mlsanityData to 7 Oct 2026

Stability of the Monge Map in Semi-Dual Optimal Transport

arXiv 2605.05569 ↗First posted 7 May 2026math.OC cs.LGcited by 0

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Abstract (arXiv)

This paper shows that the semi-dual formulation of the optimal transport problem has a degenerate saddle-point structure, and that its numerical solution is equivalent to solving a constrained optimization problem. We derive necessary and sufficient conditions for the convergence of Monge maps without requiring optimality of the dual potential. This analysis helps explain why, in practice, numerical algorithms often require more iterations to update the transport map than the potential.

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References · 15

  1. 2022Flow Matching for Generative Modelingcited by 5,426
  2. 2017Improved Training of Wasserstein GANscited by 2,223
  3. 2017GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibriumcited by 208
  4. 2019Optimal transport mapping via input convex neural networkscited by 202
  5. 2016Stochastic Optimization for Large-scale Optimal Transportcited by 188
  6. 2022Neural Optimal Transportcited by 123
  7. 2019Wasserstein-2 Generative Networkscited by 96
  8. 2021Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmarkcited by 95
  9. 2021Generative Modeling with Optimal Transport Mapscited by 80
  10. 2021Neural Monge Map estimation and its applicationscited by 40
  11. 20192-Wasserstein Approximation via Restricted Convex Potentials with Application to Improved Training for GANscited by 32
  12. 2018Semi-dual Regularized Optimal Transportcited by 26
3 more
  1. 2019(q,p)-Wasserstein GANs: Comparing Ground Metrics for Wasserstein GANscited by 10
  2. 2025Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport: A Smoothing Approach for Learning the Optimal Transport Plancited by 6
  3. 2025A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solverscited by 2

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