mlsanityData to 7 Oct 2026

Solving Schr"odinger Bridges via Maximum Likelihood

arXiv 2106.02081 ↗First posted 3 Jun 2021stat.ML cs.LGcited by 140

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Cited by 55 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

The Schr"odinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applications in machine learning such as dataset alignment and hypothesis testing. Whilst the theory behind this problem is relatively mature, scalable numerical recipes to estimate the Schr"odinger bridge remain an active area of research. We prove an equivalence between the SBP and maximum likelihood estimation enabling direct application of successful machine learning techniques. We propose a numerical procedure to estimate SBPs using Gaussian process and demonstrate the practical usage of our approach in numerical simulations and experiments.

Key facts. Cited by 140 papers in the map (data through 7 Oct 2026). Builds on 12 of its 60 references that are in the map. First posted on arXiv 3 Jun 2021.

Builds on

12 of 60 references.

References · 12

  1. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  2. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,277
  3. 2017Masked Autoregressive Flow for Density Estimationcited by 680
  4. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modelingcited by 594
  5. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 286
  6. 2020TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamicscited by 155
  7. 2020Faster Wasserstein Distance Estimation with the Sinkhorn Divergencecited by 73
  8. 2015Probabilistic Numerics and Uncertainty in Computationscited by 59
  9. 2018The data-driven Schroedinger bridgecited by 26
  10. 2019Neural Density Estimation and Likelihood-free Inferencecited by 23
  11. 2019Schr"odinger Bridge Samplerscited by 14
  12. 2017Approximate Bayes learning of stochastic differential equationscited by 11

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 5; 2022: 16; 2023: 35; 2024: 29; 2025: 40; 2026: 15.

140 papers.

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  1. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,550
  2. 2023Improving and generalizing flow-based generative models with minibatch optimal transportcited by 835
  3. 2022Fast Sampling of Diffusion Models with Exponential Integratorcited by 526
  4. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 216
  5. 2022Riemannian Score-Based Generative Modellingcited by 216
  6. 2023I2SB: Image-to-Image Schr"odinger Bridgecited by 214
  7. 2022Dual Diffusion Implicit Bridges for Image-to-Image Translationcited by 212
  8. 2023Diffusion Schr"odinger Bridge Matchingcited by 151
  9. 2022Diffusion-based Molecule Generation with Informative Prior Bridgescited by 104
  10. 2023Unpaired Image-to-Image Translation via Neural Schr"odinger Bridgecited by 96
  11. 2021Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimationcited by 95
  12. 2023Simulation-free Schr"odinger bridges via score and flow matchingcited by 81
38 more
  1. 2023Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modelingcited by 80
  2. 2023Non-Denoising Forward-Time Diffusionscited by 75
  3. 2024Optimal Flow Matching: Learning Straight Trajectories in Just One Stepcited by 74
  4. 2023On the Generalization Properties of Diffusion Modelscited by 67
  5. 2022The Schr"odinger Bridge between Gaussian Measures has a Closed Formcited by 63
  6. 2022Manifold Interpolating Optimal-Transport Flows for Trajectory Inferencecited by 63
  7. 2022Conditional Simulation Using Diffusion Schr"odinger Bridgescited by 58
  8. 2021Bayesian Learning via Neural Schr"odinger-F"ollmer Flowscited by 56
  9. 2021Simulating Diffusion Bridges with Score Matchingcited by 51
  10. 2022Entropic Neural Optimal Transport via Diffusion Processescited by 49
  11. 2022Deep Generalized Schr"odinger Bridgecited by 47
  12. 2024Sequential Controlled Langevin Diffusionscited by 47
  13. 2024Light and Optimal Schr"odinger Bridge Matchingcited by 45
  14. 2024DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transformcited by 44
  15. 2023A Computational Framework for Solving Wasserstein Lagrangian Flowscited by 43
  16. 2022Maximum Likelihood Training of Implicit Nonlinear Diffusion Modelscited by 42
  17. 2023Deep Momentum Multi-Marginal Schr"odinger Bridgecited by 40
  18. 2023Generalized Schr"odinger Bridge Matchingcited by 40
  19. 2023Aligned Diffusion Schr"odinger Bridgescited by 38
  20. 2022Stochastic Optimal Control for Collective Variable Free Sampling of Molecular Transition Pathscited by 36
  21. 2023RetroBridge: Modeling Retrosynthesis with Markov Bridgescited by 35
  22. 2023Blackout Diffusion: Generative Diffusion Models in Discrete-State Spacescited by 29
  23. 2023Schrodinger Bridges Beat Diffusion Models on Text-to-Speech Synthesiscited by 29
  24. 2025Underdamped Diffusion Bridges with Applications to Samplingcited by 28
  25. 2023GENOT: Entropic (Gromov) Wasserstein Flow Matching with Applications to Single-Cell Genomicscited by 27
  26. 2023Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputationcited by 26
  27. 2024Multi-marginal Schr"odinger Bridges with Iterative Reference Refinementcited by 26
  28. 2023Energy-guided Entropic Neural Optimal Transportcited by 25
  29. 2022Riemannian Metric Learning via Optimal Transportcited by 21
  30. 2023Transport meets Variational Inference: Controlled Monte Carlo Diffusionscited by 21
  31. 2023Augmented Bridge Matchingcited by 21
  32. 2025Adjoint Schr"odinger Bridge Samplercited by 20
  33. 2023Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmarkcited by 19
  34. 2024Dynamical Measure Transport and Neural PDE Solvers for Samplingcited by 19
  35. 2023Light Schr"odinger Bridgecited by 18
  36. 2023Tree-Based Diffusion Schr"odinger Bridge with Applications to Wasserstein Barycenterscited by 16
  37. 2023Improved Order Analysis and Design of Exponential Integrator for Diffusion Models Samplingcited by 16
  38. 2025On the Relation between Rectified Flows and Optimal Transportcited by 15

50 most cited of 140 All 140 (JSON)