mlsanityData to 7 Oct 2026

Diffusion Schr"odinger Bridge Matching

arXiv 2303.16852 ↗First posted 29 Mar 2023stat.ML cs.LGcited by 151

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Solving transport problems, i.e. finding a map transporting one given distribution to another, has numerous applications in machine learning. Novel mass transport methods motivated by generative modeling have recently been proposed, e.g. Denoising Diffusion Models (DDMs) and Flow Matching Models (FMMs) implement such a transport through a Stochastic Differential Equation (SDE) or an Ordinary Differential Equation (ODE). However, while it is desirable in many applications to approximate the deterministic dynamic Optimal Transport (OT) map which admits attractive properties, DDMs and FMMs are not guaranteed to provide transports close to the OT map. In contrast, Schr"odinger bridges (SBs) compute stochastic dynamic mappings which recover entropy-regularized versions of OT. Unfortunately, existing numerical methods approximating SBs either scale poorly with dimension or accumulate errors across iterations. In this work, we introduce Iterative Markovian Fitting (IMF), a new methodology for solving SB problems, and Diffusion Schr"odinger Bridge Matching (DSBM), a novel numerical algorithm for computing IMF iterates. DSBM significantly improves over previous SB numerics and recovers as special/limiting cases various recent transport methods. We demonstrate the performance of DSBM on a variety of problems.

Key facts. Cited by 151 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 45 of its 105 references that are in the map. First posted on arXiv 29 Mar 2023.

Builds on

45 of 105 references.

Closest ancestors · 12

Same direction, newest first

  1. 2023I2SB: Image-to-Image Schr"odinger Bridge12 Feb
  2. 2023Simple diffusion: End-to-end diffusion for high resolution images26 Jan
  3. 2023On the Importance of Noise Scheduling for Diffusion Models26 Jan
  4. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Models31 Aug
  5. 2022Applying Regularized Schr"odinger-Bridge-Based Stochastic Process in Generative Modeling15 Aug
  6. 2022Riemannian Diffusion Schr"odinger Bridge7 Jul
  7. 2022Conditional Simulation Using Diffusion Schr"odinger Bridges27 Feb
  8. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matching5 Jun
  9. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  10. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  11. 2020Denoising Diffusion Implicit Models6 Oct
  12. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 33

  1. 2022Generative Adversarial Networkscited by 5,603
  2. 2022Flow Matching for Generative Modelingcited by 5,426
  3. 2014Deep Learning Face Attributes in the Wildcited by 3,679
  4. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,550
  5. 2020Unsupervised Learning of Visual Features by Contrasting Cluster Assignmentscited by 2,845
  6. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,277
  7. 2021SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equationscited by 1,903
  8. 2019StarGAN v2: Diverse Image Synthesis for Multiple Domainscited by 1,095
  9. 2022Building Normalizing Flows with Stochastic Interpolantscited by 964
  10. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  11. 2018Computational Optimal Transportcited by 765
  12. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 286
21 more
  1. 2017Joint Distribution Optimal Transportation for Domain Adaptationcited by 280
  2. 2022Rectified Flow: A Marginal Preserving Approach to Optimal Transportcited by 277
  3. 2023Flow Matching on General Geometriescited by 260
  4. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 216
  5. 2023Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restorationcited by 164
  6. 2021Solving Schr"odinger Bridges via Maximum Likelihoodcited by 140
  7. 2021Deep Generative Learning via Schr"odinger Bridgecited by 103
  8. 2021Proximal Optimal Transport Modeling of Population Dynamicscited by 91
  9. 2022The Schr"odinger Bridge between Gaussian Measures has a Closed Formcited by 63
  10. 2021Minibatch optimal transport distances; analysis and applicationscited by 62
  11. 2021Simulating Diffusion Bridges with Score Matchingcited by 51
  12. 2019Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Driftcited by 50
  13. 2022Deep Generalized Schr"odinger Bridgecited by 47
  14. 2023Deep Momentum Multi-Marginal Schr"odinger Bridgecited by 40
  15. 2017Optimal Transport for Diffeomorphic Registrationcited by 39
  16. 2023Aligned Diffusion Schr"odinger Bridgescited by 38
  17. 2015Entropic and displacement interpolation: a computational approach using the Hilbert metriccited by 34
  18. 2018The data-driven Schroedinger bridgecited by 26
  19. 2020Learning normalizing flows from Entropy-Kantorovich potentialscited by 18
  20. 2023Transport with Support: Data-Conditional Diffusion Bridgescited by 9
  21. 2018Traversing the Schroedinger Bridge strait: Robert Fortet's marvelous proof reduxcited by 6

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 18; 2024: 36; 2025: 50; 2026: 47.

151 papers.

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  1. 2024SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformerscited by 707
  2. 2023Denoising Diffusion Bridge Modelscited by 171
  3. 2023SE(3)-Stochastic Flow Matching for Protein Backbone Generationcited by 147
  4. 2023Unpaired Image-to-Image Translation via Neural Schr"odinger Bridgecited by 96
  5. 2023Simulation-free Schr"odinger bridges via score and flow matchingcited by 81
  6. 2023Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modelingcited by 80
  7. 2025Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matchingcited by 78
  8. 2024Optimal Flow Matching: Learning Straight Trajectories in Just One Stepcited by 74
  9. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 74
  10. 2023On the Generalization Properties of Diffusion Modelscited by 67
  11. 2023Diffusion Models With Learned Adaptive Noisecited by 50
  12. 2024Diffusion Bridge Implicit Modelscited by 47
38 more
  1. 2024Neural Flow Diffusion Models: Learnable Forward Process for Improved Diffusion Modellingcited by 43
  2. 2023Image Restoration Through Generalized Ornstein-Uhlenbeck Bridgecited by 41
  3. 2023Generalized Schr"odinger Bridge Matchingcited by 40
  4. 2025FlowTok: Flowing Seamlessly Across Text and Image Tokenscited by 31
  5. 2023Schrodinger Bridges Beat Diffusion Models on Text-to-Speech Synthesiscited by 29
  6. 2023A mean-field games laboratory for generative modelingcited by 28
  7. 2023Energy-guided Entropic Neural Optimal Transportcited by 25
  8. 2024Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Samplingcited by 25
  9. 2024Consistency Diffusion Bridge Modelscited by 22
  10. 2023The Emergence of Reproducibility and Generalizability in Diffusion Modelscited by 21
  11. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processescited by 21
  12. 2023Augmented Bridge Matchingcited by 21
  13. 2024Trajectory Flow Matching with Applications to Clinical Time Series Modelingcited by 21
  14. 2025Temporal Difference Flowscited by 20
  15. 2025Adjoint Schr"odinger Bridge Samplercited by 20
  16. 2024Conditional sampling within generative diffusion modelscited by 18
  17. 2025Composition and Control with Distilled Energy Diffusion Models and Sequential Monte Carlocited by 18
  18. 2025Curly Flow Matching for Learning Non-gradient Field Dynamicscited by 17
  19. 2023Tree-Based Diffusion Schr"odinger Bridge with Applications to Wasserstein Barycenterscited by 16
  20. 2024Stochastic Optimal Control for Diffusion Bridges in Function Spacescited by 16
  21. 2024Stochastic Sampling from Deterministic Flow Modelscited by 16
  22. 2025On the Relation between Rectified Flows and Optimal Transportcited by 15
  23. 2024Theoretical guarantees in KL for Diffusion Flow Matchingcited by 14
  24. 2024CUNSB-RFIE: Context-aware Unpaired Neural Schr"odinger Bridge in Retinal Fundus Image Enhancementcited by 13
  25. 2024Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifoldscited by 13
  26. 2023Energy-Guided Continuous Entropic Barycenter Estimation for General Costscited by 12
  27. 2024Adversarial Schr"odinger Bridge Matchingcited by 12
  28. 2024Schr"odinger Bridge with Quadratic State Cost is Exactly Solvablecited by 12
  29. 2025Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schr"odinger Bridgecited by 12
  30. 2025FALCON: Few-step Accurate Likelihoods for Continuous Flowscited by 12
  31. 2024Conditioning diffusion models by explicit forward-backward bridgingcited by 11
  32. 2025Variational Regularized Unbalanced Optimal Transport: Single Network, Least Actioncited by 10
  33. 2026Discrete Adjoint Matchingcited by 10
  34. 2023Unbalanced Diffusion Schr"odinger Bridgecited by 9
  35. 2024Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transportcited by 9
  36. 2024Bridge-IF: Learning Inverse Protein Folding with Markov Bridgescited by 9
  37. 2025Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysiscited by 9
  38. 2025Bidirectional Diffusion Bridge Modelscited by 8

50 most cited of 151 All 151 (JSON)

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  1. 2024Schr"odinger Bridge Flow for Unpaired Data Translation
  2. 2025Entering the Era of Discrete Diffusion Models: A Benchmark for Schr"odinger Bridges and Entropic Optimal Transport
  3. 2025Data-to-Energy Stochastic Dynamics
  4. 2024Light and Optimal Schr"odinger Bridge Matching
  5. 2024Variational Schr"odinger Diffusion Models
  6. 2025A Unified Framework for Diffusion Bridge Problems: Flow Matching and Schr"odinger Matching into One
  7. 2024Discrete Diffusion Schr"odinger Bridge Matching for Graph Transformation
  8. 2024Reflected Schr"odinger Bridge for Constrained Generative Modeling