mlsanityData to 7 Oct 2026

On amortizing convex conjugates for optimal transport

arXiv 2210.12153 ↗First posted 21 Oct 2022cs.LG cs.AI stat.MLcited by 30

Not on the map. Why?

Abstract (arXiv)

This paper focuses on computing the convex conjugate (also known as the Legendre-Fenchel conjugate or c-transform) that appears in Euclidean Wasserstein-2 optimal transport. This conjugation is considered difficult to compute and in practice, methods are limited by not being able to exactly conjugate the dual potentials in continuous space. To overcome this, the computation of the conjugate can be approximated with amortized optimization, which learns a model to predict the conjugate. I show that combining amortized approximations to the conjugate with a solver for fine-tuning significantly improves the quality of transport maps learned for the Wasserstein-2 benchmark by Korotin et al. (2021a) and is able to model many 2-dimensional couplings and flows considered in the literature. All baselines, methods, and solvers are publicly available at http://github.com/facebookresearch/w2ot.

Key facts. Cited by 30 papers in the map (data through 7 Oct 2026). Builds on 42 of its 97 references that are in the map. First posted on arXiv 21 Oct 2022.

Builds on

42 of 97 references.

References · 42

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  3. 2015Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shiftcited by 6,979
  4. 2014Deep Learning Face Attributes in the Wildcited by 3,679
  5. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,277
  6. 2016Identity Mappings in Deep Residual Networkscited by 1,960
  7. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  8. 2015Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)cited by 1,061
  9. 2016Input Convex Neural Networkscited by 407
  10. 2014Identifying and attacking the saddle point problem in high-dimensional non-convex optimizationcited by 286
  11. 2021Learning to Optimize: A Primer and A Benchmarkcited by 206
  12. 2019Optimal transport mapping via input convex neural networkscited by 202
30 more
  1. 2016Stochastic Optimization for Large-scale Optimal Transportcited by 188
  2. 2017Large-Scale Optimal Transport and Mapping Estimationcited by 120
  3. 2019Wasserstein-2 Generative Networkscited by 96
  4. 2021Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmarkcited by 95
  5. 2020Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimizationcited by 89
  6. 2021Generative Modeling with Optimal Transport Mapscited by 80
  7. 2021Large-Scale Wasserstein Gradient Flowscited by 76
  8. 2017Continuously Differentiable Exponential Linear Unitscited by 73
  9. 2017On the regularization of Wasserstein GANscited by 65
  10. 2022Supervised Training of Conditional Monge Mapscited by 60
  11. 2021Faster Matchings via Learned Dualscited by 56
  12. 2021Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networkscited by 53
  13. 2021Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimizationcited by 50
  14. 2020Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networkscited by 49
  15. 2022Neural Optimal Transport with General Cost Functionalscited by 34
  16. 20192-Wasserstein Approximation via Restricted Convex Potentials with Application to Improved Training for GANscited by 32
  17. 2019Multi-marginal Wasserstein GANcited by 32
  18. 2020Continuous Regularized Wasserstein Barycenterscited by 32
  19. 2016A Multi-Batch L-BFGS Method for Machine Learningcited by 30
  20. 2019On Scalable and Efficient Computation of Large Scale Optimal Transportcited by 29
  21. 2022Meta Optimal Transportcited by 28
  22. 2022Learning Predictions for Algorithms with Predictionscited by 25
  23. 2017Wasserstein Divergence for GANscited by 22
  24. 2019A fast approach to optimal transport: The back-and-forth methodcited by 16
  25. 2021Riemannian Convex Potential Mapscited by 16
  26. 2022Neural Unbalanced Optimal Transport via Cycle-Consistent Semi-Couplingscited by 16
  27. 2019Adversarial Computation of Optimal Transport Mapscited by 12
  28. 2021Implicit Riemannian Concave Potential Mapscited by 10
  29. 2022Amortized Proximal Optimizationcited by 9
  30. 2020Approximately Exact Line Searchcited by 2

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 1; 2023: 8; 2024: 9; 2025: 6; 2026: 6.

30 papers.

Sort:

By area:

  1. 2023Multisample Flow Matching: Straightening Flows with Minibatch Couplingscited by 267
  2. 2024Optimal Flow Matching: Learning Straight Trajectories in Just One Stepcited by 74
  3. 2022Entropic Neural Optimal Transport via Diffusion Processescited by 49
  4. 2023The Monge Gap: A Regularizer to Learn All Transport Mapscited by 32
  5. 2023Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translationcited by 32
  6. 2023Conditional Optimal Transport on Function Spacescited by 26
  7. 2023Light Schr"odinger Bridgecited by 18
  8. 2024Neural Optimal Transport with Lagrangian Costscited by 16
  9. 2023Energy-Guided Continuous Entropic Barycenter Estimation for General Costscited by 12
  10. 2024Progressive Entropic Optimal Transport Solverscited by 10
  11. 2024Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transportcited by 9
  12. 2024Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier mapscited by 9
18 more
  1. 2023Computing high-dimensional optimal transport by flow neural networkscited by 8
  2. 2024ENOT: Expectile Regularization for Fast and Accurate Training of Neural Optimal Transportcited by 6
  3. 2025Convexity in ReLU Neural Networks: beyond ICNNs?cited by 6
  4. 2023Interpolation for Robust Learning: Data Augmentation on Wasserstein Geodesicscited by 3
  5. 2024On a Neural Implementation of Brenier's Polar Factorizationcited by 3
  6. 2024Fast and scalable Wasserstein-1 neural optimal transport solver for single-cell perturbation predictioncited by 3
  7. 2024Scalable unsupervised alignment of general metric and non-metric structurescited by 2
  8. 2025A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solverscited by 2
  9. 2025Sample and Map from a Single Convex Potential: Generation using Conjugate Moment Measurescited by 2
  10. 2025Neural Optimal Transport Meets Multivariate Conformal Predictioncited by 2
  11. 2025Multidimensional Uncertainty Quantification via Optimal Transportcited by 1
  12. 2025Deep Legendre Transformcited by 1
  13. 2026Amortized Maximum Inner Product Search with Learned Support Functionscited by 1
  14. 2026Hyperparameter Trajectory Inference with Conditional Lagrangian Optimal Transportcited by 0
  15. 2026Implicit Neural Optimal Transport via Fixed-Point Optimizationcited by 0
  16. 2026A lift for input-convex neural net trainingcited by 0
  17. 2026A Biconvex Formulation for Stable Transport of Mixture Models with a Unique Solutioncited by 0
  18. 2026Conditional Neural Optimal Transport for Predicting Cellular Phenotypes from Molecular Structurecited by 0