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Explore a novel generative modeling framework in this 51-minute conference talk that introduces an innovative approach based on discretized parabolic Monge-Ampère partial differential equations. Learn how this method emerges as a continuous limit of the Sinkhorn algorithm used in optimal transport and performs iterative refinement in the space of Brenier maps through mirror gradient descent steps. Discover the theoretical guarantees established for generative modeling through no-regret analysis, showing how iterates converge to the optimal Brenier map under various step-size schedules. Examine the technical contribution of a new Evolution Variational Inequality specifically tailored to the parabolic Monge-Ampère PDE, which connects geometry, transportation cost, and regret. Understand how this framework accommodates non-log-concave target distributions, constructs optimal sampling processes via Brenier maps, and integrates favorable learning techniques from generative adversarial networks and score-based diffusion models. Gain insights into the mathematical foundations of AI through this advanced presentation on the intersection of partial differential equations, optimal transport theory, and modern generative modeling techniques.
