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Optimal transport mapping via input convex neural networks

arXiv 1908.10962 ↗First posted 28 Aug 2019cs.LG stat.MLcited by 206

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Abstract (arXiv)

In this paper, we present a novel and principled approach to learn the optimal transport between two distributions, from samples. Guided by the optimal transport theory, we learn the optimal Kantorovich potential which induces the optimal transport map. This involves learning two convex functions, by solving a novel minimax optimization. Building upon recent advances in the field of input convex neural networks, we propose a new framework where the gradient of one convex function represents the optimal transport mapping. Numerical experiments confirm that we learn the optimal transport mapping. This approach ensures that the transport mapping we find is optimal independent of how we initialize the neural networks. Further, target distributions from a discontinuous support can be easily captured, as gradient of a convex function naturally models a em discontinuous transport mapping.

Key facts. Cited by 206 papers in the map (data through 14 Oct 2026). First posted on arXiv 28 Aug 2019.

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Citing papers per year: 2021: 25; 2022: 29; 2023: 48; 2024: 35; 2025: 34; 2026: 35.

206 papers.

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  1. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,700
  2. 2023Improving and generalizing flow-based generative models with minibatch optimal transportcited by 879
  3. 2022Rectified Flow: A Marginal Preserving Approach to Optimal Transportcited by 283
  4. 2023Multisample Flow Matching: Straightening Flows with Minibatch Couplingscited by 278
  5. 2022Neural Optimal Transportcited by 126
  6. 2024Consistency Flow Matching: Defining Straight Flows with Velocity Consistencycited by 104
  7. 2021Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmarkcited by 96
  8. 2021Proximal Optimal Transport Modeling of Population Dynamicscited by 93
  9. 2021Entropic estimation of optimal transport mapscited by 90
  10. 2022Tutorial on amortized optimizationcited by 83
  11. 2021Generative Modeling with Optimal Transport Mapscited by 82
  12. 2024Optimal Flow Matching: Learning Straight Trajectories in Just One Stepcited by 78
38 more
  1. 2021Large-Scale Wasserstein Gradient Flowscited by 77
  2. 2022Neural Conservation Laws: A Divergence-Free Perspectivecited by 70
  3. 2021Plugin Estimation of Smooth Optimal Transport Mapscited by 68
  4. 2021Low-Rank Sinkhorn Factorizationcited by 67
  5. 2023Recent Advances in Optimal Transport for Machine Learningcited by 66
  6. 2022Supervised Training of Conditional Monge Mapscited by 61
  7. 2023Posterior Collapse and Latent Variable Non-identifiabilitycited by 57
  8. 2021Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networkscited by 53
  9. 2021Variational Wasserstein gradient flowcited by 53
  10. 2021Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimizationcited by 51
  11. 2022Entropic Neural Optimal Transport via Diffusion Processescited by 50
  12. 2023A Computational Framework for Solving Wasserstein Lagrangian Flowscited by 46
  13. 2024Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifoldcited by 43
  14. 2021Improving Mini-batch Optimal Transport via Partial Transportationcited by 41
  15. 2023Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transportcited by 41
  16. 2021Neural Monge Map estimation and its applicationscited by 40
  17. 2022Neural Optimal Transport with General Cost Functionalscited by 36
  18. 2023Debias Coarsely, Sample Conditionally: Statistical Downscaling through Optimal Transport and Probabilistic Diffusion Modelscited by 33
  19. 2023Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translationcited by 33
  20. 2023The Monge Gap: A Regularizer to Learn All Transport Mapscited by 32
  21. 2022Kernel Neural Optimal Transportcited by 31
  22. 2022On amortizing convex conjugates for optimal transportcited by 31
  23. 2022Meta Optimal Transportcited by 28
  24. 2021Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysiscited by 27
  25. 2023Scalable Optimal Transport Methods in Machine Learning: A Contemporary Surveycited by 27
  26. 2024Addressing Misspecification in Simulation-based Inference through Data-driven Calibrationcited by 27
  27. 2022Towards Explaining Distribution Shiftscited by 26
  28. 2023Energy-guided Entropic Neural Optimal Transportcited by 26
  29. 2023Conditional Optimal Transport on Function Spacescited by 26
  30. 2023What's in a Prior? Learned Proximal Networks for Inverse Problemscited by 25
  31. 2024Dynamic Conditional Optimal Transport through Simulation-Free Flowscited by 25
  32. 2022Wasserstein Iterative Networks for Barycenter Estimationcited by 23
  33. 2022Optimal transport map estimation in general function spacescited by 20
  34. 2022Multivariate Quantile Function Forecastercited by 19
  35. 2023Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmarkcited by 19
  36. 2023Light Schr"odinger Bridgecited by 18
  37. 2021Riemannian Convex Potential Mapscited by 17
  38. 2021Input Convex Gradient Networkscited by 17

50 most cited of 206 All 206 (JSON)