mlsanityData to 7 Oct 2026

Conditional Optimal Transport on Function Spaces

arXiv 2311.05672 ↗First posted 9 Nov 2023math.OC math.PR stat.COcited by 26

Not on the map. Why?

Abstract (arXiv)

We present a systematic study of conditional triangular transport maps in function spaces from the perspective of optimal transportation and with a view towards amortized Bayesian inference. More specifically, we develop a theory of constrained optimal transport problems that describe block-triangular Monge maps that characterize conditional measures along with their Kantorovich relaxations. This generalizes the theory of optimal triangular transport to separable infinite-dimensional function spaces with general cost functions. We further tailor our results to the case of Bayesian inference problems and obtain regularity estimates on the conditioning maps from the prior to the posterior. Finally, we present numerical experiments that demonstrate the computational applicability of our theoretical results for amortized and likelihood-free inference of functional parameters.

Key facts. Cited by 26 papers in the map (data through 7 Oct 2026). Builds on 66 of its 122 references that are in the map. First posted on arXiv 9 Nov 2023.

Builds on

66 of 122 references.

References · 66

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  2. 2017Improved Training of Wasserstein GANscited by 2,223
  3. 2015Variational Inference with Normalizing Flowscited by 2,161
  4. 2014Conditional Generative Adversarial Netscited by 2,152
  5. 2021Image Super-Resolution via Iterative Refinementcited by 1,506
  6. 2016Variational Inference: A Review for Statisticianscited by 1,466
  7. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  8. 2021Variational Diffusion Modelscited by 1,145
  9. 2020Improved Techniques for Training Score-Based Generative Modelscited by 1,036
  10. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  11. 2018Computational Optimal Transportcited by 765
  12. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 584
38 more
  1. 2016NIPS 2016 Tutorial: Generative Adversarial Networkscited by 284
  2. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  3. 2017Advances in Variational Inferencecited by 236
  4. 2018Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flowscited by 217
  5. 2019Optimal transport mapping via input convex neural networkscited by 202
  6. 2021Conditional Image Generation with Score-Based Diffusion Modelscited by 195
  7. 2016Stochastic Optimization for Large-scale Optimal Transportcited by 188
  8. 2019Learning Likelihoods with Conditional Normalizing Flowscited by 170
  9. 2020OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transportcited by 143
  10. 2021Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmarkcited by 95
  11. 2021Generative Modeling with Optimal Transport Mapscited by 80
  12. 2023Score-based Diffusion Models in Function Spacecited by 80
  13. 2021Plugin Estimation of Smooth Optimal Transport Mapscited by 67
  14. 2015Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Modelscited by 61
  15. 2018Likelihood-free inference with emulator networkscited by 61
  16. 2018Deep Bayesian Inversioncited by 60
  17. 2022Supervised Training of Conditional Monge Mapscited by 60
  18. 2021Score-based Generative Neural Networks for Large-Scale Optimal Transportcited by 59
  19. 2018Conditional Density Estimation with Bayesian Normalising Flowscited by 54
  20. 2021Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projectionscited by 52
  21. 2022Generative Adversarial Neural Operatorscited by 51
  22. 2022Diffusion Generative Models in Infinite Dimensionscited by 49
  23. 2019Coupling techniques for nonlinear ensemble filteringcited by 48
  24. 2020Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inferencecited by 40
  25. 2023Conditional score-based diffusion models for Bayesian inference in infinite dimensionscited by 37
  26. 2014Dimension-independent likelihood-informed MCMCcited by 36
  27. 2023The Monge Gap: A Regularizer to Learn All Transport Mapscited by 32
  28. 2022Kernel Neural Optimal Transportcited by 31
  29. 2022On amortizing convex conjugates for optimal transportcited by 30
  30. 2019Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein spacecited by 25
  31. 2022Reliable amortized variational inference with physics-based latent distribution correctioncited by 25
  32. 2021Wasserstein Generative Learning of Conditional Distributioncited by 24
  33. 2022Optimal transport map estimation in general function spacescited by 20
  34. 2014The Geometric Foundations of Hamiltonian Monte Carlocited by 19
  35. 2019Subspace Detours: Building Transport Plans that are Optimal on Subspace Projectionscited by 19
  36. 2021Preconditioned training of normalizing flows for variational inference in inverse problemscited by 19
  37. 2023Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inferencecited by 19
  38. 2022On minimax density estimation via measure transportcited by 18

50 most cited of 66 All 66 (JSON)

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 1; 2024: 8; 2025: 10; 2026: 7.

26 papers.

Sort:

By area:

  1. 2023How Smooth Is Attention?cited by 44
  2. 2024Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matchingcited by 34
  3. 2024Dynamic Conditional Optimal Transport through Simulation-Free Flowscited by 24
  4. 2025Flow Matching: Markov Kernels, Stochastic Processes and Transport Planscited by 16
  5. 2025Flow Matching with Semidiscrete Couplingscited by 12
  6. 2024Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier mapscited by 9
  7. 2025Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equationscited by 9
  8. 2025Fast Flow-based Visuomotor Policies via Conditional Optimal Transport Couplingscited by 8
  9. 2024Understanding the training of infinitely deep and wide ResNets with Conditional Optimal Transportcited by 7
  10. 2024Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problemscited by 7
  11. 2025Preconditioned Langevin Dynamics with Score-Based Generative Models for Infinite-Dimensional Linear Bayesian Inverse Problemscited by 5
  12. 2025Fast filtering of non-Gaussian models using Amortized Optimal Transport Mapscited by 4
14 more
  1. 2024Sequential Conditional Transport on Probabilistic Graphs for Interpretable Counterfactual Fairnesscited by 3
  2. 2024Learning conditional distributions on continuous spacescited by 2
  3. 2025A Geometric Approach to Optimal Experimental Designcited by 2
  4. 2026Error Analysis of Bayesian Inverse Problems with Generative Priorscited by 2
  5. 2024Sequential transport maps using SoS density estimation and α-divergencescited by 1
  6. 2025Pairwise Optimal Transports for Training All-to-All Flow-Based Condition Transfer Modelcited by 1
  7. 2025Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Latercited by 1
  8. 2026Conditional Sampling via Wasserstein Autoencoders and Triangular Transportcited by 1
  9. 2026LatentFlow: A General Framework for Conditioning Stochastic Processescited by 1
  10. 2025Estimation of Stochastic Optimal Transport Mapscited by 0
  11. 2026Conditional Unbalanced Optimal Transport Maps: An Outlier-Robust Framework for Conditional Generative Modelingcited by 0
  12. 2026Optimal Transport Q-Learning for Flow Policy Steering and Accelerationcited by 0
  13. 2026Approximating Measures on Function Spaces: Transport and Truncationcited by 0
  14. 2026Improving Function Space Flow Matching with Kernel Optimal Transportcited by 0