mlsanityData to 7 Oct 2026

On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint

arXiv 1412.4430 ↗First posted 15 Dec 2014eess.SY cs.SY math-phcited by 99

Not on the map. Why?

Cited by 20 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

We take a new look at the relation between the optimal transport problem and the Schr"odinger bridge problem from the stochastic control perspective. We show that the connections are richer and deeper than described in existing literature. In particular: a) We give an elementary derivation of the Benamou-Brenier fluid dynamics version of the optimal transport problem; b) We provide a new fluid dynamics version of the Schr"odinger bridge problem; c) We observe that the latter provides an important connection with optimal transport without zero noise limits; d) We propose and solve a fluid dynamic version of optimal transport with prior; e) We can then view optimal transport with prior as the zero noise limit of Schr"odinger bridges when the prior is any Markovian evolution. In particular, we work out the Gaussian case. A numerical example of the latter convergence involving Brownian particles is also provided.

Key facts. Cited by 99 papers in the map (data through 7 Oct 2026). First posted on arXiv 15 Dec 2014.

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Citing papers per year: 2021: 5; 2022: 12; 2023: 15; 2024: 19; 2025: 17; 2026: 31.

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  1. 2022Dual Diffusion Implicit Bridges for Image-to-Image Translationcited by 212
  2. 2021Path Integral Sampler: a stochastic control approach for samplingcited by 175
  3. 2022Refining Generative Process with Discriminator Guidance in Score-based Diffusion Modelscited by 98
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  9. 2022Understanding DDPM Latent Codes Through Optimal Transportcited by 44
  10. 2023A Computational Framework for Solving Wasserstein Lagrangian Flowscited by 43
  11. 2022Maximum Likelihood Training of Implicit Nonlinear Diffusion Modelscited by 42
  12. 2022Stochastic Optimal Control for Collective Variable Free Sampling of Molecular Transition Pathscited by 36
38 more
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  4. 2023Scalable Optimal Transport Methods in Machine Learning: A Contemporary Surveycited by 27
  5. 2024Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Samplingcited by 25
  6. 2023Transport meets Variational Inference: Controlled Monte Carlo Diffusionscited by 21
  7. 2025Adjoint Schr"odinger Bridge Samplercited by 20
  8. 2023Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmarkcited by 19
  9. 2024Dynamical Measure Transport and Neural PDE Solvers for Samplingcited by 19
  10. 2023Light Schr"odinger Bridgecited by 18
  11. 2024Connecting Consistency Distillation to Score Distillation for Text-to-3D Generationcited by 18
  12. 2021On the complexity of the optimal transport problem with graph-structured costcited by 14
  13. 2024Wasserstein proximal operators describe score-based generative models and resolve memorizationcited by 14
  14. 2024Partially Observed Trajectory Inference using Optimal Transport and a Dynamics Priorcited by 13
  15. 2024Schr"odinger Bridge with Quadratic State Cost is Exactly Solvablecited by 12
  16. 2022Convergence Rates for Regularized Optimal Transport via Quantizationcited by 11
  17. 2025Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyondcited by 11
  18. 2021Learning High Dimensional Wasserstein Geodesicscited by 10
  19. 2023Doubly Regularized Entropic Wasserstein Barycenterscited by 10
  20. 2023Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assemblycited by 10
  21. 2025Distribution Prototype Diffusion Learning for Open-set Supervised Anomaly Detectioncited by 10
  22. 2025Variational Regularized Unbalanced Optimal Transport: Single Network, Least Actioncited by 10
  23. 2025Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysiscited by 9
  24. 2025Momentum Multi-Marginal Schr"odinger Bridge Matchingcited by 9
  25. 2024Soft-constrained Schrodinger Bridge: a Stochastic Control Approachcited by 8
  26. 2024Plug-in estimation of Schr"odinger bridgescited by 8
  27. 2022Neural Lagrangian Schr"odinger Bridge: Diffusion Modeling for Population Dynamicscited by 7
  28. 2026Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matchingcited by 7
  29. 2022Maximum entropy optimal density control of discrete-time linear systems and Schr"odinger bridgescited by 6
  30. 2025Neural Flow Samplers with Shortcut Modelscited by 5
  31. 2025On the Hopf-Cole Transform for Control-affine Schr"odinger Bridgecited by 5
  32. 2023Scalable Motion Style Transfer with Constrained Diffusion Generationcited by 4
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  35. 2026Discrete Adjoint Schr"odinger Bridge Samplercited by 4
  36. 2026Efficient Generative Modeling beyond Memoryless Diffusion via Adjoint Schr"odinger Bridge Matchingcited by 4
  37. 2026Scalable Maximum Entropy Reinforcement Learning for Diffusion Policies via Adjoint Matchingcited by 4
  38. 2021Differential Motion Evolution for Fine-Grained Motion Deformation in Unsupervised Image Animationcited by 3

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