Seminar: A Global Geometric View of Optimal Transport最优传输的全局几何观点

Time: Friday, December 8th, 2023 , 15:00-16:00

Speaker: David Xianfeng Gu, State University of New York at Stony Brook

顾险峰博士在清华大学获得计算机科学学士学位，后在哈佛大学师从世界著名的微分几何大师：丘成桐先生，获得硕士和博士学位。顾博士目前是纽约州立大学石溪分校计算机科学系和应用数学系的帝国创新教授。丘先生和顾博士共同创立了一个跨学科领域：计算共形几何，并将其应用于工程和医学科学的多个领域。顾博士在2005年获得美国国家科学基金会Career奖，2006年获得国家自然科学基金委员会杰出海外青年学者奖，2013年在国际华人数学家大会（ICCM）上获得晨兴应用数学金奖。

Dr. David Xianfeng Gu got his bachelor degree in computer science from Tsinghua university, his master and PhD from Harvard university, supervised by the world famous differential geometer: Prof. Shing-Tung Yau. Dr. Gu is currently an empire innovation professor in the computer science department and applied mathematics department in the State University of New York at Stony Brook. Prof. Yau and Dr. Gu founded an interdisciplinary field: Computational Conformal Geometry, and applied it for many fields in engineering and medical sciences. Dr. Gu got NSF Career award in 2005, NSFC Outstanding Overseas Young Scholar award in 2006, Morningside gold medal in applied mathematics in ICCM 2013. 

Globally, in 1994 Gelfand geometrizes the triangulations of a point configuration, such that all coherent triangulations form a convex polytope, the so-called secondary polytope, which is the triangulation of all triangulations. The space of the solutions to the semi-discrete optimal transport problem, namely all the Brenier potentials, has a natural cell decomposition, we call it the secondary power diagram, which is the power diagram of all power diagrams. We show the secondary power diagram is the dual of the secondary polytope. This global geometric view leads to novel computational algorithms to solve the optimal transport problem.

最优运输在人工智能中的生成模型中扮演着重要角色。这次演讲聚焦于最优运输与凸微分几何之间的内在联系。在最优运输中的Brenier理论等同于凸几何中Minkowski-Alexandroff理论，它们都归结为解决一种Monge-Ampere型偏微分方程。这揭示了最优运输中的许多几何对称性。