# Understanding DDPM Latent Codes Through Optimal Transport

arXiv: https://arxiv.org/abs/2202.07477 · HTML: https://mlsanity.com/paper/2202.07477/

## Key facts

- Cited by 45 papers in the map (data through 14 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.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 27 of 27)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2108.02938/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Label-Efficient Semantic Segmentation with Diffusion Models](https://mlsanity.com/paper/2112.03126/) (2021)
- [SRDiff: Single Image Super-Resolution with Diffusion Probabilistic Models](https://mlsanity.com/paper/2104.14951/) (2021)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Measuring Sample Quality with Stein's Method](https://mlsanity.com/paper/1506.03039/) (2015)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [On Efficient Optimal Transport: An Analysis of Greedy and Accelerated Mirror Descent Algorithms](https://mlsanity.com/paper/1901.06482/) (2019)
- [Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives](https://mlsanity.com/paper/1708.09165/) (2017)

## Built on by (most cited first, 30 of 45)

- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [DiffEdit: Diffusion-based semantic image editing with mask guidance](https://mlsanity.com/paper/2210.11427/) (2022)
- [A Survey on Generative Diffusion Model](https://mlsanity.com/paper/2209.02646/) (2022)
- [ReNoise: Real Image Inversion Through Iterative Noising](https://mlsanity.com/paper/2403.14602/) (2024)
- [Understanding Hallucinations in Diffusion Models through Mode Interpolation](https://mlsanity.com/paper/2406.09358/) (2024)
- [Unifying Diffusion Models' Latent Space, with Applications to CycleDiffusion and Guidance](https://mlsanity.com/paper/2210.05559/) (2022)
- [Maximum Likelihood Training of Implicit Nonlinear Diffusion Models](https://mlsanity.com/paper/2205.13699/) (2022)
- [DiffFashion: Reference-based Fashion Design with Structure-aware Transfer by Diffusion Models](https://mlsanity.com/paper/2302.06826/) (2023)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputation](https://mlsanity.com/paper/2305.07247/) (2023)
- [AID: Attention Interpolation of Text-to-Image Diffusion](https://mlsanity.com/paper/2403.17924/) (2024)
- [Out-of-Distribution Detection with a Single Unconditional Diffusion Model](https://mlsanity.com/paper/2405.11881/) (2024)
- [Data Attribution for Text-to-Image Models by Unlearning Synthesized Images](https://mlsanity.com/paper/2406.09408/) (2024)
- [The Emergence of Reproducibility and Generalizability in Diffusion Models](https://mlsanity.com/paper/2310.05264/) (2023)
- [IMPUS: Image Morphing with Perceptually-Uniform Sampling Using Diffusion Models](https://mlsanity.com/paper/2311.06792/) (2023)
- [Gradpaint: Gradient-Guided Inpainting with Diffusion Models](https://mlsanity.com/paper/2309.09614/) (2023)
- [FreeMorph: Tuning-Free Generalized Image Morphing with Diffusion Model](https://mlsanity.com/paper/2507.01953/) (2025)
- [Diffusion models for Gaussian distributions: Exact solutions and Wasserstein errors](https://mlsanity.com/paper/2405.14250/) (2024)
- [Formulating Discrete Probability Flow Through Optimal Transport](https://mlsanity.com/paper/2311.03886/) (2023)
- [Optimal Transport-Guided Conditional Score-Based Diffusion Models](https://mlsanity.com/paper/2311.01226/) (2023)
- [DDMM-Synth: A Denoising Diffusion Model for Cross-modal Medical Image Synthesis with Sparse-view Measurement Embedding](https://mlsanity.com/paper/2303.15770/) (2023)
- [Masked Completion via Structured Diffusion with White-Box Transformers](https://mlsanity.com/paper/2404.02446/) (2024)
- [Image generation with shortest path diffusion](https://mlsanity.com/paper/2306.00501/) (2023)
- [HARIVO: Harnessing Text-to-Image Models for Video Generation](https://mlsanity.com/paper/2410.07763/) (2024)
- [Image Embedding for Denoising Generative Models](https://mlsanity.com/paper/2301.07485/) (2022)
- [NoisePrints: Distortion-Free Watermarks for Authorship in Private Diffusion Models](https://mlsanity.com/paper/2510.13793/) (2025)
- [Improving Diffusion-Based Generative Models via Approximated Optimal Transport](https://mlsanity.com/paper/2403.05069/) (2024)
- [Efficiently Access Diffusion Fisher: Within the Outer Product Span Space](https://mlsanity.com/paper/2505.23264/) (2025)
- [Finite Difference Flow Optimization for RL Post-Training of Text-to-Image Models](https://mlsanity.com/paper/2603.12893/) (2026)

## 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.