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Abstract:

We introduce a novel generative modeling framework based on a discretized parabolic Monge-Ampère PDE, which emerges as a continuous limit of the Sinkhorn algorithm commonly used in optimal transport. Our method performs iterative refinement in the space of Brenier maps using a mirror gradient descent step. We establish theoretical guarantees for generative modeling through the lens of no-regret analysis, demonstrating that the iterates converge to the optimal Brenier map under a variety of step-size schedules. As a technical contribution, we derive a new Evolution Variational Inequality tailored to the parabolic Monge-Ampère PDE, connecting geometry, transportation cost, and regret. Our framework accommodates non-log-concave target distributions, constructs an optimal sampling process via the Brenier map, and integrates favorable learning techniques from generative adversarial networks and score-based diffusion models. As direct applications, we illustrate how our theory paves new pathways for generative modeling and variational inference.

Citation

Nabarun Deb, and Tengyuan Liang. 2026. “No-Regret Generative Modeling via Parabolic Monge-Ampère PDE.” The Annals of Statistics, 54(5), pp. 2260-2285.

@article{DebLiang2025,
  title = {No-Regret Generative Modeling via Parabolic Monge-Amp\`ere PDE},
  author = {Deb, Nabarun and Liang, Tengyuan},
  journal = {The Annals of Statistics},
  volume = {54},
  number = {5},
  pages = {2260--2285},
  year = {2026},
  month = oct,
  publisher = {Institute of Mathematical Statistics},
  issn = {0090-5364},
  doi = {10.1214/26-AOS2635},
  url = {http://dx.doi.org/10.1214/26-AOS2635},
}

This work is part 0 of a series on distributional shifting and shrinkage; see part 1 , part 2 , and part 3 .