# No-Regret Generative Modeling via Parabolic Monge-Amp`ere PDE

arXiv: https://arxiv.org/abs/2504.09279 · HTML: https://mlsanity.com/paper/2504.09279/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Apr 2025.

## Built on by (most cited first, 4 of 4)

- [Distributional Shrinkage I: Universal Denoiser Beyond Tweedie's Formula](https://mlsanity.com/paper/2511.09500/) (2025)
- [Distributional Shrinkage II: Higher-Order Scores Encode Brenier Map](https://mlsanity.com/paper/2512.09295/) (2025)
- [A Markov Chain Approach to Preference Alignment](https://mlsanity.com/paper/2606.22652/) (2026)
- [Sample complexity bounds for categorical Markov random fields via Discrete Diffusions](https://mlsanity.com/paper/2610.02128/) (2026)

## Abstract (arXiv)

We introduce a novel generative modeling framework based on a discretized parabolic Monge-Amp`ere 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-Amp`ere 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.