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
Abstract:
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.
Biography:
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.