
  10. 2024An Unconditional Representation of the Conditional Score in Infinite-Dimensional Linear Inverse Problemscited by 1
  11. 2026Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spacescited by 1
  12. 2026Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inferencecited by 1
  13. 2026GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domainscited by 1
  14. 2026Discretization and Statistical Consistency of Functional Flow Matchingcited by 1
  15. 2024Deep Optimal Sensor Placement for Black Box Stochastic Simulationscited by 0
  16. 2025FuncGenFoil: Airfoil Generation and Editing Model in Function Spacecited by 0
  17. 2026Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensionscited by 0
  18. 2026Stochastic Interpolants in Hilbert Spacescited by 0
  19. 2026Score-based diffusion models for severely ill-posed problems in diffuse optical tomographycited by 0
  20. 2026Infinite-dimensional generative diffusions via Doob's h-transformcited by 0
  21. 2026SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximationcited by 0
  22. 2026Christoffel-DPS: Optimal sensor placement in diffusion posterior sampling for arbitrary distributionscited by 0
  23. 2026Dimension-Uniform Discretization Analysis of Preconditioned Annealed Langevin Dynamics for Multimodal Gaussian Mixturescited by 0
  24. 2026Kolmogorov Regression for Robust Diffusion Policiescited by 0
  25. 2026Learning Informative Prior with Infinite-Dimensional Continuous Normalizing Flow for Bayesian Inverse Problemcited by 0
  26. 2026Approximating Measures on Function Spaces: Transport and Truncationcited by 0

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  1. 2022Diffusion Generative Models in Infinite Dimensions
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  3. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Models
  4. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models
  5. 2025Analysis of Diffusion Models for Manifold Data
  6. 2022Diffusion Models: A Comprehensive Survey of Methods and Applications
  7. 2024Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions
  8. 2023Diffusion Models for Generative Artificial Intelligence: An Introduction for Applied Mathematicians