mlsanityData to 7 Oct 2026

Understanding DDPM Latent Codes Through Optimal Transport

arXiv 2202.07477 ↗First posted 14 Feb 2022stat.ML cs.AI cs.LGcited by 44

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion models have recently outperformed alternative approaches to model the distribution of natural images, such as GANs. Such diffusion models allow for deterministic sampling via the probability flow ODE, giving rise to a latent space and an encoder map. While having important practical applications, such as estimation of the likelihood, the theoretical properties of this map are not yet fully understood. In the present work, we partially address this question for the popular case of the VP SDE (DDPM) approach. We show that, perhaps surprisingly, the DDPM encoder map coincides with the optimal transport map for common distributions; we support this claim theoretically and by extensive numerical experiments.

Key facts. Cited by 44 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 27 of its 56 references that are in the map. First posted on arXiv 14 Feb 2022.

Builds on

27 of 56 references.

Closest ancestors · 8

Same direction, newest first

  1. 2021ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models6 Aug
  2. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  3. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  4. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  5. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  6. 2020Denoising Diffusion Implicit Models6 Oct
  7. 2020Denoising Diffusion Probabilistic Models19 Jun
  8. 2020Improved Techniques for Training Score-Based Generative Models16 Jun

Other references · 19

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  2. 2018A Style-Based Generator Architecture for Generative Adversarial Networkscited by 5,986
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  4. 2018Large Scale GAN Training for High Fidelity Natural Image Synthesiscited by 2,609
  5. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,277
  6. 2021SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equationscited by 1,903
  7. 2021Image Super-Resolution via Iterative Refinementcited by 1,506
  8. 2020Training Generative Adversarial Networks with Limited Datacited by 1,184
  9. 2019StarGAN v2: Diverse Image Synthesis for Multiple Domainscited by 1,095
  10. 2021Label-Efficient Semantic Segmentation with Diffusion Modelscited by 495
  11. 2021SRDiff: Single Image Super-Resolution with Diffusion Probabilistic Modelscited by 415
  12. 2016A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluationcited by 236
7 more
  1. 2017Near-linear time approximation algorithms for optimal transport via Sinkhorn iterationcited by 219
  2. 2020OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transportcited by 143
  3. 2014On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpointcited by 99
  4. 2015Measuring Sample Quality with Stein's Methodcited by 94
  5. 2021Large-Scale Wasserstein Gradient Flowscited by 76
  6. 2019On Efficient Optimal Transport: An Analysis of Greedy and Accelerated Mirror Descent Algorithmscited by 60
  7. 2017Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectivescited by 53

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 7; 2023: 14; 2024: 16; 2025: 6; 2026: 1.

44 papers.

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  1. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,550
  2. 2022Diffusion Models in Vision: A Surveycited by 1,008
  3. 2022DiffEdit: Diffusion-based semantic image editing with mask guidancecited by 643
  4. 2022A Survey on Generative Diffusion Modelcited by 238
  5. 2024ReNoise: Real Image Inversion Through Iterative Noisingcited by 127
  6. 2024Understanding Hallucinations in Diffusion Models through Mode Interpolationcited by 87
  7. 2022Unifying Diffusion Models' Latent Space, with Applications to CycleDiffusion and Guidancecited by 67
  8. 2022Maximum Likelihood Training of Implicit Nonlinear Diffusion Modelscited by 42
  9. 2023DiffFashion: Reference-based Fashion Design with Structure-aware Transfer by Diffusion Modelscited by 33
  10. 2023Scalable Optimal Transport Methods in Machine Learning: A Contemporary Surveycited by 27
  11. 2023Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputationcited by 26
  12. 2024AID: Attention Interpolation of Text-to-Image Diffusioncited by 24
32 more
  1. 2024Out-of-Distribution Detection with a Single Unconditional Diffusion Modelcited by 22
  2. 2024Data Attribution for Text-to-Image Models by Unlearning Synthesized Imagescited by 22
  3. 2023The Emergence of Reproducibility and Generalizability in Diffusion Modelscited by 21
  4. 2023IMPUS: Image Morphing with Perceptually-Uniform Sampling Using Diffusion Modelscited by 20
  5. 2023Gradpaint: Gradient-Guided Inpainting with Diffusion Modelscited by 17
  6. 2025FreeMorph: Tuning-Free Generalized Image Morphing with Diffusion Modelcited by 17
  7. 2024Diffusion models for Gaussian distributions: Exact solutions and Wasserstein errorscited by 15
  8. 2023Formulating Discrete Probability Flow Through Optimal Transportcited by 12
  9. 2023Optimal Transport-Guided Conditional Score-Based Diffusion Modelscited by 11
  10. 2023DDMM-Synth: A Denoising Diffusion Model for Cross-modal Medical Image Synthesis with Sparse-view Measurement Embeddingcited by 10
  11. 2024Masked Completion via Structured Diffusion with White-Box Transformerscited by 9
  12. 2023Image generation with shortest path diffusioncited by 8
  13. 2024HARIVO: Harnessing Text-to-Image Models for Video Generationcited by 8
  14. 2022Image Embedding for Denoising Generative Modelscited by 7
  15. 2024Improving Diffusion-Based Generative Models via Approximated Optimal Transportcited by 5
  16. 2025Efficiently Access Diffusion Fisher: Within the Outer Product Span Spacecited by 5
  17. 2025NoisePrints: Distortion-Free Watermarks for Authorship in Private Diffusion Modelscited by 5
  18. 2023Attribute Based Interpretable Evaluation Metrics for Generative Modelscited by 4
  19. 2024Explanatory Model Monitoring to Understand the Effects of Feature Shifts on Performancecited by 4
  20. 2025Improved Immiscible Diffusion: Accelerate Diffusion Training by Reducing Its Miscibilitycited by 4
  21. 2026Finite Difference Flow Optimization for RL Post-Training of Text-to-Image Modelscited by 4
  22. 2023Straighter Flow Matching via a Diffusion-Based Coupling Priorcited by 3
  23. 2024Rethinking the Diffusion Models for Numerical Tabular Data Imputation from the Perspective of Wasserstein Gradient Flowcited by 3
  24. 2025Next Tokens Denoising for Speech Synthesiscited by 3
  25. 2024Theoretical research on generative diffusion models: an overviewcited by 2
  26. 2024Solving Prior Distribution Mismatch in Diffusion Models via Optimal Transportcited by 2
  27. 2024Preference Alignment for Diffusion Model via Explicit Denoised Distribution Estimationcited by 2
  28. 2024There and Back Again: On the relation between Noise and Image Inversions in Diffusion Modelscited by 1
  29. 2023Dior-CVAE: Pre-trained Language Models and Diffusion Priors for Variational Dialog Generationcited by 0
  30. 2023Touring sampling with pushforward mapscited by 0
  31. 2024Semi-Discrete Optimal Transport: Nearly Minimax Estimation With Stochastic Gradient Descent and Adaptive Entropic Regularizationcited by 0
  32. 2025Decreasing Entropic Regularization Averaged Gradient for Semi-Discrete Optimal Transportcited by 0

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  3. 2024Denoising Diffusion Probabilistic Models in Six Simple Steps
  4. 2023Faster Diffusion: Rethinking the Role of the Encoder for Diffusion Model Inference
  5. 2022Efficient Diffusion Models for Vision: A Survey
  6. 2023Open-DDVM: A Reproduction and Extension of Diffusion Model for Optical Flow Estimation
  7. 2024SDXS: Real-Time One-Step Latent Diffusion Models with Image Conditions
  8. 2022Few-shot Image Generation with Diffusion Models