mlsanityData to 14 Oct 2026

Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory

arXiv 2110.11291 ↗First posted 21 Oct 2021stat.ML cs.LG math.APcited by 219

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Cited by 98 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

Schr"odinger Bridge (SB) is an entropy-regularized optimal transport problem that has received increasing attention in deep generative modeling for its mathematical flexibility compared to the Scored-based Generative Model (SGM). However, it remains unclear whether the optimization principle of SB relates to the modern training of deep generative models, which often rely on constructing log-likelihood objectives.This raises questions on the suitability of SB models as a principled alternative for generative applications. In this work, we present a novel computational framework for likelihood training of SB models grounded on Forward-Backward Stochastic Differential Equations Theory - a mathematical methodology appeared in stochastic optimal control that transforms the optimality condition of SB into a set of SDEs. Crucially, these SDEs can be used to construct the likelihood objectives for SB that, surprisingly, generalizes the ones for SGM as special cases. This leads to a new optimization principle that inherits the same SB optimality yet without losing applications of modern generative training techniques, and we show that the resulting training algorithm achieves comparable results on generating realistic images on MNIST, CelebA, and CIFAR10. Our code is available at https://github.com/ghliu/SB-FBSDE.

Key facts. Cited by 219 papers in the map (data through 14 Oct 2026). Builds on 41 of its 72 references that are in the map. First posted on arXiv 21 Oct 2021.

Builds on

41 of 72 references.

References · 41

  1. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  2. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,713
  3. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  5. 2018Neural Ordinary Differential Equationscited by 3,779
  6. 2021Improved Denoising Diffusion Probabilistic Modelscited by 3,761
  7. 2019Analyzing and Improving the Image Quality of StyleGANcited by 3,588
  8. 2015Trust Region Policy Optimizationcited by 2,580
  9. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,354
  10. 2020Training Generative Adversarial Networks with Limited Datacited by 1,186
  11. 2020HiFi-GAN: Generative Adversarial Networks for Efficient and High Fidelity Speech Synthesiscited by 1,154
  12. 2021Variational Diffusion Modelscited by 1,153
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  1. 2020Improved Techniques for Training Score-Based Generative Modelscited by 1,043
  2. 2021Maximum Likelihood Training of Score-Based Diffusion Modelscited by 672
  3. 2020NVAE: A Deep Hierarchical Variational Autoencodercited by 637
  4. 2021Score-based Generative Modeling in Latent Spacecited by 636
  5. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modelingcited by 602
  6. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 594
  7. 2017Solving high-dimensional partial differential equations using deep learningcited by 556
  8. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 292
  9. 2020Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Imagescited by 264
  10. 2021Gotta Go Fast When Generating Data with Score-Based Modelscited by 219
  11. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matchingcited by 183
  12. 2021On Fast Sampling of Diffusion Probabilistic Modelscited by 170
  13. 2021Solving Schr"odinger Bridges via Maximum Likelihoodcited by 141
  14. 2019BIVA: A Very Deep Hierarchy of Latent Variables for Generative Modelingcited by 109
  15. 2021Deep Generative Learning via Schr"odinger Bridgecited by 104
  16. 2019AutoGAN: Neural Architecture Search for Generative Adversarial Networkscited by 94
  17. 2020Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimationcited by 88
  18. 2021Diffusion Normalizing Flowcited by 77
  19. 2020Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Modelscited by 66
  20. 2019WaveFlow: A Compact Flow-based Model for Raw Audiocited by 51
  21. 2019Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Driftcited by 50
  22. 2020Refining Deep Generative Models via Discriminator Gradient Flowcited by 45
  23. 2020VFlow: More Expressive Generative Flows with Variational Data Augmentationcited by 38
  24. 2021Styleformer: Transformer based Generative Adversarial Networks with Style Vectorcited by 27
  25. 2018An Optimal Control Approach to Deep Learning and Applications to Discrete-Weight Neural Networkscited by 26
  26. 2019Discriminator optimal transportcited by 25
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  29. 2021Dynamic Game Theoretic Neural Optimizercited by 2

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Citing papers per year: 2021: 2; 2022: 18; 2023: 44; 2024: 56; 2025: 55; 2026: 44.

219 papers.

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  6. 2022Riemannian Score-Based Generative Modellingcited by 219
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  24. 2023Schrodinger Bridges Beat Diffusion Models on Text-to-Speech Synthesiscited by 30
  25. 2025Underdamped Diffusion Bridges with Applications to Samplingcited by 29
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  27. 2023GENOT: Entropic (Gromov) Wasserstein Flow Matching with Applications to Single-Cell Genomicscited by 28
  28. 2023Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputationcited by 27
  29. 2024Multi-marginal Schr"odinger Bridges with Iterative Reference Refinementcited by 27
  30. 2024Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Samplingcited by 27
  31. 2023Energy-guided Entropic Neural Optimal Transportcited by 26
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  33. 2024Self-Consistent Recursive Diffusion Bridge for Medical Image Translationcited by 26
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  35. 2023Transport meets Variational Inference: Controlled Monte Carlo Diffusionscited by 21
  36. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processescited by 21
  37. 2023Neural Diffusion Modelscited by 21
  38. 2023Augmented Bridge Matchingcited by 21

50 most cited of 219 All 219 (JSON)