mlsanityData to 7 Oct 2026

Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization

arXiv 2102.01752 ↗First posted 2 Feb 2021cs.LG stat.MLcited by 50

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Abstract (arXiv)

Wasserstein barycenters provide a geometric notion of the weighted average of probability measures based on optimal transport. In this paper, we present a scalable algorithm to compute Wasserstein-2 barycenters given sample access to the input measures, which are not restricted to being discrete. While past approaches rely on entropic or quadratic regularization, we employ input convex neural networks and cycle-consistency regularization to avoid introducing bias. As a result, our approach does not resort to minimax optimization. We provide theoretical analysis on error bounds as well as empirical evidence of the effectiveness of the proposed approach in low-dimensional qualitative scenarios and high-dimensional quantitative experiments.

Key facts. Cited by 50 papers in the map (data through 7 Oct 2026). Builds on 18 of its 43 references that are in the map. First posted on arXiv 2 Feb 2021.

Builds on

18 of 43 references.

References · 18

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,277
  3. 2018Computational Optimal Transportcited by 765
  4. 2016Input Convex Neural Networkscited by 407
  5. 2013Fast Computation of Wasserstein Barycenterscited by 216
  6. 2019Optimal transport mapping via input convex neural networkscited by 202
  7. 2016Stochastic Optimization for Large-scale Optimal Transportcited by 188
  8. 2017Large-Scale Optimal Transport and Mapping Estimationcited by 120
  9. 2019Wasserstein-2 Generative Networkscited by 96
  10. 2020Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networkscited by 49
  11. 2015Scalable Bayes via Barycenter in Wasserstein Spacecited by 44
  12. 2020Continuous Regularized Wasserstein Barycenterscited by 32
6 more
  1. 2019On Scalable and Efficient Computation of Large Scale Optimal Transportcited by 29
  2. 2018Decentralize and Randomize: Faster Algorithm for Wasserstein Barycenterscited by 21
  3. 2019Wasserstein Style Transfercited by 21
  4. 2020Estimating Barycenters of Measures in High Dimensionscited by 18
  5. 2017Parallel Streaming Wasserstein Barycenterscited by 14
  6. 2019Mixability of Integral Losses: a Key to Efficient Online Aggregation of Functional and Probabilistic Forecastscited by 7

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 11; 2022: 10; 2023: 11; 2024: 6; 2025: 9; 2026: 3.

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  1. 2022Neural Optimal Transportcited by 123
  2. 2021Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmarkcited by 95
  3. 2021Large-Scale Wasserstein Gradient Flowscited by 76
  4. 2024Optimal Flow Matching: Learning Straight Trajectories in Just One Stepcited by 74
  5. 2021Low-Rank Sinkhorn Factorizationcited by 66
  6. 2023Recent Advances in Optimal Transport for Machine Learningcited by 66
  7. 2021Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networkscited by 53
  8. 2021Variational Wasserstein gradient flowcited by 53
  9. 2022Entropic Neural Optimal Transport via Diffusion Processescited by 49
  10. 2021Neural Monge Map estimation and its applicationscited by 40
  11. 2021Improving Mini-batch Optimal Transport via Partial Transportationcited by 40
  12. 2022Neural Optimal Transport with General Cost Functionalscited by 34
38 more
  1. 2021Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descentcited by 32
  2. 2022Kernel Neural Optimal Transportcited by 31
  3. 2022On amortizing convex conjugates for optimal transportcited by 30
  4. 2022Meta Optimal Transportcited by 28
  5. 2023Energy-guided Entropic Neural Optimal Transportcited by 25
  6. 2022Wasserstein Iterative Networks for Barycenter Estimationcited by 23
  7. 2023Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmarkcited by 19
  8. 2023Tree-Based Diffusion Schr"odinger Bridge with Applications to Wasserstein Barycenterscited by 16
  9. 2024Estimating Barycenters of Distributions with Neural Optimal Transportcited by 14
  10. 2026Learning Continuous Wasserstein Barycenter Space for Generalized All-in-One Image Restorationcited by 14
  11. 2023Energy-Guided Continuous Entropic Barycenter Estimation for General Costscited by 12
  12. 2023Doubly Regularized Entropic Wasserstein Barycenterscited by 10
  13. 2024Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transportcited by 9
  14. 2023Light Unbalanced Optimal Transportcited by 8
  15. 2023Computing high-dimensional optimal transport by flow neural networkscited by 8
  16. 2021Variational Wasserstein Barycenters with c-Cyclical Monotonicitycited by 7
  17. 2021Learning to Generate Wasserstein Barycenterscited by 6
  18. 2023Uncovering Challenges of Solving the Continuous Gromov-Wasserstein Problemcited by 6
  19. 2025Learning of Population Dynamics: Inverse Optimization Meets JKO Schemecited by 6
  20. 2023Generating Synthetic Datasets by Interpolating along Generalized Geodesicscited by 5
  21. 2021Iterative Alignment Flowscited by 4
  22. 2022Learning Optimal Transport Between two Empirical Distributions with Normalizing Flowscited by 4
  23. 2024Analysis of Kernel Mirror Prox for Measure Optimizationcited by 4
  24. 2024Relative Translation Invariant Wasserstein Distancecited by 3
  25. 2025BaryIR: Learning Multi-Source Unified Representation in Continuous Barycenter Space for Generalizable All-in-One Image Restorationcited by 3
  26. 2025Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimizationcited by 2
  27. 2025Meta Optimality for Demographic Parity Constrained Regression via Post-Processingcited by 2
  28. 2025Schr"odinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentrescited by 2
  29. 2022Learning with Stochastic Orderscited by 1
  30. 2024GeOT: A spatially explicit framework for evaluating spatio-temporal predictionscited by 1
  31. 2025RAID: An In-Training Defense against Attribute Inference Attacks in Recommender Systemscited by 1
  32. 2025Offline Reinforcement Learning with Wasserstein Regularization via Optimal Transport Mapscited by 1
  33. 2025Neural Hamilton--Jacobi Characteristic Flows for Optimal Transportcited by 1
  34. 2025Wasserstein Gradient Flows for Scalable and Regularized Barycenter Computationcited by 1
  35. 2026Binding Multiple Modalities via Multimodal Wasserstein Barycentercited by 1
  36. 2022Connecting adversarial attacks and optimal transport for domain adaptationcited by 0
  37. 2023Computational Guarantees for Doubly Entropic Wasserstein Barycenters via Damped Sinkhorn Iterationscited by 0
  38. 2026Implicit Neural Optimal Transport via Fixed-Point Optimizationcited by 0