mlsanityData to 7 Oct 2026

Deep Generative Learning via Schr"odinger Bridge

arXiv 2106.10410 ↗First posted 19 Jun 2021cs.LG cs.CVcited by 103

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Cited by 47 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

We propose to learn a generative model via entropy interpolation with a Schr"odinger Bridge. The generative learning task can be formulated as interpolating between a reference distribution and a target distribution based on the Kullback-Leibler divergence. At the population level, this entropy interpolation is characterized via an SDE on [0,1] with a time-varying drift term. At the sample level, we derive our Schr"odinger Bridge algorithm by plugging the drift term estimated by a deep score estimator and a deep density ratio estimator into the Euler-Maruyama method. Under some mild smoothness assumptions of the target distribution, we prove the consistency of both the score estimator and the density ratio estimator, and then establish the consistency of the proposed Schr"odinger Bridge approach. Our theoretical results guarantee that the distribution learned by our approach converges to the target distribution. Experimental results on multimodal synthetic data and benchmark data support our theoretical findings and indicate that the generative model via Schr"odinger Bridge is comparable with state-of-the-art GANs, suggesting a new formulation of generative learning. We demonstrate its usefulness in image interpolation and image inpainting.

Key facts. Cited by 103 papers in the map (data through 7 Oct 2026). Builds on 45 of its 58 references that are in the map. First posted on arXiv 19 Jun 2021.

Builds on

45 of 58 references.

References · 45

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  3. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  4. 2020Denoising Diffusion Implicit Modelscited by 9,655
  5. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  6. 2018A Style-Based Generator Architecture for Generative Adversarial Networkscited by 5,986
  7. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
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33 more
  1. 2018Large Scale GAN Training for High Fidelity Natural Image Synthesiscited by 2,609
  2. 2017Improved Training of Wasserstein GANscited by 2,223
  3. 2015Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networkscited by 2,048
  4. 2016Density estimation using Real NVPcited by 2,029
  5. 2016WaveNet: A Generative Model for Raw Audiocited by 1,710
  6. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,618
  7. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  8. 2019Generating Diverse High-Fidelity Images with VQ-VAE-2cited by 1,376
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  11. 2014NICE: Non-linear Independent Components Estimationcited by 1,052
  12. 2020Improved Techniques for Training Score-Based Generative Modelscited by 1,036
  13. 2018Self-Attention Generative Adversarial Networkscited by 809
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  15. 2017Masked Autoregressive Flow for Density Estimationcited by 680
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  19. 2015Adversarial Autoencoderscited by 359
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  21. 2018WaveGlow: A Flow-based Generative Network for Speech Synthesiscited by 308
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  23. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 286
  24. 2016Mode Regularized Generative Adversarial Networkscited by 123
  25. 2019Maximum Mean Discrepancy Gradient Flowcited by 123
  26. 2019Deep Network Approximation Characterized by Number of Neuronscited by 120
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  28. 2018Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusionscited by 77
  29. 2019Deep Generative Learning via Variational Gradient Flowcited by 32
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 14; 2023: 29; 2024: 16; 2025: 31; 2026: 11.

103 papers.

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  1. 2022Flow Matching for Generative Modelingcited by 5,426
  2. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,550
  3. 2022Diffusion Models in Vision: A Surveycited by 1,008
  4. 2023Improving and generalizing flow-based generative models with minibatch optimal transportcited by 835
  5. 2022Fast Sampling of Diffusion Models with Exponential Integratorcited by 526
  6. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 216
  7. 2023Reduce, Reuse, Recycle: Compositional Generation with Energy-Based Diffusion Models and MCMCcited by 216
  8. 2022Dual Diffusion Implicit Bridges for Image-to-Image Translationcited by 212
  9. 2023Diffusion Schr"odinger Bridge Matchingcited by 151
  10. 2022Diffusion-based Molecule Generation with Informative Prior Bridgescited by 104
  11. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Modelscited by 102
  12. 2023Unpaired Image-to-Image Translation via Neural Schr"odinger Bridgecited by 96
38 more
  1. 2023Simulation-free Schr"odinger bridges via score and flow matchingcited by 81
  2. 2023Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modelingcited by 80
  3. 2023Non-Denoising Forward-Time Diffusionscited by 75
  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. 2024Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processescited by 61
  8. 2021Bayesian Learning via Neural Schr"odinger-F"ollmer Flowscited by 56
  9. 2023Diffusion Models With Learned Adaptive Noisecited by 50
  10. 2022Entropic Neural Optimal Transport via Diffusion Processescited by 49
  11. 2022Deep Generalized Schr"odinger Bridgecited by 47
  12. 2024Rethinking Score Distillation as a Bridge Between Image Distributionscited by 36
  13. 2023RetroBridge: Modeling Retrosynthesis with Markov Bridgescited by 35
  14. 2024Learning stochastic dynamics from snapshots through regularized unbalanced optimal transportcited by 30
  15. 2023Schrodinger Bridges Beat Diffusion Models on Text-to-Speech Synthesiscited by 29
  16. 2023A mean-field games laboratory for generative modelingcited by 28
  17. 2023Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputationcited by 26
  18. 2024Multi-marginal Schr"odinger Bridges with Iterative Reference Refinementcited by 26
  19. 2023Mathematical analysis of singularities in the diffusion model under the submanifold assumptioncited by 20
  20. 2023Neural Diffusion Modelscited by 20
  21. 2025Adjoint Schr"odinger Bridge Samplercited by 20
  22. 2023Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmarkcited by 19
  23. 2022First Hitting Diffusion Models for Generating Manifold, Graph and Categorical Datacited by 18
  24. 2023Generative modeling for time series via Schr"odinger bridgecited by 18
  25. 2023Stage-by-stage Wavelet Optimization Refinement Diffusion Model for Sparse-View CT Reconstructioncited by 18
  26. 2023Light Schr"odinger Bridgecited by 18
  27. 2023Improved Order Analysis and Design of Exponential Integrator for Diffusion Models Samplingcited by 16
  28. 2024Convergence Analysis of Flow Matching in Latent Space with Transformerscited by 16
  29. 2024Improving Probabilistic Diffusion Models With Optimal Diagonal Covariance Matchingcited by 13
  30. 2025Audio Super-Resolution with Latent Bridge Modelscited by 13
  31. 2024Schr"odinger Bridge with Quadratic State Cost is Exactly Solvablecited by 12
  32. 2023Dynamic Flows on Curved Space Generated by Labeled Datacited by 10
  33. 2023Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assemblycited by 10
  34. 2023Transport with Support: Data-Conditional Diffusion Bridgescited by 9
  35. 2024Towards Robust Event-based Networks for Nighttime via Unpaired Day-to-Night Event Translationcited 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. 2023Light Unbalanced Optimal Transportcited by 8

50 most cited of 103 All 103 (JSON)