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.

翻译：我们提出了一种基于离散化抛物型蒙日-安培偏微分方程的新型生成建模框架，该方程作为最优输运中常用Sinkhorn算法的连续极限而出现。我们的方法利用镜像梯度下降步骤在Brenier映射空间中进行迭代精化。通过无遗憾分析的视角，我们为生成建模建立了理论保证，证明了在多种步长调度方案下迭代过程收敛到最优Brenier映射。作为技术贡献，我们推导出一种针对抛物型蒙日-安培偏微分方程的新进化变分不等式，将几何、输运代价与遗憾联系起来。我们的框架能够处理非对数凹目标分布，通过Brenier映射构建最优采样过程，并融合了生成对抗网络与基于分数的扩散模型中的有利学习技术。作为直接应用，我们展示了该理论如何为生成建模与变分推断开辟新路径。