mlsanityData to 14 Oct 2026

Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputation

arXiv 2305.07247 ↗First posted 12 May 2023cs.LGcited by 27

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

The Schr"odinger bridge problem (SBP) is gaining increasing attention in generative modeling and showing promising potential even in comparison with the score-based generative models (SGMs). SBP can be interpreted as an entropy-regularized optimal transport problem, which conducts projections onto every other marginal alternatingly. However, in practice, only approximated projections are accessible and their convergence is not well understood. To fill this gap, we present a first convergence analysis of the Schr"odinger bridge algorithm based on approximated projections. As for its practical applications, we apply SBP to probabilistic time series imputation by generating missing values conditioned on observed data. We show that optimizing the transport cost improves the performance and the proposed algorithm achieves the state-of-the-art result in healthcare and environmental data while exhibiting the advantage of exploring both temporal and feature patterns in probabilistic time series imputation.

Key facts. Cited by 27 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 42 of its 66 references that are in the map. First posted on arXiv 12 May 2023.

Builds on

42 of 66 references.

Closest ancestors · 8

Same direction, newest first

  1. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  2. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  3. 2022Conditional Simulation Using Diffusion Schr"odinger Bridges27 Feb
  4. 2022Understanding DDPM Latent Codes Through Optimal Transport14 Feb
  5. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  6. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  7. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  8. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 34

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,713
  2. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  4. 2017Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networkscited by 879
  5. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 594
  6. 2021CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputationcited by 582
  7. 2016Recurrent Neural Networks for Multivariate Time Series with Missing Valuescited by 554
  8. 2020Neural Controlled Differential Equations for Irregular Time Seriescited by 506
  9. 2021Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecastingcited by 337
  10. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 292
  11. 2018BRITS: Bidirectional Recurrent Imputation for Time Seriescited by 290
  12. 2020Scalable Gradients for Stochastic Differential Equationscited by 271
22 more
  1. 2017Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysiscited by 243
  2. 2021Multi-Time Attention Networks for Irregularly Sampled Time Seriescited by 238
  3. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 219
  4. 2019Latent ODEs for Irregularly-Sampled Time Seriescited by 183
  5. 2019GP-VAE: Deep Probabilistic Time Series Imputationcited by 172
  6. 2019High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processescited by 148
  7. 2021Solving Schr"odinger Bridges via Maximum Likelihoodcited by 141
  8. 2021Deep Generative Learning via Schr"odinger Bridgecited by 104
  9. 2022Neural Conservation Laws: A Divergence-Free Perspectivecited by 70
  10. 2022Statistical Efficiency of Score Matching: The View from Isoperimetrycited by 64
  11. 2022The Schr"odinger Bridge between Gaussian Measures has a Closed Formcited by 63
  12. 2019Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Driftcited by 50
  13. 2020Uncertainty-Aware Variational-Recurrent Imputation Network for Clinical Time Seriescited by 33
  14. 2018The data-driven Schroedinger bridgecited by 26
  15. 2020A Contour Stochastic Gradient Langevin Dynamics Algorithm for Simulations of Multi-modal Distributionscited by 26
  16. 2021Stability of Entropic Optimal Transport and Schr"odinger Bridgescited by 26
  17. 2022On the Convergence Rate of Sinkhorn's Algorithmcited by 16
  18. 2014Stochastic bridges of linear systemscited by 14
  19. 2022Stability of Schr"odinger Potentials and Convergence of Sinkhorn's Algorithmcited by 13
  20. 2023Quantitative contraction rates for Sinkhorn's algorithm: beyond bounded costs and compact marginalscited by 13
  21. 2021Continuous Latent Process Flowscited by 10
  22. 2023Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approachcited by 7

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 4; 2024: 13; 2025: 3; 2026: 7.

27 papers.

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  1. 2023The probability flow ODE is provably fastcited by 146
  2. 2024UniTS: A Unified Multi-Task Time Series Modelcited by 94
  3. 2024Deep Learning for Multivariate Time Series Imputation: A Surveycited by 57
  4. 2024TSI-Bench: Benchmarking Time Series Imputationcited by 33
  5. 2025Non-stationary Diffusion For Probabilistic Time Series Forecastingcited by 27
  6. 2024Sports-Traj: A Unified Trajectory Generation Model for Multi-Agent Movement in Sportscited by 19
  7. 2023Light Schr"odinger Bridgecited by 18
  8. 2024SSD-TS: Exploring the Potential of Linear State Space Models for Diffusion Models in Time Series Imputationcited by 18
  9. 2024MTSCI: A Conditional Diffusion Model for Multivariate Time Series Consistent Imputationcited by 13
  10. 2024Reflected Schr"odinger Bridge for Constrained Generative Modelingcited by 12
  11. 2023BayOTIDE: Bayesian Online Multivariate Time series Imputation with functional decompositioncited by 10
  12. 2024Variational Schr"odinger Diffusion Modelscited by 9
15 more
  1. 2024How Deep is your Guess? A Fresh Perspective on Deep Learning for Medical Time-Series Imputationcited by 5
  2. 2024Conditional Lagrangian Wasserstein Flow for Time Series Imputationcited by 4
  3. 2024Rethinking the Diffusion Models for Numerical Tabular Data Imputation from the Perspective of Wasserstein Gradient Flowcited by 3
  4. 2024Recurrent Interpolants for Probabilistic Time Series Predictioncited by 2
  5. 2025Optimal Stochastic Trace Estimation in Generative Modelingcited by 2
  6. 2023Deep sub-ensembles meets quantile regression: uncertainty-aware imputation for time seriescited by 1
  7. 2025Causal View of Time Series Imputation: Some Identification Results on Missing Mechanismcited by 1
  8. 2026Spatial-Temporal Feedback Diffusion Guidance for Controlled Traffic Imputationcited by 1
  9. 2026Pathwise Learning of Stochastic Dynamical Systems with Partial Observationscited by 1
  10. 2026ABC: Any-Subset Autoregression via Non-Markovian Diffusion Bridges in Continuous Time and Spacecited by 1
  11. 2024Modular Deep Learning for Multivariate Time-Series: Decoupling Imputation and Downstream Taskscited by 0
  12. 2026Analyzing and Improving Diffusion Models for Time-Series Data Imputation: A Proximal Recursion Perspectivecited by 0
  13. 2026PAMF: Prior-Aware Multimodal Fusion for Incomplete Time Series Datacited by 0
  14. 2026TimeES: Probabilistic and Deterministic Time Series Forecasting via Evolutionary Spectracited by 0
  15. 2026MissPath-FM: Flow Matching with Structured Priors for Partially Observed Time Seriescited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026Foundations of Schr"odinger Bridges for Generative Modeling
  2. 2025A Closed-Form Framework for Schr"odinger Bridges Between Arbitrary Densities
  3. 2024Soft-constrained Schrodinger Bridge: a Stochastic Control Approach
  4. 2023Simulation-free Schr"odinger bridges via score and flow matching
  5. 2024Schr"odinger Bridge Flow for Unpaired Data Translation
  6. 2025Data-to-Energy Stochastic Dynamics
  7. 2023Generalized Schr"odinger Bridge Matching
  8. 2025Incorporating Pre-trained Diffusion Models in Solving the Schr"odinger Bridge Problem