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

Understand

We introduce a novel generative modeling framework based on a discretized parabolic Monge-Amp\`{e}re 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\`{e}re PDE, connecting geometry, transportation cost, and regret.