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Existence, Duality, and Cyclical monotonicity for weak transport costs

arXiv 1809.05893 ↗First posted 16 Sep 2018math.OC math.CA math.PRcited by 21

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Abstract (arXiv)

The optimal weak transport problem has recently been introduced by Gozlan et. al. We provide general existence and duality results for these problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in the spirit of cyclical monotonicity. As an application we extend the Brenier-Strassen Theorem of Gozlan-Juillet to general probability measures on Rd under minimal assumptions. A driving idea behind our proofs is to consider the set of transport plans with a new (`adapted') topology which seems better suited for the weak transport problem and allows to carry out arguments which are close to the proofs in the classical setup.

Key facts. Cited by 21 papers in the map (data through 7 Oct 2026). First posted on arXiv 16 Sep 2018.

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Citing papers per year: 2021: 1; 2022: 4; 2023: 5; 2024: 4; 2025: 4; 2026: 3.

21 papers.

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  1. 2022Neural Optimal Transportcited by 123
  2. 2022Entropic Neural Optimal Transport via Diffusion Processescited by 49
  3. 2022Neural Optimal Transport with General Cost Functionalscited by 34
  4. 2022Kernel Neural Optimal Transportcited by 31
  5. 2023Energy-guided Entropic Neural Optimal Transportcited by 25
  6. 2023Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmarkcited by 19
  7. 2023Light Schr"odinger Bridgecited by 18
  8. 2024Estimating Barycenters of Distributions with Neural Optimal Transportcited by 14
  9. 2023Energy-Guided Continuous Entropic Barycenter Estimation for General Costscited by 12
  10. 2024Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transportcited by 9
  11. 2023Uncovering Challenges of Solving the Continuous Gromov-Wasserstein Problemcited by 6
  12. 2025Taming Flow Matching with Unbalanced Optimal Transport into Fast Pansharpeningcited by 6
9 more
  1. 2021A novel notion of barycenter for probability distributions based on optimal weak mass transportcited by 5
  2. 2024Scalable Simulation-free Entropic Unbalanced Optimal Transportcited by 5
  3. 2026Muon Dynamics as a Spectral Wasserstein Flowcited by 3
  4. 2025Hybrid Generative Modeling for Incomplete Physics: Deep Grey-Box Meets Optimal Transportcited by 1
  5. 2025Neural Hamilton--Jacobi Characteristic Flows for Optimal Transportcited by 1
  6. 2025Wasserstein Gradient Flows for Scalable and Regularized Barycenter Computationcited by 1
  7. 2024Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximizationcited by 0
  8. 2026Implicit Neural Optimal Transport via Fixed-Point Optimizationcited by 0
  9. 2026From Cone Geometry to Monge Structure: Local Defects and Global Regret in High-Dimensional Optimal Transportcited by 0