mlsanityData to 14 Oct 2026

Formulating Discrete Probability Flow Through Optimal Transport

arXiv 2311.03886 ↗First posted 7 Nov 2023cs.LG cs.AI math.OCcited by 12

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Continuous diffusion models are commonly acknowledged to display a deterministic probability flow, whereas discrete diffusion models do not. In this paper, we aim to establish the fundamental theory for the probability flow of discrete diffusion models. Specifically, we first prove that the continuous probability flow is the Monge optimal transport map under certain conditions, and also present an equivalent evidence for discrete cases. In view of these findings, we are then able to define the discrete probability flow in line with the principles of optimal transport. Finally, drawing upon our newly established definitions, we propose a novel sampling method that surpasses previous discrete diffusion models in its ability to generate more certain outcomes. Extensive experiments on the synthetic toy dataset and the CIFAR-10 dataset have validated the effectiveness of our proposed discrete probability flow. Code is released at: https://github.com/PangzeCheung/Discrete-Probability-Flow.

Key facts. Cited by 12 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 41 of its 58 references that are in the map. First posted on arXiv 7 Nov 2023.

Builds on

41 of 58 references.

Closest ancestors · 13

Same direction, newest first

  1. 2022Score-based Continuous-time Discrete Diffusion Models30 Nov
  2. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  3. 2022A Continuous Time Framework for Discrete Denoising Models30 May
  4. 2022Understanding DDPM Latent Codes Through Optimal Transport14 Feb
  5. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  6. 2021Vector Quantized Diffusion Model for Text-to-Image Synthesis29 Nov
  7. 2021Unleashing Transformers: Parallel Token Prediction with Discrete Absorbing Diffusion for Fast High-Resolution Image Generation from Vector-Quantized Codes24 Nov
  8. 2021Structured Denoising Diffusion Models in Discrete State-Spaces7 Jul
  9. 2021Cascaded Diffusion Models for High Fidelity Image Generation30 May
  10. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  11. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  12. 2020Denoising Diffusion Implicit Models6 Oct
1 more
  1. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 28

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  2. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,585
  3. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,355
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  5. 2018Neural Ordinary Differential Equationscited by 3,779
  6. 2021GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Modelscited by 3,614
  7. 2022Prompt-to-Prompt Image Editing with Cross Attention Controlcited by 2,431
  8. 2022Video Diffusion Modelscited by 2,305
  9. 2022Imagen Video: High Definition Video Generation with Diffusion Modelscited by 1,778
  10. 2021Image Super-Resolution via Iterative Refinementcited by 1,511
  11. 2022Human Motion Diffusion Modelcited by 1,080
  12. 2022Diffusion-LM Improves Controllable Text Generationcited by 1,027
16 more
  1. 2022DiffEdit: Diffusion-based semantic image editing with mask guidancecited by 647
  2. 2021Argmax Flows and Multinomial Diffusion: Learning Categorical Distributionscited by 574
  3. 2022LION: Latent Point Diffusion Models for 3D Shape Generationcited by 553
  4. 2022DiffuSeq: Sequence to Sequence Text Generation with Diffusion Modelscited by 450
  5. 2022MCVD: Masked Conditional Video Diffusion for Prediction, Generation, and Interpolationcited by 309
  6. 2022Flexible Diffusion Modeling of Long Videoscited by 294
  7. 2022MM-Diffusion: Learning Multi-Modal Diffusion Models for Joint Audio and Video Generationcited by 227
  8. 2021Autoregressive Diffusion Modelscited by 210
  9. 2023MeshDiffusion: Score-based Generative 3D Mesh Modelingcited by 199
  10. 2021ImageBART: Bidirectional Context with Multinomial Diffusion for Autoregressive Image Synthesiscited by 122
  11. 2022Generative Flow Networks for Discrete Probabilistic Modelingcited by 114
  12. 2020Learning Discrete Energy-based Models via Auxiliary-variable Local Explorationcited by 30
  13. 2023Semantic Diffusion Network for Semantic Segmentationcited by 30
  14. 2023Stochastic Multi-Person 3D Motion Forecastingcited by 28
  15. 2021Beyond In-Place Corruption: Insertion and Deletion In Denoising Probabilistic Modelscited by 23
  16. 2022Diffusion bridges vector quantized Variational AutoEncoderscited by 16

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 6; 2025: 5; 2026: 1.

12 papers.

  1. 2024Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Designcited by 305
  2. 2024View-decoupled Transformer for Person Re-identification under Aerial-ground Camera Networkcited by 23
  3. 2024Tackling the Singularities at the Endpoints of Time Intervals in Diffusion Modelscited by 13
  4. 2025Discrete Neural Flow Samplers with Locally Equivariant Transformercited by 13
  5. 2025Training-Free Class Purification for Open-Vocabulary Semantic Segmentationcited by 9
  6. 2024Energy-Based Modelling for Discrete and Mixed Data via Heat Equations on Structured Spacescited by 5
  7. 2025Efficiently Access Diffusion Fisher: Within the Outer Product Span Spacecited by 5
  8. 2025Improved Immiscible Diffusion: Accelerate Diffusion Training by Reducing Its Miscibilitycited by 4
  9. 2025Masked Diffusion Models as Energy Minimizationcited by 3
  10. 2024Solving Prior Distribution Mismatch in Diffusion Models via Optimal Transportcited by 2
  11. 2024Discrete Modeling via Boundary Conditional Diffusion Processescited by 0
  12. 2026On the Error-Correcting Effects of Stochasticity in Discrete Diffusioncited by 0

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