mlsanityData to 14 Oct 2026

Geometric Neural Diffusion Processes

arXiv 2307.05431 ↗First posted 11 Jul 2023stat.ML cs.LGcited by 10

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Denoising diffusion models have proven to be a flexible and effective paradigm for generative modelling. Their recent extension to infinite dimensional Euclidean spaces has allowed for the modelling of stochastic processes. However, many problems in the natural sciences incorporate symmetries and involve data living in non-Euclidean spaces. In this work, we extend the framework of diffusion models to incorporate a series of geometric priors in infinite-dimension modelling. We do so by a) constructing a noising process which admits, as limiting distribution, a geometric Gaussian process that transforms under the symmetry group of interest, and b) approximating the score with a neural network that is equivariant w.r.t. this group. We show that with these conditions, the generative functional model admits the same symmetry. We demonstrate scalability and capacity of the model, using a novel Langevin-based conditional sampler, to fit complex scalar and vector fields, with Euclidean and spherical codomain, on synthetic and real-world weather data.

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 55 of its 89 references that are in the map. First posted on arXiv 11 Jul 2023.

Builds on

55 of 89 references.

Closest ancestors · 23

Same direction, newest first

  1. 2023Reflected Diffusion Models10 Apr
  2. 2023∞-Diff: Infinite Resolution Diffusion with Subsampled Mollified States31 Mar
  3. 2023Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation8 Mar
  4. 2023Continuous-Time Functional Diffusion Processes1 Mar
  5. 2023Score-based Diffusion Models in Function Space14 Feb
  6. 2022Diffusion Generative Models in Infinite Dimensions1 Dec
  7. 2022Score-based Denoising Diffusion with Non-Isotropic Gaussian Noise Models21 Oct
  8. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  9. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  10. 2022Blurring Diffusion Models12 Sep
  11. 2022Soft Diffusion: Score Matching for General Corruptions12 Sep
  12. 2022Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise19 Aug
11 more
  1. 2022Riemannian Diffusion Models16 Aug
  2. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  3. 2022Wavelet Score-Based Generative Modeling9 Aug
  4. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  5. 2022Subspace Diffusion Generative Models3 May
  6. 2022Riemannian Score-Based Generative Modelling6 Feb
  7. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  8. 2021Cascaded Diffusion Models for High Fidelity Image Generation30 May
  9. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  10. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  11. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 32

  1. 2017Attention Is All You Needcited by 60,361
  2. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  4. 2021Image Super-Resolution via Iterative Refinementcited by 1,511
  5. 2022RePaint: Inpainting using Denoising Diffusion Probabilistic Modelscited by 1,430
  6. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,323
  7. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 746
  8. 2019General E(2)-Equivariant Steerable CNNscited by 446
  9. 2018Conditional Neural Processescited by 426
  10. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 354
  11. 2018Neural Processescited by 311
  12. 2019Attentive Neural Processescited by 283
20 more
  1. 2020Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densitiescited by 250
  2. 2022Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problemcited by 236
  3. 2023SE(3) diffusion model with application to protein backbone generationcited by 219
  4. 2014Theoretical guarantees for approximate sampling from smooth and log-concave densitiescited by 208
  5. 2015Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithmcited by 197
  6. 2022e3nn: Euclidean Neural Networkscited by 174
  7. 2022Generative Modelling With Inverse Heat Dissipationcited by 167
  8. 2016High-dimensional Bayesian inference via the Unadjusted Langevin Algorithmcited by 146
  9. 2019Convolutional Conditional Neural Processescited by 142
  10. 2020Efficiently Sampling Functions from Gaussian Process Posteriorscited by 127
  11. 2021Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifoldscited by 73
  12. 2018geomstats: a Python Package for Riemannian Geometry in Machine Learningcited by 71
  13. 2019Sequential Neural Processescited by 55
  14. 2019The Functional Neural Processcited by 43
  15. 2023Autoregressive Conditional Neural Processescited by 33
  16. 2020Equivariant Learning of Stochastic Fields: Gaussian Processes and Steerable Conditional Neural Processescited by 29
  17. 2022Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact casecited by 25
  18. 2022The Neural Process Family: Survey, Applications and Perspectivescited by 23
  19. 2021Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernelscited by 19
  20. 2023Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spacescited by 15

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 3; 2024: 4; 2025: 1; 2026: 2.

10 papers.

  1. 2024On conditional diffusion models for PDE simulationscited by 60
  2. 2023A framework for conditional diffusion modelling with applications in motif scaffolding for protein designcited by 22
  3. 2023Functional Diffusioncited by 13
  4. 2024Generating Physical Dynamics under Priorscited by 6
  5. 2023Learning multi-modal generative models with permutation-invariant encoders and tighter variational objectivescited by 4
  6. 2024Spectral Convolutional Conditional Neural Processescited by 4
  7. 2024Structure Preserving Diffusion Modelscited by 3
  8. 2025Neural Bridge Processescited by 0
  9. 2026Revisiting Neural Processes via Fourier Transform and Volterra Seriescited by 0
  10. 2026Continuous-Time Machine Learning: A Unified Mathematical Perspectivecited by 0

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