mlsanityData to 7 Oct 2026

Diffusion on the Probability Simplex

arXiv 2309.02530 ↗First posted 5 Sep 2023cs.LG stat.MLcited by 14

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion models learn to reverse the progressive noising of a data distribution to create a generative model. However, the desired continuous nature of the noising process can be at odds with discrete data. To deal with this tension between continuous and discrete objects, we propose a method of performing diffusion on the probability simplex. Using the probability simplex naturally creates an interpretation where points correspond to categorical probability distributions. Our method uses the softmax function applied to an Ornstein-Unlenbeck Process, a well-known stochastic differential equation. We find that our methodology also naturally extends to include diffusion on the unit cube which has applications for bounded image generation.

Key facts. Cited by 14 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 12 of its 14 references that are in the map. First posted on arXiv 5 Sep 2023.

Builds on

12 of 14 references.

Closest ancestors · 6

Same direction, newest first

  1. 2022Categorical SDEs with Simplex Diffusion26 Oct
  2. 2022Diffusion Models: A Comprehensive Survey of Methods and Applications2 Sep
  3. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  4. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  5. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  6. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 6

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  2. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  4. 2018Neural Ordinary Differential Equationscited by 3,733
  5. 2022Make-A-Video: Text-to-Video Generation without Text-Video Datacited by 1,725
  6. 2021Diff-TTS: A Denoising Diffusion Model for Text-to-Speechcited by 156

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 6; 2025: 5; 2026: 3.

14 papers.

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  1. 2024Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Designcited by 294
  2. 2024Unlocking Guidance for Discrete State-Space Diffusion and Flow Modelscited by 124
  3. 2024Dirichlet Flow Matching with Applications to DNA Sequence Designcited by 110
  4. 2024How Discrete and Continuous Diffusion Meet: Comprehensive Analysis of Discrete Diffusion Models via a Stochastic Integral Frameworkcited by 48
  5. 2025Fast Solvers for Discrete Diffusion Models: Theory and Applications of High-Order Algorithmscited by 47
  6. 2024Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexitycited by 42
  7. 2025Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approachcited by 18
  8. 2024Hotspot-Driven Peptide Design via Multi-Fragment Autoregressive Extensioncited by 9
  9. 2025TFG-Flow: Training-free Guidance in Multimodal Generative Flowcited by 4
  10. 2025Simplex-to-Euclidean Bijections for Categorical Flow Matchingcited by 3
  11. 2026Simplex Relaxation for Discrete Diffusioncited by 2
  12. 2025A Unification of Discrete, Gaussian, and Simplicial Diffusioncited by 1
2 more
  1. 2026A Continuous-Time Reinforcement Learning Framework for Fine-Tuning Discrete Diffusion Modelscited by 1
  2. 2026Tweedie's Formulae and Diffusion Generative Models Beyond Gaussiancited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026Simplex Diffusion Models
  2. 2024Non-Normal Diffusion Models
  3. 2023Information-Theoretic Diffusion
  4. 2026A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models
  5. 2023Infinite-Dimensional Diffusion Models
  6. 2023Generalised Diffusion Probabilistic Scale-Spaces
  7. 2025Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction
  8. 2025Diffusion Models: A Mathematical Introduction