[IMS Seminar] A Global Geometric View of Optimal Transport

ON2023-12-07TAG: ShanghaiTech UniversityCATEGORY: Lecture

Topic: A Global Geometric View of Optimal Transport

Speaker:  Associate Professor David Xianfeng Gu, Department of Computer Science,  State University of New York at Stony Brook (Stony Brook)

Date and time: 15:00–16:00, December 8

Venue: Room S408 of IMS


Optimal  transport plays an important role in generative models in Artificial  Intelligence. This talk focuses on the intrinsic relations between  optimal transport and convex differential geometry. The Brenier theory  in optimal transport is equivalent to Minkowski-Alexandrov theory in  convex geometry, both of them are reduced to solve a Monge-Ampere type  PDE.  This discovers the many geometric symmetries in optimal transport.

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.


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.