mlsanityData to 14 Oct 2026

Spectral Diffusion Processes

arXiv 2209.14125 ↗First posted 28 Sep 2022stat.ML cs.LGcited by 25

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion models have proven to be a flexible and effective framework for modelling probability distributions on finite-dimensional spaces. However, many physical modelling problems such as time series are naturally described over function spaces. In this work we apply diffusion models to such stochastic processes. To do so we consider a spectral representation of the data, obtained using a kernel, thereby dissociating the stochastic part of the processes from their space-time structure. As a result, the stochasticity of the processes is entirely encoded in the spectral coefficients, which we truncate and model using standard finite-dimensional diffusion models. By truncating the representation in the spectral domain we ensure our resulting model defines valid stochastic processes, thereby naturally satisfying consistency and exchangeability criteria. Projecting our spectral diffusion models back to the original input space, we show that for any given marginals our approach corresponds to a diffusion model with correlated noise, with explicit covariance matrix given by the kernel. We demonstrate our method's effectiveness for modelling various multimodal datasets as well as conditional sampling by amortising our models with respect to a context set.

Key facts. Cited by 25 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 32 of its 71 references that are in the map. First posted on arXiv 28 Sep 2022.

Builds on

32 of 71 references.

Closest ancestors · 9

Same direction, newest first

  1. 2022Blurring Diffusion Models12 Sep
  2. 2022Soft Diffusion: Score Matching for General Corruptions12 Sep
  3. 2022Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise19 Aug
  4. 2022Wavelet Score-Based Generative Modeling9 Aug
  5. 2022Subspace Diffusion Generative Models3 May
  6. 2021Cascaded Diffusion Models for High Fidelity Image Generation30 May
  7. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  8. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  9. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 23

  1. 2017Attention Is All You Needcited by 60,361
  2. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  3. 2021Image Super-Resolution via Iterative Refinementcited by 1,511
  4. 2018Conditional Neural Processescited by 426
  5. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 384
  6. 2018Neural Processescited by 311
  7. 2021Skillful Precipitation Nowcasting using Deep Generative Models of Radarcited by 311
  8. 2019Attentive Neural Processescited by 283
  9. 2022Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problemcited by 236
  10. 2022Generative Modelling With Inverse Heat Dissipationcited by 167
  11. 2019Convolutional Conditional Neural Processescited by 142
  12. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 135
11 more
  1. 2019Functional Variational Bayesian Neural Networkscited by 127
  2. 2016Variational Fourier features for Gaussian processescited by 85
  3. 2020Meta-Learning Stationary Stochastic Process Prediction with Convolutional Neural Processescited by 76
  4. 2019Sequential Neural Processescited by 55
  5. 2022Neural Diffusion Processescited by 50
  6. 2019The Functional Neural Processcited by 43
  7. 2021Convolutional conditional neural processes for local climate downscalingcited by 29
  8. 2020A Kernel Two-Sample Test for Functional Datacited by 28
  9. 2022Practical Conditional Neural Processes Via Tractable Dependent Predictionscited by 25
  10. 2022The Neural Process Family: Survey, Applications and Perspectivescited by 23
  11. 2022Multi-fidelity Hierarchical Neural Processescited by 15

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 1; 2023: 4; 2024: 6; 2025: 7; 2026: 7.

25 papers.

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  1. 2023Score-based Diffusion Models in Function Spacecited by 82
  2. 2025A Fourier Space Perspective on Diffusion Modelscited by 52
  3. 2022Modeling Temporal Data as Continuous Functions with Stochastic Process Diffusioncited by 39
  4. 2023Infinite-Dimensional Diffusion Modelscited by 39
  5. 2023Continuous-Time Functional Diffusion Processescited by 39
  6. 2023Conditional score-based diffusion models for Bayesian inference in infinite dimensionscited by 38
  7. 2024Time Series Diffusion in the Frequency Domaincited by 37
  8. 2025FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modelingcited by 20
  9. 2024Stochastic Optimal Control for Diffusion Bridges in Function Spacescited by 16
  10. 2024Conditioning non-linear and infinite-dimensional diffusion processescited by 15
  11. 2024Infinite-dimensional Diffusion Bridge Simulation via Operator Learningcited by 7
  12. 2025Probability-Flow ODE in Infinite-Dimensional Function Spacescited by 5
13 more
  1. 2025Scale-Adaptive Generative Flows for Multiscale Scientific Datacited by 5
  2. 2026Physics-informed diffusion models in spectral spacecited by 5
  3. 2024Spectral Convolutional Conditional Neural Processescited by 4
  4. 2024SigDiffusions: Score-Based Diffusion Models for Time Series via Log-Signature Embeddingscited by 4
  5. 2025A Review of Diffusion-based Simulation-Based Inference: Foundations and Applications in Non-Ideal Data Scenarioscited by 1
  6. 2025GFT-GCN: Privacy-Preserving 3D Face Mesh Recognition with Spectral Diffusioncited by 0
  7. 2025Preserving Spectral Structure and Statistics in Diffusion Modelscited by 0
  8. 2026Stochastic Interpolants in Hilbert Spacescited by 0
  9. 2026Generative Neural Operators through Diffusion Last Layercited by 0
  10. 2026Structured State-Space Regularization for Generation-Friendly Image Tokenizationcited by 0
  11. 2026Revisiting Neural Processes via Fourier Transform and Volterra Seriescited by 0
  12. 2026Low-Pass Flow Matchingcited by 0
  13. 2026MercerFlow: Flow Matching in a Kernel-Induced Latent Space for Probabilistic Forecastingcited by 0

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  1. 2026Spectral Diffusion Models on the Sphere
  2. 2025Spectral Analysis of Diffusion Models with Application to Schedule Design
  3. 2026First Learn, Then Memorize: The Spectral Bias of Diffusion Models
  4. 2026Parallel Complex Diffusion for Scalable Time Series Generation
  5. 2026Revisiting Spectral Representations in Generative Diffusion Models
  6. 2025Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction
  7. 2026A Decomposable Forward Process in Diffusion Models for Time-Series Forecasting
  8. 2022Diffusion Generative Models in Infinite Dimensions